Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Using cognitive task analysis to capture expert instruction in division of fractions
(USC Thesis Other)
Using cognitive task analysis to capture expert instruction in division of fractions
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
Running head: COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 1
USING COGNITIVE TASK ANALYSIS TO CAPTURE EXPERT INSTRUCTION IN
DIVISION OF FRACTIONS
By
Douglas C. Wieland
A Dissertation Presented to the
FACULTY OF THE USC ROSSIER SCHOOL OF EDUCATION
UNIVERSTIY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF EDUCATION
May, 2015
Copyright 2015 Douglas C. Wieland
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 2
Dedication
This dissertation is dedicated, in no particular order, to my father Bill, my wife Tamiko,
and my mother Carolyn.
To my father, a 1932 graduate of this university, who once patiently explained to me that
trout rest facing the current, and thus, when angling, it is important to wade in an upstream
direction. This, he said, increases one’s chances of sneaking up on them from behind.
To my wife, who understands me implicitly, and accordingly, understood going in that
putting up with me during this process would involve incredible patience, support and
understanding on her part. Tamiko, you provided all of that and more, and I owe you a debt of
gratitude.
And to my mother, who encouraged me to attempt this task and who provided invaluable
support. Mom, your unmitigated disdain for suffering fools is matched only by your immoderate
fondness for leisurely dining. The next meal’s on me.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 3
Acknowledgments
I would like to express my sincere appreciation to Dr. Kenneth Yates for his keen
intellect, for his willingness to make himself available for consultation whenever I needed a
helping hand, for his unflagging good humor, and for his ability to make a daunting task seem
somehow manageable. Thanks to him, I was able to successfully complete this process.
I also wish to sincerely thank Dr. Sandra Kaplan, who early on in this process, and for
reasons that I cannot possibly imagine, took a genuine interest in me as a learner, and who
accordingly provided invaluable insight and guidance in helping me to select a dissertation topic.
Throughout this process, Dr. Kaplan has assisted me with her incredible experience, profound
wisdom and, and above all, grace.
Last, and certainly not least, I extend my heartfelt gratitude to Dr. Artineh Samkian for
her willingness to share with me her expertise in conducting research, for her constant
encouragement, for the tireless effort she expended in reviewing this manuscript, and for her
cheerful disposition and her ability to make it all look so easy.
I thank you all!
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 4
Table of Contents
List of Tables 7
List of Figures 8
List of Abbreviations 9
Abstract 10
Chapter One: Overview of the Study 11
Statement of the Problem 11
Purpose of the Study 14
Methodology of the Study 14
Definition of Terms 15
Organization of the Study 17
Chapter Two: Literature Review 18
United States K-12 Mathematics Achievement: An
International Context 18
United States Algebra Achievement: A National Context 22
Overall Significance of Algebra 25
Components of Secondary Algebra 26
Foundations of Secondary Algebra 27
Fluency with Fractions as Critical Foundation
for Success in Algebra 29
A Grade-Level Framework for Instruction in Fractions 30
Third Grade 30
Fourth Grade 30
Fifth Grade 31
Sixth Grade 31
Seventh Grade 31
Division of Fractions 31
Cognitive Components of Division of Fractions 32
How Division of Fractions is Taught 37
The Invert and Multiply Approach 37
The Complex Fraction Approach 38
The Common Denominator Approach 38
Teacher Deficiencies: Instruction in Division of Fractions 49
Inadequate Teacher Subject Matter Knowledge 49
Teacher Inability to Gauge Student Understanding 40
Inappropriate Instructional Practice 40
How Teachers are Trained 41
Summary 42
Knowledge Types 42
Declarative Knowledge 43
Procedural Knowledge 43
Automaticity 43
Expertise 44
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 5
Characteristics of Experts 44
Building Expertise 45
Consequences of Expertise 46
Expert Omissions 46
Cognitive Task Analysis (CTA) 47
Definition of CTA 47
Brief History of CTA 47
CTA Methodology 47
Taxonomies of Knowledge Elicitation Techniques 48
Pairing Knowledge Elicitation with Knowledge
Representation Analysis 48
Effectiveness of CTA 48
Benefits of CTA for Instruction 49
Conclusion 49
Chapter Three: Methodology 51
Overview 51
Participants 51
Data Collection for Question 1 52
Phase 1: Collect Preliminary Knowledge 53
Phase 2: Identify Knowledge Representations 53
Phase 3: Apply Knowledge Elicitation Techniques 54
Instrumentation 54
Interviews 54
Phase 4: Data Analysis 55
Coding 55
Inter-rater Reliability 55
SME Protocol and Verification 56
Phase 5: Formatting the Results 56
Gold Standard Protocol 56
Summary 56
Data Analysis for Question 2 57
Spreadsheet Analysis 57
Chapter Four: Results 59
Overview of Results 59
Research Questions 59
Question 1 59
Inter-rater Reliability 59
Flowchart Analysis 59
Gold Standard Protocol 60
Recalled Action and Decision Steps 63
Action and Decision Steps Contributed by
Each SME 64
Additional Action and Decision Steps Captured
During Follow-up Interviews and SME Review of
Preliminary Gold Standard Protocol 66
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 6
Alignment of SMEs in Describing the Same
Action and Decision Steps 67
Question 2 68
Total Knowledge Omissions 68
Analysis of Action and Decision Step Omissions 70
Chapter Five: Discussion 71
Overview of the Study 71
Process of Conducting the Cognitive Task Analysis 73
Selection of Experts 73
Data Collection 76
Data Analysis 79
Discussion of Findings 81
Research Question 1 81
Action Steps versus Decision Steps 81
Differences in Recall among SMEs 84
Additions, Deletions, and Revisions Captured during
Review of Initial Individual Protocols and of the
Preliminary Gold Standard Protocol 87
Review of Initial Gold Standard Protocol 87
Review of Preliminary Gold Standard Protocol 90
Alignment of SMEs vis-à-vis Total Action and Decision Steps 91
Research Question 2 92
Expert Knowledge Omissions 92
Limitations 93
Confirmation Bias 94
Internal Validity 95
External Validity 95
Implications 96
Future Research 97
Conclusion 99
References 101
Appendix A: Cognitive Task Analysis Interview Protocol 117
Appendix B: Inter-rater Reliability Code Sheet 120
Appendix C: Job Aid for Developing a Gold Standard Protocol 121
Appendix D: Flowchart for SME A Individual Protocol 123
Appendix E: Gold Standard Protocol 145
Appendix F: Incremental Coding Spreadsheet 169
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 7
List of Tables
Table 1: Cumulative Action and Decision Steps Captured for each SME
In the Initial Individual Protocols 64
Table 2: Additional Expert Knowledge Captured during Follow-up
Interviews and SME Review of Preliminary Gold Standard Protocol 66
Table 3: Number and Percentage of Total Action and Decision Steps: Highly
Aligned, Partially Aligned, and Slightly Aligned 67
Table 4: Total Expert Knowledge Omissions by SME as Compared to the
Final Gold Standard Protocol 69
Table 5: Criteria for Selection of Expert Mathematics Teachers 74
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 8
List of Figures
Figure 1: PISA Overall Mathematics Results for Selected Countries, 2009 and 2012 19
Figure 2: TIMSS Combined Average Mathematics Results for Selected Countries,
2007 and 2011 20
Figure 3: TIMSS Algebra Results for Selected Countries, 2007 and 2011 22
Figure 4: Visual Representation of the 3i = 3r CTA Method 57
Figure 5: A Sequential Representation of the Process of Aggregating Action
Steps from SME C and SME A onto an Action Step from SME B 61
Figure 6: Total Action and Decision Steps in the Gold Standard Protocol 65
Figure 7: Total SME Knowledge Omissions as Compared to the Final
Gold Standard Protocol 70
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 9
List of Abbreviations
CCSS: Common Core State Standards
CDM: Critical Decision Method
CPP: Concepts, Processes, and Principles
CTA: Cognitive Task Analysis
GSP: Gold Standard Protocol
IRB: Institutional Review Board
K–12: Kindergarten through Twelfth Grade
NAEP: National Assessment of Educational Progress
NCTM National Council of Teachers of Mathematics
OECD: Organisation for Economic Co-operation and Development
PGSP: Preliminary Gold Standard Protocol
SME: Subject Matter Expert
TIMSS Trends in International Mathematics and Science Study
3i + 3r: Three Independent Interviews + Three Reviews
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 10
Abstract
This study applies Cognitive Task Analysis (CTA), a method for eliciting the automated,
unconscious knowledge and skills of experts, to capture expertise in teaching K-12 mathematics.
The purpose of this study was to conduct a CTA with middle school teachers who have been
identified as experts, to capture the knowledge and skills they use when providing instruction in
the division of fractions by fractions. Also, this study investigated whether experts’ knowledge
omissions in this field would conform to those in other fields, which can approach 70%. CTA
methods in this study included semi-structured interviews with three middle school mathematics
teachers. The study’s findings indicated that these experts recalled, on average, 39.14% of the
action and decision steps compared to the gold standard protocol, while omitting, on average,
60.86% of such steps. The study’s implications are that the degree of omissions among expert
middle school teachers are similar to those of experts in other fields. Additionally, the greater
degree of knowledge capture provided by the use of multiple experts, compared to that for a
single expert, indicates that the use of CTA for the development of teacher preparation and
professional development programs shows promise when compared to current models, which
rely primarily on individual experts.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 11
CHAPTER ONE: OVERVIEW OF THE STUDY
Statement of the Problem
Compared to their peers in foreign countries, American K-12 students display less than
stellar mathematical achievement. The 2009 Program for International Student Assessment
(PISA) assessed fifteen year-olds from Organization for Economic Cooperation and
Development (OECD) countries in broad math content measures such as space and shape,
quantity, uncertainty and data, and change and relationships (National Center for Education
Statistics (NCES), 2013a). American students that year had an average score in math of 487,
compared to an average of 496 for all participating OECD countries (NCES, 2009). In 2012,
U.S. students fared no better, with an average math score of 481 (NCES, 2013a). This compares
to an OECD average of 494, and falls far below the scores of the highest achieving nations, such
as Shanghai, China, with a 613, and Singapore, with a score of 573 (NCES, 2012). U.S. students
even trailed typical low-performers such as Slovenia, with an average of 501, and The Czech
Republic, which scored 499 (NCES, 2012).
An international assessment that focuses more specifically on students’ ability in specific
content areas is the Trends in International Mathematics and Science Study (TIMSS), which
assessed eighth graders from more than forty countries in numbers, algebra, geometry, and data
and chance (NCES, 2007). In 2007 American students achieved an average overall math score
of 508, just slightly above the international average of 500, and well below traditional math
powerhouse countries, such as Singapore, with an average score of 593, and Honk Kong which
averaged 572 (NCES, 2007). Scores of American eighth graders on the 2011 TIMSS were little
better, with American students averaging a score of 509 in math overall, compared to the
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 12
international average of 500 (NCES, 2011). Again, Asian countries outperformed all others,
with Korea achieving an overall score of 613, and Taiwan a score of 609 (NCES, 2011).
In terms of specific content, one area of math where U.S. students replicate their overall
mediocrity is in the domain of algebra. In 2009, U.S. students’ average score in algebra on the
TIMSS was 507, compared to the international average of 500, and well below Korea (608),
Japan (567), Hong Kong (575), and Singapore (614) (NCES, 2011). American students’ algebra
performance on the 2011 TIMSS was slightly better, with a score of 512, but again, well below
Korea at 617, Japan at 570, Hong Kong at 583 and Singapore at 614 (NCES, 2011).
The importance of algebra cannot be underestimated. The National Council of Teachers
of Mathematics has categorized algebra as a towering accomplishment that is critical to
mathematical work (2000). The National Math Advisory Panel (NMAP) identified algebra as a
gateway to and necessity for more advanced math course work in high school (2008). The
Common Core State Standards (National Governors’ Council (NGA), 2010) stress the
importance of students’ ability to comprehend abstract situations and represent them
symbolically. And perhaps most significantly, the authors of the Principles and Standards for
School Mathematics (NCTM, 2000) posit that, algebra constitutes a major component of the
school mathematical curriculum and serves to unify it.
One domain of elementary and middle school math that is considered foundational for
learning algebra is the study of fractions (Fuchs et al., 2014). The National Mathematics
Advisory Panel (NMAP, 2008) identified fluency with fractions as a critical steppingstone to
algebra. An analysis of two nationally representative data sets, one in Britain and the other in the
United States, indicated that elementary students’ understanding of fractions predicted their
knowledge of algebra in high school (Siegler & Pike, 2013). Kieren (1976) concluded that
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 13
students must master fractional number concepts to sufficiently learn algebraic concepts.
Finally, a survey of 1,000 U.S. Algebra 1 teachers identified fractions as one of the most
significant weaknesses in students’ preparation for algebra coursework (Hoffer, Venkataraman,
Hedberg, & Shagle, 2007).
For students, one particularly vexing component of understanding fractions involves
division of fractions. Sharp and Welder (2014) identify division of fractions as a common area
of struggle in seventh grade math. Trafton and Zawojewski (1984, p. 20) describe division of
fractions as a “troublesome” endeavor for many students. Coughlin (2010/2011) labels division
of fractions one of the most complex tasks in elementary math, while Cengiz and Rathouz (2011)
consider the procedure one of the most rote-like and least understood elementary math concepts.
Many of the difficulties that students face with division of fractions can be traced to
deficiencies in teachers’ mathematical knowledge (Koichu, Harel, & Manaster, 2013). Many
teachers make use of the invert and multiply algorithm, but are unable to explain the principle
involved (Borko et al., 1992). Ma (1999) found that many U.S. teachers were unable to create
division of fraction word problems, due to insufficient conceptual understanding. Triosh (2000)
found that among pre-service teachers of mathematics, there was incomplete understanding of
students’ fraction misconceptions. If it were possible to remediate these deficiencies in
instruction and understanding, and capture the conceptual and procedural knowledge of teachers
expert in teaching division of fractions, a schematic representation could be developed (Crandall,
Klein &Hoffman, 2006) that could be used to train teachers.
Unfortunately, because the procedural knowledge possessed by experts is largely
automated, and thus unconscious (Clark & Estes, 1996), highly skilled practitioners can omit up
to 70% of such knowledge when asked to describe the complex tasks in which they are expert
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 14
(Clark, Feldon, van Merrienboer, Yates, & Early, 2008). One method of eliciting the knowledge
and processes used by experts is Cognitive Task Analysis (CTA) (Tofel-Grehl & Feldon, 2013).
CTA is grounded in data that suggest interviewing multiple experts increases the body of
knowledge and process data from roughly 30%, when one expert is interviewed, to 75% or more
when three experts are interviewed (Clark et al., 2008). Researchers can then translate this
information into instructional guides that can be used to train novices (Clark et al., 2008). As a
result this study proposes to adopt this methodology in capturing and synthesizing the expert
declarative and procedural knowledge necessary to create a training guide for teaching division
of fractions.
Purpose of the Study
The purpose of this study is to conduct a CTA with middle school teachers who have
been identified as experts, to capture the knowledge and skills they use when providing
instruction in the division of fractions by fractions.
The research questions that guide this study are:
1. What are the action and decision steps that expert middle school math teachers
recall when they describe how they teach the division of fractions by fractions?
2. What percent of action and/or decision steps, when compared to a gold standard,
do expert middle school math teachers omit when they describe how they teach
the division of fractions by fractions?
Methodology of the Study
This study’s methodology involved conducting a Cognitive Task Analysis to capture the
knowledge and skills of middle school teachers from school districts in Southern California who
were identified as subject matter experts (SMEs) in providing instruction in division of fractions
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 15
by fractions. Three SMEs were selected; each participated in the interviews while all three also
verified the aggregate data collected. The CTA followed a five-step process:
1) a preliminary phase for building general familiarity with the instructional process;
2) the identification of declarative and procedural knowledge, in addition to any
organizational schemes used in applying these knowledge types;
3) a knowledge elicitation phase employing semi-structured interviews;
4) a data analysis phase, involving coding of interview transcripts, determining inter-rater
reliability, and individual SME protocol verification;
5) the development of a gold standard protocol that was used to identify expert omissions
and that can serve as a training guide for novice teachers.
Definition of Terms
The following are definitions of terms related to Cognitive Task Analysis as suggested by
Zepeda McZeal (2014).
Adaptive expertise: When experts can rapidly retrieve and accurately apply
appropriate knowledge and skills to solve problems in their fields or expertise; to possess
cognitive flexibility in evaluating and solving problems (Gott, Hall Pokorny, Dibble, &
Glaser, 1993; Hatano & Inagaki, 2000).
Automaticity: An unconscious fluidity of task performance following sustained and
repeated execution; results in an automated mode of functioning (Anderson, 1996; Ericsson,
2004).
Automated knowledge: Knowledge about how to do something; operates outside of
conscious awareness due to task repetition (Wheatley & Wegner, 2001).
Cognitive load: Simultaneous demands placed on working memory during information
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 16
processing that can present challenges to learners (Sweller, 1988).
Cognitive tasks: Tasks that require mental effort and engagement to perform (Clark &
Estes, 1996).
Cognitive task analysis: Knowledge elicitation techniques for extracting implicit and
explicit knowledge from multiple experts for use in instruction and instructional design (Clark
et al., 2008; Schraagen, Chipman, & Shalin, 2000).
Conditional knowledge: Knowledge about why and when to do something; a type of
procedural knowledge to facilitate the strategic application of declarative and procedural
knowledge to problem solve (Paris, Lipson, & Wixson, 1983).
Declarative knowledge: Knowledge about why or what something is; information that
is accessible in long-term memory and consciously observable in working memory
(Anderson, 1996a; Clark & Elen, 2006).
Expertise: The point at which an expert acquires knowledge and skills essential for
consistently superior performance and complex problem solving in a domain; typically
develops after a minimum of 10 years of deliberate practice or repeated engagement in
domain-specific tasks (Ericsson, 2004).
Procedural knowledge: Knowledge about how and when something occurs; acquired
through instruction or generated through repeated practice (Anderson, 1982; Clark & Estes,
1996).
Subject matter expert: An individual with extensive experience in a domain who can
perform tasks rapidly and successfully; demonstrates consistent superior performance or
ability to solve complex problems (Clark et al., 2008).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 17
Organization of the Study
Chapter Two of this study reviews the literature in two sections: the first section of the
review examines the literature relevant to instruction in division of fractions by fractions,
while the second section examines the literature relevant to Cognitive Task Analysis and its
use as a knowledge elicitation technique. Chapter Three describes the study methodology
and the manner in which the research approach addresses the research questions. Chapter
Four is a review of the study results, and compares these results to each of the research
questions. Chapter Five includes a discussion of the findings, an analysis of the implications
of the results vis-à-vis instruction in the division of fractions by fractions and CTA, a
discussion of the study’s limitations, and a consideration of the implications for future
research.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 18
CHAPTER TWO: LITERATURE REVIEW
United States K-12 Mathematics Achievement: An International Context
An examination of one internationally administered assessment reveals that, compared to
their grade level peers in other Organisation for Economic Co-operation and Development
(OECD) countries, American students score at or below international averages for mathematics
achievement. This assessment, the Program for International Student Assessment (PISA), has
been administered every three years since 2000 by the OECD in over 60 OECD and non-OECD
countries (National Center for Education Statistics (NCES), 2013a). The PISA assesses fifteen
year-olds in mathematics, reading and science fluency (NCES, 2013a). The mathematics portion
of the PISA measures achievement in four, broad content areas: space and shape, quantity,
uncertainty and data, and change and relationships (NCES, 2013a). In addition it provides a
measure of students’ proficiency in three mathematical process skills: employing, formulating,
and interpreting (NCES, 2013a). Overall scores reported represent an average of these four
content and three process measures, expressed on a scale of from 0 to 1000 (NCES, 2013a). In
2009, a nationally representative sample of 5,233 U.S. students had an average overall math
score of 487, which compares to an OECD average of 496, and falls far below Korea at 546,
Finland at 541, and Switzerland at 534 (NCES, 2009). Overall, in 2009 the overall score for U.S.
students was surpassed by 25 of the 34 participating OECD countries (NCES, 2009). These
results are presented in Figure 1.
PISA scores in 2012, the most recent administration of the test, were equally disappointing,
with a nationally representative sample of 6,111 U.S students posting an overall average of 481,
which compares to an OECD overall average of 494 (NCES, 2012). Again, U.S. scores were
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 19
well below those of higher-achieving countries, such as Korea with 554, Japan with 536, and
Switzerland with 531 (NCES, 2012). And again, U.S. students’ were outperformed by most of
the participating countries: 23 of the 34 OECD nations had higher average scores than the U.S.
(NCES, 2012). These results are presented in Figure 1.
Figure1. PISA overall mathematics scores for Korea, Switzerland, and OECD average, as
compared to the United States. Scale: 0 to 1,000.
Another major international assessment, The Trends in International Mathematics and
Science Study (TIMSS), is administered by the International Association for Evaluation of
Educational Achievement (IAE), an independent international research cooperative with
approximately 70 member countries (NCES, 2013b). The TIMSS has been administered every
four years since 1995 to fourth and eighth graders, and unlike the PISA, the TIMSS assesses
students in more traditional, specific content areas: fourth graders are assessed in numbers,
geometric shapes and measures, and data display, while eighth graders are assessed in numbers,
algebra, geometry, and data and chance (NCES, 2013b). Results that combine the various
content area scores into a combined average are reported on a scale of 0 to 1000. Focusing in on
eighth graders reveals a picture of U.S. underachievement similar to that depicted by PISA
assessments. A nationally representative sample of 7,377 U.S. eighth graders in 2007 averaged
554
534
496
487
554
531
494
481
440
460
480
500
520
540
560
Korea Switzerland OECD Average United States
2009 2012
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 20
508 in math, slightly above the international average of 500 (NCES, 2007), yet well below high
performing nations such as Taipei at 598, Korea at 597, Singapore at 593, Hong Kong at 572,
and Japan at 570 (NCES, 2007).
In 2011, the most recent administration of the TIMSS, 10,477 nationally representative U.S.
eight graders again failed to make headway against the international math average of 500,
scoring a mere 9 points above it (NCES, 2011). Again, U.S. eighth graders were significantly
outperformed by Asian school systems: Korea averaged 613, Singapore 611, Taipei 609, Hong
Kong 586, and Japan averaged 570 (NCES, 2011). An examination of these results leads to an
obvious conclusion: in the mathematics portions of both the PISA and the TIMSS, American
middle and high school students achieve at levels just below or only slightly above international
averages, while at the same time, U.S. average math scores are far below the scores of the
highest performing school systems. Figures for both 2007 and 2011 are presented in Figure 2.
Figure 2. TIMSS combined average math scores, Taipei, Singapore, Hong Kong, Japan, and
the international average, as compared to the United States, 2007 and 2011.
598
593
572
570
508
500
609
611
586
570
509
500
450
470
490
510
530
550
570
590
610
630
Taipei Singapore Hong Kong Japan United States International
Avg.
2007 2011
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 21
Considering the lackluster overall math achievement of American students on these
international assessments, it is not surprising then that U.S. students also display mediocre
performance in several of the individual content areas that are assessed. Using the traditional
content clusters that are disaggregated by the TIMSS, one domain in particular, algebra, bears
examination. In the 2007 TIMSS, U.S. eighth graders produced results in algebra that mirror
their overall math achievement for that year, scoring an average of 507, compared to the
international algebra average of 500, while again, they fell well behind many Asian school
systems, such as Taipei at 629, Korea at 608, and Singapore at 591 (NCES, 2007). Results for
2011 were only slightly better: U.S. students averaged 512 in algebra, while the international
average was again 500 (NCES, 2011). And, once more, Asian systems far outpaced American
students, with Korea, Singapore, Taipei, and Hong Kong averaging 613, 611, 609, and 586,
respectively (NCES, 2011). In terms of algebra achievement, American eighth-graders replicate
their overall math performance, being barely able to distinguish themselves from the
international average and unable to match the results of the highest performing school systems.
The TIMSS algebra results for Taipei, Singapore, Korea and the United States are summarized in
Figure 3.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 22
Figure 3. TIMSS algebra results for Taipei, Korea, Singapore, and the international average,
as compared to the United States, 2007 and 2011.
A finer-grained examination of K-12 algebra performance, within the United States, and,
compared across multiple grade levels, helps further define the scope of this deficiency in
algebra achievement.
United States Algebra Achievement: A National Context
In the U.S. the National Assessment of Educational Progress (NAEP) is the largest ongoing,
nationally representative study of student achievement in a variety of subjects, including math,
reading, science and writing (NCES, 2013c). Known as the Nation’s Report Card, the NAEP
was first administered nationally in 1969 (NCES, 2013c). However, it wasn’t until 1990 that
administration of the NAEP took its present form. It is now administered continually, roughly
every two years, to fourth, eighth and twelfth-graders during a January to March testing window
(NCES, 2013c). In the most recent administration in 2013, roughly 376,000 fourth graders,
341,000 eighth graders and 92,000 twelfth graders from across the nation took part (NCES,
2013c).
629
608
591
507
500
609
613
611
512
500
450
470
490
510
530
550
570
590
610
630
650
Taipei Korea Singapore United States International Avg.
2007 2011
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 23
The math portion of the NAEP assesses students in five content areas: number properties and
operations; measurement; geometry; data analysis, statistics and probability; and algebra (NCES,
2013c). Scores are disaggregated for each of the content areas, and for each of the three tested
grade levels, and reported on a scale of 0 to 500 in fourth and eighth grade, and 0 to 300 in
twelfth grade (NCES, 2013c). The NCES further assigns these scaled scores to one of three
performance bands: basic, proficient, or advanced (NCES, 2013c).
NCES descriptions of what students should know and be able to do at each of the
performance bands evidence a broad continuum of mathematical performance. Generally,
students performing at the basic level show some evidence of understanding the mathematical
concepts and procedures for a particular content strand, while students scoring at the proficient
level can more consistently apply conceptual and procedural knowledge when solving problems
(NCES, 2013c). Meanwhile students performing at the advanced level can both apply such
knowledge, and also integrate, synthesize, and/or make generalizations about their mathematical
understanding (NCES, 2013c).
Based on the marked differences in mathematical understanding depicted by these
descriptions, it is somewhat disheartening then to examine the disaggregated algebra scores for
American fourth, eighth and twelfth graders. In 2009, for algebra, U.S. fourth-graders scored
basic with 244 (NCES, 2013c). In 2011 they scored basic, also with 244, and in 2013 they
scored basic again, with 245 (NCES, 2013c). The cut score (the minimum qualifying result) for
proficient, meanwhile, was 249 (NCES, 2013c), meaning that for those three administrations of
the NAEP, American fourth-graders were performing at the higher ranges of the basic
performance band, yet still not scoring high enough to evidence mathematical understanding that
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 24
could be categorized as proficient – namely, the ability to consistently apply their conceptual and
procedural algebraic knowledge.
Algebra scores for eighth and twelfth-graders were roughly comparable. In 2009, U.S.
eighth-graders scored basic at 287, in 2011 they scored basic at 289, and in 2013 they scored
basic again at 290, with a cut score for proficient of 299 (NCES, 2013c). Twelfth-graders in
2009 scored basic with a 155, did not participate in 2011, and in 2013 scored basic as well, with
155, compared to a cut score for proficient of 176 (NCES, 2013c). However, upon closer
examination, the scores for all three grade levels and the corresponding proficient cut scores
reveal a worrisome trend: U.S. students’ algebra scores tend to gravitate from the higher reaches
of the basic performance band in the elementary grades, to the middle and lower portions as they
matriculate into middle and high school. To wit, in fourth grade, U.S. students are scoring
approximately 5 points below the proficient cut line, in eighth grade they are scoring about 10
points below, and by the time they reach twelfth-grade, they are performing at roughly 20 points
below the line. Not surprisingly, these results are in line with the conclusions of the National
Mathematics Advisory Panel, which reported that the decline in overall math achievement
among U.S. students begins in late middle school, where for a majority of students, algebra
courses are introduced into the curriculum (2008).
What implications, then, can be drawn from the levels of performance outlined above? How
does the underperformance of U.S. students in this one content domain affect their ability to
make sense of the mathematics that make up the other content domains? And how does below-
average achievement in algebra affect students’ ability to be successful in college and/or career?
To begin to explore the answers to these questions, it is necessary to examine the role that
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 25
algebra plays in the K-12 mathematics curriculum, and its importance in the wider sphere of
academic and professional aspirations.
Overall Significance of Algebra
As defined by the Oxford English Dictionary algebra is, “The part of mathematics which
investigates the relations and properties of numbers or other mathematical structures by means of
general symbols” (“Algebra”, 2002, p.52). Unlike arithmetic, which involves performing
numeric operations that generally ignore the features of and relations between associated
mathematical expressions, algebra takes as its focus the abstract and structural representations of
numbers and the relations between and among them (Tolar, Lederberg, & Fletcher, 2008).
According to the National Council of Teachers of Mathematics (NCTM) algebra is a “…way of
thinking and a set of concepts and skills that enable students to generalize, model, and analyze
mathematical relationships” (2000, p.1). The NCTM further characterizes algebra as critical to
mathematical work, because it is a major content component and serves to unify the school
mathematics curriculum (2000). When the National Mathematics Advisory Panel was created by
executive order in 2006, its mission was to foster greater understanding and achievement in math
by American students (2008). Undergirding this mission was a mandate to support talent and
creativity, ensure continued American competitiveness, encourage innovation and help
government give students the education they need (NMAP, 2008). It is not surprising, then, that
given the importance of algebra to mathematics in general, the NMAP (2008) included as a
major focus of its research endeavor the essential components that constitute the K-12 algebra
curriculum and the course work that serves as prerequisite.
Taking algebra in middle and high school confers a number of benefits on students. It is
widely recognized as a gateway course for student access to, increased rates of enrollment in,
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 26
understanding of, and overall success in higher levels of study in mathematics, as well as science
(American Institute for Research, 2006; Haas, 2005; Kaput, 2000; Matthews & Farmer, 2008;
U.S. Department of Education, 1997; Walker & Senger, 2007; Wang & Goldschmidt, 2003). It
is seen as a predictor of high school graduation (American Institutes for Research, 2006).
Middle school Algebra I and high school Algebra II are also recognized as foundational
preparation courses for college entrance exams and, thus, as gateway courses to higher education
(Evan, Gray & Olchefske, 2006; Johnson, 2010; Schiller & Hunt, 2003; Swail et al., 2004).
Specifically, students who take algebra are more likely than those who do not to enroll in four-
year colleges (Adelman, 1994; Schneider et al., 1994) and to continue on to graduation
(American Institutes for Research, 2006). Furthermore, the NCTM concluded that competence
in algebra is an important adult skill, both for academic and vocational pursuits (2000).
Given then the importance of algebra, as a cornerstone of the overall math curriculum, as a
key to success in advanced math, as a predictor of achievement in high school, and as crucial to
college and career success, it would behoove one to investigate the underlying components of the
algebra curriculum itself. To wit, what are the major topics and content strands that constitute
middle and high school algebra?
Components of Secondary Algebra
The NCTM recommends that a prekindergarten through twelfth grade mathematics
curriculum enable students to function proficiently in four areas of algebra (NCTM, 2000).
These are, 1) understanding patterns, relations, and functions; 2) using algebraic symbols to
represent and analyze mathematical structures and situations; 3) representing and understanding
quantitative relationships through the use of mathematical models; and 4) the ability to analyze
change in various contexts (NCTM, 2000).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 27
The Common Core State Standards for mathematics specify four overarching areas of high
school algebra (NGA, 2010). The first involves identifying structure in expressions, which
includes the ability to interpret and create expressions to solve problems. The second is fluency
with polynomial arithmetic and rational functions. This relates to the ability to understand the
relations between zeroes and factors of polynomials, and using polynomials to solve problems.
The third area is the ability to create equations that describe numbers or relationships. The
fourth area involves reasoning with equations and inequalities. This covers the ability to solve
inequalities, equations and systems of equations, and proficiency in using reasoning in solutions
and the ability to justify that reasoning (NGA, 2010).
The National Mathematics Advisory Panel has created a list of what it deems to be the major
topics of school algebra. These are, 1) symbols and expressions, a topic which includes
polynomials and rational expressions; 2) linear equations, encompassing number lines and
graphing; 3) quadratic equations; 4) six major kinds of functions, (linear, quadratic, polynomial,
nonlinear, logarithmic and trigonometric); 5) the algebra of polynomials, including roots,
complex numbers, and binomial coefficients; and 6) combinations and finite probability (NMAP,
2008).
Given the breadth and scope of these descriptions of what constitutes secondary algebra, it is
important then to gain an understanding of the prerequisite knowledge and conceptual
understanding students in elementary mathematics must possess before they first encounter
algebra in middle school.
Foundations of Secondary Algebra
Stacey and MacGregor (1997) describe a number of specific, rather narrowly defined
conceptual understandings students must possess to be successful in middle and high school
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 28
algebra. These include an understanding of the quantity zero, and its relation to both addition
and multiplication, as well as an understanding of the quantity one, and its role in multiplication
(Stacey & MacGregor, 1997). The authors also posit that success in algebra requires that
students have a sound understanding of the concept of reciprocals.
A number of authors address the prerequisite understanding necessary before students are
ready to tackle algebra from a somewhat broader conceptual perspective (Bay-Williams, 2001;
Edwards, 2000; NCTM, 2008; Stacey & MacGregor, 1997). These include an understanding of
the properties of numbers (Stacey & MacGregor, 1997), and specifically, the commutative,
distributive, and associative properties (Edwards, 2000). In addition, students’ understanding of
numbers must extend beyond positive whole numbers to negative integers and positive and
negative rational numbers (Stacey & MacGregor, 1997). Stacey & MacGregor (1997) further
contend that students must have conceptual understanding of the equal sign and its connection to
the notion of equality. Also deemed important is a basic understanding of numerical patterns
(Bay-Williams, 2001), which includes the ability to describe and generalize about them (NCTM,
2008). Furthermore, the NCTM (2008) holds that as a precursor to algebra coursework, students
need to be able to identify mathematical relationships.
Viewing the conceptual understanding critical to success in secondary algebra from an even
broader, almost overarching perspective, the NCTM (2008) and the NMAP (2008) have distilled
what constitutes a host of specific knowledge and understandings into three main competencies.
These are fluency with whole numbers (NCTM, 2008; NMAP, 2008), selected aspects of
measurement and geometry, and fluency with fractions (NMAP, 2008).
Taking as an example just one of these core competencies, fluency with fractions, one can
see encapsulated in it many of the more specific competencies outlined previously. For example,
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 29
competence in fractions draws upon: the importance of understanding the relation of the quantity
one to multiplication (Edwards, 2000), the necessity of being able to grasp the significance of the
equal sign (Stacey & MacGregor, 1997), the need for an understanding of the properties of
numbers (Stacey & MacGregor, 1997; Edwards, 2000), the importance of an understanding of
and an ability to describe and make generalizations about patterns (Bay-Williams, 200; NCTM,
2008), the need for an ability to understand and manipulate reciprocals (Edwards, 2000), the
necessity of being able to identify relationships (NCTM, 2008), and the importance of an
understanding of whole numbers and integers that extends to rational numbers (Stacey &
MacGregor, 1997). As a result, because so many requisite conceptual understandings are
components of fractional fluency, it is no wonder then that the NMAP regards it as a core
foundation for success in algebra.
Fluency with Fractions as Critical Foundation for Success in Algebra
A number of other researchers also point to competency in fractions as a critical
precursor to the study of algebra (Fuchs et al., 2014; Hoffer, Venkataraman, Hedberg, & Shagle,
2007; Kieren, 1976; NMAP, 2008; Siegler & Pike, 2013). According to Kieren (1976), students
must master the concepts of fractional numbers in order to be prepared to learn algebraic
concepts. The NMAP (2008) contends that the teaching of fractions introduces students to two
of the integral aspects of algebra: manipulating numbers through symbolic notation and the
concept of generality. An analysis of two nationally representative data sets, one in Britain and
the other in the United States, indicated that elementary students’ understanding of fractions
predicted their knowledge of algebra in high school (Siegler & Pike, 2013). A recent survey of
1,000 Algebra I teachers identified fractions as one of the most significant weaknesses in
students’ preparation for algebra coursework (Hoffer, et al., 2007). Furthermore, Fuchs et al.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 30
(2014) characterized the study of fractions in elementary and middle school as foundational for
learning algebra.
So given then the consensus that exists as to the importance of a well-grounded
understanding of fractions for the study of secondary algebra, what is the recommended grade-
by-grade framework for instruction in fractions, from its introduction in elementary school to the
onset of formal algebra in eighth grade? An examination of two key artifacts, the CCSS for
Mathematics and the NMAP Benchmarks for the Critical Foundations, provides a roadmap for
answering that question.
A Grade-Level Framework for Instruction in Fractions
Both the NMAP and the CCSS delineate, by grade level, the content strands for teaching
fractions.
Third grade. Traditionally, students are introduced to formal instruction in fractions in
the third grade (NGA 2010; NMAP, 2008). Instruction focuses on the concepts of parts of a
whole, simple equivalence, and visual modeling (NGA, 2010). Later in the year, students learn
to understand whole numbers as fractions, locate fractions on a number line, and to compare
fractions with like denominators (NGA, 2010).
Fourth grade. In fourth grade, students build on the concepts from the previous year,
such as how to identify and represent fractions, both with models and on number lines (NMAP,
2008). Students begin to generate equivalents through multiplication, and learn to compare
fractions with unlike numerators or denominators (NGA, 2010). Students are introduced to
mixed numbers, and begin addition and subtraction of both fractions and mixed numbers with
like denominators (NGA, 2010). The second semester of fourth grade finds students multiplying
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 31
fractions by whole numbers, and beginning to grapple with decimal-fraction equivalence (NGA,
2010).
Fifth grade. Students continue their work comparing, adding and subtracting fractions
(NMAP, 2008), and move on to addition and subtraction with unlike denominators (NGA, 2010).
In fifth grade, students are introduced to the concept of fractions as division, and use this
understanding to yield fractions from the division of whole numbers (NGA, 2010). Whereas the
CCSS (2010) recommends that students in this grade are also introduced to multiplication of
fractions by fractions, the NMAP reserves this strand until sixth grade (NMAP, 2008).
Sixth grade. In sixth grade, instruction focuses on the multiplication of fractions by
fractions, as well as the beginning of instruction of the concept of division of fractions by
fractions, with learning activities that emphasize the underlying meaning of these two operations
(NGA, 2010; NMAP, 2008).
Seventh grade. In the last year before the introduction of formal algebra, instruction
focuses on consolidating understanding of the four operations as applied to fractions, and
features the introduction of negative fractions (NMAP, 2008). In addition, students extend their
understanding of addition, subtraction, multiplication and division to rational numbers, including
decimals (NGA, 2010).
From among this progression of recommended instruction involving fraction concepts,
perhaps one area stands out as most challenging for a large percentage of students. This involves
the division of fractions by fractions.
Division of Fractions
A significant body of research points to the division of fractions by fractions as a
particularly problematic endeavor for many students (Cengiz and Rathouz, 2011; Coughlin
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 32
2010/2011; Fendel, 1987; Ma, 1999; Ott, Snook, & Gibson, 1991; Payne, 1976; Sharp & Welder
2014; Tirosh, 2000; Trafton & Zawojewski, 1984). Sharp and Welder (2014) point to the
division of fractions by other fractions as a common area of struggle in seventh grade math.
Trafton and Zawojewski describe the understandings involved in division of fractions as a
“troublesome” endeavor for many students (1984, p. 20). Coughlin (2010/2011) labels the
division of fractions by fractions one of the most complex tasks in elementary math, while
multiple authors consider the procedure one of the most rote-like, mechanistic, and least
understood elementary math concepts (Cengiz and Rathouz, 2011; Fendel, 1987; Payne, 1976;
Tirosh, 2000). Ott, Snook, & Gibson (1991) posit that many students lack a clear understanding
of the meaning of division involving fractions, and that, furthermore, most students are incapable
of correctly interpreting and articulating the results of such calculations. Finally, Ma (1999,
p.55) contends that, "[d]ivision by fractions, the most complicated operation with the most
complex numbers, can be considered as a topic at the summit of arithmetic.”
An analysis of the cognitive mechanisms that must come into play when students are
confronted with problems involving division of fractions by fractions reveals the difficulties
inherent in this particular mathematical task, and helps to explain why it is so challenging for
such a significant percentage of U.S. students.
Cognitive Components of Fractions and Division
Generally, there is a developmental progression by which children are able to make sense
of fraction concepts. This begins with the concept of partitioning a unit whole or region into
component pieces of the same size (Sharp & Adams, 2002). Initially, students find success with
the skill of halving, wherein they seek to partition a whole into two, or any power of two (e.g. 4,
8, 16 etc.) pieces/denominators (Sharp & Adams, 2002). Halving leads directly to the next stage,
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 33
evenness, where students become comfortable with partitioning unit wholes into other even
pieces/denominators (Sharp & Adams, 2002). By the time children have developed the ability to
grapple with oddness, they are able to partition unit wholes or regions into any number of pieces,
which includes odd numbers of pieces/denominators (Sharp & Adams, 2002). In sum, children
begin work with fractions by contemplating values such as
1
2,
,
1
4
, or
1
8
, and progressing to the stage
where they have become comfortable understanding and representing fractions such as
1
3
,
1
6
, and
1
9
(Sharp & Adams, 2002). More importantly, what ultimately emerges from this developmental
process is the key understanding that, as the number of parts/denominators increases, the smaller
each becomes (Gabriel et al., 2013; Siegler & Pyke, 2013).
In addition to partitioning, there are several other cognitive sub-constructs involved in
making sense of fractions (Bottge, Ma, Gassaway, Butler, & Toland, 2014). One of these, ratio,
involves the idea that fractions represent a comparison between the numerator and denominator,
and if each is multiplied by the same quantity, that comparative relationship does not change
(Bottge, et al, 2014). The operator principle introduces students to the idea that a fraction such
as
3
5
can be seen as 3 x
1
5
of a unit whole, or as
1
5
x 3 unit wholes (Bottge, et al., 2014). The
quotient sub-construct requires students to understand that a fraction is not two separate values
(numerator and denominator), but rather a single value: the quotient that results from dividing
those two separate values (Bottge et al., 2014). Finally, measure involves the concept that a
fraction is both a number, and an interval between one point, usually zero, and another (Bottge et
al., 2014).
Through introduction to two of these sub-constructs, quotient and measure, children
grapple for the first time with the notion of rational numbers, defined as any numbers that can be
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 34
expressed as the quotient of two integers (positive and negative whole numbers), in the form
𝑎 𝑏
(Gabriel et al., 2013). Perhaps the major difficulty for students at this juncture in their cognitive
development is the notion of whole number bias (Ni & Zhou, 2005), which reflects the fact that,
up to this point, students in elementary mathematics have worked almost exclusively with whole
numbers (Siegler & Pyke, 2013). As a result they tend, naturally, yet often mistakenly, to ascribe
to rational numbers many of the properties of whole numbers (Siegler & Pyke, 2013).
This highlights an important distinction. Whole numbers and integers (which, unlike
whole numbers, also include negative numbers) represent discrete values (Gabriel et al., 2013).
This means that whole numbers and integers have unique successors: between any two
consecutive whole numbers or integers there is no other value (Gabriel et al., 2013). Rational
numbers, on the other hand, represent continuous, or dense, values (Vamvakoussi & Vosniadou,
2011). In other words, no rational number has a unique successor (Vamvakoussi & Vosniadou,
2011). Therefore, between any two rational numbers, there is an infinitude of other rational
numbers (Gabriel et al., 2013). In addition, any rational number can be represented by an
infinitude of other rational numbers (Gabriel et al., 2013). In other words the value represented
by
1
2
can also be expressed as
2
4
,
3
6
,
4
8
,
5
10
, and so on, to
∞
∞
(Gabriel et al., 2013).
All of these notions come into play when students begin to grapple with operations
involving fractions. The result is a multiplicity of often counterintuitive relationships. Common
denominators are required to add and subtract fractions, but, generally, are not required to
multiply and divide them (Siegler & Pyke, 2013). Only the numerator comes into play in
fraction addition and subtraction, while with division and multiplication, both numerator and
denominator are operated upon (Siegler & Pyke). And perhaps most challenging of all for
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 35
students, multiplication of fractions does not necessarily result in a greater value, while fraction
division does not necessarily yield a smaller value (Vamvakoussi & Vosniadou, 2011).
From a cognitive perspective then, and taking into consideration each of these aspects of
rational numbers, division of fractions by fractions may be the most counterintuitive, confusing
and challenging undertaking for elementary students in working with this subset of numbers.
Returning to whole numbers, models of division typically adopt either a partitive or a quotative
orientation (Koichu, Harel, & Monaster, 2013). In a partitive approach, also sometimes referred
to as sharing, a learner partitions the dividend into the number of groups indicated by the divisor,
and then counts the number of items in each group (Koichu et al., 2013). So in the problem 8÷4,
one would distribute the dividend, 8, among the 4 groups comprising the divisor, yielding 2
items in each group. In a quotative approach, commonly known as long division, the learner
simply counts the number of times the divisor, 4, can be subtracted from the dividend 8 (Koichu
et al., 2013). Thus, in the same problem, 8÷4, one would determine how many times the divisor
4 could be subtracted from the dividend 8, the result of which, again, is 2.
When applied to division of fractions, each of these approaches becomes much more
complicated (Li, 2008) and can, for many learners, appear nonsensical (Rizvi & Lawson, 2007).
For the problem
1
8
÷
1
2
, a partitive approach would require one to partition, or share, the quantity
1
8
among
1
2
groups, which, at face value, appears to be a non sequitur. In a quotative approach,
one would need to determine how many times
1
2
could be subtracted from
1
8
, an operation that
would appear to result in a negative number, whilst the correct answer must be positive. To
move beyond these apparent contradictions and fully grasp the conceptual nature of fraction
division, one needs to contemplate a fundamental difference between whole numbers and
rationals. To wit, whereas whole numbers and integers answer the question “how many”,
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 36
rational numbers are concerned with the question “how much” (Vamvakoussi & Vosniadou,
2011). Thus in the problem
1
8
÷
1
2
, it is much harder to conceptualize how many times
1
2
can be
subtracted from, or fit into,
1
8
, than it is to conceptualize how much of
1
2
can fit into
1
8
. This latter
approach leads to the solution,
1
4
. In other words,
1
4
of
1
2
fits into
1
8
, because
1
2
is the same as
4
8
, and
1
4
of those four-eighths (four parts, each part comprising one-eighth) is indeed one-eighth.
Rizvi and Lawson (2007), further distill these distinctions into a greater, overarching
framework – namely, that a conceptual understanding of fraction division requires students to
make connections between the concept of division and the concept of ratio, or rate. This
involves being able to perceive the multiplicative relationship that exists among the dividend, the
divisor, and the quotient (Rizvi & Lawson, 2007). This means that learners need to understand
that a problem such as 15 ÷ 3 represents a ratio between the dividend, 15, and the divisor, 3, and
that those two values are just one pair in an infinite set of other pairs connected by the same ratio
(Rizvi & Lawson, 2007). To understand what that ratio is, one needs to ask, “If 15 is for 3, then
how many are for 1?” (Rizvi & Lawson, 2007). This leads to the conclusion that, according to
the underlying ratio, or multiplicative relationship, 10 would correspond to 2, and 5 would
correspond to 1. In this manner, learners can conceptually come to an understanding that 15 ÷ 3
= 5. Extending this orientation to fraction division,
1
8
÷
1
2
can be analyzed as, “
1
8
is for
1
2
so how
much is for 1?” Doubling
1
2
would yield 1, so one would then simply also double
1
8
which results
in
2
8
, or
1
4
.
To conclude, in light of the level of complexity inherent in developing conceptual
understanding of fraction division, it is little wonder that this domain of elementary mathematics
poses such a challenge to a significant percentage of students. Obviously, instructional practice
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 37
vis-à-vis this important topic must go beyond superficial and mechanistic approaches so that
students can build the deep, conceptual understanding necessary to have success with these
concepts, fundamental as they are to algebraic understanding. One conclusion, then, is obvious:
an examination of current instructional practices vis-à-vis fraction division is necessary to
analyze their efficacy in building conceptual understanding.
How Division of Fractions is Taught
Among a number of instructional approaches for teaching division of fractions, perhaps
three are the most commonly used. These are the invert and multiply approach, the complex
fraction approach, and the common denominator approach (Li, 2009).
The invert and multiply approach. One common approach to division of fractions by
fractions is the invert and multiply procedure. In this approach, students are taught to invert the
divisor and then multiply it by the dividend. In other words,
𝑎 𝑏 ÷
𝑐 𝑑 becomes
𝑎 𝑏 ×
𝑑 𝑐 . The rationale
for this approach is that division and multiplication are inverse properties (Li, 2008).
Additionally, the inverse of any number, a,
is
1
𝑎 . Thus, in the invert and multiply procedure, both the dividend and the operation are
inversed, which serves to retain the mathematical balance of the original expression (Chabe,
1963; Li, 2008). The logic of this approach can be seen when dividing whole numbers. For
example, in the equation, 6 ÷ 3 = 2, one can obtain the same quotient, 2, by inverting the
dividend and inversing the operation. In other words, 6 ÷ 3 becomes 6×
1
3
=
6
3
= 2. This same
rationale extends to fractions. For example,
𝑎 𝑏 ÷
𝑐 𝑑 is the same as
𝑎 𝑏 ×
1
𝑐 𝑑 , an expression in which
the dividend is
𝑎 𝑏
and the divisor is
1
𝑐 𝑑 . If we then multiply both numerator and denominator of
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 38
the expression’s divisor,
1
𝑐 𝑑 , by the same value,
𝑑 𝑐 , the denominator’s inverse, we get an equivalent
fraction with a denominator of 1:
𝑎 𝑏 ×
1×
𝑑 𝑐 𝑐 𝑑 ×
𝑑 𝑐 =
𝑎 𝑏 ×
𝑑 𝑐 1
=
𝑎 𝑏 ×
𝑑 𝑐 (Li, 2008). Using a concrete
example,
2
3
÷
4
5
, the procedure would be the same. Thus,
2
3
÷
4
5
=
2
3
×
1
4
5
=
2
3
×
1×
5
4
4
5
×
5
4
=
2
3
×
5
4
1
=
2
3
×
5
4
=
10
12
.
The complex fraction approach. The complex fraction approach for dividing fractions
draws on the division properties of whole numbers (Li, 2008; Novillis, 1979). Just as 6 ÷ 3 is
the same as
6
3
, so can
𝑎 𝑏 ÷
𝑐 𝑑 be expressed as
𝑎 𝑏 𝑐 𝑑 . And much like the invert and multiply approach,
the complex fraction procedure makes use of equivalent fractions to transform elements of the
expression. Thus, the expression
𝑎 𝑏 ÷
𝑐 𝑑 becomes
𝑎 𝑏 𝑐 𝑑 . And, multiplying both the numerator and
denominator by the same value,
𝑑 𝑐 , the denominator’s inverse, yields an equivalent expression:
𝑎 𝑏
×
𝑑 𝑐 𝑐 𝑑 ×
𝑑 𝑐 . Since the two fractions in the denominator are inverses, they yield a product of 1. Thus, the
expression becomes
𝑎 𝑏 ×
𝑑 𝑐 1
which is equal to
𝑎 𝑏 ×
𝑑 𝑐 . Turning again to a concrete example,
2
3
÷
4
5
=
2
3
4
5
=
2
3
×
5
4
4
5
×
5
4
=
2
3
×
5
4
1
=
2
3
×
5
4
=
10
12
.
The common denominator approach. The common denominator approach builds on
students’ familiarity with least common multiples, a strategy commonly taught in elementary
math for adding and subtracting fractions with unlike denominators. In general, the common
denominator approach can be expressed as:
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 39
𝑎 𝑏 ÷
𝑐 𝑑 =
𝑎 ×𝑑 𝑏 ×𝑑 ÷
𝑐 ×𝑏 𝑑 ×𝑏 =
𝑎𝑑
𝑏𝑑
÷
𝑐𝑏
𝑏𝑑
=
𝑎𝑑 ÷𝑐𝑏
1
= ad ÷ 𝑏𝑑 =
𝑎𝑑
𝑏𝑑
(Gregg & Gregg, 1983; Li, 2008). Unlike
the previous two approaches, this technique does not rely on inverting the operation from
division to multiplication. Using, once again, a concrete example,
2
3
÷
4
5
, this approach would
involve identifying the least common multiple of the two denominators, 3 and 5, which would be
15. Thus,
2
3
÷
4
5
=
2×5
3×5
÷
4×3
5×3
=
10÷12
15÷15
=
10÷12
1
= 10 ÷ 12 =
10
12
.
In the hands of a competent instructor, any of these three techniques could serve as a framework
within which to build the deep conceptual understanding necessary for students to make meaning
with respect to division of fractions. Unfortunately, a number of studies point to inadequacies on
the part of teachers as a leading cause for the difficulties many students face when dealing with
the division of fractions by fractions (Gearhart & Saxe, 2004; Holmes, 2012; LeSage, 2012; Ma,
1999; Matthews & Ding, 2011; Shulman, 1987; Tirosh, 2000).
Teacher Deficiencies: Instruction in Division of Fractions
Studies detailing inadequacy on the part of math teachers describe three underlying
problems: inadequate teacher subject matter knowledge, teachers’ failure to grasp student
understanding, and poor instructional practice.
Inadequate teacher subject matter knowledge. LeSage (2012) found that many
teachers of fraction division lack conceptual understanding of rational numbers, and of the
challenges these pose to students, and as a result, teachers pass these misunderstandings on to
students. Holmes (2012) found a lack of deep subject matter knowledge among teachers of
division of fractions, while Shulman (1987) found teachers of fractions lack not only subject
matter knowledge, but also what he termed pedagogical content knowledge, which is a mental
storehouse of topic-specific examples and clarifications. Ball, Thames, and Phelps (2008) further
delineated such knowledge into two sub-categories: one of which they term knowledge of content
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 40
and teaching. A significant number of mathematics teachers exhibit deficiencies in this type of
knowledge, which is a measure of the degree to which teachers’ knowledge about mathematics
effectively interacts with their design of instruction (Ball et al., 2008).
Teacher inability to gauge student understanding. Shulman (1987) also addressed the
inability of teachers of math to consider the needs and interests of their students. He posited that
ineffective teachers of subjects such as the division of fractions are unable to blend content and
pedagogy into representations that conform to the idiosyncratic abilities of their students
(Shulman, 1987). Tirosh (2000) found that many teachers of division of fractions lack
knowledge of their students’ misconceptions, which can include mistakes that are algorithm-
based, student beliefs about rational numbers that draw mistakenly from the properties of whole
numbers, or lack of prior knowledge. Ball et al. (2008), again drawing on and refining Shulman’s
(1987) notion of pedagogical content knowledge, posited that many teachers lack knowledge of
content and students, an ability to understand student interests and motivations, as well as the
ways in which student thinking emerges, is incomplete, or is marked by misconceptions. .
Inappropriate instructional practice. Matthews and Ding (2011) described
misconceptions on the part of teachers in formulating problems. Specifically, they found that
many teachers adopted a measurement (how many groups) perspective when creating division of
fractions word problems, when a partitive (how many in each group) approach was called for
(Matthews & Ding, 2011). According to Ma (1999), most U.S. math teachers of fraction
division fail to even use story problems, which are deemed critical for helping students make
connections between the underlying concepts and their own lives. Ma (1999) attributed this
failure to incomplete understanding on the part of teachers as to the meaning of fraction division,
and an inability to connect this topic to other models in math.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 41
Such teacher deficiencies become an even more critical issue when considered from the
perspective of the current movement for the implementation of standards-based curricula, such
as those embodied in the Principles and Standards for School Mathematics and the Common
Core State Standards. As such, it is instructive to consider current models for teacher training.
How Teachers are Trained
To enable teachers to implement standards-based curricula, school districts are
increasingly required to pay greater attention to the manner in which they support all teachers,
whether they are new to the profession or seasoned veterans (Darling-Hammond, 2004; Polly et
al., 2014). Crucial to this endeavor has been an emphasis on providing ongoing professional
development that builds teachers’ capacity (Darling-Hammond, 2004). Such professional
development has been found to be most effective when it develops teachers’ knowledge of
content and pedagogy (Garet, Porter, Desimone, Birman, & Yoon, 2001; Heck, Banilower,
Weiss, & Rosenberg, 2008), provides teachers with a sense of ownership of their professional
learning (Loucks-Horsley, Stiles, Mundry, Love, & Hewson, 2010), and features ongoing
support in the form of both a workshop model as well as classroom-centered experiences
(Loucks-Horsley et al., 2010; Polly & Hannafin, 2010). Because it has been found that adults are
largely incapable of assuming responsibility for and enacting their own professional learning
(Downey, Steffy, English, Frase, & Poston, (2004), school districts often rely on subject matter
experts in content knowledge and pedagogy to conduct their professional development initiatives.
However, according to Feldon and Clark (2006), when experts self-report on the
knowledge and skills critical to their expertise, the results are often incomplete, inaccurate, and
contain errors and omissions that can impede the ability of trainees to perform a target task. It has
been found that the extent of such expert omissions can reach 70% of the knowledge and skills
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 42
crucial for replicating expert performance (Feldon & Clark, 2006). As a result, trainees often seek
to fill the voids created by such incomplete information with information of their own, which is
often riddled with fallacies and misconceptions (Feldon & Clark, 2006).
Summary
There is a record of underachievement among U.S. students in algebra, with origins that
can be traced to the middle school grades. Furthermore, the difficulty inherent in understanding
the conceptual nature of mathematics in general, and algebra and fractions in particular, makes
teaching these concepts problematical for a significant number of teachers. Because the current
model for teacher professional development relies on subject matter experts, and due to the
extent of knowledge omissions by such experts during training of novices, this study used
cognitive task analysis (CTA), a method of eliciting expert knowledge, to capture the expertise
of teachers proficient in teaching division of fractions, a foundational competency for algebra
achievement.
To more fully understand how cognitive task analysis makes possible the capture of such
expert knowledge, it is necessary first to examine the nature of knowledge itself, as well as the
manner in which knowledge becomes automated. Furthermore, it is important to explore how
expertise is characterized, how expert knowledge is developed, and the implications of expertise
vis-à-vis knowledge omissions.
Knowledge Types
The purpose of education is to replicate knowledge (Jackson, 1985). Researchers have
categorized knowledge into two types: declarative and procedural (Ambrose, Bridges, DiPietro,
Lovett, & Norman, 2010; Anderson & Krathwohl, 2001; Clark & Estes, 1996)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 43
Declarative Knowledge
Declarative knowledge is information about why, what, and that, and is typified by its
conscious quality and the speed with which it can be learned and modified (Clark & Estes,
1996). Most knowledge comes into cognition in declarative form, is committed to long-term
memory (Anderson & Finchman, 1994), and serves to help humans handle novel tasks (Clark &
Estes, 1996).
Procedural Knowledge
Procedural knowledge is knowing how and when (Anderson & Krathwohl, 2001),
consists of IF/THEN propositions (Anderson, 1982), is goal-oriented, and promotes problem
solving (Corbett & Anderson, 1995). Conditional knowledge, which is a type of procedural
knowledge, involves knowledge of when, as well as why, provides a rationale for various actions
(Paris, Lipson, & Wixson, 1983), and modulates the fact-to-action (declarative-to-procedural)
process (Anderson, 1982).
Automaticity
Automaticity is the process by which declarative and procedural task knowledge becomes
automated and unconscious in nature, as a result of repeated performance and deliberate practice
(Ericsson, Krampe, & Tesch-Romer, 1993). Four stages of automaticity have been identified: a
cognitive stage, in which a learner can complete a task with initial instruction, an associative
stage, in which the learner works through the procedure and acquires relevant declarative
knowledge and requires less cueing, an autonomous stage, in which verbal cueing is no longer
necessary, and a fourth stage, in which subject matter experts (SMEs) add their own innovations
(Anderson, 1996; Ericsson, Krampe, & Tesch-Romer, 1993).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 44
Automated knowledge helps to reduce cognitive overload, by freeing up limited working
memory (Kirschner, Sweller, & Clark, 2006), which enables the expert to attend to novel tasks
and deploy strategies to solve problems (Clark, 1999). However, automated processes often
initiate without prompting, and then run to completion (Feldon, 2007). This leads to a double-
edged sword: although the expert has ample working memory, automated processes are resistant
to change, and require considerable monitoring in order to modify or eliminate them (Clark,
2008; Wheatley & Wegner, 2001).
Expertise
An expert is one who is uncommonly accurate and reliable in making judgments, displays
superior skill and economy of effort in task completion, and is able to deal effectively with
certain types of rare or problematic cases (Chi, 2006).
Characteristics of Experts
Expertise is typified by extensive, highly structured domain knowledge, a command of
effective strategies for domain-specific problem solving, and expanded working memory within
which elaborated schemas allow for the rapid storage, retrieval and manipulation of information
(Chi, 2006). Viewed through the lens of a relative approach, expertise does not require innate
talent per se; rather it is a level of proficiency that novices can achieve (Chi, 2006). However,
expertise is task specific, and does not transfer from domain to domain (Bedard & Chi, 1992).
Experts outperform novices in the essential skills of a domain (Feldon, 2007). These
skills are a product of experience-based domain knowledge (Feldon, 2007), which can include
principles, concepts, and connections (Bedard& Chi, 1992). Experts can draw upon knowledge
structures that facilitate the recall of problem states and allow them to engage in forward
reasoning (Bedard & Chi, 1992).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 45
Experts view problems differently than novices (Bedard & Chi, 1992). They can see
beyond function and simple schemas, can identify relevance among a number of cues, instill
meaning in ill-defined problems, and are able to select and match strategies to problems (Bedard
& Chi, 1992).
In general, experts possess a superior functional capacity of working memory (Feldon,
2007), through selective encoding of relevant information and mechanisms (Ericsson &
Lehmann, 1996). They can more rapidly attend to, encode, manipulate, and decode domain-
relevant information in working memory than can non-experts (Feldon, 2007). In addition, this
superior memory function is found in both short and long-term working memory (Ericsson &
Lehmann, 1996).
Building Expertise
Expertise is acquired as a result of continuous and deliberate practice. Alexander’s
(2003) Model of Domain Learning posits that its development involves a journey from
acclimation, whereby learners adapt to an unfamiliar domain and task, to competence, whereby
learners demonstrate a foundational body of knowledge, cohesive and principled in structure, to
proficiency, whereby the expert has developed a synergy among the various cognitive
components of his expertise.
Crucial to the development of expertise is deliberate practice, which is characterized by
repeated performance of the task, as well as an innate motivation on the part of the learner to
attend to the task and exert effort to improve performance (Ericsson, Krampe, & Tesch-Romer,
1993). In addition, the learner needs immediate informative feedback and knowledge of task
performance during practice (Ericsson, Krampe, & Tesch-Romer, 1993). It is also important to
limit daily practice time, to avoid exhaustion (Ericsson, Krampe, & Tesch-Romer, 1993).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 46
One hindrance to building expertise presents itself as learners reach the automated phase
of task competence, because performance at this point often reaches a stable plateau, and no
further improvement in performance occurs (Ericsson, K. A., 2004). The challenge for the
learner is to avoid such arrested development through an orderly and deliberate approach to
practice that includes task monitoring, planning and analysis, with the aim of identifying changes
that can be integrated into one’s performance (Ericsson, K.A., 2004).
Consequences of Expertise
As new knowledge becomes automated and unconscious, experts are often unable to
completely and accurately recall the knowledge and skills that comprise their expertise (Chi,
2006; Feldon, 2007). Experts often are overly-confident, overlook details, make inaccurate
predictions and offer faulty advice (Chi, 2006). In addition, as their skills improve, experts’ self-
report errors and omissions tend to increase, while their accuracy of introspection decreases
(Feldon, 2007). Furthermore, because experts’ schemas are adapted to problem solving, they can
fail to articulate relevant cues, and can unintentionally fabricate consciously reasoned
explanations for their automated behaviors (Feldon, 2007).
Expert Omissions
Because automaticity and the accuracy of self-reporting have been found to be negatively
correlated, experts in an instructional role may unintentionally leave out information that learners
must master when learning procedural skills (Feldon, 2004). In fact, experts may leave out up to
70% of the critical information necessary to perform a task, forcing novices to fill in the blanks
with error-prone trial-and-error methods (Clark et al., 2011). The automated nature of
knowledge causes procedural steps to blend together in experts’ mind and makes it difficult for
them to share the complex thought processes of technical skill execution (Clark et al., 2011).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 47
Cognitive Task Analysis (CTA)
Definition of CTA
Cognitive task analysis refers to a variety of methods for eliciting and representing the
knowledge and skills of expert practitioners when they perform complex tasks and solve difficult
problems. CTA, an extension of traditional task analysis, uses a variety of interview and
observation strategies to identify the knowledge, thought processes and goals that underlie
observable task performance, as well as overt and covert cognitive functions (Chipman, 2000;
Clark, Feldon, van Merrienboer, Yates, & Early, 2008).
Brief History of CTA
CTA can be traced to the advent of applied psychology in the 1880s, the growth of time
and motion studies in the early 20
th
century, and the study of complex machine systems in the
mid-20
th
century (Militello & Hoffman, 2008). It emerged as a result of the study of social,
psychological and cognitive activities in the workplace in the 1960s, and became prominent in
the 1980s when the study of knowledge acquisition fueled a demand for expert systems and other
applications of artificial intelligence (Hoffman & Woods, 2000).
CTA Methodology
A number of researchers have identified discrete stages through which a typical cognitive
task analysis would proceed (Chipman, Schraagen, and Shalin, 2000; Clark, Feldon, van
Merrienboer, Yates, & Early, 2008). These are (a) a preliminary, data collection phase; (b)
identification of knowledge representations; (c) application of knowledge elicitation methods;
(d) ) expert review and analysis of elicited knowledge; and (e) formatting of results for the
desired application (Chipman, Schraagen, and Shalin, 2000; Clark, Feldon, van Merrienboer,
Yates, & Early, 2008).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 48
Taxonomies of Knowledge Elicitation Techniques
Knowledge elicitation, a subset of knowledge acquisition, is the process of extracting the
domain specific knowledge that underlies human performance (Cooke, 1994). One researcher
has identified four categories: 1) observations of task performance, 2) various interview
techniques, 3) process tracing of sequential behavioral events, and 4) methods to elicit the
structure of domain-related concepts (Cooke, 1994; Cooke, 1999). Wei and Salvendy (2004)
identify a fifth family – formal models. However, since these typologies are based on processes,
analysts may struggle to select an appropriate CTA approach if the desired result is a particular
type of knowledge (Yates, 2007).
Pairing Knowledge Elicitation with Knowledge Representation/Analysis
Yates (2007) identified the most frequently used CTA methods and the knowledge types
associated with them. The author found it more appropriate to examine CTA as a pairing of
knowledge elicitation with an analysis/representation technique, and to classify CTA methods in
terms of desired outcome, rather than process (Yates, 2007) .
Effectiveness of CTA
Cognitive task analysis is regarded as a necessary component of research in complex
cognitive work, since its use allows for the identification of the explicit and implicit knowledge
of experts, which supports effective and efficient training (Hoffman & Militello, 2009). It is
seen as an optimal method for capturing knowledge because it emphasizes aspects of tasks that
are important to the learner, facilitates understanding of abstract knowledge across domains, and
provides a framework for abstract problem solving (Means & Gott, 1988). In educational and
work settings, CTA assists researchers in identifying subtle skills, perceptual differences, and
procedures (Crandall, Klein, & Hoffman, 2006).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 49
Research has shown that using cognitive task analysis is more cost effective and efficient
than other models (Clark, Feldon, van Merrienboer, Yates, & Early, 2008; Clark & Estes, 1996).
CTA can reduce total training days by nearly half (Clark, Feldon, van Merrienboer, Yates, &
Early, 2008), while producing results comparable to conventional training methods that take
longer (Clark & Estes, 1996).
Benefits of CTA for Instruction
Several studies indicate that instruction based on cognitive task analysis is superior to
other instructional models (Hoffman & Militello, 2009; Crandall, Klein, & Hoffman, 2006;
Clark, Yates, Early, & Moulton, 2010). CTA can allow the data analyst to identify the explicit
and implicit knowledge of experts, including domain content, concepts and principles, schemas,
reasoning and heuristics, and mental models, all of which can support effective and efficient
cognitive training, scenario design, cognitive feedback, and on-the-job training (Crandall, Klein ,
& Hoffman,2006; Hoffman & Militello, 2009). In addition, CTA leads to guided instruction that
is more structured and successful than learning that is based on media, games, or discovery
(Clark, Yates, Early, & Moulton, 2010). Overall, CTA has been shown to be effective in
capturing expertise and informing instruction in a wide range of professions, including health
technicians (Clark, 2014), nursing (Crandall & Gretchell-Reiter, 1993), physicians (Fackler et
al., 2009), and education (Crandall, Klein, & Hoffman, 2006).
Conclusion
In summary, educators face inherent difficulties in understanding the conceptual nature of
mathematics in general, and algebra and fractions in particular. As a result, providing effective
instruction in these concepts poses a problematical endeavor for a significant number of teachers.
In response, the standard paradigm for remediating teacher inadequacies in these areas has been
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 50
predicated on professional development that relies primarily on subject matter experts. However,
and as previously outlined, researchers point to an extent of knowledge omissions by such
experts during training of novices that is considerable (Clark & Feldon, 2004). Therefore, this
study proposes to capture the expertise of teachers proficient in teaching the division of fractions
by fractions through the use cognitive task analysis (CTA), a method of eliciting expert
knowledge, that research shows to be superior to other models in providing training ((Hoffman
& Militello, 2009; Crandall, Klein, & Hoffman, 2006; Clark, Yates, Early, & Moulton, 2010).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 51
CHAPTER THREE: METHODOLOGY
Overview
The purpose of this study was to conduct a Cognitive Task Analysis to determine the
knowledge and skills, represented by the action and decision steps, as well as other knowledge,
that expert middle school math teachers (subject matter experts, or SMEs) employ when they
describe how they teach the division of fractions by fractions. As explicated in Chapter Two of
this study, the division of fractions comprises a high-leverage prerequisite for competency in
algebra, which itself, and also, as detailed in Chapter Two, is a core prerequisite for achievement
in secondary level mathematics overall. Based on the definition of expertise, the researcher
assumed that these subject matter experts possessed highly automated declarative and procedural
knowledge that was often unconscious. Thus, the researcher assumed it would be difficult for
these SMEs to describe accurately and in detail the what, why, how, and when of teaching the
division of fractions by fractions.
To wit, the research questions that guided the study were:
1. What are the action and decision steps that expert middle school math teachers
recall when they describe how they teach the division of fractions by
fractions?
2. What percent of action and/or decision steps, when compared to a gold
standard, do expert middle school math teachers omit when they describe how
they teach the division of fractions by fractions?
Participants
This study identified teachers from three Southern California school districts who are
expert in teaching division of fractions by fractions. Each of these teachers had at least five
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 52
years of recent and continual experience teaching division of fractions by fractions, and this
experience was recognized as successful by school and/or district administrators. The researcher
explained to these administrators that such successful experience would entail deep content
knowledge, best practices pedagogy, assessment-driven practice, and an ability to instill in
students a balance of conceptual understanding and procedural fluency. Additionally, each of
these subject matter experts (SMEs) have experienced a wide variety of contexts, settings,
problems, and applications in their work with teaching division of fractions by fractions, and did
not have experience as trainers or instructors in teaching division of fractions by fractions (Yates,
2007). They were also selected on the basis of being verbal, cooperative, and available and
willing to participate in audio-recorded, in-person interviews (Yates & Clark, 2011), as
determined by the researcher in initial phone conversations.
The researcher made an effort to recruit a fourth SME for the purposes of reviewing the
final Gold Standard Protocol. However, due to the great difficulty the researcher encountered in
identifying teachers with expertise in division of fractions, the researcher was unable to secure
this fourth expert; as a result each of the three SMEs were asked to review the Preliminary Gold
Standard Protocol as a complete and accurate aggregation of their individual protocols.
Data Collection for Question 1: What are the action and decision steps that expert middle
school math teachers recall when they describe how they teach the division of fractions by
fractions?
In order to elicit experts’ knowledge of teaching division of fractions by fractions, this
study adopted the five-stage protocol of cognitive task analysis described by Clark et al. (2008)
in which researchers:
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 53
1. Collect preliminary knowledge through unstructured interviews, observations,
and document analysis.
2. Identify knowledge representations, through the use of flow charts, concept maps
or semantic nets.
3. Apply focused knowledge elicitation techniques which can vary based on the
type of knowledge required.
4. Analyze and verify the data acquired through coding and review by SMEs.
5. Format the results for the desired application.
In this study, this five-stage process was implemented as outlined in the following sections.
Phase 1: Collect preliminary knowledge. The researcher is an elementary school
teacher, with a general knowledge of mathematics instruction. Because the study involved
middle school mathematics, the researcher conducted a thorough literature review to collect
preliminary information and build a more general understanding of teaching division of fractions
by fractions.
Phase 2: Identify knowledge representations. Conducting the literature review on
Cognitive Task Analysis allowed the researcher to gain an understanding of the nature of both
declarative and procedural knowledge. The researcher also participated, with other researchers
and under the guidance of a senior researcher, in practice activities designed to elucidate the
differences between these two knowledge types. These practice activities also helped the
researcher to identify action steps, decision steps, and conceptual knowledge types, such as
concepts, processes, and principles. Familiarity with these knowledge types and action/decision
steps was critical to creating the interview protocol.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 54
Phase 3: Apply knowledge elicitation techniques.
Instrumentation. This study employed the semi-structured interview protocol described
by Clark, Pugh, Yates, Early and Sullivan (2008) which is based on a series of questions aimed
at eliciting (a) conditions/indications; (b) processes; (c) action and decision steps; (d) standards
of time and quality; (e) equipment needed; (f) pedagogical reasoning; (g) conceptual
understanding, as exemplified by the vocabulary and symbols an expert teacher would need to
know. The action and decision steps constitute the critical information a novice would need to
perform the target task. Action steps begin with a verb, and state what a person should do. An
example would be, “Direct students to compare their answers with a partner.” Decision steps
take the form of IF/THEN propositions, and usually provide two alternative courses of action.
An example would be “IF student provides correct answer, THEN go to step 3.2. IF students
does not provide correct answer, THEN provide remediation after class.”
This interview protocol was an adaptation of the critical decision method (Hoffman,
Crandall, & Shadbolt, 1998), or CDM, that employs cognitive probes to understand how experts
assess situations and make decisions during task execution. The protocol, which appears in
Appendix A, was also based on the PARI (precursors, actions, results, interpretations)
methodology that involves interviewing experts about the aspects of a task that are associated
with declarative, procedural and strategic knowledge (Hall, Gott, & Pokorny, 1995).
Interviews. Following IRB approval from the University of Southern California, three
SMEs in teaching division of fractions by fractions were interviewed according to the semi-
structured protocol described above. Each SME was interviewed and, with SME permission,
audio-recorded for approximately 90 minutes. A follow-up interview with each SME also lasted
for approximately 90 minutes, and a final phone conversation to review the preliminary protocol
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 55
lasted approximately 30 minutes. Thus, in aggregate, the researcher spent about three and a half
hours in conversation with each SME. The interview protocol was designed to capture the
explicit action steps, as well as the implicit, non-observable decision steps, judgments and other
cognitive processes that are associated with expert instruction in division of fractions by
fractions.
Phase 4: Data analysis. Audio recording of the interviews, coupled with verbatim
transcription, provided by a professional transcriber, gave the researcher the ability to acquire,
through multiple read-throughs of the transcripts, a deeper, richer understanding of what was
revealed in the interview by the subject matter expert, as opposed to not recording and
transcribing, which would have required the researcher to rely on memory only.
Coding. Once each interview recording was transcribed, the transcripts were coded
according to an a priori scheme, based on Clark’s (2006) concepts, processes and principles
method. Examples of the codes used include “main procedure, “action step”, and “decision
step.” The coding scheme was also used for calculating inter-rater reliability and is included as
part of Appendix B.
Inter-rater reliability. Both the study researcher and a fellow researcher independently
coded the transcription of one of the SME interviews to determine consistency of coding
between researchers. The two coded transcripts were then compared for inter-rater reliability.
An inter-rater reliability was calculated as a percentage of agreement between the two coders,
and appears in Appendix B. According to Hoffman, Crandall, and Shadbolt (1998), an inter-
rater reliability of 85% or higher indicates that the coding process is consistent and reliable
among different coders. The results of the inter-rater reliability, expressed as a percentage, was
93%, and are also included in Chapter Four.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 56
SME protocol and verification. Each coded interview transcript was used to generate a
step-by-step cognitive task analysis protocol for teaching division of fractions by fractions. Each
protocol was then reviewed by the SME from whose interview transcript it was generated.
Phase 5: Formatting the results.
Gold standard protocol. Once each subject matter expert reviewed, verified, and as
necessary, corrected their individual protocol, a synthesis of the three individual protocols was
used to generate an aggregate Preliminary Gold Standard Protocol (PGSP). This aggregate
protocol was developed by first identifying the clearest, most complete, and most articulately
worded individual protocol. Each action and decision step from the other two individual
protocols was then compared to those of this ideal protocol. If any language, action or decision
steps were found to be the same across individual protocols, then they were attributed to both
SMEs. If any language, action, or decision steps were found to be more accurate or complete
compared to either of the other two protocols, then the action or decision step was modified to
reflect that and attributed to both SMEs. If any action or decision step was unique, and not listed
in the ideal individual protocol, then it was added to the ideal individual protocol, to facilitate the
building of an aggregated Preliminary Gold Standard Protocol. See Appendix C for a description
of the steps involved in creating a GSP. The completed Preliminary Gold Standard Protocol was
then reviewed by each of the three SMEs for their review and, when necessary, correction.
Summary. The five phase process described above is also referred to as the 3i = 3r
method (Flynn, 2012), which stands for three initial interviews and three reviews. A visual
representation of this method appears in Figure 4.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 57
Figure 4. Visual representation of the 3i + 3r CTA method.
Data Analysis for Question 2: What percent of action and/or decision steps, when compared
to a gold standard, do expert middle school math teachers omit when they describe how they
teach the division of fractions by fractions?
Spreadsheet analysis. Once the Gold Standard Protocol was finalized, each of the action
and decision steps of this final version was transferred to an Excel spreadsheet. Each individual
SME protocol was analyzed and compared to the GSP action and decision steps listed in the
spreadsheet. Four spreadsheet columns, one each for the GSP and the three individual protocols,
allowed the researcher to visually represent and assess the degree to which the individual
protocols either conformed to the action and decision steps in the GSP or contained omissions.
Specifically, each row of the spreadsheet served to represent one of the action or decision steps
Researcher conducts semi-structured interviews
SME A SME B SME C
Individual Protocol Individual Protocol Individual Protocol
SME Protocol Review SME Protocol Review SME Protocol Review
Preliminary Gold Standard Protocol (PGSP)
SME A PGSP Review SME B PGSP Review SME C PGSP Review
Gold Standard Protocol (GSP)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 58
revealed by the GSP. If an individual SME protocol contained a GSP action or decision step, a
“1” was placed in the corresponding cell for that SME; an omission was represented by a “0”.
Using the spreadsheet, the researcher was able to calculate the number and percentage of total
agreements and total omissions of each individual protocol.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 59
CHAPTER FOUR: RESULTS
Overview of Results
This study focuses on the declarative and procedural knowledge of three middle school
math teachers, expert in teaching the division of fractions by fractions. This knowledge, which
was captured using CTA methodology, takes the form of objectives, standards, cues, conceptual
understanding, and action and decision steps. This chapter presents a data analysis of the results
of that CTA study, organized by research question.
Research Questions
Question 1
What are the action and decision steps that expert middle school math teachers recall
when they describe how they teach the division of fractions by fractions?
Inter-rater reliability. As described in Chapter Three, the researcher and a researcher
colleague derived inter-rater reliability through independent coding of one of the SME interview
transcripts. Following coding, the two researchers tallied the number of coded items that were in
agreement and divided that number by the total number of coded items. This inter-rater
reliability was 93%. The tally sheet used to compute this percentage appears in Appendix B.
Based on the relatively high inter-rater agreement represented by this value, the researcher then
coded the remaining two SME interview transcripts without the assistance of a second coder, and
then created an individual protocol for each SME.
Flowchart analysis. The researcher next used SME A’s individual protocol to create a
flowchart of the action and decision steps captured in the interview. The goal of the
flowcharting process was to determine whether the action and decision steps recalled by the
SME in the initial interview represented a logical progression, and additionally, whether there
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 60
were any decision steps that did not lead to an appropriate action step. A number of questions
vis-à-vis the sequence of steps captured in the initial interview revealed themselves during
flowcharting. Specifically, several decision steps from the initial interview did not result in a
progression to an action step, effectively terminating the progression of the protocol. The
flowchart appears in Appendix D. The researcher then addressed these questions during the
follow-up interview, which informed the creation of the final SME protocol. This process of
initial interview, development of initial protocol, flowcharting, follow-up interview, and creation
of final protocol for SME A allowed the researcher insight into how to more effectively conduct
both the initial and follow-up interviews for SMEs B and C.
Gold standard protocol. As explained in Chapter Three, the researcher analyzed each
of the three SME individual protocols to create an aggregate, preliminary gold standard protocol
for teaching the division of fractions by fractions. This analysis revealed a continuum of
protocols, from most complete to least complete. SME B’s protocol was found to be the most
complete, SME C’s protocol was identified as slightly less complete, while SME A’s protocol
was found to be the least complete. As a result, SME B’s individual protocol served as the
foundational protocol in developing a preliminary gold standard. Each of the action and decision
steps for SME B were compared to those for SME C. Where those steps were identical in
meaning, attribution was given to both SMEs. In those cases where an action or decision step in
SME C’s protocol was not present in SME B’s, then that step was added to the foundational
protocol, and attribution was given to SME C only. Once this process was complete, the
individual protocol for SME A was also similarly aggregated into the emerging preliminary gold
standard protocol. An example of how each of the SMEs contributed to the preliminary gold
standard protocol is illustrated in Figure 5.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 61
Figure 5. A sequential representation of the process of aggregating action steps from SME C and
SME A onto an action step from SME B’s foundational individual protocol, resulting in an
action step as it appears in the GSP.
Once the preliminary gold standard protocol was developed, the researcher emailed an
explanation of the aggregation process along with a copy of the preliminary protocol to SMEs A,
B, and C. Once the three SMEs had had time to review the preliminary protocol, the researcher
contacted each of them individually by phone, for approximately 30 minutes, to discuss the
SME B – Action Step:
Review that we can
convert a fraction
division problem into
a complex fraction (B)
SME C – Additions (in
Bold) to SME B’s Action
Step:
Review that converting a
fraction division problem
into a complex fraction
results in multiplying the
first fraction by the
reciprocal of the second
fraction, a fact we also
discovered when we
divided fractions by
whole numbers using
pictorial representations
(B, C)
SME A – Additions (in
italics) to PGSP (As step
reads in final GSP)
Review that converting a
fraction division problem
into a complex fraction
results in multiplying the
first fraction by the
reciprocal of the second
fraction, a fact we also
discovered when we
divided fractions by whole
numbers and whole
numbers by fractions using
manipulatives and pictorial
representations (A, B, C)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 62
protocol and then make additions, modifications, and deletions based on their input. The result is
the final gold standard protocol.
This final gold standard protocol, which is attached as Appendix E, is a distillation of the
action and decision steps that expert middle school math teachers employ in teaching the division
of fractions by fractions, and serves, furthermore, as the response to Research Question One.
This final protocol consists of a sequence of twelve procedures that the three SMEs who
participated in this study identified as necessary for teaching the division of fractions by
fractions. These twelve procedures are:
1. Review concepts of multiplication.
2. Review concepts of division.
3. Teach operations with integers.
4. Teach number domains.
5. Review representations of fractions.
6. Review addition and subtraction of fractions and mixed numbers.
7. Review multiplication of fractions and mixed numbers.
8. Teach division of whole numbers by fractions.
9. Teach division of fractions by whole numbers.
10. Teach division of fractions by fractions.
11. Teach division of mixed numbers by mixed numbers.
12. Teach division of fraction word problems.
The following sections contain a description of the disaggregated results for each of the research
questions.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 63
Recalled action and decision steps. Actions steps refer to behaviors that are observable.
Decision steps take the form of unobservable cognitive processes. These unobservable processes
act as cues and/or prompts that allow subject matter experts to decide among alternative courses
of action, based on evaluative and interpretive analysis. The sum total of action and decision
steps captured from subject matter experts that appears in a final gold standard protocol makes
up the information necessary for novice practitioners to replicate expert performance. To answer
Research Question One, the researcher analyzed the action and decision steps contributed by the
individual SMEs, in order to quantify the number of such steps attributable to each.
To analyze the number of action and decision steps captured by each SME, the researcher
entered each step from the final gold standard protocol in individual rows of a Microsoft Excel
spreadsheet. The spreadsheet appears in Appendix F. The first column of the spreadsheet
provides space for coding of the step number, beginning with “1”. The second column provides
space for coding each step as either “A” for action or “D” for decision step. The third column
contains the wording of the action or decision step, while columns four, five, and six are labeled
with the identifiers “SME A”, “SME B”, and SME C”, reflecting the order in which the subject
matter experts were originally interviewed. Each time a SME contributed an action or decision
step that appears in the final gold standard protocol, a “1” was entered in the spreadsheet cell
where the row containing that step and the column bearing that SMEs identifier intersect.
Additionally, if an action or decision step appearing in the final protocol was not attributable to a
particular SME, a “0” was placed in that SME’s identifier column. As an example, for the row
corresponding to column three, action step 5, “Introduce the example 2 x 3”, a “1” appears in the
same row in the column labeled “SME A”, while “0” appears in the columns for SME B and
SME C, indicating that SME A was the sole contributor of that action step.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 64
A spreadsheet formula was used to total the number of action and decision steps for each
SME. These totals appear at the bottom of each of the SME identifier columns. Action and
decision step totals for the individual SMEs appear in Table 1.
Table 1
Cumulative Action and Decision Steps Captured for each SME in the Initial Individual Protocols
Steps _______________________
Action Steps Decision Steps Total Steps
SME A
110
20
130
SME B 279 106 385
SME C
Total
199 28 227
742
Action and decision steps contributed by each SME. The total numbers of action and
decision steps recalled by each SME are summarized in Table 1. As a clarification, there were
multiple cases in which two or more SMEs recalled the same action or decision steps. Because
the final gold standard protocol represents non-repeating action and decision steps only, the total
number of such steps in this final gold standard protocol, 632, is less than the total of 742 steps
for all three SMEs.
Figure 6 provides a graphic representation of the action and decision steps reported in
Table 1.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 65
Figure 6. Total action and decision steps from the CTA study captured by individual SMES as
they appear in the final gold standard protocol. Compare to gold standard protocol non-repeating
action and decisions steps: action steps – 490, decision steps – 142, total action and decision
steps – 632.
Collectively, as reported in the final gold standard protocol, the three SMEs described
632 action and decision steps. On an individual basis, however, the range of total action and
decision steps reported by the three SMEs varied from a low of 130, or 20.57% of the total steps
in the final gold standard protocol, to a high of 385, or 60.92% of the total gold standard protocol
steps. For each of the three SMEs, individually, there were more action than decision steps
reported, and implications of this are discussed in detail in Chapter 5. SME A recalled 110 action
steps compared to 20 decisions steps, SME B recalled 279 action steps and 106 decision steps,
while SME C recalled 199 action steps compared to 28 decision steps. In terms of action steps,
SME B reported the most, comprising 56.94% of the action steps in the final protocol, while
SME A recalled the least, accounting for just 22.45% of the final protocol action steps. A similar
110
279
199
20
106
28
130
385
227
0
40
80
120
160
200
240
280
320
360
400
SME A SME B SME C
Action Steps Decision Steps Total Steps
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 66
disparity characterizes the decision steps. To wit, SME B recalled the most decision steps, at
74.65% of the total decision steps in the final protocol, while SME A reported the least,
comprising 14.08% of final protocol decision steps.
Additional action and decision steps captured during follow-up interviews and SME
review of preliminary gold standard protocol. To provide a more nuanced answer to
Research Question One, the researcher additionally tabulated the number of action and decision
steps that were added, deleted or modified during the follow-up interviews and as a result of the
review by the three SMEs of the preliminary gold standard protocol. These figures are shown in
Table 2.
Table 2
Additional Expert Knowledge Captured, in the Form of Action and Decision Steps, During
Follow-up Interviews and SME Review of the Preliminary Gold Standard Protocol
Additional Expert Knowledge Captured
Action Steps Decision Steps_____
Added Modified Deleted Added Modified Deleted
SME A 24 21 12 7 2 3
SME B 80 0 0 34 0 0
SME C 30 13 1 0 1 0
All three SMEs made multiple additions to action steps, while only two, SME A and
SME B, made additions to decision steps during the follow-up interviews and preliminary gold
standard review. Of the three, SME A provided the widest variety of additional expert
knowledge, contributing a total of 31 additions, 23 modifications, and 15 deletions, whereas
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 67
SME B provided the greatest quantity of additional knowledge, with 80 additional action steps
and 34 additional decision steps. The implications of this capture of additional knowledge are
discussed in Chapter5.
Alignment of SMEs in describing the same action and decision steps. The
spreadsheet analysis also allowed the researcher to gauge the degree to which each of the action
and decision steps in the final gold standard protocol indicated alignment among the three SMEs.
For each step, a column at the far right of the spreadsheet provided a cell where the researcher
could enter one of three values, “1”, “2”, or “3”. If an action or decision step was attributed to
one SME only, then a “1”, signifying slight alignment, was entered. If an action or decision step
was attributed to two SMEs, then a “2”, signifying partial alignment, was entered, whereas if an
action or decision step was attributed to all three SMEs, then a “3” was entered, indicating high
alignment among the three SMEs. The results of this analysis appear in Table 3.
Table 3
Number and Percentage of Total Action and Decision Steps: Highly Aligned, Partially Aligned,
and Slightly aligned.
_____________________________________________________________________________
Frequency Percentage
Highly Aligned 21 3.32%
Partially Aligned 68 10.76%
Slightly Aligned 543 85.92%
On a collective basis, there were 21 action or decision steps that were highly aligned
among the three SMES, 68 that were partially aligned, and 543 that were slightly aligned.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 68
Percentagewise, 3.32% of the steps were highly aligned, 10.76% were partially aligned, and
85.92% were slightly aligned. The steps in the final gold standard protocol that were highly
aligned were steps 254-256 and 258-259, involving review fraction addition and division, steps
422 and 428, involving making meaning of the division of a whole number by a fraction, steps
459, 510, 563, and 609-615, involving the standard fraction division procedure for inverting the
divisor and then multiplying, and step 585, involving division of mixed numbers. The
implications of these alignment totals also are discussed in Chapter 5.
Question 2
What percent of action and/or decision steps, when compared to a gold standard, do
expert middle school math teachers omit when they describe how they teach the division of
fractions by fractions?
Total knowledge omissions. The spreadsheet analysis also allowed the researcher to
determine the number of action and decision steps individual SMEs omitted when they recalled
the expert knowledge necessary to teach the division of fractions by fractions. If an action or
decision step was included in the final gold standard protocol, but was not attributed to an
individual SME, then a “0” was entered in the same column that was used to determine
knowledge alignment among the SMEs, as described in the previous section. The spreadsheet
included a formula to total the number of such omissions for each SME, and additionally divided
that figure by the total number of action and decision steps that comprise the final gold standard
protocol, which served to calculate an omission percentage. A summary of the action and
decision step omissions for each of the three SMEs appears in Table 4.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 69
Table 4
Total Expert Knowledge Omissions by SME as Compared to the Final Gold Standard Protocol
Steps Omitted________________________
Total Action & Action Decision
Decision Steps Steps Steps
Omitted % Omitted % Omitted %
SME A 502 79.43% 380 77.55% 122 85.92%
SME B 247 39.08% 211 43.06% 36 25.35%
SME C 405 64.08% 291 59.39% 114 80.28%
Mean
Omissions 107.33 60.86% 294.0 60.00% 90.66 63.85%
Range 255 169 86
SD 105.09 69.02 38.79
Taking all three SMEs into consideration, there were an average of 107.33 total action
and decision step omissions (SD ± 105.09) when recalling how to teach the division of fractions
by fractions. For action steps alone, the three SMEs, on average omitted 294.0 steps (SD ±
69.02), while for decision steps alone, the three SMEs averaged 49.67 omissions (SD ± 38.79).
On an individual basis, there was significant variance among the three SMEs in terms of
knowledge omissions. Total individual action and decisions step omissions ranged from a low of
39.08% to a high of 79.43%. Total individual action step omissions ranged from a low of
43.06% to a high of 77.55%, while total individual decision step omissions ranged from a low of
25.35% to a high of 85.92%. The implications of these knowledge omissions are addressed in
Chapter 5.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 70
Analysis of action and decision step omissions. The action and decision step omissions
for the three SMEs, as compared to the final gold standard protocol, are presented in Figure 7.
Figure 7
Total SME Knowledge Omissions as Compared to the Final Gold Standard Protocol
Figure 7. Total individual action and decision step omissions for SME A, SME B, and SME C.
Compare to gold standard protocol non-repeating action and decision steps: action steps – 490,
decision steps – 142, total action and decision steps – 632.
The next chapter presents an overview of the study, a discussion of findings, as well as
limitations of the present study, its implications, and possible avenues for future research.
380
211
291
122
36
114
502
247
405
0
100
200
300
400
500
600
SME A SME B SME C
Omitted Action Steps Omitted Decision Steps Total Omitted Action and Decision Steps
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 71
CHAPTER FIVE: DISCUSSION
Overview of the Study
The purpose of this study was to use cognitive task analysis to capture the knowledge and
skills, represented by the action and decision steps, that middle school mathematics teacher
experts recall when they teach the division of fractions by fractions. As explicated in Chapter
Two of this study, the division of fractions comprises a high-leverage prerequisite for
competency in algebra, which itself, and also, as detailed in Chapter Two, is a core prerequisite
for achievement in secondary level mathematics overall. Additionally, this study sought to
determine the number and percentage of action and decision steps that teacher experts in this
instructional process omitted during their recall.
As shown in the literature review, there is a record of underachievement by K-12 students
in mathematics in the United States. This record of underachievement has roots in students’
understanding of algebra, of which the division of fractions by fractions is a critical component.
In recent years, there has been a movement to establish standards-based curricula to address such
underachievement in mathematics, as well as in other subject areas. To enable teachers to
implement these curricula, school districts are increasingly required to pay greater attention to
the manner in which they provide instructional support, both for novice teachers and seasoned
veterans (Darling-Hammond, 2004; Polly et al., 2014). Instrumental to this endeavor has been
an emphasis on providing ongoing professional development that builds teachers’ capacity
(Darling-Hammond, 2004). Currently, many school districts rely on subject matter experts in
content knowledge and pedagogy to conduct these professional development programs (Darling-
Hammond, 2004; Polly et al., 2014).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 72
Unfortunately, the use of such experts poses a number of problems. As experts’ new
knowledge becomes automated and unconscious, they are often unable to completely and
accurately recall the knowledge and skills that comprise their expertise (Chi, 2006; Feldon,
2007). Experts often are overly-confident, overlook details, make inaccurate predictions and
offer faulty advice (Chi, 2006). In addition, as their skills improve, experts’ self-report errors
and omissions tend to increase, while their accuracy of introspection decreases (Feldon, 2007).
Furthermore, because experts’ schemas are adapted to problem solving, they can fail to articulate
relevant cues, and can unintentionally fabricate consciously reasoned explanations for their
automated behaviors (Feldon, 2007).
Because automaticity and the accuracy of self-reporting have been found to be negatively
correlated, experts in an instructional role may unintentionally leave out information that learners
must master when learning procedural skills (Feldon, 2004). In fact, experts may omit up to
70% of the critical information necessary to perform a task (Clark et al., 2011). The automated
nature of knowledge causes procedural steps to blend together in experts’ minds and makes it
difficult for them to share the complex thought processes of technical skill execution (Clark et
al., 2011). As a result, teacher professional development that is guided by such experts remains
an imperfect model.
Studies indicate that instruction based on cognitive task analysis is superior to other
instructional models (Clark, Yates, Early, & Moulton, 2010; Crandall, Klein, & Hoffman, 2006;
Hoffman & Militello, 2009). CTA can allow the data analyst to identify the explicit and implicit
knowledge of experts, which can support effective and efficient cognitive training, scenario
design, cognitive feedback, and on-the-job training (Crandall, Klein, & Hoffman, 2006; Hoffman
& Militello, 2009). In addition, CTA leads to guided instruction that is more structured and
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 73
successful than other types of learning (Clark, Yates, Early, & Moulton, 2010). To that end, this
study sought to use CTA to capture the knowledge and skills of math practitioners’ expertise in
teaching the division of fractions by fractions. The resulting gold standard protocol could serve
to inform professional development for pre-service and in-service middle-school mathematics
teachers.
This chapter presents a discussion of the process of conducting the task analysis, a
discussion of the findings, an analysis of the study’s implications, and an exploration of avenues
for future research.
Process of Conducting the Cognitive Task Analysis
Selection of Experts
A key feature of a sound qualitative research study is purposeful selection of interview
subjects. Subjects selected purposefully are best positioned to help the researcher answer the
research questions because they have the most knowledge to impart (Creswell, 2009; Merriam,
2009). In the field of cognitive task analysis, researchers have identified several criteria that can
assist in such purposeful selection of experts. These are at least 3-5 years of recent, consistent
and successful task performance, with 10 or more years being optimal; history of performance in
a wide and varied array of settings; and, no experience as trainers or instructors in the task
(Clark, 2014; Clark et al., 2008; Feldon, 2007).
Identifying expert teachers can be a more complex endeavor. Beyond simply appraising
years of experience, diversity of experience, peer recommendation and student test scores, it is
necessary to consider a host of other variables. For example, Smith and Strahan (2004)
identified six broad predictors of teaching expertise: (a) a sense of confidence; (b) a view of the
classroom as a learning community; (c) the ability to develop nurturing, trusting relationships
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 74
with students; (d) a student-centered approach; (e) professional contributions; and, (f) content
mastery.
Taking all of these criteria into consideration, the researcher approached district
administrators and school principals in 11 Southern California school districts, and over 100
individual schools, seeking expert math teachers. It soon became clear that finding math
teachers that meet these criteria is extremely difficult, time consuming and produces few suitable
candidates. Although the number of middle school math teachers employed in all of the 11
districts and more than 100 schools stands at somewhere between 400 and 500, it took the
researcher approximately five months of continual search and outreach to locate just 3 teachers
that matched many, but not all of the above criteria. Table 5 summarizes the selection criteria
against which each SME was evaluated.
Table 5
Criteria for Selection of Expert Mathematics Teachers
SME A SME B SME
5 years teaching experience Yes Yes Yes
10 or more years’ teaching experience No Yes Yes
Diversity of experience
(Little, Some, Much) Little Much Much
Recommendation/District administrator Yes No No
Recommendation/School site administrator No Yes No
Recommendation/Peer No No Yes
Student achievement results No No No
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 75
As can be seen from Table 5, beyond measuring years of experience, other quantifiable
criteria, such as student achievement data, did not come into play. The reason for this is simple:
Perhaps most difficult of all was finding teachers with quantifiable measures of student
achievement data. Few districts were either willing or able to provide such data, or to identify
teachers on such a basis. In the end, because of this challenge, the selection of experts for this
study excluded student achievement data. This conforms to the experience of McZeal (2014)
who struggled to obtain achievement data in a study of special education teachers, and is in
contrast to Mutie (2015) who was able to base selection of expert secondary math teachers partly
on student achievement data.
This lack of the use of achievement data in the present study can be put into context. For
example, Shanteau (1992) defines professional expertise as enhanced quality of task performance
that is a function of additional experience. In certain fields of endeavor, increases in years of
experience lead to a quantifiable degree of quality of task performance. For example, chess
masters record more wins than losses or draws; expert test pilots have superior records of test
mission successes, with few if any mishaps; expert insurance analysts are directly responsible for
minimizing loss claims. Does this mean, then, that expertise in teaching leads to quantifiable
gains in student achievement, or conversely, that a superior record of student achievement is an
indicator of teaching expertise?
Because of the number of variables that contribute to expertise in teaching, it is a moot
point as to whether quantifiable measures, such as student achievement results alone, give an
accurate assessment of a teacher’s expertise. In fact, Sternberg and Horvath (1995) found that
this criterion, achievement data, as well as many of the other commonly accepted measures of
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 76
teaching expertise, including years of experience, provide inconclusive evidence of success.
Both the number of variables that exist in the profession of teaching, and the difficulty in
identifying expertise is akin to the nature of other professions, where research has shown that
years of experience do not correlate with quality of task performance. One such field is
psychotherapy, where research has shown no measurable correlation between years of
experience and accuracy or skill (Tracey, Wampold, Lidhtenberg, & Goodyear, 2014). In
addition, much as in teaching, where student learning is partly dependent on the motivation of
students to learn, a psychotherapist’s ability to effect patient outcomes also rests partly on the
desire and motivation of said patient to participate earnestly in treatment (Tracey et al., 2014).
The implication is that present methods for identifying expert teachers are imperfect at
best. One problem lies in the ways professional expertise, including that among teachers, is
identified. The typical course of action is to identify particular traits, attributes, or metrics, and
then to seek individuals that conform to these. Such a confirmatory approach often leads to
searches that uncover partial evidence only, since the tendency is to ignore evidence that refutes
expertise (Tracey et al., 2014). An alternative approach adopts a disconfirmatory stance, in
which one seeks to identify traits or metrics that would render a professional inexpert (Tracey, et
al., 2014). Such an approach yields more complete, less biased information, and leads to better
decision making (Tracey et al., 2014). The use of such an approach could serve as an
appropriate model for the identification of expertise in education. It should be noted that the
current study did not adopt such an approach.
Data Collection
The researcher has 15 years’ experience as an elementary school teacher, and as such, has
no experience in teaching middle school mathematics. As a result, a primary means of
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 77
conducting the initial two stages in the CTA process, collect preliminary knowledge and identify
knowledge representations (Clark et al., 2008) involved a literature review of secondary
mathematics content and pedagogy in general, and algebra and fractions content and pedagogy in
particular. In this way, the researcher was able to gain a general appreciation of the types of
knowledge possessed by middle school teacher experts in the division of fractions by fractions,
and of the major tasks and subtasks they typically perform while delivering instruction.
Data collection took the form of a multi-stage, semi-structured interview technique based
on the Concepts, Processes and Principles (CPP) procedure (Clark et al., 2008), a model
designed to capture the automated, unconscious knowledge experts acquire through extended
experience. The interview protocol appears in Appendix A.
During the course of the interviews, it became apparent that each SME approached the
target task in markedly different ways. For example, SME A’s approach to the division of
fractions relied heavily on review of the prior knowledge necessary to make sense of this target
task. In addition, this SME stressed the importance of students acquiring procedural fluency in
the invert and multiply approach, but was less concerned with their acquiring conceptual
understanding. SME B also recalled a considerable number of action and decision steps
involving review of prior knowledge, but unlike the other two SMEs, also stressed the
importance of helping students build connections between different number types, such as
integers and rational numbers. This SME was also the only one of the three to employ a
complex fraction procedural approach as a bridge to conceptual understanding. SME C recalled
the fewest action and decision steps involving review of critical prior knowledge, but was most
heavily invested in helping students acquire conceptual understanding, as exemplified by a
reliance on realia, pictorial representations, and the use of real life examples.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 78
The researcher found the semi-structured approach, which allows for the use of
extemporaneous probes, an ideal format for responding to these differences in approach to the
target task. Each time a SME began to describe a feature of instruction that differed from
features described in the literature, or features described by the other two SMEs, it was possible
to use probes to tease out much of the detail, complexity, and underlying logic inherent in such
features. This would have been impossible with a structured approach, in which there is
essentially no leeway for deviating from a script of questions. The researcher’s experience
seems to confirm Merriam (2009) contention that a semi-structured approach best enables an
interviewer “…to respond to the situation at hand, to the emerging worldview of the respondent,
and to new ideas on the topic” (p.90).
Contrary to the view of Crandall, Klein and Hoffman (2006), who recommend that the
interviewer take detailed, hand-written notes during the interview, and then transcribe these to a
text file, the researcher audiotaped each interview. In addition, the researcher took general notes,
yet only when necessary, to record impressions, underscore a point made by the SME, or to note
emerging themes. This primary reliance on audiotaping became an important feature of this
study. To wit, eschewing the need to attend to intensive note taking, the researcher was able to
lessen cognitive load and free up working memory (Sweller, 1988). In this way, the researcher
avoided becoming distracted from the dynamics of the interview (Clark et al., 2008), and with a
lightened cognitive load, could respond thoughtfully with the judicious use of extemporaneous
probes to the data as it emerged, while also ensuring that everything the SME articulated was
preserved on tape for later analysis (Merriam, 2009).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 79
Following each interview, the researcher sent the audio files to a professional
transcription service, which provided verbatim transcriptions of the interviews. A discussion of
the researcher’s use of these transcripts for data analysis appears in the next section.
Data Analysis
After gaining permission from each of the three SMEs, the researcher audiotaped the
interviews in their entirety, and then sent the audio files to a professional transcription service,
where they were converted to typed, verbatim transcripts. This is contrary to the practice of
Zepeda-McZeal (2014), who created edited transcripts, containing only information pertinent to
the procedural steps of the target task. This researcher preferred using verbatim transcripts,
finding that reading through such scripts, while simultaneously referring to his own interview
notes, brought forth vivid mental images of each interview, of the SME’s tone and facial
expressions, of sensory impressions, and other non-verbal nuances of the interview. This added
immeasurably to the researcher’s ability to maintain a sense of proximity to the interviews when
later reconstructing the transcripts into the individual protocols by adding the nuances of the
interview recalled in memory and the resulting details of the actions and decisions recalled by
the SME. The researcher’s experience also runs counter to the position taken by Crandall, Klein
and Hoffman (2006) that a verbatim transcript, “…gives you the words, but leaves everything
else out.” (p. 280). The verbatim transcripts were then analyzed, coded, and converted by the
researcher into initial protocols of action and decision steps.
There are various ways to verify an initial SME protocol. This researcher met in person
with each SME and presented his or her protocol for review. This runs contrary to the work of
Clark et al. (2008), who recommend giving the protocols of each SME to one of the other SMEs
for review, and conforms to the work of Crispin (2010), Embrey (2012), Flynn (2012) and
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 80
Zepeda-McZeal, who also asked each SME to review his or her own protocol. The researcher’s
rationale for adopting this approach is twofold. In the first place, giving a protocol of one SME
to another SME for review would have been too time consuming. Secondly, and more
importantly, looking at an initial protocol, which is essentially a detailed list of action and
decision steps, can be a bit disorienting for SMEs. Such a representation differs markedly from
the structure of the interview conversation, and this researcher felt it best to allow each SME to
reconcile his or her recollection of the interview with the completed protocol, and accordingly to
verity it for accuracy and integrity.
As mentioned earlier in this chapter, the researcher encountered great difficulty in
identifying, contacting, and securing the cooperation of subject matter experts. Because it took
anywhere from one month to six weeks to identify each additional expert, the researcher ended
up adopting a specific pattern of data analysis. Following each interview, the researcher
arranged for transcription of the audio file, received a transcript, coded and then converted the
transcript into an individual task protocol. Then the researcher met with the SME for a follow-
up interview during which the SME suggested revisions. Based on these recommendations, the
researcher then produced a revised individual protocol. All this time, the researcher was also
engaged in the process of locating additional SMEs. As it happened, for each protocol, no
additional SMEs were located before the researcher had had time to complete this entire process
of interviewing, coding, developing a protocol, conducting a follow-up interview, and then
producing a revised individual protocol. As a result, the researcher was able to concentrate
exclusively on each SME’s version of the target task. Interview transcripts did not sit around
“gathering dust” while additional interviews were being arranged and conducted, the researcher
did not have to juggle in his mind alternate SME versions of the task procedures, nor did the
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 81
researcher ever experience a sense of “distance” from the immediacy of the interview
experience. The researcher accordingly was able, as closely as possible, to experience an
emergent qualitative research study (Merriam 2009). Data collection and analysis evolved
rhythmically, unbrokenly, and, at times, approached simultaneity (Merriam, 2009). It would be
interesting to compare the results of this study to other CTA studies where data collection and
analysis took place in a more fragmented, less continuous manner.
Discussion of Findings
No formal hypotheses were developed for this research study. Instead, the study was
guided by two main research questions.
Research Question 1
What are the action and decision steps that expert middle school math teachers recall
when they describe how they teach the division of fractions by fractions?
Action steps versus decision steps. Each of the three SMEs recalled more action steps
than decision steps. As reported in Chapter Four: Results, SME A recalled 110 action steps
compared to 20 decision steps, SME B recalled 279 action steps versus 106 decision steps, and
SME C recalled 199 action steps as compared to 28 decision steps. On a collective basis, the
three SMEs recalled an average of 196 action steps versus an average of 51.34 decision steps.
Decision steps, by their very nature, involve unobservable, cognitive processes. Through
repeated task performance, experts’ execution of these processes becomes automated (Anderson,
1996; Feldon, 2007), which makes it difficult for experts to consciously explain how they go
about deciding on a course of action (Clark & Elen, 2006; Clark & Estes, 1996). Therefore, the
preponderance of reported action steps over decision steps in the present study appears to
conform to this research on automaticity (Clark, 2014). In addition, the results from the three
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 82
SMEs in this study are in line with other CTA studies, which have also reported a greater number
of action steps than decision steps, such as Canillas’ (2010) study of a central venous catheter
placement, Crispen’s (2010) study of an open cricothyrotomy procedure, Embry’s (2012) study
of tracheal extubation, and Zepeda-McZeal’s (2014) study of reading instruction in a special
education environment.
Of additional interest is the degree to which the differences between the numbers of
action and decision steps, both individually and collectively, compare to such difference in other
CTA studies. One analytical tool for making such a comparison, percentage difference,
calculates a value that is a function of two compared values, in this case, the number of action
steps and the number of decision steps. This value is derived by subtracting a relative value (the
number of decision steps) from a reference number (the number of action steps). The resulting
value is divided by the average of the relative and reference numbers, and then multiplied by 100
to convert it to a percentage (Zepeda-McZeal, 2014). In the current study, the three SMEs, on
average, collectively recalled 116.97% more action steps than decision steps. Individually, SME
A recalled 138.46% more action steps, SME B recalled 89.87% more action steps, and SME C
recalled 150.66% more action than decision steps.
Compared to other CTA studies, these figures appear, at first, to be excessively large. In
a study of instruction among special education teachers, Zepeda-McZeal (2014) reported a
collective average difference of 20.56%, and individual differences that ranged from 15.73% to
27.78%. In a study involving a medical procedure, open cricothyrotomy, Tolano-Leveque
(2010) reported a collective average difference between action and decision steps of 37.95%,
while in a study of K-12 principals, Hammitt (2015) reported a collective average difference of
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 83
13.53%. Clearly, the percentage differences reported in the current study are significantly
greater, and do not seem to conform to the results of other studies.
However, taking into consideration the nature of the tasks under study, and of the number
of total reported action and decision steps, the large percentage differences reported in the
present study can be put into context. In the present study, the final gold standard protocol
contains 632 total action and decision steps, and 348 more action steps than decision steps. In
the other studies cited, the total number of action and decision steps is significantly smaller, and
the overall difference between action and decision steps is as well. For example, in the study by
Zepeda-McZeal (2014), the final gold standard protocol contains 179 total action and decision
steps, with just 21 more action than decision steps. In Hammitt’s (2015) study, the final gold
standard protocol contains 196 total action and decision steps, with a mere 14 more action steps
than decision steps. It may be that the tasks in the two studies referenced, in one case instruction
in informational text in a special education environment (Zepeda-McZeal, 2014), and in the
other, observations by K-12 principals of instructional practice (Hammitt, 2015), do not lend
themselves to comparison to the task analyzed in the current study, instruction in fraction
division, either because of differences in the nature of the tasks, and/or because of the disparities
in the number of action and decision steps in the tasks. This possibility is also suggested by
Hoffman (1987), where the extent of knowledge extraction with experts was found to vary
according to task complexity.
One CTA study that also involved mathematics instruction sheds some light on this
question. Mutie (2015) studied four instructors expert in teaching quadratic equations to middle
school eighth-graders. Much like the task in the present study, the task in that study also
involved a large number of action and decision steps, with a concomitantly large disparity
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 84
between types of steps. In the final gold standard protocol, Mutie (2015) captured 404 total
action and decision steps, while there were 234 more action steps than decision steps. In terms
of percentage difference, the four SMEs in that study collectively recalled, on average, 118.52%
more action than decision steps. Individually, SME A recalled 134.02% more action steps, SME
B recalled 113.51% more actions steps, SME C recalled 110.06% more action steps, and SME D
recalled 130.94% more action steps than decision steps. In light of the similarities in the
percentage differences between the present study and that of Mutie (2015), and in the similarities
in the numbers of steps and in the disparities of types of steps, the results of the present study
seem to conform to, in at least one case, studies that analyze tasks with similar characteristics.
Differences in recall among SMEs. SME A recalled 130 action and decision steps, the
fewest among the three SMEs. This SME was recommended, on the basis of instructional
performance, by a district director of secondary mathematics, possessed five years of experience
teaching middle school mathematics, and entered teaching mid-career, following several years
working in an unrelated field. The relatively low rate of recall of this SME might be attributable
to a couple of factors. First of all, five years is seen as the low end of the range for the
development of expertise, with ten years typically regarded as a more predictive time span
(Ericsson, 2004; Ericsson, Krampe, & Tesch-Romer, 1993). Thus, this SME may have lacked
the same degree of expertise displayed by the other two SMEs, who possessed, respectively,
twenty-one years’ experience, and twenty-two years’ experience. Furthermore, although five
years of task experience is viewed as sufficient for practitioners to attain task automaticity, this
time frame also represents a point at which performance can, under certain conditions, reach a
stable plateau (Ericsson, 2004). Without any additional time, beyond five years of practice, in
which to engage in a conscious, orderly and deliberate approach to improvement, this SME may
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 85
have been rooted in this transitional stage of expertise, one that is typified by arrested
development of further skill acquisition (Ericsson, 2004).
SME B recalled the most action and decision steps among the three SMEs, with a total of
385. This SME was recommended on the basis of instructional performance by the school
principal, and had twenty-one years’ experience teaching middle school mathematics. The fact
that this SME provided the most recall of combined action and decision steps can be viewed as
running counter to research findings. In general, because of the length of time this expert had
been developing conscious declarative knowledge of task execution, the more automated in
nature this knowledge should have become (Feldon, 2007). Because such knowledge is often
difficult for experts to articulate, due to its non-conscious, automated nature (Kirschner, Sweller,
& Clark, 2006) it seems counterintuitive that this SME’s recall of expertise was so extensive.
Additionally, and more specifically, this SME reported far more decision steps than either of the
other two SMEs: 106 decision steps versus 20 decision steps for SME A and 28 decision steps
for SME C. As delineated earlier in this chapter, decision steps, by their very nature, involve
unobservable, cognitive processes. Through repeated task performance, experts’ execution of
such cognitive processes also is subject to becoming automated (Anderson, 1996; Feldon, 2007),
making it difficult for such experts to consciously explain how they go about making decisions
pursuant to a course of action (Clark & Elen, 2006; Clark & Estes, 1996). Thus, this SME’s
superior recall of this variety of expertise also presents a contradiction. Yet it is also possible
that individual SMEs have different understandings of how much review of prior knowledge is
necessary when providing instruction, and thus, could have different conceptions of when a task
actually starts or ends.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 86
One explanation for the degree to which SME B was able to provide recall of expertise,
compared to the other two SMEs, may lie in the fact that this SME was, during the course of this
study, involved with a group of educators in the collaborative design of a forthcoming middle
school mathematics textbook. Involvement in such a process, which involves working with
other experts to delineate the knowledge inherent in a specific curriculum, including the division
of fractions, would seem to be representative of the deliberate and orderly nature of practice that
research has identified as being essential to the improvement of performance (Ericsson, 2004).
Such a motivation to improve performance (Ericsson, Krampe, & Tesch-Romer, 1993) would no
doubt immerse an educator in deep and thoughtful consideration of the nature of each
instructional task in that curriculum, and of each and every element necessary to execute those
tasks, thus allowing greater recall of attendant expertise as reported in this study. Such mental
effort is reminiscent of the proficiency stage of Alexander’s (2003) Model of Domain Learning,
in which practitioners make concerted effort to contribute new knowledge to a field of endeavor.
This deep, thoughtful, and conscious consideration brought to the task by this SME can also be
compared to Anderson’s (1996) associative stage of automaticity, in which a learner must still
consciously work through the steps of a procedure during task execution, as not all declarative
and procedural task knowledge has been fully automated. Perhaps this SME, as a result of
participation in a textbook committee, had actually regressed from full to partial automaticity.
SME C possessed twenty-two years’ experience teaching middle school mathematics,
was recommended based on her instructional performance by a district administrator, and
recalled 227 total action and decision steps. Although this SME recalled more action steps than
SME A, this expert recalled significantly fewer steps than SME B, a fact which seems to
conform more readily to the research literature on expertise. To wit, with such extensive task
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 87
experience, this SME may have possessed deeply ingrained, automated procedural knowledge,
resulting in a concomitant decrease in the accuracy and completeness of recall, coupled with an
increase in self-report errors and omissions (Clark et al., 2011; Feldon, 2007).
Additions, deletions, and revisions captured during review of initial individual
protocols and of the preliminary gold standard protocol. As described earlier, following the
drafting of an initial, individual protocol, the researcher met with each SME for a review. The
researcher described the coding and protocol development process, and then presented the SME
his or her protocol. In each case, the SME read over the protocol and then made revisions, a
process that was slightly different for each SME. Revisions took three forms: additions,
modifications, and deletions. Additions involved an entirely new step, modifications involved
changes in the language of pre-existing steps, and deletions involved the complete removal of a
pre-existing step. Following this review process, the revised individual protocols were
aggregated into a preliminary gold standard protocol, which was also presented to each SME for
review.
Review of initial individual protocols. SME A conducted the most thorough, systematic
review of the initial protocol, compared to the other two experts. This expert carefully read over
each page, line by line, stopping often to pencil in notes in the margins. Following this
approximately 45 minute read-through, SME A carefully went over each of these marginalia
with the researcher. One result was the addition of 24 action and 7 decision steps. In almost all
cases, these additions involved enhancements to checks for student understanding, usually in the
form of polling individual students and having the teacher orally reinforce concepts. Most of
these were action steps, with the decision steps comprising either moving on if students
understood, or conducting a review if comprehension was weak.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 88
SME A made the most modifications, 23, most of which were action steps. The majority
of these involved minor revisions to the manner in which the researcher had worded these steps,
with little change in semantics. Of interest was this SME’s choice to move an eight step sub-
procedure, involving conceptual understanding of division of a fraction by a whole number, to an
earlier juncture in the larger procedure, commenting, “We’ve already created understanding of
this.” This attests to the care and degree of mental effort this SME brought to the review
process.
SME A also made 15 deletions. Most of these were action and decision steps that the
researcher had inferred from the interview transcript. They mostly involved reinforcing a
concept with additional independent practice, a course of action the SME found superfluous.
This is of interest, in that it lies in contrast to the numerous steps involving checks for
understanding this expert added, as described earlier.
SME B’s approach to this process was markedly different. This expert listened to the
researcher’s description of the protocol development process, was given the protocol, read over
each of its 16 pages methodically, yet rapidly, taking just over 15 minutes, and made no written
annotations or asked any questions. However, following this read-through, SME B commented
that in the two weeks since the initial interview, “…in my work with a committee to write a
textbook, I realized that I left out a big piece where we need to relate fractions to decimals.”
This SME then laid out an additional 80 action steps and 30 decision steps designed to convey
that fractions and decimals both are representations of parts of a whole. In most cases, these
additional steps involved instruction aimed at helping students perform fraction operations, and
then replicate those operations by converting the fractions to decimals and repeating the
operations.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 89
While SME B added the most additional expertise of the three SMEs, this expert made no
modifications and no deletions. This SME seemed preoccupied at the beginning of this second
interview. To wit, the SME answered some preliminary questions perfunctorily, appeared mildly
disinterested during the researcher’s description of the process of developing the protocol, and,
when prompted, indicated that she had no questions. The expert’s somewhat cursory read-
through of the protocol seemed to confirm this sense of preoccupation. It is possible that SME B
was concerned mainly with imparting the additional expertise that work with a textbook
committee had uncovered, and that this concern distracted the expert from a thorough review of
the initial protocol. There might have been better capture of additional expertise pursuant to the
initial protocol, had the researcher contacted this SME beforehand by phone, explained the
purpose of the second interview, and then emailed the initial protocol for the SME to review at
leisure. Although this course of action runs counter to the findings of other such studies
(Hammitt, 2015; Zepeda-McZeal, 2014), this SME may have been an atypical case requiring
atypical measures.
Among the three experts, SME C contributed the least additional decision-step expertise.
This involved a single modification to a decision step involving how to provide remediation to
struggling students. The majority of this expert’s additions of expertise involved 30 action steps,
and of these, the majority involved two categories: directing the teacher to make use of real-
world objects (realia), such as apples, pizzas, bagels and sheets of paper to reinforce conceptual
understanding, and, asking students to formulate real-life examples of the concepts under
consideration. This SME also made 13 modifications to action steps, and the majority of these
involved changes to the types of realia employed.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 90
This expert’s focus on conceptual understanding, either through the use of realia or of
real-life examples is consistent with the fact that this SME was the only one of the three to
include division of fraction word problems in the initial protocol. Such word problems are
regarded as essential in helping students make sense of mathematical concepts (NMAP, 2008).
This SME remarked that too few teachers understand the importance of such conceptual
understanding and the result is that numerous teachers, “… complain to me that they taught this
concept two weeks ago, and the students have already forgotten it.” This SME, with the most
years of teaching experience of the three, may exemplify some of the more advanced
characteristics of expertise: the possession of a profoundly organized body of knowledge, with
well-developed schemas, and the ability to create and present to students mental models, which
can be quickly and efficiently retrieved from long term memory (Bedard & Chi, 1992).
Review of the preliminary gold standard protocol. Subsequent to these follow-up
interviews, the researcher revised each individual protocol, and then aggregated these three into a
single preliminary gold standard protocol. Each SME was contacted by phone, and agreed to
review an emailed copy of this preliminary gold standard. Once each SME had indicated by
return email that he or she had had time to review it, a second phone call was made to discuss
any possible revisions.
At this point, there was essentially, no further revision. Aside from a few questions about
how the other SMEs had approached the instructional task, or for an explication of the rationales
involved in some of the procedures that had been suggested by other SMEs, each expert
indicated that the protocol, as it stood, was acceptable to him or her.
This runs counter to the experience of the researcher during the initial follow-up
interviews, during which, as has been described, each SME read over the protocol and made not
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 91
insubstantial revisions. As a result, perhaps, and as also suggested by Zepeda-McZeal (2014),
the researcher may have been able to capture additional expertise for the preliminary gold
standard protocol through the use of an in-person review. However, choosing to accept the
avowal by each of the three SMEs that the preliminary protocol was complete and could be used
to inform instruction, the researcher chose not to pursue an in-person interview format, and thus,
the preliminary gold standard protocol basically transmogrified into the final gold standard
protocol, with no further additions, revisions, or deletions. As a result, it is impossible to say that
the final gold standard protocol represents a complete listing of the action and decision steps
necessary to teach the division of fractions by fractions.
Alignment of SMEs vis-à-vis total action and decision steps. As described in Chapter
4, this study included an analysis of the degree to which there was alignment among the three
SMEs in their recall of total action and decision steps. As reported, just 21, or 3.32% of the total
steps in the final gold standard protocol were highly aligned, meaning that they were recalled by
each of the three SMEs. Additionally, a mere 68, or 10.76% of the total action and decision
steps in the final protocol were recalled by two of the SMEs in the study, and were thus
classified as partially aligned. The majority of recalled action and decision steps, 543 steps,
comprising 85.92% of the total, were slightly aligned, meaning they were attributable to a single
SME only.
As research shows, the evidence-based, highly automated and unconscious knowledge of
experts is difficult to articulate, and experts’ self-report errors and omissions tend to increase as
skills improve (Feldon, 2007; Kirschner, Sweller, & Clark, 2006). Because the highly developed
and adaptive schemas of such experts can interfere with the accurate recall of problem situations
(Feldon, 2007), CTA methods that rely on multiple experts to elicit expertise have been found to
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 92
be an effective means of informing instruction (Clark & Elen, 2006). As this study’s analysis of
action and decision step alignment shows, reliance on a single subject matter expert to inform
instruction could result in an imperfect approach in teaching teachers how to provide instruction
in the division of fractions by fractions. This is borne out by a number of recent studies in the
fields of medical instruction and K-12 instruction (Bartholio, 2010; Canillas, 2010; Mutie, 2015;
Zepeda-McZeal, 2014), in which the use of multiple experts led to significant increases in
knowledge capture.
Research Question 2
What percent of action and/or decision steps, when compared to a gold standard, do
expert middle school math teachers omit when they describe how they teach the division of
fractions by fractions?
Expert knowledge omissions. Research indicates that experts’ unconscious, automated
knowledge is difficult to articulate, and this leads to omissions by such experts during recall
(Kirschner, Sweller, & Clark, 2006). Therefore, the researcher conducted an omission analysis
for this study by comparing the action and decision steps in the individual protocol for each SME
to the action and decision steps in the final gold standard protocol. On average, the three SMEs
collectively omitted 60.86% of total action and decision steps, 60.0% of action steps, and
63.85% of decision steps, with a difference between action and decision step omissions of
3.85%. The small difference between these average action step and decision step omissions
conforms to the findings of other recent CTA studies. For example, Mutie (2015) found a
difference between average action and decision steps of 3.10%, Hammitt (2015) found a
difference of 0.34%, while Zepeda-McZeal derived a difference between average action and
decision step omissions of 1.15%.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 93
The researcher further analyzed the extent to which each SME individually omitted
combined action and decision steps. SME A omitted 79.43% of the action and decision steps in
the final gold standard protocol, SME B omitted 39.08% of the action and decision steps in the
final protocol, while SME C omitted 64.08% of the combined action and decision steps in the
final gold standard protocol. The mean of these omissions is 60.86%, with a standard deviation
of 20.37%. Of interest to the researcher was the degree to which these omissions either did or
did not conform to the research finding that experts may omit up to 70% of action and decision
steps when prompted to describe task execution (Clark, et al., 2011; Feldon, 2004).
The researcher performed a one-sample, two-tailed t-test to determine whether the
omission percentages found in the study conform to the hypothesized omission value of 70%, or
whether the findings were due to chance. The t-test computation used the following values: n = 3
participants, sample mean of omissions = .6086, with a sample standard deviation (SD) = .2037,
and a population mean = .70. The resulting t value was -0.7773, while the two-tailed p value
based on an alpha level of α = 0.05 was .5183. The magnitude of the p value indicates that the
findings of combined action and decision steps for the three SMEs in this study conform to the
research finding that experts may omit up to 70% of critical action and decision steps during
recall, and were not the result of chance (Clark et al., 2011; Feldon, 2004).
Limitations
The results of the current study were consistent with those of other CTA studies seeking
to capture and analyze expert knowledge recall and omissions. Following is a discussion of the
limitations inherent in this study.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 94
Confirmation Bias
Confirmation bias is the tendency among researchers to give greater credit to information
that aligns with their own preconceptions, whether or not the information is actually true (Corbin
& Strauss, 2008). As Clark (2014) further notes, in the realm of cognitive task analysis, this bias
is seen when knowledge analysts are experienced in the domain of the task under study, and can
lead to the analyst unconsciously editing knowledge captured from SMEs. The researcher, at the
time of the present study, had 15 years of instructional experience in K-12 settings, specifically
at the elementary level. Although those 15 years of instructional experience did include
mathematics instruction, the researcher was not overly familiar with the middle school
mathematics that were the focus of the study. Thus the researcher sought to bootstrap the
information necessary to become familiar with the domain of middle school mathematics,
primarily through a search of the literature (Schraagen et al., 2000). The researcher’s intent was
to acquire domain familiarity commensurate with of that of an accomplished novice (Schraagen,
et al., 2000), with the aim of minimizing the tendency that would be characteristic of a more
accomplished expert to filter or edit the knowledge captured.
It should additionally be noted that the researcher does have 15 years’ experience in
teaching general fraction concepts. As a result, during interviews, when SMEs were describing
activities involving review in such general concepts, as a precursor to describing more
specialized instruction in the target task, the researcher strove to avoid making judgments, based
on those 15 years of experience that might have resulted in filtering, editing, or otherwise
altering the data.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 95
Internal Validity
Maxwell (2005) cautions that the aim of research is to capture reality, and that this can be
a daunting, elusive goal. In the present study, internal validity represents the degree to which
reality, namely the actual enactment of the target task by the SMEs on a daily basis in their
classrooms, matches the descriptions of such practice as delineated in the gold standard protocol
(Merriam, 2009). To that end, internal validity for this study would involve triangulation
through the use of multiple data sources (Merriam, 2009): one source involving observation of
the three SMEs teaching the target task, and a second source consisting of an analysis of the
number of action and decision steps found in the gold standard manifested in their classroom
practice. For the purposes of the present study, such an internal validity analysis by observation
was not performed, and thus presents a limitation of the study.
External Validity
External validity is a measure of the degree to which the results of a study are
generalizable to other settings (Merriam, 2009). Threats to external validity in the present study
include the small sample size (n = 3), and the fact that the sampling of experts was non-random
and included only teachers from three somewhat adjacent Southern California school districts. It
remains moot, therefore, whether the gold standard protocol produced in this study would be
applicable to other teachers in school districts in other regions and/or states.
Yet Merriam (2009) takes a somewhat contrarian, yet intriguing, position. Because case
studies such as this are generally rich in qualitative description, they provide for the reader
opportunities to apply what is gleaned from such description to similar cases, possibly serving as
the basis for teacher education or evaluation (Eisner, 1991; Merriam, 2009). Additionally, it is
always possible to apply lessons learned from particular cases, such as this, to others that are
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 96
similar, although it should also be noted that generalizability is not the immediate aim of case
studies (Erickson, 1986; Merriam, 2009). Nevertheless, future CTA studies that aim to elicit the
declarative and procedural knowledge of middle school teachers expert in the division of
fractions by fractions couldn’t help but serve to increase the present study’s external validity by
employing a larger, more diverse, and ideally, randomly selected sample of subject matter
experts.
Implications
The current movement to establish standards-based curricula to address
underachievement among K-12 students has led school districts to pay greater attention to the
manner in which they provide ongoing professional development to build teachers’ capacity.
The current model, in which districts rely on subject matter experts in content knowledge and
pedagogy to conduct professional development programs (Darling-Hammond, 2004; Polly et al.,
2014), has been found to be problematic, as research shows that such experts may
unintentionally leave out up to 70% of the information that learners must master when learning
new tasks (Feldon, 2004). Several studies indicate that instruction based on cognitive task
analysis is superior to other instructional models (Canillas, 2010; Clark, Yates, Early, &
Moulton, 2010; Crandall, Klein, & Hoffman, 2006; Hoffman & Militello, 2009; Zepeda-McZeal,
2014). CTA can allow data analysts to identify the explicit and implicit knowledge of experts,
which can support effective and efficient cognitive training, scenario design, cognitive feedback,
and on-the-job training (Crandall, Klein, & Hoffman, 2006; Hoffman & Militello, 2009).
Similarly, the findings of the current study provide support for the use of CTA to inform teacher
professional development in K-12 instructional tasks, such as the division of fractions by
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 97
fractions, an area of the mathematics curriculum that has been identified as critical to overall
mathematics achievement (NMAP, 2008).
There is, however, a major caveat to the use of this study’s final gold standard protocol as
the basis for teacher professional development in the division of fractions by fractions. It must
be stressed that developing a gold standard protocol through the use of CTA methods is
predicated on the educational context within which the subject matter experts chosen operate.
Thus, attempting to generalize such a gold standard protocol to other educational contexts must
be done with caution. Differences in how individual students learn and how individual teachers
deliver instruction would preclude the possibility of such a gold standard protocol representing
the definitive method of best practice in division of fractions by fractions. In addition, the
concerns discussed earlier in this chapter regarding the paucity of further input by the three
SMEs when asked to review the preliminary gold standard protocol further leads to the
conclusion that the gold standard protocol developed in the current study cannot be regarded as
representing 100% of the action and decision steps comprising best practices instruction in the
division of fractions by fractions.
Future Research
There appear to be no other CTA studies that have investigated the focus task of this
study: instruction in the division of fractions by fractions. A search of the literature on cognitive
task analysis and this branch of mathematics was inconclusive. This tends to suggest, therefore,
that an avenue for future research might be to use the gold standard protocol developed in the
current study as the basis for a randomized experimental study with middle school teachers
tasked with instruction in the division of fractions by fractions. The intent would be to compare
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 98
the learning gains realized among students receiving instruction in traditional methods as
compared to students receiving instruction in accordance with the gold standard protocol.
Of further interest, and as mentioned earlier, this study employed a specific method for
expert review of the preliminary gold standard protocol. To wit, once this protocol had been
aggregated, the three SMEs responsible for its creation were tasked with its final review. As
reported, this review resulted in no additional capture of expertise. Two avenues for future
research present themselves as a result of this outcome. First of all, the review process did not
involve an in-person discussion between the researcher and each individual SME. A future study
in which such an in-person review was incorporated might shed light on whether or not such
practice led to increased capture of expertise. Additionally, future research might employ an
alternative review method, one in which review of the preliminary protocol were conducted not
by the SMEs responsible for the preliminary gold standard, but rather, by an independent expert.
It might be instructive to determine whether the use of such a fourth expert also could lead to an
increase in capture of expertise.
Also, this study drew attention to the large disparities in percentage differences between
recalled action steps versus decision steps among the three SMEs, as compared to other CTA
studies (Hammitt, 2015; Tolano-Leveque, 2010; Zepeda-McZeal, 2014). As discussed, it is
possible that these differences were a result of differences in the nature of the task in this study
versus the nature of the tasks in the comparison studies. Although, upon further analysis, these
percentage differences were found to be similar to the percentage differences in another CTA
study that also involved a mathematics instructional task (Mutie, 2015), future research might
investigate whether other CTA studies have investigated K-12 mathematics instruction, and to
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 99
what extent such percentage differences between recalled action and decision steps in those
studies compare to those in the present study.
Finally, for the purposes of assessing internal validity, there are two avenues for future
research. In the short term, either the current researcher, or another researcher, might conduct an
analysis of the actual, classroom practice of the three SMEs that were the focus of this study.
The aim would be to determine the degree to which that practice conforms to the action and
decision steps reported in the final gold standard protocol. In the long term, future CTA studies
focused on the division of fractions might also include observation of SME instructional practice
to validate the credibility of any resulting final protocols.
Conclusion
This study builds on the current body of knowledge concerning the use of cognitive task
analysis to capture the knowledge and skills of experts involved in complex tasks, and
additionally, to analyze the omissions such practitioners make when recalling their expertise. As
mentioned, this is the first known CTA study involving instruction in the division of fractions by
fractions in a middle school setting. With respect to knowledge omissions among experts, this
study found an average of omissions of action and decision steps among three subject matter
experts of just over 60%, which conforms statistically to established research findings that
experts may omit up to 70% of critical knowledge and skills when asked to recall their expertise
(Feldon, 2004). Compared to the amount of knowledge and skills that were captured from the
three SMEs individually, the extent of such knowledge and skills that was captured from these
three experts in aggregate confirms the superiority of the use of multiple experts for knowledge
capture, and of the use of CTA methods. The resulting final gold standard protocol from this
study could serve as the basis for a professional development program for novice and veteran
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 100
middle school teachers alike, with the aim of providing a model of teacher capacity building that
more successfully improves student mathematics achievement, as compared to current models,
which rely on individual subject matter experts.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 101
References
Adelman, Clifford. 1994. Lessons of a Generation: Education and Work in the Lives of the High
School Class of 1972. New York: Jossey-Bass.
Alexander, P.A. (2003). The development of expertise: The journey from acclimation to
proficiency. Educational Researcher, 32(8), 10-14.
Algebra. (2002). Shorter Oxford English Dictionary (Vol. 1, p. 52). New York: Oxford
University Press, Inc.
Algebra: What, When, and for Whom. (2008). A Position of the National Council of Teachers of
Mathematics. National Council of Teachers of Mathematics: Reston, VA.
Ambrose, S. A., Bridges, M. W., DiPietro, M., Lovett, M. C., & Norman, M. K. (2010). How
learning works, 7 research-based principles for smart teaching. San Francisco, CA:
Jossey-Bass.
American Institutes for Research. (2006). A call for middle school reform - The research and its
implications. Washington, DC: Microsoft.
Anderson, J.R. (1982). Acquisition of cognitive skill. Psychological Review, 89(4), 369-406.
Anderson, J. R. (1996). The Architecture of Cognition. Mahwah, NJ: Lawrence Erlbaum
Associates.
Anderson, L.W., & Krathwohl (Eds.). (2001). A Taxonomy for learning, teaching, and
assessing: A revision of Bloom's taxonomy of educational objectives. New York:
Longman.
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What
makes it special? Journal of Teacher Education, 59(5), 389.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 102
Bartholio, C. W. (2010). The use of cognitive task analysis to investigate how many experts
must be interviewed to acquire the critical information needed to perform a central
venous catheter placement (Doctoral dissertation). Retrieved from
http://digitallibrary.usc.edu/cdm/ref/collection/p15799coll127/id/385767
Bedard, J. & Chi, M.T.H. (1992). Expertise. Current Directions in Psychological Science, 1(4),
135-139.
Bottge, B. A., Ma, X., Gassaway, L., Butler, M., & Toland, M. D. (2014). Detecting and
correcting fractions computation error patterns. Exceptional Children, 80(2), 237-255.
Brown, A. S. (2008). Presidents Panel urges more resources for mathematics education.
Mechanical Engineering, 130(5), 9.
Canillas, E. N. (2010). The use of cognitive task analysis for identifying the critical
information omitted when experts describe surgical procedures (Unpublished
doctoral dissertation). University of Southern California, Los Angeles.
Carpenter, T. P., Moser, J. M., & Bebout, H. C. (1988). Representation of addition and
subtraction word problems. Journal for Research in Mathematics Education, 19, 345-357.
Cengiz, N., & Rathouz, M. (2011). Take a bite out of fraction division. Mathematics Teaching in
the Middle School, 17(3), 146-153.
Chabe, Alexander M. (2003). Rationalizing Inverting and Multiplying. Arithmetic
Teacher, 10, 272-73.
Chao, C-J., & Salvendy, G. (1994). Percentage of procedural knowledge acquired as a
function of the number of experts from whom knowledge is acquired for diagnosis,
debugging, and interpretation of tasks. International Journal of Human-Computer
Interaction, 6(3), 221-233.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 103
Chen, Z. (1999). Schema induction in children's analogical problem solving. Journal of
Educational Psychology, 91, 703-715.
Chi, M. T. H. (2006). Two approaches to the study of experts’ characteristics. In K. A. Ericsson,
N. Charness, P. J. Feltovich, & R. R. Hoffman (Eds.), The Cambridge handbook of
expertise and expert performance (pp. 21–30). New York: Cambridge University Press.
Chi, M., Glaser, R., & Rees, E. (1982). Expertise in problem solving. In R. Sternberg
(Ed.), Advances in the psychology of human intelligence (pp. 7–75). Hillsdale, NJ:
Erlbaum.
Chipman, S.F. (2000). Introduction to Cognitive Task Analysis. In J.M. Schraagen, S.F.
Chipman & V.L. Shalin (Eds.), Cognitive Task Analysis (pp. 24-36). Mahwah, NJ:
Lawrence Erlbaum Associates.
Chipman, S. F., Schraagen, J. M., & Shalin, V. L. (2000). Introduction to cognitive task analysis.
In J. M. Schraagen, S. F. Chipman, & V. L. Shalin (Eds.), Cognitive task analysis (pp. 3-
23). Mahwah, NJ: Lawrence Erlbaum Associates.
Clark, R.E. (1999). Ying and yang: Cognitive motivational processes in multimedia learning
environments. In J. van Merrienboer (ed) Cognition and multimedia Design. Herleen,
Netherlands: Open University Press.
Clark, R.E. (2008). Resistance to change: Unconscious knowledge and the challenge of
unlearning. In D.C. Berliner, & H. Kupermintz (Eds.), Changing institutions,
environments, and people. Mahwah, NJ: Lawrence Erlbaum Associates.
Clark, R.E. (2014). Cognitive task analysis for expert-based instruction in healthcare. in Spector,
J.M. Merrill, M.D. Elen, J. and Bishop, M.J. (Eds.). Handbook of Research on
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 104
Educational Communications and Technology, 4th Edition. New York: Simon and
Schuster Macmillan.
Clark, R., & Elen, J. (2006). When less is more: Research and theory insights about
instruction for complex learning. In J. Elen & R.E. Clark (Eds.), Handling complexity
in learning environments: Theory and research (pp. 283-295). New York, NY:
Elsevier.
Clark, R., & Estes, F. (1996). Cognitive Task Analysis for Training. International Journal of
Educational Research, 25(5), 403-417.
Clark, R. E., Feldon, D., van Merrienboer, J., Yates, K., & Early, S. (2008). Cognitive Task
Analysis. In J.M. Spector, M.D. Merrill, J.J.G. van Merrienboer & M.P. Driscoll (Eds.),
Handbook of research on educational communications and technology (3rd ed., pp.
577-593). Mahwah, NJ: Lawrence Erlbaum Associates.
Clark, R. E., Pugh, C. M., Yates, K. A., Inaba, K., Green, D. J., & Sullivan, M. E. (2011). The
use of cognitive task analysis to improve instructional descriptions of procedures. The
Journal of Surgical Research, doi: 101016/j.jss2011.09.003
Clark, R. E., Yates, K., Early, S., & Moulton, K. (2010). An analysis of the failure of electronic
media and discovery-based learning: Evidence for the performance benefits of guided
training methods. In K. H. Silber & R. Foshay, (Eds.). Handbook of training and
improving workplace performance, Volume 1 Instructional design and training delivery
(pp. 263-297). New York: John Wiley and Sons.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 105
Cooke, N. J. (1994). Varieties of knowledge elicitation techniques. International Journal of
Human-Computer Studies, 41, 801-849.
Cooke, N. J. (1999). Knowledge elicitation. In F. T. Durso (Ed.), Handbook of Applied
Cognition (pp. 479-509). New York: Wiley.
Corbett, A. T., Anderson, J. R. (1995). Knowledge Tracing: modeling the acquisition of
procedural knowledge. User Modeling and User-Adapted Interaction, 4, 253-278.
Coughlin, H. A. (2011). Dividing fractions: What is the divisor's role? Mathematics Teaching in
the Middle School, 16(5), 280.
Crandall, B., & Getchell-Reiter, K. (1993). Critical decision method: A technique for eliciting
concrete assessment indicators from the intuition of NICU nurses. Advances in Nursing
Science, 16(1), 47-51.
Crandall, B., Klein, G., &Hoffman, R. R. (2006). Working minds: A practitioner’s guide to
cognitive task analysis. Cambridge, MA: MIT Press.
Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed methods
approaches. Thousand Oaks, CA: SAGE Publications.
Crispen, P. D. (2010). Identifying the point of diminishing marginal utility for cognitive task
analysis surgical subject matter expert interviews (Doctoral dissertation). Available from
ProQuest Dissertations and Theses database. (UMI No. 3403725)
Darling-Hammond, L. (2004). Standards, accountability, and school reform. The Teachers
College Record, 106(6), 1047-1085.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 106
Desoete, A., Roeyers, H., & De Clercq, A. (2003). Can offline metacognition enhance
mathematical problem solving? Journal of Educational Psychology, 95(1), 188-200.
Downey, C. J., Steffy, B. E., English, F. W., Frase, L. E., & Poston, W. K., Jr., (2004). The
three-minute classroom walk-through: Changing school supervisory practice one
teacher at a time. Thousand Oaks, CA: Corwin Press.
Eisner, E. W. (1991). The enlightened eye: Qualitative inquiry and the enhancement of
educational practice. Old Tappan, NJ: Macmillan.
Embry, K. K. (2012). The use of cognitive task analysis to capture expertise for tracheal
extubation training in anesthesiology (Doctoral dissertation). Retrieved from
http://digitallibrary.usc.edu/cdm/ref/collection/p15799coll127/id/678652
Erickson, F. (1986). Qualitative methods in research on teaching. In M. C. Whittrock (Ed.),
Handbook of research on teaching (3
rd
ed.), (pp119-161). Old Tappan, NJ: Macmillan.
Ericsson, K. A. (2004). Invited Address: Deliberate practice and the acquisition and maintenance
of expert performance in medicine and related domains. Academic Medicine, 79(10), s70-
s81
Ericsson, K. A., Krampe, R. T., & Tesch-Römer, C. (1993). The role of deliberate practice in the
acquisition of expert performance. Psychological Review, 100(3), 363-406.
Ericsson, K. A., & Lehmann, A. C. (1996). Expert and exceptional performance: Evidence of
maximal adaptation to task constraints. Annual Review of Psychology, 47(1), 273-305.
Evan, A., Gray, T. A., & Olchefske, J. (2006). The gateway to student success in mathematics
and science. Washington, DC: American Institutes for Research.
Fackler, J. C., Watts, C., Grome, A., Miller, T., Crandall, B., & Pronovost, P. (2009). Critical
care physician cognitive task analysis: an exploratory study. Critical Care, 13(2), R33.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 107
Feldon, D. F. (2004). Inaccuracies in expert self-report: Errors in the description of strategies
for designing psychology experiments (Doctoral dissertation, University of Southern
California).
Feldon, D. (2007). Implications of research on expertise for curriculum and pedagogy.
Educational Psychology Review, 19(2), 91-110.
Fendel, D. M. (1987). Understanding the structure of elementary school mathematics. Newton,
MA: Allyn & Bacon.
Flynn, C. L. (2012). The relative efficiency of two strategies for conducting cognitive task
analysis. (Doctoral dissertation). Retrieved from
http://gradworks.umi.com/35/61/3561771.html
Fuchs, L. S., Schumacher, R. F., Sterba, S. K., Long, J., Namkung, J., Malone, A., Changas, P.
(2014). Does working memory moderate the effects of fraction intervention? An
aptitude–treatment interaction. Journal of Educational Psychology, 106(2), 499-514.
Gabriel, F., Coché, F., Szucs, D., Carette, V., Rey, B., & Content, A., (2013). A compnential
view of children’s difficulties in learning fractions. Frontiers in Psychology, 4, 715.
Garet, M., Porter, A., Desimone, L., Briman, B., & Yoon, K. (2001). What makes professional
development effective? Analysis of a national sample of teachers. American Educational
Research Journal, 38, 915-945.
Gearhart, M., & Saxe, G. B. (2004). When teachers know what students know: Integrating
mathematics assessment. Theory into Practice, 43(4), 304-313.
Glaser, R., & Chi, M. T. H. (1988). Overview. In M. T. H. Chi, R. Glaser, & M. Farr (Eds.), The
Nature of Expertise (p. xv-xxviii). Mahwah, NJ: Lawrence Erlbaum Associates.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 108
Gregg, J., & Gregg, D. U. (2007). Measurement and Fair Sharing Models for Dividing Fractions.
Mathematics Teaching in the Middle School, 12, 490-96
Gott, S. P., Hall, E. P., Pokorny, R. A., Dibble, E, & Glaser, R. (1993). A naturalistic study of
transfer: Adaptive expertise in technical domains. In D. K. Detterman & R. J.
Sternberg (Eds.), Transfer on trial: Intelligence, cognition, and instruction (pp. 255-
288). Norwood, NJ: Ablex.
Haas, M. (2005). Teaching methods for secondary algebra: A meta-analysis of findings. National
Association of Secondary School Principals. NASSP Bulletin, 89(642), 24-46.
Hammitt, C. S. (2015). Using cognitive task analysis to capture how expert principals conduct
informal classroom walk-throughs and provide feedback to teachers (Unpublished doctoral
dissertation). University of Southern California, Los Angeles, CA.
Hatano, G. & Inagaki (2000). Practice makes a difference: Design principles for adaptive
expertise. Presented at the Annual Meeting of the American Education Research
Association. New Orleans, LA: April, 2000.
Heck, D. J., Banilower, E. R., Weiss, I. R., & Rosenberg, S. L. (2008). Studying the effects of
professional development: The case of the NSF's local systemic change through teacher
enhancement initiative. Journal for Research in Mathematics Education, 39(2), 113-152.
Hoffman, R. R. (1987). The problem of extracting the knowledge of experts from the perspective
of experimental psychology. AI Magazine, 8(2), 53-67.
Hoffman, R. & Militello, L. (2009). Perspectives on cognitive task analysis: Historical origins
and modern communities of practice. New York: Psychology Press. Hoffman, R. &
Militello, L. (2009). Perspectives on cognitive task analysis: Historical origins and
modern communities of practice. New York: Psychology Press.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 109
Hoffman R. & Woods D. ( 2000). Studying cognitive systems in context: Preface to the special
section. Hoffman, R. R., & Woods, D. D. (2000). Studying cognitive systems in context:
Preface to the special section. Human Factors, 42 (1), 1-7.
Holmes, V. (2012). Depth of teachers' knowledge: Frameworks for teachers' knowledge of
mathematics. Journal of STEM Education: Innovations and Research, 13(1), 55-71
Hutchinson, N. L. (1993). Effects of cognitive strategy instruction on algebra problem solving of
adolescents with learning disabilities. Learning Disability Quarterly, 16, 34-63.
Jackson, P. W. (1985). Private lessons in public schools: Remarks on the limits of adaptive
instruction. In M. C. Wang & H. J. Walberg (Eds.), Adapting instruction to individual
differences (pp. 66–81). Berkeley, CA: McCutchan.
Johnson, A. (2010). Teaching mathematics to culturally and linguistically diverse learners.
Boston: Pearson Education, Inc.
Kalyuga, S., Chandler, P., Tuovinen, J., & Sweller, J. (2001). When problem solving is superior
to studying worked examples. Journal of Educational Psychology, 93, 579-588.
Kaput, J. J. (2000). Teaching and learning a new algebra with understanding. Dartmouth,
MA: National Center for Improving Student Learning and Achievement in Mathematics
and Science.
Kirschner, P. A, Sweller, J., & Clark, R. E. (2006). Why minimally guided instruction does not
work: An analysis of the failure of constructivist, discovery, problem based, experiential,
and inquiry based teaching. Educational psychologist, 41(2), 75-86.
Klein, G. A., Calderwood, R., and MacGregor, D. (1989), Critical decision method for
eliciting knowledge. IEEE Trans. Syst. Man Cybernet., 21, 1018-1026.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 110
Koichu, B., Harel, G., & Manaster, A. (2013). Ways of thinking associated with mathematics
teachers' problem posing in the context of division of fractions. Instructional
Science, 41(4), 681-698.
LeSage, A. (2012). Adapting math instruction to support prospective elementary
teachers. Interactive Technology and Smart Education, 9(1), 16-32.
Lester, F. K., Garofalo, J., & Kroll, D. L. (1989). Self-confidence, interest, beliefs, and
metacognition: Key influences on problem-solving behavior. In D. B. McLeod & V. M.
Adams (Eds.), Affect and mathematical problem solving (pp. 75-88). New York:
Springer-Verlag.
Li, Y. (2008). What do students need to learn about division of fractions? Mathematics Teaching
in the Middle School, 13(9), 546-552.
Lopez-Real, F. (2006). A new look at a Polya problem. Mathematics Teaching, 196, 12-15.
Loucks-Horsley, S., Stiles, K. E., Mundry, S., Love, N., & Hewson, P. W. (2010). Designing
professional development for teachers of science and mathematics (3rd ed.). Thousand
Oaks, CA: Corwin.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of
fundamental mathematics in China and the United States. New York: Dale Seymour
Publications.
Matthews, M., & Ding, M. (2011). Common mathematical errors of pre-service elementary
school teachers in an undergraduate course. Mathematics and Computer
Education, 45(3), 186-196.
Matthews, M. S., & Farmer, J. L. (2008). Factors affecting the algebra I achievement of
academically talented learners. Journal of Advanced Academics, 19(3), 472-501.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 111
Maxwell, J. A (2005). Qualitative research design: An interactive approach (2
nd
ed.).
Thousand Oaks, CA: Sage.
Mayer, R. E. (1999). The promise of educational psychology: Vol. I. Learning in the content
areas. Upper Saddle River, NJ: Merrill Prentice Hall.
Means, B., & Gott, S. (1988). Cognitive task analysis as a basis for tutor development:
Articulating abstract knowledge representations. In J. Psotka, L. D. Massey, & S. A.
Mutter (Eds.), Intelligent Tutoring Systems: Lessons Learned (pp. 35-58). Hillsdale, NJ:
Lawrence Erlbaum Associates, Inc.
Merriam, S. B. (2009). Qualitative research: A guide to design and implementation. San
Francisco: Jossey-Bass.
Militello, L. G., & Hoffman, R. R. (2008). The forgotten history of cognitive task analysis.
Proceedings of the Human Factors and Ergonomics Society Annual Meeting, 52, 383-
387.
Montague, M., Applegate, B., & Marquard, K. (1993). Cognitive strategy instruction and
mathematical problem-solving performance of students with learning disabilities.
Learning Disabilities Research & Practice, 8, 223-232.
Mutie, A. M. (2015). The use of cognitive task analysis to capture expert instruction in teaching
mathematics (Unpublished doctoral dissertation). University of Southern California, Los
Angeles, CA.
National Center for Education Statistics. (2007). TIMSS 2007 Results. Retrieved December, 27,
2013 from www.nces.ed.gov/timss
National Center for Education Statistics. (2009). Selected findings from PISA 2009. Retrieved
December, 27, 2013 from www.nces.ed.gov/pisa
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 112
National Center for Education Statistics. (2011). TIMSS 2011 Results. Retrieved December, 27,
2013 from www.nces.ed.gov/timss
National Center for Education Statistics. (2012). Selected findings from PISA 2012. Retrieved
December, 27, 2013 from www.nces.ed.gov/pisa
National Center for Education Statistics. (2013a). Frequently asked questions. Retrieved
December, 27, 2013 from www.nces.ed.gov/pisa
National Center for Education Statistics. (2013b). Frequently asked questions. Retrieved
December, 27, 2013 from www.nces.ed.gov/timss
National Center for Education Statistics. (2013c). Frequently asked questions. Retrieved
December, 27, 2013 from www.nces.ed.gov/naep
National Council of Teachers of Mathematics. (2000). Principles and standards for school
mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2008). Algebra: What, when, and for whom? A
position of the National Council of Teachers of Mathematics. Reston, VA: Author
National Governors Association Center for Best Practices & Council of Chief State School
Officers. (2010). Common Core State Standards for mathematics. Washington, DC:
Authors.
National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the
National Mathematics Advisory Panel, U.S. Department of Education: Washington, DC.
Ni, Y. J., & Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The
origins and implications of whole number bias. Educational Psychologist, 40, 27–52.
Novillis, C. F. (1979. Why Teach Elementary School Students the Division Meaning of
Fractions? School Science and Mathematics, 79, 705-8
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 113
Ott, J. M., Snook, D. L., & Gibson, D. L. (1991). Understanding partitive division of
fractions. The Arithmetic Teacher, 39(2), 7.
Paris, S.G., Lipson, M.Y., & Wixson. (1983). Becoming a strategic reader. Contemporary
Educational Psychology 8, 293-316.
Parmar, R. S., Cawley, J. E, & Frazita, R. R. (1996). Word problem-solving by students with and
without mild disabilities. Exceptional Children, 62, 415-429.
Payne, J. N. ( 1976). Review of research on fractions. In R. Lesh (Ed.), Number and
measurement (pp. 145-188). Athens: University of Georgia.
Polly, D., & Hannafin, M. J. (2010). Reexamining technology's role in learner-centered
professional development. Educational Technology Research and Development, 58(5),
557-571.
Rizvi, N. F., & Lawson, M. J. (2007). Prospective teachers' knowledge: Concept of
division. International Education Journal, 8(2), 377-392.
Schiller, Kathryn S. and Donald Hunt. 2003. "Disadvantaged Students' Mathematics Course
Trajectories in High School." Paper presented at the Annual Meeting of the American
Educational Research Association, April, Chicago.
Schneider, Barbara, Christopher B. Swanson, and Catherine Riegle-Crumb. 1998. "Opportunities
for Learning: Course Sequences and Positional Advantages." Social
Psychology of Education, 2, 25-53.
Schraagen, J. M., Chipman, S. F., & Shalin, V. L. (2000). Cognitive Task Analysis. Mahwah,
NJ: Lawrence Erlbaum Associates.
Shanteau, J. (1992). Competence in experts: The role of task characteristics. Organizational
Behavior and Human Decision Processes, 53, 252–266.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 114
Sharp, J., & Adams, B. (2002). Children's constructions of knowledge for fraction division after
solving realistic problems. The Journal of Educational Research, 95(6), 333.
Sharp, J., & Welder, R. M. (2014). Reveal limitations through fractions: Division problem
posing. Mathematics Teaching in the Middle School, 19(9), 540.
Shulman, L.S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard
Educational Review, 57(1), 1-22.
Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding
of fractions. Developmental Psychology, 49(10), 1994-2004.
Smith, T. W., & Strahan, D. (2004). Toward a prototype of expertise in teaching. A descriptive
case study. Journal of Teacher Education, 55(4), 357-371.
Swail, W. S., Cabrera, A. F., Lee, C., & Williams, A. (2005). Latino students & the educational
pipeline: Pathways to the bachelor’s degree for Latino students. Washington, DC:
Educational Policy Institute.
Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive
Science, 12(2), 257-285.
Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: The
case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-
25.
Tofel-Grehl, C., & Feldon, D. F. (2013). Cognitive task analysis-based training: A meta-analysis
of studies. Journal of Cognitive Engineering and Decision Making, 7(3), 293-304.
Tolano-Leveque, M. (2010). Using cognitive task analysis to determine the percentage of critical
information that experts omit when describing a surgical procedure (Unpublished
doctoral dissertation). University of Southern California, Los Angeles
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 115
Tolar, T. D., Lederberg, A. R., & Fletcher, J. M. (2009). A structural model of algebra
achievement: Computational fluency and spatial visualization as mediators of the effect
of working memory on algebra achievement. Educational Psychology, 29(2), 239-266.
Tracey, T. J. G., Wampold, B. E., Lichtenberg, J. W., & Goodyear, R. K. (2014, January 6).
Expertise in Psychotherapy: An Elusive Goal? American Psychologist. Advance online
publication.
Trafton, P. R., & Zawojewski, J. S. (1984). Teaching rational number division: A special
problem. Arithmetic Teacher, 31(6), 20-22.
Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions?
Aspects of secondary school students' understanding of rational numbers and their
notation. Cognition and Instruction, 28(2), 181-209.
Van de Walle, J. A. (2004). Elementary and middle school mathematics: Teaching
developmentally (5
th
ed.). Boston: Allyn & Bacon.
Walker, S., & Senger, E. (2007). Using technology to teach developmental African-American
Algebra students. Journal of Computers in Mathematics and Science Teaching, 26, 217-
231.
Wang, J., & Goldschmidt, P. (2003). Importance of middle school mathematics on high school
students' mathematics achievement. Journal of Educational Research, 97, 3-19.
Wei, J. and Salvendy, G. (2004). The cognitive task analysis methods for job and task design:
review and reappraisal. Behaviour & Information Technology, 23(4), 273–299.
Wheatley, T., & Wegner, D. M. (2001). Automaticity of action, psychology of. In N. J. Smelser
& P. B. Baltes (Eds.), International encyclopedia of the social and behavioral sciences
(pp. 991-993). Oxford: Elsevier Science Ltd.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 116
Yates (2007). Towards A Taxonomy of Cognitive Task Analysis Methods: A Search for Cognition
and Task Analysis Interactions. (Doctoral dissertation). Retrieved from
http://search.proquest.com/docview/304828231?accountid=14749.
Yates, K. A., Sullivan, M.E., & Clark, R. E. (2011, March). Integrated studies in the use of
cognitive task analysis to capture surgical expertise for central venous catheter
placement and open cricothyrotomy. Symposium conducted at the meeting of the
Association for Surgical Education in Boston, Massachusetts.
Zepeda-McZeal, D. (2014). Using cognitive task analysis to capture expert reading
instruction in informational text for students with mild to moderate learning
disabilities (Unpublished doctoral dissertation). University of Southern California,
Los Angeles, CA.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 117
Appendix A
Cognitive Task Analysis Interview Protocol
Begin the Interview: Meet the Subject Matter Expert (SME) and explain the purpose of the
interview. Ask the SME for permission to record the interview. Explain to the SME the
recording will be only used to ensure that you do not miss any of the information the SME
provides.
Name of task(s): How to teach division of fractions by fractions
Performance Objective:
Ask: “What is the objective of teaching how to divide fractions by fractions?” “What action
verb should be used?”
Step 1:
Objective: Capture a complete list of student learning outcomes for teaching division of fraction
by fractions.
A. Ask the Subject Matter Expert (SME) to list student outcomes when these tasks are
complete. Ask them to make the list as complete as possible
B. How is the student assessed on these outcomes?
Step 2:
Objective: Provide practice exercises that are authentic to the context in which the tasks are
performed
A. Ask the SME to list all the contexts in which these tasks are performed (e.g. classroom;
small group; whole group; addition vs. subtraction, etc.)
B. Ask the SME how the tasks would change for context/setting
Step 3:
Objective: Identify main steps or stages to accomplish the task
C. Ask SME the key steps or stages required to accomplish the task.
D. Ask SME to arrange the list of main steps in the order they are performed, or if there is
no order, from easiest to difficult.
Step 4:
Objective: Capture a list of “step by step” actions and decisions for each task
A. Ask the SME to list the sequence of actions and decisions necessary to complete the task
and/or solve the problem
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 118
Ask: “Please describe how you accomplish this task step-by-step, so a novice trainee
could perform it.”
For each step the SME gives you, ask yourself, “Is there a decision being made by the
SME here?” If there is a possible decision, ask the SME.
If SME indicates that a decision must be made…
Ask: “Please describe the most common alternatives (up to a maximum of three) that
must be considered to make the decision and the criteria trainees should use to decide
between the alternatives”.
Step 5:
Objective: Identify prior knowledge and information required to perform the task.
A. Ask SME about the prerequisite knowledge and other information required to perform the
task.
1. Ask the SME about Cues and Conditions
Ask: “For this task, what must happen before someone starts the task? What prior task,
permission, order, or other initiating event must happen? Who decides?”
2. Ask the SME about New Concepts and Processes
Ask: “Are there any concepts or terms required of this task that may be new to the
trainee?”
Concepts – terms mentioned by the SME that may be new to the novice
teacher
Ask for a definition and at least one example
Processes - How something works
If the trainee is operating equipment, or working on a team that may or may not
be using equipment, ask the SME to “Please describe how the team and/or the
equipment work - in words that novices will understand. Processes usually consist
of different phases and within each phase, there are different activities – think of
it as a flow chart”
Ask: “Must trainees know this process to do the task?” “Will they have to use it
to change the task in unexpected ways?”
IF the answer is NO, do NOT collect information about the process.
3. Ask the SME about Equipment and Materials
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 119
Ask: “What equipment and materials are required to succeed at this task in
routine situations? Where are they located? How are they accessed?
4. Performance Standard
Ask: “How do we know the objective has been met? What are the criteria, such
as time, efficiency, quality indicators (if any)?”
5. Sensory experiences required for task
Ask: “Must trainees see, hear, smell, feel, or taste something in order to learn
any part of the task? For example, are there any parts of this task they could not
perform unless they could smell something?”
Step 6:
Objective: Identify routine or simple problems that can be solved by using the procedure.
A. Ask the SME to describe at least one routine or simple problem and two to three
complex problems that the trainee should be able to solve if they can perform each of
the tasks on the list you just made.
Ask: “Of the task we just discussed, describe at least one routine problem and two to
three complex problems that the trainee should be able to solve IF they learn to perform
the task”.
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 120
Appendix B
Inter-rater Reliability Code Sheet for SME A
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 121
Appendix C
Job Aid for Developing a Gold Standard Protocol
Richard Clark and Kenneth Yates (2010,
Proprietary)
The goals of this task are to 1) aggregate CTA protocols from multiple experts to create a
“gold standard protocol” and 2) create a “best sequence” for each of the tasks and steps you
have collected and the best description of each step for the design of training.
Trigger: After having completed interviews with all experts and capturing all goals, settings,
triggers, and all action and decision steps from each expert – and after all experts have edited
their own protocol.
Create a gold standard protocol
STEPS: Actions and Decisions
1. For each CTA protocol you are aggregating, ensure that the transcript line number
is present for each action and decision step.
a. If the number is not present, add it before going to Step 2.
2. Compare all the SME’s corrected CTA protocols side-by-side and select one
protocol (marked as P1) that meets all the following criteria:
a. The protocol represents the most complete list of action and decision steps.
b. The action and decision steps are written clearly and succinctly.
c. The action and decision steps are the most accurate language and terminology.
3. Rank and mark the remaining CTA protocols as P2, P3, and so forth, according to
the same criteria.
4. Starting with the first step, compare the action and decision steps of P2 with P1
and revise P1 as follows:
a. IF the step in P2 has the same meaning as the step in P1, THEN add “(P2)” at the
end of the step.
b. IF the step in P2 is a more accurate or complete statement of the step in P1,
THEN revise the step in P1 and add “(P1, P2)” at the end of the step.
c. IF the step in P2 is missing from P1, THEN review the list of steps by adding the
step to P1 and add “(P2N)”* at the end of the step.
5. Repeat Step 4 by comparing P3 with P1, and so forth for each protocol.
6. Repeat Steps 4 and 5 for the remaining components of the CTA report such as
triggers, main procedures, equipment, standards, and concepts to create a
“preliminary gold standard protocol” (PGSP).
7. Verify the PGSP by either:
a. Asking a senior SME, who has not been interviewed for a CTA, to review
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 122
the PGSP and note any additions, deletions, revisions, and comments.
b. Asking each participating SME to review the PGSP, and either by hand or
using MS Word Track Changes, note any additions, deletions, revisions, or
comments.
i. IF there is disagreement among the SMEs, THEN either
1. Attempt to resolve the differences by communicating with
the SMEs, OR
2. Ask a senior SME, who has not been interviewed for a CTA,
to review and resolve the differences.
8. Incorporate the final revisions in the previous Step to create the “gold standard protocol”
(GSP).
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 123
Begin procedure 1:
Review concepts of
multiplication
Access prior knowledge
by reviewing area models
of multiplication
Introduce the example:
2x3=6
Model drawing procedure:
2 rows long enough to
later accommodate the
columns
Ask, “What do we
multiply the 2 rows by?”
Call on students whose
hands are not raised
Procedure 1 continued on
Page 2
Decision
point
Begin and end
process
Legend
Action step
Off-page connector
Appendix D
Flowchart for SME A Individual Protocol
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 124
Procedure continued from
Page 1
Does the student
have the correct
answer?
Ask other students
whose hands
aren’t raised, until
you get correct
answer
Hold class discussion to
consolidate understanding
Model how to divide the
2 horizontal rows into 3
vertical columns
Ask students to now
independently create a
new area model with a
new multiplication
problem
Circulate and visually
check student models
Procedure 1 continues on
Page 3
No
Yes
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 125
Procedure 1 continues
from Page 2
Ask students, “What shape
have we been creating?”
Do students
demonstrate
understanding?
Give student an area
model worksheet for
homework
Conduct discussion on
characteristics of rectangles
No
Yes
Introduce several
multiplication problems
with 2 equal multiplicands
Procedure 1 continues on
Page 4
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 126
Procedure 1 continues from
Page 3
Ask students to create on
area model with equal and
one with unequal
multiplicands
Check models visually
Lead discussion on
relationships between
squares and rectangles
Visually display on board
all squares with sides in
length from 1 to 10 units
Discuss that squares are a
subset of rectangles, and
thus all squares are
rectangles
Do students
demonstrate
understanding?
Give students a
worksheet for
homework
No
Yes
Procedure 1 continues on
Page 5
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 127
Procedure 1 continues from
Page 4
Introduce a new problem
with unequal multiplicands
Ask students to compare
their models with a
neighbor
Lead discussion on
similarities/differences of
rectangles with unequal
multiplicands, but same
number of squares
Reinforce that 3x4 and 4x3
models, although shaped
differently, yield the same
number of boxes, 12
Equate this with notion of
groups: 3x4 is the same as
4x3
Procedure 1 continues on
Page 6
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 128
Procedure 1 continues from
Page 5
Connect previous notion to
understanding that “3
groups of 4” is identical to
“4 groups of 3”
Explain that this relationship
between multiplicands is
known as the commutative
property of multiplication
End Procedure 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 129
Begin Procedure 2:
Review concepts of
division
Explain that division is the
inverse of multiplication so
we must work backwards in
applying multiplication to
division
Explain this involves
changing the product of our
multiplication problem so
that it is now the dividend
of our division problem
Demonstrate with 2x3=6,
and working backwards, this
is related to the equation
6÷3=2
Pose the problem, “What is
10÷5?”
Did student
provide the correct
answer, “2”?
Repeat step 2a,
this page, with
a new problem
Ask student how she derived
the answer
Yes
No
Procedure 2 continues on
Page 8
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 130
Procedure 2 continues from
Page 7
What answer does
student provide?
Student says
she decided
how many
times 5 fits into
10
Student says she
broke the dividend
into the number of
groups indicated by
the divisor, and
determined how many
were in each group
Explain that because
multiplication if commutative,
there are 2 ways to think about
division
Explain that preferred method
is to determine how many
times the divisor fits into the
dividend, just as in long
division
Pose new problem, such as
8÷4
Procedure 2 continues on
Page 9
See step 2c on
Page 9
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 131
Procedure 2 continues from
Page 8
Ask student to explain how
she derived the answer
What answer does
student provide?
Student says
she determined
how many
times the
divisor fit into
the dividend
Student says she
broke the dividend
into the number of
groups indicated by
the divisor and
determined how
many were in each
group
See step 2b on Page
8
Continue to reinforce that
the goal of division is not to
determine how many equal
groups are indicated by the
divisor, but rather, to
determine how many times
the divisor fits into the
dividend
Assign simple division
problems with no
remainder, such as 12÷4
Ask students to provide a
written description of how
they derived answer
Do student
descriptions show
understanding of
“how many times
concept?
No
Assign long division
problems
Are students fluent
doing this?
End Procedure 2
No
Yes
Yes
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 132
Begin Procedure 3: review of
fraction concepts
Distribute manipulative
fraction bars
Ask student pairs to create
fractions with same length as
1
Check that students are
creating fractions such as
2
2
or
3
3
Ask, “How many halves make
1, and how many thirds make
1?"
Reinforce characteristics of
fractions equal to 1
Does observation
indicate students
understand?
No
Yes
Review addition of fractions
with common denominators
Demonstrate examples, then
assign similar problems that
add to 1, such as
1
2
+
1
2
=
2
2
Procedure 3 continues on
Page 11
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 133
Procedure 3 continues from
Page 10
Does observation indicate
students understand?
See step
3g on
Page 10
Review multiplication of
fractions
Demonstrate examples that
add to 1, such as
1
2
x
2
1
=
2
2
Assign similar problems
Does observation indicate
students understand?
No
Review fraction operations
with uncommon denominators
Ask students to use fraction
bars to find fractions with
uncommon denominators that
have the same length, such as
1
2
and
2
4
Remind students denominators
of both fractions share 2 as a
common factor
Procedure 3 continues on Page
12
Yes
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 134
Procedure 3 continues from
Page 11
Demonstrate that we can
simplify (reduce)
2
4
by
dividing both the numerator
and denominator by 2
Direct students to simplify
other fraction examples
Do students
demonstrate
understanding?
Reteach Greatest
Common Factor
method
Review concept that
simplifying fractions results
in an equivalent fraction
Review concept that we can
also multiply both the
numerator and denominator
by the same number to yield
an equivalent fraction
Do students demonstrate
understanding?
See step 3g on
Page 11
Review concept of common
denominators
Procedure 3 continues on
Page 13
Yes
No
No
Yes
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 135
Procedure 3 continues from
Page 12
Ask students to use fraction
bars to find a common
denominator that could be
used to represent 2 fractions
with uncommon
denominators, such as
1
2
and
1
3
,
which can also be represented
by
3
6
and
2
6
Walk around and visually
check for understanding
Demonstrate on board with
fraction bars how to line up
both
1
2
and
1
3
with
3
6
and
2
6
Assign new pairs of fractions
and ask students to use
fraction bars to find equivalent
fractions with a single
common denominator
Do students demonstrate
understanding?
No
End Procedure 3
Yes
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 136
Begin Procedure 4: review
of concept of inverse
relationships
Access prior knowledge: fact
families are any two numbers
and their product
Reinforce that an inverse
operation undoes the original
operation
Write example on board, e.g.
3x4=6
Explain that to invert a
multiplication fact and create
a division fact, the following
format must be used
Always turn the product (12
in this case) into the dividend
The second multiplicand
(here, the 4) always becomes
the divisor
Demonstrate the resulting
inverse: 12÷4=3
Reinforce: outside numbers in
multiplication equation
switch places, while the
second multiplicand remains
in the middle
Give examples/students solve
Do students demo understanding? No
Procedure 4 continues on
Page15
Yes
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 137
Procedure 4 continues from
Page 14
Review concept that division
should always involve not
dividing the dividend into the
number of groups indicated
by the divisor, but should
involve determining how
many times the divisor fits
into the dividend
Assign examples on board,
students solve, and then
provide a written description
of how they performed the
operation
What do written
descriptions indicate?
Student
divided the
dividend into
the number of
groups
indicated by
the divisor
Student
determined
how many
times the
divisor fit into
the dividend
End Procedure 4
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 138
Access prior knowledge:
any number multiplied by
its reciprocal equals 1
Access prior knowledge:
reciprocals of whole
numbers
Access prior knowledge:
reciprocals of fractions
Procedure 5 continues on
Page 17
Begin Procedure 5: review
concept of reciprocals
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 139
Procedure 5 continues from
Page 16
Now relate concept of
reciprocals to division
Explain that we are going to
need to learn to divide
without using divison
Confirm that we know that
6÷2=3
Ask how to derive 6÷2=3
without dividing and do not
provide hint to multiply
Solicit student suggestions to
gauge level of understanding
Explain that our review of
fractions revealed that 6x
1
2
=
6
2
Remind students we know to
simplify fractions by dividing
the numerator and
denominator by same number
Demonstrate that
6
2
÷
2
2
=
3
1
,
which is equal to 3
Remind students that instead
of dividing, we multiplied the
dividend by the reciprocal of
the divisor
Assign problems: students
divide by multiplying by
reciprocal of divisor
Do students demo understanding?
End Procedure 5
No
Yes
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 140
Begin Procedure 6: teach
division of fractions
Begin with division of whole
numbers by fractions
Begin with example such as
1÷
1
2
Ask, “How many times does
1
2
fit into 1?”
Do students understand answer
is 2?
Ask
students to
model
answer
with
fraction
bars
Access prior knowledge: we
know, in division, we can also
multiply by the reciprocal
Pose division of whole
number by fraction problem,
to see if students remember
how to multiply by reciprocal
Have students provide written
description of how they
derived answer
Procedure 6 continues on Page
19
Does answer indicate student
multiplied by reciprocal of divisor?
No
No
Yes
Yes
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 141
Procedure 6 continues from
Page 18
Continue with fractions
divided by whole numbers
Access prior knowledge: use
simple example, such as
1
2
÷4
Ask students to consider what
portion of the 4 fits into
1
2
Follow up by asking students
to determine how many
halves we get if we divide 4
into halves
Do students understand
there are 2 halves in 1,
and thus 8 halves in 4?
Ask students to
model answer
using fraction
bars
Procedure 6 continues on
page 20
Yes
No
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 142
Procedure 6 continues from
Page 19
Confirm answer is 8 halves,
and ask what 1 of those 8
halves represents
Confirm that one of those
halves represents
1
8
of 4
Remind students that in
division, we can multiply by
the reciprocal
Demonstrate that
1
2
÷4 is the
same as
1
2
x
1
4
=
1
8
Explain the solution signifies
that
1
8
of 4 fits into
1
2
Pose new problem, such as
1
3
÷5
Ask students to solve and
write what the answer,
1
15
signifies
Do students demo
understanding?
No
Yes
Convene whole-class
discussion on advantages of
multiplying by reciprocal
Procedure 6 continues on Page
21
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 143
Procedure 6 continues from
Page 20
Transition to division of
fractions by fractions
Begin with simple example,
such as
1
2
÷
1
4
Check for understanding: Do
most students understand two
one-fourths fit into
1
2
?
No
Yes
Entertain a discussion:
less important to
understand concept than
the procedure of
multiplying by reciprocal
Confirm answer is two one-
fourths
Transition to procedural
solution method: aim is to use
multiply by reciprocal method
Demonstrate how to solve
1
2
÷
1
4
Demonstrate that the
reciprocal of the divisor is 4
and that
1
2
x4=
4
2
Procedure 6 continues on
Page 22
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 144
Procedure 6 continues from
Page 21
Remind students to reduce by
dividing both the numerator
and denominator by 2
Confirm answer, which is 2,
signifying 2 one-fourths
Assign new problem, such as
1
3
÷
1
4
Do students
demonstrate
procedural
understanding?
No
See step 6h on
page 21
Yes
Stress once more that from
now on, we will approach
division from a multiply by
the reciprocal perspective
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 145
Appendix E
Gold Standard Protocol
Final Gold Standard Protocol
Task: To teach the division of fractions by fractions in 6
th
, 7
th
, and 8
th
grades.
Definitions
Muir Time: the final period of the day during which students at “JM” Middle School either begin
homework in their homerooms, or are directed to come to the class of a teacher who is providing
remediation in a specific subject.
Example of Muir Time: a student spends Muir Time in the classroom of his/her social studies
teacher, reviewing material that was incorrect on a recent test.
Non-example of Muir Time: a student neither works on homework in his/her homeroom nor
reports to another teacher’s class for remediation, but rather, attends a Student Council meeting.
Procedures
1. Review concept of multiplication
2. Review concepts of division
3. Teach operations with integers
4. Teach number domains
5. Review representations of fractions
6. Review addition and subtraction of fractions and mixed numbers
7. Review multiplication of fractions and mixed numbers
8. Teach division of whole numbers by fractions
9. Teach division of fractions by whole numbers
10. Teach division of fractions by fractions
11. Teach division of mixed numbers by mixed numbers
12. Teach division of fraction word problems
1. Review concept of multiplication
1.1. Explain that to model multiplication, we will construct rectangles (A)
1.1.1. Introduce notion of rectangles as area models for multiplication (A)
1.1.2. Review concept that area models are arrays with rows and columns (A)
1.1.2.1. Explain that the rows and columns correspond to the two multiplicands
1.1.3. Introduce the example of 2 x 3 (A)
1.1.3.1. Model how to draw the rows: 2 (first multiplicand) rectangles on top of
each other (234-5) and that the 2 rectangles need to be long enough to
accommodate the columns that correspond to the other multiplicand, 3 (A)
1.1.3.2. Ask, “What do we multiply those 2 rows by?” (A)
1.1.3.3. Call on students whose hands are not raised (A)
1.1.3.3.1. IF student has correct answer, THEN go to step 1.1.3.4 (A)
1.1.3.3.2. IF student has incorrect answer, THEN call on another student
whose hand isn’t raised, until a student gives the correct answer (A)
1.1.3.4. Check for understanding with discussion (A)
1.1.4. Model how to divide those two boxes by 3, the other multiplicand, so that there
are 3 columns (A)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 146
1.1.4.1. Demonstrate how to count the number of boxes created by the 2 rows and
3 columns, which is 6 (A)
1.1.4.2. Reinforce the point of the exercise that 2 rows by 3 columns is a graphical
representation of 2x3=6 (A)
1.2. Repeat above sequence with a new multiplication problem with unequal multiplicands;
students independently create area models corresponding to that problem (A)
1.3. Wander around and check visually (A)
1.4. Ask students, following several additional problems area models with unequal
multiplicands, “What shape have we been creating?” (A)
1.4.1. IF students demonstrate understanding, THEN step 1.5 (A)
1.4.2. IF students are unclear, THEN give them a worksheet for homework (A)
1.5. Reinforce that all examples have been rectangles (A)
1.6. Ask, “What does it mean to be a rectangle?” (A)
1.7. Ask, “How do you know a rectangle when you see one?” (A)
1.8. Challenge students to create a definition of a rectangle (2 pairs of parallel sides, the
opposites being equal in length, and 4 right angles) (A)
1.9. Reinforce that what you have built is a rectangle (A)
1.10. Introduce several multiplication problems with 2 equal multiplicands (A)
1.11. Ask students to create one area model with unequal multiplicands and then one
area model with equal multiplicands (A)
1.12. Check visually (A)
1.12.1. Build a connection by asking, “What plane figure is created when all sides are
equal?” (A)
1.12.2. Ask, after student response of “square” is elicited, “Is a square not also a
rectangle?” (A)
1.12.3. Reinforce concept that a square is a rectangle with 4 equal sides (A)
1.12.4. Display visual on board during discussion. (A)
1.12.5. Display all squares whose sides range in length from 1 to 10 units. (A)
1.12.6. Discuss that squares are a subset of rectangles (A)
1.12.7. Discuss here that all squares are rectangles (A)
1.12.7.1. IF students are struggling to conceptualize these understandings,
problems. THEN give students a worksheet (A)
1.12.8. IF students understand concepts, THEN step 1.13 (A)
1.13. Introduce a new problem with unequal multiplicands (e.g. 3x4) (A)
1.13.1. Ask students to compare their models with a neighbor (A)
1.13.2. Ask students to decide if their area model looks like their neighbor’s (A)
1.13.2.1. Showcase students whose models differ (e.g. 3x4 area models versus 4x3
area models) (A)
1.13.2.2. Circulate and ask students with different models to draw their models on
board and then have a discussion (A)
1.13.2.3. Ask, “If our multiplication problem is 3x4, how many boxes do we expect
in our area model,” the answer, of course to which is 12 (A)
1.13.2.4. Challenge students to pair with someone else with a differently shaped
area model and establish that both have the same number of boxes (A)
1.13.2.5. Explain that 3x4 and 4x3 area models, although they are shaped
differently, yield the same number of boxes (A)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 147
1.13.2.5.1. Equate this with the notion of groups: a 3x4 area model is the same
multiplication problem as a 4x3 area model (A)
1.13.2.5.2. Establish a connection to the understanding that “3 groups of 4 is
the same idea as 4 groups of 3” and explain that this relationship between
multiplicands is known as the commutative property of multiplication (A)
2. Review concepts of division
2.1. Explain that division is the inverse of multiplication so we have to work backwards in
applying multiplication to division (A)
2.1.1. Explain this involves changing the product of our multiplication problem so that it
is now the dividend of our division problem (A)
2.1.2. Demonstrate that 2x3=6, and working backward, this is related to the equation
6÷3=2 (A)
2.1.3. Pose division problem with a quotient greater than one, such as 12÷2 (A, C)
2.1.3.1. IF students derive correct answer, 6, THEN step 2.1.4 (A)
2.1.3.2. IF students seem unclear, THEN step 2.1 (A)
2.1.4. Ask students, “What is the meaning of this problem?” (A, C)
2.1.5. Direct students to discuss this with a neighbor (C)
2.1.6. Check for understanding by calling on students randomly (A, C)
2.1.6.1.1. IF student explains that she decided how many times 2 fits into 12,
THEN go to step 2.1.8 (A)
2.1.6.1.2. IF student says she broke the dividend into the number of groups
indicated by the divisor and determined how many were in each group
THEN step 2.1.6.2 (A)
2.1.6.2. Realize that this is a pattern of student thought that needs to be rewired
(A)
2.1.6.3. Explain that because multiplication is commutative there are two ways to
think about division (A)
2.1.6.3.1. Explain that one way is to break the dividend into the number of
groups indicated by the divisor and then determine the size of each group
(A)
2.1.6.3.2. Explain that this is a method that we are going to avoid (A)
2.1.6.3.3. Explain that the other method is to determine how many times the
divisor fits into the dividend, just as we do in long division and then show
an example on board (A, C)
2.1.6.3.4. Establish the expectation that this is the way students need to begin
to think about division (A, C)
2.1.7. Pose new problem, such as 12÷2 (A, C)
2.1.7.1. Ask students to explain what the meaning of this problem is (A,C)
2.1.7.1.1. IF student says she broke the dividend into the number of groups
indicated by the divisor and determined how many were in each group
THEN step 2.1.6.2 (A)
2.1.7.1.2. IF student says she determined how many times the divisor fit into
the dividend, THEN step 2.1.8 (A)
2.1.8. Ask students to formulate a real-life example of the problem (C)
2.1.9. Direct students to discuss this with a partner (C)
2.1.10. Check for understanding by calling on students randomly (C)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 148
2.1.11. Highlight correct examples, such as, “If Juan has $12, how many $2 bills would
be needed to match that amount?” (C)
2.2. Continue to reinforce that the goal of division is not to determine how many equal
groups are indicated by the divisor, but rather, to determine how many times the divisor
fits into the dividend, as in long division (A, C)
2.3. Conclude lesson with a similar problem, to which students must provide a written
answer (A, C)
2.4. Review all answers (A, C)
2.4.1. IF more than 3 answers indicate misunderstanding, THEN begin next class
session with a review of the concept of the meaning of division (A, C)
2.4.2. IF only 2 or 3 answers indicate misunderstanding, THEN sit with these students
during next class session and provide remediation (A, C)
2.5. Pose division problem with a quotient less than one, such as 1÷2 (C)
2.5.1. Ask students, “What is the meaning of this problem?” (A,C)
2.5.2. Direct students to discuss this with a neighbor (C)
2.5.3. Check for understanding by calling on students randomly (C)
2.5.4. Entertain a discussion centering on the concept that the problem involves
determining how many groups of 2 are in 1 (A, C)
2.5.5. Ask students to formulate a real-life example of the problem (C)
2.5.6. Direct students to discuss this with a partner (C)
2.5.7. Check for understanding by calling on students randomly (C)
2.5.8. Highlight correct examples, such as, “If Juan has 1 bagel, how many groups of 2
bagels are in that 1 bagel?” (C)
2.6. Pose another similar problem, such as 1÷3 (C)
2.6.1. Ask students, “What is the meaning of this problem?” (A, C)
2.6.2. Direct students to discuss this with a neighbor (C)
2.6.3. Check for understanding by calling on students randomly (C)
2.6.4. Entertain a discussion centering on the concept that the problem involves
determining how many groups of 3 are in 1 (A, C)
2.6.5. Ask students to formulate a real-life example of the problem (C)
2.6.6. Direct students to discuss this with a partner (C)
2.6.7. Check for understanding by calling on students randomly (C)
2.6.8. Highlight correct examples, such as, “If Rachel has 1 bagel, how many groups of
3 bagels are in that 1 bagel (C)
2.7. Conclude lesson with a similar problem, to which students must provide a written
answer (C)
2.8. Review all answers (C)
2.8.1. IF a majority of answers indicate misunderstanding, THEN begin next class
session with a review of the concept of the meaning of division (C)
2.8.2. IF only 2 or 3 answers indicate misunderstanding, THEN sit with these students
during next class session and provide remediation (C)
3. Teach operations with integers
3.1. Teach integer operations using a number line (B)
3.1.1. Demonstrate addition of a negative: 4 + -2 involves starting at 4 and going back 2
on number line to 2 (B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 149
3.1.2. Demonstrate addition of a positive: -2 + 7 involves starting at -2 and going
forward 7 to 5 (B)
3.1.3. Demonstrate subtraction of a positive: -4 – 2 involves starting at -4 and going
back 2 to -6 (B)
3.1.4. Demonstrate subtraction of a negative: -4 – (-2) is the opposite of -4 -2, so we go
forward 2 to -2 (B)
3.1.5. Assign similar problems for homework (B)
3.1.6. Visually inspect one or two key problems on following day using clipboard (B)
3.1.6.1. IF student demonstrates understanding, THEN go to step 3.2 (B)
3.1.6.2. IF student’s answers do not demonstrate understanding, THEN check in
with student before end of class to provide brief review (B)
3.2. Teach integer operations using manipulatives (B)
3.2.1. Distribute integer chips (B)
3.2.2. Explain that the black side indicates positive and the red side indicates negative
(B)
3.2.3. Demonstrate addition (B)
3.2.3.1. Show that -4 + 2 can be represented as four red chips and two black chips
(B)
3.2.3.2. Match one red chip and one black chip to indicate zero, based on the
inverse property of addition (B)
3.2.3.3. Repeat for the other black chip to yield a total of two zeroes, which yields
a sum of -2 (B)
3.2.4. Demonstrate subtraction (B)
3.2.4.1. Show that -4 – 2 can be represented as four red chips (B)
3.2.4.2. Explain that it’s impossible to match negatives and positives because you
only have 4 black chips (B)
3.2.4.3. Explain that identity property of addition says that adding zero to a
number results in no change to that number (B)
3.2.4.4. Demonstrate placing two zero pairs (one black and one red chip) next to
the four red chips (B)
3.2.4.5. Demonstrate that you can now subtract the two positive (black) chips and
the resulting value is -6 (B)
3.2.5. Assign similar problems for homework (B)
3.2.6. Visually inspect one or two key problems on following day using clipboard (B)
3.2.6.1. IF student demonstrates understanding, THEN go to step 3.2.7
3.2.6.2. IF student’s answers do not demonstrate understanding, THEN check in
with student before end of class to provide brief review (B)
3.2.7. Demonstrate multiplication (B)
3.2.7.1. Show that 3 x -2 can be represented as three groups of two red chips,
which is -6 (B)
3.2.7.2. Show that -3 x -2 would be the opposite (additive inverse) of 3 x -2 (B)
3.2.7.3. Demonstrate that we can derive the opposite of 3 x -2 by simply flipping
the integer chips over, yielding 6 positives, which signifies that -3 x -2 equals
positive 6 (B)
3.2.8. Demonstrate division (B)
3.2.8.1. Show that -6 ÷ 2 can be represented as six red chips (B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 150
3.2.8.2. Ask, “How many groups of positive 2 can I make out of these 6
negatives?” (B)
3.2.8.3. Explain that the answer is -3 (B)
3.2.8.4. Show next that -6 ÷ -2 would be the opposite (additive inverse) of -6 ÷ 2
by flipping the six red tiles (-6) over and then showing that the answer is 3
positive groups (B)
3.2.8.5. Assign similar problems for homework (B)
3.2.8.6. Visually inspect one or two key problems on following day using
clipboard (B)
3.2.8.6.1. IF student demonstrates understanding, THEN go to step 4
3.2.8.6.2. IF student’s answers do not demonstrate understanding, THEN
check in with student before end of class to check for understanding (B)
3.2.8.6.2.1. IF student demonstrates understanding, THEN go to step 3
(B)
3.2.8.6.2.2. IF student does not demonstrate understanding, direct her
to come to class during Muir Time and provide remediation (B)
4. Teach number domains
4.1. Explain that before we can begin operations with fractions, we need to understand the
different types of numbers (B)
4.2. Review that whole numbers are 0,1,2,3,4…to infinity (B)
4.3. Review that integers are whole numbers and their opposites, such as 1 and -1, 2 and -2,
etc. (B)
4.4. Teach that in between the whole numbers and integers are other numbers (B)
4.5. Explain that these are called rational numbers, defined as any number that can be written
as a fraction (B)
4.6. Explain that whole numbers and integers are also rational numbers, because they can
also be written as fractions (B)
4.7. Explain that decimals are also rational numbers, because they can be written as fractions
(B)
4.7.1. Display the example of 0.7 (B)
4.7.2. Ask students to write on their whiteboards the name of the place the 7 occupies
(B)
4.7.2.1. IF student understands it is in the tenths place, THEN go to step 4.7.3 (B)
4.7.2.2. If student does not understand it is in the tenths place, check in with
student before end of class to provide brief review (B)
4.7.3. Ask students to write how 0.7 is read on their whiteboards (B)
4.7.3.1. IF student understands it is read as “seven tenths”, THEN go to step 4.7.4
(B)
4.7.3.2. IF student does not understand it is read as “seven tenths”, then check in
with student before end of class to provide brief review (B)
4.7.4. Ask students to write 0.7 as a fraction on their whiteboards (B)
4.7.4.1. IF student understands it is written as
7
10
THEN go to step 4.7.5 (B)
4.7.4.2. IF student does not understand it is written as
7
10
THEN check in with
student before end of class and provide brief review (B)
4.7.5. Ask students to write 0.03 as a fraction on their whiteboards (B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 151
4.7.5.1. IF student understands it is written as
3
100
then go to step 4.7.6 (B)
4.7.5.2. IF student does not understand it is written as
3
100
THEN check in with her
before end of class and provide brief review (B)
4.7.6. Ask students to write 0.004 as a fraction on their whiteboards (B)
4.7.6.1. IF student understands it is written as
4
1000
THEN go to step 4.7.7 (B)
4.7.6.2. IF student does not understand it is written as
4
1000
THEN check in with
her before end of class and provide brief review (B)
4.7.7. Review that the place of the number on the right in any decimal determines what
the denominator will be when it is converted to a fraction (B)
4.8. Review procedure for converting a fraction to a decimal (B)
4.8.1. Review that we need a power of 10 in the denominator in order to convert many
fractions to a decimal (B)
4.8.2. Demonstrate that a fraction such as
3
5
needs to be converted to an equivalent
fraction with a power of 10, in this case 10, in the denominator (B)
4.8.3. Demonstrate multiplying
3
5
x
2
2
to yield
6
10
(B)
4.8.4. Demonstrate that this converts to the decimal 0.6 (B)
4.8.5. Assign similar problems (B)
4.8.6. Visually check for understanding (B)
4.8.6.1. IF students demonstrate understanding, THEN go to step 4.8.7 (B)
4.8.6.2. IF student does not demonstrate understanding, THEN check in with her
before end of class to provide brief review (B)
4.8.7. Review that some fractions, such as
1
16
, have denominators that cannot be
converted to a power of 10 (B)
4.8.8. Review that with fractions such as these, we can divide the numerator by the
denominator, such that 1÷ 16 = .0625 (B)
4.8.9. Assign similar problems (B)
4.8.9.1. IF student demonstrates understanding, THEN go to step 4.8.10 (B)
4.8.9.2. IF student does not demonstrate understanding, THEN check in with
student before end of class to provide brief review (B)
4.8.10. Review that some fractions, such as
1
3
yield non-terminating, repeating decimals
(B)
4.8.11. Demonstrate that, again, we can divide the numerator by the denominator, such
that 1 ÷ 3 = 0.3333
̅
, which repeats and does not terminate, as indicated by the
superscript (B)
4.8.12. Assign similar problems (B)
4.8.12.1. IF student demonstrates understanding, THEN go to step 4.8.13 (B)
4.8.12.2. IF student does not demonstrate understanding, THEN check in with her
before end of class and provide brief review (B)
4.8.13. Assign homework problems for decimal/fraction concepts (B)
4.8.14. Visually inspect one or two key problems on following day using clipboard (B)
4.8.14.1. IF student demonstrates understanding, THEN go to step 4.9 (B)
4.8.14.2. IF student does not demonstrate understanding, THEN check in with her
before end of class to check for understanding (B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 152
4.8.14.2.1. IF student demonstrates understanding, THEN go to step 4.9 (B)
4.8.14.2.2. IF student does not demonstrate understanding, THEN direct her to
come to class during Muir Time and provide remediation (B)
4.9. Explain that not all rational numbers can be integers or whole numbers (B)
4.10. Explain that a number that cannot be written as a fraction (a non-repeating, non-
terminating decimal, such as √12), is an irrational number (B)
5. Review representations of fractions
5.1. Explain that, although this review involves fractions as parts of a whole, later we will be
dealing with fraction concepts involving ratios and proportions (B)
5.2. Remind students that the concept of fractions as parts of a whole requires us to split up
that whole (B)
5.3. Distribute fraction circle manipulatives (B)
5.4. Ask students to represent
3
8
(B)
5.4.1. Visually check for understanding (B)
5.4.1.1. IF student demonstrates understanding, THEN go to step 5.5 (B)
5.4.1.2. IF student does not demonstrate understanding, THEN check in with
student before end of class to provide brief review (B)
5.5. Ask students to represent
1
2
(B)
5.5.1. Visually check for understanding (B)
5.5.1.1. IF student demonstrates understanding, THEN go to step 5.6 (B)
5.5.1.2. IF student does not demonstrate understanding, THEN check in with
student before end of class to provide brief review (B)
5.6. Ask students to show another way to make
1
2
(B)
5.7. Explain to students that representing
1
2
by its equivalents, such as
2
4
or
3
6
, exemplifies the
identity property of multiplication, which states that multiplying any number by a form
of 1 yields that same number (B)
5.8. Ask students, “How do you write an equivalent fraction for
1
2
?” (B)
5.9. Remind students that you do that by multiplying by a form of 1, such as
2
2
, resulting in
the equivalent fraction,
2
4
(B)
5.10. Assign similar problems (B)
5.10.1. Visually check for understanding (B)
5.10.1.1. IF student demonstrates understanding, THEN go to step 5.11 (B)
5.10.1.2. IF student does not demonstrate understanding, THEN check in with
student before end of class to provide brief review (B)
5.11. Distribute fraction strip manipulatives (B)
5.12. Ask students, in groups, or with partners to represent
1
2
(B)
5.12.1. Visually check for understanding (B)
5.12.1.1. IF student demonstrates understanding, THEN go to step 5.13 (B)
5.12.1.2. IF student does not demonstrate understanding, THEN check in with
student before end of class to provide brief review (B)
5.13. Ask students to demonstrate as many ways as possible to represent
1
2
(B)
5.13.1. Visually check for understanding (B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 153
5.13.1.1. IF student demonstrates understanding, THEN go to step 5.14 (B)
5.13.1.2. IF student does not demonstrate understanding, THEN check in with
student before end of class to provide brief review (B)
5.14. Remind students that dividing any number by a form of 1 is identical to the
identity property of multiplication: the result in both cases is the original number (B)
5.15. Distribute fraction circle manipulatives (B)
5.16. Ask students to represent a fraction such as
4
8
in simplest terms (B)
5.17. Visually check for understanding (B)
5.17.1. IF student demonstrates understanding, THEN go to step 6 (B)
5.17.2. IF student does not demonstrate understanding, THEN check in with student
before end of class to check for understanding (B)
5.17.2.1. IF student demonstrates understanding of all elements of step 5, THEN go
to step 6 (B)
5.17.2.2. IF student does not demonstrate understanding of all elements of step 4,
THEN direct her to come to class during Muir time and provide remediation
(B)
6. Review addition and subtraction of fractions and mixed numbers
6.1. Review addition and subtraction with like denominators (A, B)
6.2. Distribute manipulative fraction strips and circles (A, B)
6.2.1. Ask students to model a problem such as
1
2
+
1
2
with strips and circles (B)
6.2.2. Ask students to demonstrate that the answer, two halves, is visually equivalent to
one whole (B)
6.2.3. Ask students to divide by a fraction equal to 1, such as
2
2
, to also yield one whole
(A, B)
6.2.4. Ask students to model a problem such as
3
8
+
1
8
(B)
6.2.5. Ask students to demonstrate that the answer, four eighths, is visually equivalent to
1
2
(B)
6.2.6. Ask students to divide by a fraction equal to 1, such as
4
4
, to also yield
1
2
(A, B)
6.2.6.1. IF students demonstrate understanding, then go to step 6.2.7 (B)
6.2.6.2. IF student appears unclear, then check in with her before end of class to
provide brief review (B)
6.2.7. Ask students to model a problem, such as
5
8
-
1
8
(B)
6.2.8. Ask students to demonstrate that the answer is visually equivalent to
1
2
(B)
6.2.9. Ask students to divide by a fraction equal to 1, such as
4
4
, to also yield
1
2
(A, B)
6.2.9.1. IF students demonstrate understanding, then go to step 6.3 (B)
6.2.9.2. IF student appears unclear, THEN check in with her before end of class to
provide brief review (B)
6.3. Review algorithm for adding and subtracting fractions (A, B)
6.3.1. Ask students to add two fractions with like denominators, such as
1
4
+
1
4
(A, B)
6.3.2. Visually check that students are adding numerators to yield
2
4
(A, B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 154
6.3.3. Ask students to divide by a fraction equal to 1, such as
2
2
, to put in simplest terms,
1
2
(A, B)
6.3.4. Ask students to subtract two fractions with like denominators, such as
3
6
-
1
6
(B)
6.3.5. Ask students to divide by a fraction equal to 1, such as
2
2
, to put in simplest terms,
1
3
(A, B)
6.3.5.1. Assign similar problems for homework (B)
6.3.5.2. Visually inspect one or two key problems on following day using
clipboard (B)
6.3.5.2.1. IF student demonstrates understanding, THEN go to step 6.4 (B)
6.3.5.2.2. IF student’s answers do not demonstrate understanding, THEN
check in with student before end of class to check for understanding (B)
6.3.5.2.2.1. IF student demonstrates understanding, THEN go to step
6.4 (B)
6.3.5.2.2.2. IF student does not demonstrate understanding, direct her
to come to class during Muir Time and provide remediation (B)
6.4. Review addition and subtraction of fractions with unlike denominators (A, B, C)
6.4.1. Demonstrate adding two fractions with unlike denominators, such as
1
2
+
1
4
(A, B,
C)
6.4.2. Review that we can’t add them unless the denominators are the same (A, B, C)
6.4.3. Review that the identify property of multiplication means that we can multiply
1
2
by a form of one,
2
2
to yield a fraction with a denominator of 4,
2
4
(A, B)
6.4.4. Review that we can now add
2
4
+
1
4
=
3
4
(A, B, C)
6.4.5. Assign similar problems (A, B, C)
6.4.5.1. IF students demonstrate understanding, then go to step 6.4.6 (B)
6.4.5.2. IF student appears unclear, then check in with her before end of class to
provide brief review (B)
6.4.6. Demonstrate subtracting two fractions with unlike denominators, such as
1
2
-
1
4
(B)
6.4.7. Review that the identify property of multiplication means that we can multiply
1
2
by a form of one,
2
2
to yield a fraction with a denominator of 4,
2
4
(A,B)
6.4.8. Review that we can now subtract
2
4
-
1
4
=
1
4
(B)
6.4.9. Assign similar problems (B)
6.4.9.1. IF students demonstrate understanding, then go to step 6.4.10 (B)
6.4.9.2. IF student appears unclear, then check in with her before end of class to
provide brief review (B)
6.4.10. Assign addition and subtraction of fractions with unlike denominators for
homework (B)
6.4.11. Visually inspect one or two key problems on following day using clipboard (B)
6.4.11.1. IF student demonstrates understanding, THEN go to step 6.5 (B)
6.4.11.2. IF student’s answers do not demonstrate understanding, THEN check in
with student before end of class to check for understanding (B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 155
6.4.11.2.1. IF student demonstrates understanding, THEN go to step 6.5 (B)
6.4.11.2.2. IF student does not demonstrate understanding, direct her to come
to class during Muir Time and provide remediation (B)
6.5. Review addition and subtraction of mixed numbers with like denominators (B)
6.5.1. Demonstrate 2
1
2
+ 4
1
2
using hand-drawn circles to represent the wholes and
fractions (B)
6.5.2. Review that the two fractions add to one whole, which we must add to the other
six wholes, to yield 7 (B)
6.5.3. Demonstrate 1
3
5
+ 2
4
5
using hand-drawn circles (B)
6.5.4. Review that this yields 3 wholes and seven-fifths (B)
6.5.5. Review through use of fraction strips that
7
5
is the same as one whole and
2
5
(B)
6.5.6. Demonstrate that we now add the 1 whole to the other 3 whole to yield 4 whole,
and then add
2
5
, resulting in 4
2
5
(B)
6.5.7. Demonstrate with fraction strips the corollary of this concept: 1
2
5
can be converted
to the improper fraction,
7
5
(B)
6.5.8. Demonstrate 6
1
8
- 4
5
8
using hand-drawn circles (B)
6.5.9. Review that one whole can be represented by any number over itself, such as
6
6
or
7
7
or
8
8
(B)
6.5.10. Review that we need to borrow one whole from the 6, which we can represent as
8
8
(B)
6.5.11. Review that we must add the
8
8
we borrowed to the
1
8
to yield
9
8
(B)
6.5.12. Demonstrate that we can now subtract to yield 1
4
8
(B)
6.5.13. Review that we need to simplify the
4
8
by dividing by a form of one,
4
4
, yielding
1
1
2
(B)
6.5.14. Assign similar problems (B)
6.5.15. Visually check for understanding (B)
6.5.15.1. IF students demonstrate understanding, then go to step 6.5.16 (B)
6.5.15.2. IF student appears unclear, then check in with her before end of class to
provide brief review (B)
6.5.16. Assign addition and subtraction of mixed numbers with like denominators for
homework (B)
6.5.17. Visually inspect one or two key problems on following day using clipboard (B)
6.5.17.1. IF student demonstrates understanding, THEN go to step 6.6 (B)
6.5.17.2. IF student’s answers do not demonstrate understanding, THEN check in
with student before end of class to check for understanding (B)
6.5.17.2.1. IF student demonstrates understanding, THEN go to step 6.6 (B)
6.5.17.2.2. IF student does not demonstrate understanding, direct her to come
to class during Muir Time and provide remediation (B)
6.6. Review addition and subtraction of mixed numbers with unlike denominators (B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 156
6.6.1. Demonstrate 1
1
2
+ 2
1
4
using hand-drawn circles (B)
6.6.2. Review that we cannot add fractions unless we have the same denominators (A,
B, C)
6.6.3. Review that we can multiply by a form of one,
2
2
, to yield 1
2
4
(B)
6.6.4. Demonstrate adding 1
2
4
+ 2
1
4
to yield 3
3
4
(B)
6.6.5. Demonstrate subtracting 3
1
2
- 1
1
4
using hand-drawn circles (B)
6.6.6. Review that we cannot subtract fractions unless we have the same denominators
(B)
6.6.7. Review that we can multiply by a form of one,
2
2
, to yield 3
2
4
(A, B)
6.6.8. Demonstrate subtracting 3
2
4
- 1
1
4
to yield 2
1
4
(B)
6.6.9. Assign similar problems (B)
6.6.10. Visually check for understanding (B)
6.6.10.1. IF students demonstrate understanding, then go to step 6.6.11 (B)
6.6.10.2. IF student appears unclear, then check in with her before end of class to
provide brief review (B)
6.6.11. Assign addition and subtraction of mixed numbers with unlike denominators for
homework (B)
6.6.12. Visually inspect one or two key problems on following day using clipboard (B)
6.6.12.1. IF student demonstrates understanding, THEN go to step 6.7 (B)
6.6.12.2. IF student’s answers do not demonstrate understanding, THEN check in
with student before end of class to check for understanding (B)
6.6.12.2.1. IF student demonstrates understanding, THEN go to step 6.7 (B)
6.6.12.2.2. IF student does not demonstrate understanding, direct her to come
to class during Muir Time and provide remediation (B)
6.7. Review that because rational numbers can be written as both fractions and decimals, we
can add and subtract fractions and mixed numbers by converting them to decimals (B)
6.7.1. Demonstrate this for fraction addition using an example such as
1
4
+
1
4
=
1
2
(B)
6.7.2. Review that
1
4
needs to be converted to a fraction with a denominator that is a
power of 10 (B)
6.7.3. Demonstrate that multiplying
1
4
by
25
25
yields
25
100
, which in decimal form is .25 (B)
6.7.4. Review that to add .25 + .25 we must line up the decimals and places, then add,
yielding a sum of .5 (B)
6.7.5. Demonstrate that .5 is the same as five tenths, or
5
10
, which can be reduced to
simplest terms as
1
2
(B)
6.7.6. Demonstrate that we can derive the same answer, .5, by converting our answer in
1
4
+
1
4
=
1
2
(B)
6.7.7. Demonstrate that if we convert
1
2
to a fraction with a power of 10 in the
denominator, we get
5
10
(B)
6.7.8. Review that
5
10
is the same as .5 (B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 157
6.7.9. Reinforce idea that regardless of whether we add rational numbers as fractions or
decimals, the answer is the same (B)
6.7.10. Assign similar problems(B)
6.7.10.1. IF students demonstrates understanding, THEN go to step 6.7.11 (B)
6.7.10.2. IF student does not demonstrate understanding, THEN check in with her
before end of class and provide brief review (B)
6.7.11. Demonstrate same concept for subtraction using an example such as
3
4
-
1
4
=
2
4
, or
1
2
(B)
6.7.12. Demonstrate that multiplying
3
4
and
1
4
by
25
25
yields
75
100
and
25
100
, which in decimal
form are .75 and .25 (B)
6.7.13. Review that to subtract .75 - .25 we must line up the decimals and places, then
subtract, yielding a difference of .5 (B)
6.7.14. Demonstrate that .5 is the same as five tenths, or
5
10
, which can be reduced to
simplest terms as
1
2
(B)
6.7.15. Demonstrate that we can derive the same answer, .5, by converting our answer in
3
4
-
1
4
=
1
2
(B)
6.7.16. Demonstrate that if we convert
1
2
to a fraction with a power of 10 in the
denominator, we get
5
10
(B)
6.7.17. Review that
5
10
is the same as .5 (B)
6.7.18. Reinforce idea that regardless of whether we subtract rational numbers as
fractions or decimals, the answer is the same (B)
6.7.19. Assign similar problems (B)
6.7.19.1. IF students demonstrates understanding, THEN go to step 6.7.20 (B)
6.7.19.2. IF student does not demonstrate understanding, THEN check in with her
before end of class and provide brief review (B)
6.7.20. Demonstrate same concept for addition of mixed numbers using an example such
as 1
1
4
+ 1
1
4
= 2
2
4
, or 2
1
2
(B)
6.7.21. Review that
1
4
needs to be converted to a fraction with a denominator that is a
power of 10 (B)
6.7.22. Demonstrate that multiplying
1
4
by
25
25
yields
25
100
, which in decimal form is .25 (B)
6.7.23. Review that to add 1.25 + 1.25 we must line up the decimals and places, then add,
yielding a sum of 2.5 (B)
6.7.24. Demonstrate that 2.5 is the same as two and five tenths, or 2
5
10
, which can be
reduced to simplest terms as 2
1
2
(B)
6.7.25. Demonstrate that we can derive the same answer, 2.5, by converting our answer in
1
1
4
+ 1
1
4
= 2
1
2
(B)
6.7.26. Demonstrate that if we convert 2
1
2
to a fraction with a power of 10 in the
denominator, we get 2
5
10
(B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 158
6.7.27. Review that 2
5
10
is the same as 2.5 (B)
6.7.28. Reinforce idea that regardless of whether we add rational numbers as fractions or
decimals, the answer is the same (B)
6.7.29. Assign similar problems (B)
6.7.29.1. IF students demonstrates understanding, THEN go to step 6.7.30 (B)
6.7.29.2. IF student does not demonstrate understanding, THEN check in with her
before end of class and provide brief review(B)
6.7.30. Demonstrate same concept for subtraction of mixed numbers using an example
such as 2
3
4
- 1
1
4
= 1
2
4
, or 1
1
2
(B)
6.7.31. Demonstrate that multiplying
3
4
and
1
4
by
25
25
yields
75
100
and
25
100
, which in decimal
form are .75 and .25 (B)
6.7.32. Review that to subtract 2.75 - 1.25 we must line up the decimals and places, then
subtract, yielding a difference of 1.5 (B)
6.7.33. Demonstrate that 1.5 is the same as one and five tenths, or 1
5
10
, which can be
reduced to simplest terms as 1
1
2
(B)
6.7.34. Demonstrate that we can derive the same answer, 1.5, by converting our answer in
2
3
4
- 1
1
4
= 1
1
2
(B)
6.7.35. Demonstrate that if we convert 1
1
2
to a fraction with a power of 10 in the
denominator, we get 1
5
10
(B)
6.7.36. Review that 1
5
10
is the same as 1.5 (B)
6.7.37. Reinforce idea that regardless of whether we subtract rational numbers as
fractions or decimals, the answer is the same (B)
6.7.38. Assign similar problems (B)
6.7.38.1. IF students demonstrates understanding, THEN go to step 6.7.39 (B)
6.7.38.2. IF student does not demonstrate understanding, THEN check in with her
before end of class and provide brief review (B)
6.7.39. Assign homework problems for decimal/fraction equivalence (B)
6.7.40. Visually inspect one or two key problems on following day using clipboard (B)
6.7.40.1. IF student demonstrates understanding, THEN go to step 7
6.7.40.2. IF student does not demonstrate understanding, THEN check in with her
before end of class to check for understanding (B)
6.7.40.2.1. IF student demonstrates understanding, THEN go to step 7 (B)
6.7.40.2.2. IF student does not demonstrate understanding, THEN direct her to
come to class during Muir Time and provide remediation (B)
7. Review multiplication of fractions and mixed numbers
7.1. Review multiplication of fractions (B)
7.1.1. Demonstrate multiplying a fraction by a whole number, such as
1
3
x 2 (B)
7.1.2. Demonstrate we can approach this through repeated addition (B)
7.1.3. Review that
1
3
x 2 means two one-thirds, or
1
3
+
1
3
=
2
3
(B)
7.1.4. Demonstrate that we can also convert the whole number, 2, to a fraction by giving
the 2 a denominator, 1, yielding
2
1
(B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 159
7.1.5. Demonstrate that
1
3
x 2 is actually
1
3
x
2
1
, or
2
3
(B)
7.1.6. Review that in multiplication of fractions, the algorithm requires that we multiply
the numerators by the numerators and the denominators by the denominators (B)
7.1.7. Extend this understanding to multiplication of fractions by fractions
7.1.8. Demonstrate that a problem such as
1
3
x
1
2
also involves multiplying the
numerators by the numerators and the denominators by the denominators, yielding
1
6
(B)
7.1.9. Assign similar problems (B)
7.1.10. Visually check for understanding (B)
7.1.10.1. IF students demonstrate understanding, then go to step 7.1.11 (B)
7.1.10.2. IF student appears unclear, then check in with her before end of class to
provide brief review (B)
7.1.11. Assign multiplication of fractions for homework (B)
7.1.12. Visually inspect one or two key problems on following day using clipboard (B)
7.1.12.1. IF student demonstrates understanding, THEN go to step 7.1.13 (B)
7.1.12.2. IF student’s answers do not demonstrate understanding, THEN check in
with student before end of class to check for understanding (B)
7.1.12.2.1. IF student demonstrates understanding, THEN go to step 7.1.13
(B)
7.1.12.2.2. IF student does not demonstrate understanding, direct her to come
to class during Muir Time and provide remediation (B)
7.1.13. Demonstrate multiplying a mixed number by a mixed number, such as 2
1
4
x 2
1
6
(B)
7.1.14. Review that we must convert each mixed number into an improper fraction before
we can multiply (B)
7.1.15. Review that 2
1
4
can be represented as
9
4
and 2
1
6
can be represented as
13
6
(B)
7.1.16. Demonstrate that we multiply the numerators by the numerators and the
denominators by the denominators, yielding
117
24
(B)
7.1.17. Review that we need to simplify, yielding 4 whole and 21 twenty-fourths (B)
7.1.18. Review that
21
24
can be simplified by dividing by a form of one,
3
3
, yielding
7
8
(B)
7.1.19. Display completed product, 4
7
8
(B)
7.1.20. Assign similar problems (B)
7.1.21. Visually check for understanding (B)
7.1.21.1. IF students demonstrate understanding, then go to step 7.1.22 (B)
7.1.21.2. IF student appears unclear, then check in with her before end of class to
provide brief review (B)
7.1.22. Assign multiplication of mixed numbers for homework (B)
7.1.23. Visually inspect one or two key problems on following day using clipboard (B)
7.1.23.1. IF student demonstrates understanding, THEN go to step 8 (B)
7.1.23.2. IF student’s answers do not demonstrate understanding, THEN check in
with student before end of class to check for understanding (B)
7.1.23.2.1. IF student demonstrates understanding, THEN go to step 8 (B)
7.1.23.2.2. IF student does not demonstrate understanding, direct her to come
to class during Muir Time and provide remediation (B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 160
8. Teach division of whole numbers by fractions
8.1. Hold up a whole piece of paper and pose question that represents the problem 1÷
1
2
,
such as “If this piece of paper represents a whole cake, how can I divide it by one-
halves?” (B, C)
8.2. Direct students to discuss the question (B, C)
8.3. Circulate to check for understanding (B, C)
8.4. Call on volunteers who were able to understand that the solution is two halves (B, C)
8.5. Distribute fraction strips (C)
8.6. Relate the scenario, “If I have one piece of paper, how many halves are there in that one
piece?” (C)
8.6.1. Ask students to work with a partner to model scenario with manipulatives (C)
8.6.2. Circulate among students to determine whether they are creating concrete
representations of 1 ÷
1
2
= 2 (C)
8.6.3. Check for understanding by calling on students randomly (C)
8.6.4. Validate correct answers (C)
8.6.5. Demonstrate a correct representation on whiteboard using magnetic fraction strips
(A, C)
8.6.6. Ask students in pairs to formulate a real-life example of the problem, 1 ÷
1
2
= 2
(C)
8.6.7. Circulate among students to check for understanding (C)
8.6.8. Call on volunteers who were able to formulate correct examples, such as, “If I
have a whole pizza, how many halves are contained therein?” (C)
8.6.9. Ask students to work with a partner to explain what the problem 1 ÷
1
2
= 2 means
(C)
8.6.10. Circulate to check for understanding (C)
8.6.11. Call on volunteers who were able to articulate that the problem means, “How
many one-halves are contained in one whole” (A, B, C)
8.7. Direct students to work with a partner to create a real-life example of the problem, 1÷
1
4
(C)
8.8. Circulate among students to check for understanding (C)
8.9. Call on volunteers who were able to articulate examples, such as, “If I have one dollar,
how many quarters are contained in that dollar?” (C)
8.10. Ask students to work with a partner to explain what the problem 1÷
1
4
means (C)
8.11. Circulate to check for understanding (C)
8.12. Call on volunteers who were able to articulate that the problem means, “How
many one-fourths are contained in one whole” (A, B, C)
8.12.1. Ask students to work with a partner to model scenario with manipulatives (C)
8.12.2. Circulate among students to determine whether they are creating concrete
representations of 1 ÷
1
4
= 4 (C)
8.12.3. Check for understanding by calling on students randomly (C)
8.12.4. Validate correct answers (C)
8.12.5. Demonstrate a correct representation on whiteboard using magnetic fraction strips
(C)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 161
8.12.5.1. IF teacher observation indicates student understands how to create
concrete representations of problems involving whole numbers divided by
fractions, THEN step 8.13 (C)
8.12.5.2. IF teacher observation indicates student does not understand, THEN
provide remediation with similar examples after class (C)
8.13. Distribute math paper and pencils
8.14. Relate the scenario, “If I have one pizza and want to share it among 3 people, how
much will each person get?” (C)
8.14.1. Direct students to work with a partner to draw either a number line or area model
representation of the scenario (C)
8.14.2. Circulate among students to determine whether they are creating pictorial
representations of 1 ÷
1
3
= 3 (C)
8.14.3. Ask students to share their drawings with other neighbors (C)
8.14.4. Check for understanding by calling on students randomly, asking them to describe
their drawings (C)
8.14.5. Validate correct answers that represent 1 ÷
1
3
= 3 (C)
8.14.6. Demonstrate a correct representation on whiteboard by drawing both an area
model and a number line depicting 1 ÷
1
3
= 3 (C)
8.15. Relate next scenario, “If I have one bagel and want to share it among 4 people,
how much will each person get?” (C)
8.15.1. Direct students to work with a partner to draw either a number line or area model
representation of the scenario (C)
8.15.2. Circulate among students to determine whether they are creating pictorial
representations of 1 ÷
1
4
= 4 (C)
8.15.3. Ask students to share their drawings with other neighbors (C)
8.15.4. Check for understanding by calling on students randomly, asking them to describe
their drawings (C)
8.15.5. Validate correct answers that represent 1 ÷
1
4
= 4 (C)
8.15.6. Demonstrate a correct representation on whiteboard by drawing both an area
model and a number line depicting 1 ÷
1
4
= 4 (C)
8.15.6.1. IF teacher observation indicates student understands how to create
pictorial representations of problems involving whole numbers divided by
fractions, THEN step 8.16 (C)
8.15.6.2. IF teacher observation indicates student does not understand, THEN
provide remediation after class (C)
8.16. Ask students to consider their answers to previous problems, such as 1 ÷
1
2
= 2 , 1
÷
1
3
= 3 , and 1 ÷
1
4
= 4 (C)
8.16.1. Ask students to pair with a partner (C)
8.16.2. Ask students to look for patterns in these problems (A, C)
8.16.3. Ask students to formulate a procedural rule that can be followed (A, C)
8.16.4. Circulate among students to check for understanding (A, C)
8.16.5. Ask students that were able to formulate a rule to share it with the class (A, C)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 162
8.16.6. Validate that the rule is to multiply the whole number by the reciprocal of the
divisor (A, B, C)
8.16.7. Assign similar problems (A, C)
8.16.8. Direct students to solve using the procedural rule and to write an explanation of
the procedure they used (A, C)
8.16.9. Circulate among students, and check written explanations for understanding (A,
C)
8.16.9.1. IF student demonstrates understanding, THEN step 9 (A, C)
8.16.9.2. IF student does not demonstrate understanding, THEN provide
remediation after class (A, C)
9. Teach division of fractions by whole numbers
9.1. Display an apple that is cut into two pieces, asking, “If I want to share one of these
halves of an apple between two people, how much will each person get?” (C)
9.2. Call on student volunteers (C)
9.3. Confirm correct answers by demonstrating that the answer is one-fourth of the whole
apple for each person (C)
9.4. Distribute fraction strips (C)
9.5. Relate the same scenario, “If I have one half an apple and want to share it between two
people, how much will each person get?” (C)
9.5.1. Ask students to work with a partner to model scenario with manipulatives (C)
9.5.2. Circulate among students to determine whether they are creating concrete
representations of
1
2
÷ 2 =
1
4
(C)
9.5.3. Check for understanding by calling on students randomly (C)
9.5.4. Validate correct answers (C)
9.5.5. Demonstrate a correct representation on whiteboard using magnetic fraction strips
(A, C)
9.6. Relate next scenario, “If I have one half an apple and want to share it among three
people, how much will each person get?” (C)
9.7. Display an apple that is cut into two pieces, asking, “If I want to share one of these
halves of an apple among three people, how much will each person get?” (C)
9.8. Call on student volunteers (C)
9.9. Confirm correct answers by demonstrating that the answer is one-sixth of the whole
apple for each person (A, C)
9.9.1. Ask students to work with a partner to model scenario with manipulatives (C)
9.9.2. Circulate among students to determine whether they are creating concrete
representations of
1
2
÷ 3 =
1
6
(C)
9.9.3. Check for understanding by calling on students randomly (C)
9.9.4. Validate correct answers (C)
9.9.5. Demonstrate a correct representation on whiteboard using magnetic fraction strips
(A, C)
9.9.5.1. IF teacher observation indicates student understands how to create
concrete representations of problems involving fractions divided by whole
numbers, THEN step 9.10 (C)
9.9.5.2. IF teacher observation indicates student does not understand, THEN
provide remediation after class (C)
9.10. Distribute math paper and pencils (C)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 163
9.11. Relate the scenario, “If I have one third of a candy bar and want to share it
between 2 people, how much will each person get?” (C)
9.12. Draw both an area model and a number line representation on whiteboard of
problem (C)
9.13. Demonstrate how both of these representations are similar to the previous fraction
strip representation (C)
9.13.1. Direct students to work with a partner to draw either a number line or area model
representation of the scenario (C)
9.13.2. Circulate among students to determine whether they are creating pictorial
representations of
1
3
÷ 2 (C)
9.13.3. Ask students to share their drawings with other neighbors (C)
9.13.4. Check for understanding by calling on students randomly, asking them to describe
their drawings (C)
9.13.5. Validate correct answers that represent
1
3
÷ 2 =
1
6
(C)
9.14. Relate next scenario, “If I have one fourth of a bagel and want to share it between
2 people, how much will each person get?” (C)
9.14.1. Direct students to work with a partner to draw either a number line or area model
representation of the scenario (C)
9.14.2. Circulate among students to determine whether they are creating pictorial
representations of
1
4
÷ 2 (C)
9.14.3. Ask students to share their drawings with other neighbors (C)
9.14.4. Check for understanding by calling on students randomly, asking them to describe
their drawings (C)
9.14.5. Validate correct answers that represent
1
4
÷ 2 =
1
8
(C)
9.14.6. Demonstrate a correct representation on whiteboard by drawing both an area
model and a number line depicting
1
4
÷ 2 =
1
8
(C)
9.14.6.1. IF teacher observation indicates student understands how to create
pictorial representations of problems involving fractions divided by whole
numbers, THEN step 9.15 (C)
9.14.6.2. IF teacher observation indicates student does not understand, THEN
provide remediation after class (C)
9.15. Ask students to consider their answers to previous problems, such as
1
2
÷ 2 =
1
4
,
1
2
÷ 3 =
1
6
, and
1
4
÷ 2 =
1
8
(C)
9.15.1. Ask students to pair with a partner (C)
9.15.2. Ask students to look for patterns in the previous problems (A, C)
9.15.3. Ask students to formulate a procedural rule that can be followed (A,C)
9.15.4. Circulate among students to check for understanding (A, C)
9.15.5. Ask students that were able to formulate a rule to share it with the class (A, C)
9.15.6. Validate that the rule is to multiply the fraction by the reciprocal of the whole
number (A, B, C)
9.15.7. Assign similar problems, such as
1
5
÷ 2 (A,C)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 164
9.15.8. Direct students to solve using fraction strips, then a pictorial representation, and
then to validate using the procedural rule, and include a written description of the
written description (A, C)
9.15.9. Circulate among students and check strips, pictures, and written descriptions for
understanding (A, C)
9.15.9.1. IF student demonstrates understanding, THEN step 10 (A, C)
9.15.9.2. IF student does not demonstrate understanding, THEN provide
remediation after class (A, C)
10. Teach division of fractions by fractions
10.1. Teach concrete representation (C)
10.2. Distribute fractions strips (C)
10.2.1. Pose problem such as
3
4
÷
1
4
(C)
10.2.2. Remind students that the problem involves determining how many fourths are in
three fourths (A, B, C)
10.2.3. Use magnetic fraction strips on whiteboard to demonstrate that the answer is 3:
there are 3 one-fourths in three-fourths (C)
10.2.4. Direct students to model the same problem at their desks (C)
10.2.5. Circulate among students to check for understanding (C)
10.2.6. Pose similar problem, such as
1
2
÷
1
4
(C)
10.2.7. Direct students to model problem with fraction strips or linking cubes (C)
10.2.8. Circulate among students to check for understanding (C)
10.2.9. Demonstrate on board that the answer is 2: there are two one-fourths in one half
(C)
10.2.10. Pose similar problems (130-131)
10.2.11. Circulate among students to check for understanding (77-78)
10.2.11.1. IF student demonstrates understanding, THEN step 10.3 (C)
10.2.11.2. IF students does not demonstrate understanding, THEN provide
remediation after class (C)
10.3. Teach pictorial representation (B, C)
10.3.1. Pose problem, such as
3
4
÷
1
4
(B, C)
10.3.2. Remind students that the problem involves determining how many fourths are in
three fourths (A, B, C)
10.3.3. Draw an area model and a number line on whiteboard to demonstrate that there
are three one-fourths in three fourths (B, C)
10.3.4. Direct students to model the same problem at their desks (B, C)
10.3.5. Circulate among students to check for understanding (B, C)
10.3.6. Direct students to validate the answer using the procedural rule (C)
10.3.7. Circulate among students to check for understanding (B, C)
10.3.8. Call on volunteers who were able to multiply three-fourths by the reciprocal of
one-fourth, 4, to yield 3 (B, C)
10.3.9. Pose similar problem, such as
1
2
÷
1
4
(C)
10.3.10. Direct students to model problem with fraction strips, and then area
models or number lines, and then validate with the procedural rule (C)
10.3.11. Circulate among students to check for understanding (C)
10.3.12. Demonstrate on board that there are two fourths in one half (C)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 165
10.3.13. Pose similar problems (C)
10.3.14. Circulate among students to check for understanding (C)
10.3.14.1. IF student demonstrates understanding, THEN step 10.4 (C)
10.3.14.2. IF students does not demonstrate understanding, THEN provide
remediation after class (C)
10.4. Teach complex fraction approach (B)
10.4.1. Remind students that division can be represented through fractions: the dividend
can be represented as numerator, and the divisor as denominator (B)
10.4.2. Demonstrate that
3
4
÷
1
4
can be represented as a complex fraction, in the form
3
4
1
4
(B)
10.4.3. Remind students that anything divided by one is itself (B)
10.4.4. Remind students that the identity property states that multiplying a value by 1
does not change that value (B)
10.4.5. Demonstrate that multiplying
3
4
1
4
x
4
1
4
1
is both multiplying by 1 and also going to
create a 1 in the denominator, based on the inverse property of multiplication (690-
B)
10.4.6. Draw the outline of a “1” around
4
1
4
1
(B)
10.4.7. Remind students that because the denominator is 1, we are left with
3
4
x
4
1
(B)
10.4.8. Remind students that the algorithm for multiplying fractions, by which we
multiply the numerators by numerators and denominators by denominators, means
that the product is going to be
12
4
(B)
10.4.9. Remind students to reduce by dividing by a form of 1, in this case
4
4
(B)
10.4.10. Explain that the answer is 3, which is the same answer derived through the
pictorial method (B)
10.4.11. Assign similar problems asking students to solve using complex fractions
(B)
10.4.12. Visually check for understanding (B)
10.4.12.1. IF students demonstrate understanding, then go to step 10.4.13 (B)
10.4.12.2. IF student appears unclear, then check in with her before end of class to
provide brief review (B)
10.4.13. Review that converting a fraction division problem into a complex fraction
results in multiplying the first fraction by the reciprocal of the second fraction, a fact
we also discovered when we divided whole numbers by fractions and fractions by
whole numbers using manipulatives and pictorial representations (A, B, C)
10.4.14. Administer assessment in which students must solve similar problems and
also write an explanation of why they can derive the same answer by simply
multiplying by the reciprocal (B)
10.4.14.1. IF student answers demonstrate understanding, THEN go to step 10.5 (B)
10.4.14.2. IF student does not demonstrate understanding, direct her to come to class
during Muir Time and provide remediation (B)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 166
10.5. Teach alternate method for dividing fractions by fractions, which involves
converting the fractions to decimals, and then dividing the decimals (B)
10.5.1. Demonstrate conceptual nature of process with an example using currency
10.5.1.1. Demonstrate using the example
1
2
÷
1
4
(B)
10.5.1.2. Review that we must first convert both
1
2
and
1
4
to equivalent fractions with
denominators that are powers of 10 or 100 (B)
10.5.1.3. Demonstrate multiplying
1
2
and
1
4
by
5
5
and
25
25
respectively, to yield
5
10
and
25
100
(B)
10.5.1.4. Review that these new fractions convert to decimals of .50 and .25 (B)
10.5.1.5. Connect these two values to the concept of money by explaining that .50
can be viewed as fifty cents and .25 can be viewed as twenty five cents (B)
10.5.1.6. Demonstrate that .50 ÷ .25 is equivalent to asking, “How many quarters
make up fifty cents?” (B)
10.5.1.7. Demonstrate that the answer is “2” (B)
10.5.2. Demonstrate procedural process (B)
10.5.2.1. Review that
1
2
÷
1
4
can be represented by decimals as .50 ÷ .25 (B)
10.5.2.2. Review that we can represent this in long division as .25). 50
̅ ̅ ̅ ̅ ̅
(B)
10.5.2.3. Review that we must shift both the decimal point in the divisor and the
decimal point in the dividend two places to the right before we can divide,
yielding 25)50
̅ ̅ ̅ ̅
(B)
10.5.2.4. Demonstrate that the quotient becomes 2, the same answer we obtained in
the money example (B)
10.5.2.5. Assign similar problems (B)
10.5.2.6. Visually check for understanding (B)
10.5.2.6.1. IF student demonstrates understanding, THEN step 11 (B)
10.5.2.6.2. IF student does not demonstrate understanding, THEN direct her to
come to class during Muir Time and provide remediation (B)
11. Teach division of mixed numbers by mixed numbers
11.1. Pose problem such as 2
1
2
÷ 1
1
4
(A, B, C)
11.2. Direct students to create a concrete representation with fraction strips (C)
11.2.1. Circulate among students to check for understanding (C)
11.2.2. Ask students with correct answers to recreate their representations on whiteboard
with magnetic fraction strips (C)
11.2.3. Validate that correct answer is 2 (C)
11.2.4. Assign similar problems (C)
11.2.4.1. IF student demonstrates understanding, THEN step 11.2.5 (C)
11.2.4.2. IF student does not demonstrate understanding, THEN provide
remediation after class (C)
11.2.5. Direct students to create a pictorial representation of same problem (B, C)
11.2.6. Circulate among students to check for understanding (B, C)
11.2.7. Ask students with correct answers to recreate their area models, circle models, or
number lines on whiteboard (B, C)
11.2.8. Validate that correct drawings depict 2
1
2
÷ 1
1
4
= 2 (B, C)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 167
11.2.9. Assign similar problems (B, C)
11.2.9.1. IF student demonstrates understanding, THEN step 11.3 (B, C)
11.2.9.2. IF student does not demonstrate understanding, THEN provide
remediation after class (B, C)
11.3. Direct students to solve problem using the complex fraction method (B)
11.3.1. Remind students that we need to convert each term to an improper fraction,
yielding
5
2
÷
5
4
(B)
11.3.2. Demonstrate that we can convert this to a complex fraction,
5
2
5
4
(B)
11.3.3. Remind students that we can multiply by
4
5
4
5
to change the denominator to 1 (B)
11.3.4. Remind students that converting to a complex fraction and then converting the
denominator to 1 is same as multiplying by the reciprocal (B)
11.3.5. Assign similar problems (B)
11.3.6. Visually check for understanding (B)
11.3.6.1. IF students demonstrate understanding, then go to step 11.4 (B)
11.3.6.2. IF student does not demonstrate understanding, direct her to come to class
during Muir Time and provide remediation (B)
11.4. Direct students to solve problem using the procedural method (A, B, C)
11.4.1. Circulate among students to check for understanding (A, B, C)
11.4.2. Ask students with correct answers to recreate their algorithms on whiteboard (A,
B, C)
11.4.3. Validate that correct procedure involves converting 2
1
2
to
5
2
and 1
1
4
to
5
4
, and then
multiplying
5
2
by the reciprocal of
5
4
to yield
5
2
x
4
5
=
20
10
or 2 (A, B, C)
11.4.4. Assign similar problems (A, B, C)
11.4.4.1. IF student demonstrates understanding, THEN step 12 (A, B, C)
11.4.4.2. IF student does not demonstrate understanding, THEN provide
remediation after class (A, B, C)
12. Teach division of fraction word problems
12.1.1. Pose problem such as, “If a recipe calls for
1
2
cup of flour for one batch of bread,
and I have
3
4
cup of flour, how many batches of the recipe can I make?” (C)
12.1.2. Direct students to model the problem with manipulatives (C)
12.1.3. Circulate among students to check for understanding (C)
12.1.4. Ask students with correct representations to recreate their representations on
whiteboard with magnetic fraction strips (C)
12.1.5. Validate correct concrete representation of
3
4
÷
1
2
(C)
12.1.6. Direct students to model the problem pictorially (C)
12.1.7. Circulate among students to check for understanding (C)
12.1.8. Ask students with correct representations to recreate their representations on
whiteboard with number lines or area models (C)
12.1.9. Validate correct pictorial representation of
3
4
÷
1
2
(C)
12.1.10. Direct students to solve the problem with the previously discovered
procedure (C)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 168
12.1.11. Circulate among students to check for understanding (C)
12.1.12. Ask students with correct procedure to write their number sentences on the
board (C)
12.1.13. Validate correct number sentence is
3
4
÷
1
2
=
3
4
x
2
1
=
6
4
= 1
1
2
(C)
12.1.14. Assign similar problems, asking students to create both a concrete and
pictorial representation, and to solve using the number sentence procedure (C)
12.1.15. Circulate among students to check for understanding (C)
12.1.15.1. IF student demonstrates understanding, THEN end task (C)
12.1.15.2. IF student does not demonstrate understanding, THEN provide
remediation after class (C)
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 169
SME Steps Alignment
Step Type
Final Gold Standard Protocol
Analysis A B C A D 1, 2, 3
Procedure 1: Review concept
of multiplication
40 6
1 A
1.1 Explain that to model
multiplication, we will construct
rectangles (A)
1 0 0 1
2 A
1.1.1 Introduce notion of
rectangles as area models for
multiplication (A)
1 0 0 1
3 A
1.1.2 Review concept that
area models are arrays with rows
and columns (A)
1 0 0 1
4 A
1.1.2.1 Explain that the
rows and columns correspond to
the two multiplicands
1 0 0 1
5 A
1.1.3 Introduce the
example of 2 x 3 (A)
1 0 0 1
Appendix F
Incremental Coding Spreadsheet
Spreadsheet Analysis: Gold Standard Protocol Procedures, Action and Decision Steps
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 170
6 A
1.1.3.1 Model how to draw
the rows: 2 (first multiplicand)
rectangles on top of each other
(234-5) and that the 2 rectangles
need to be long enough to
accommodate the columns that
correspond to the other
multiplicand, 3 (A)
1 0 0 1
7 A
1.1.3.2 Ask, “What do we
multiply those 2 rows by?” (A)
1 0 0 1
8 A
1.1.3.3 Call on students
whose hands are not raised (A)
1 0 0 1
9 D
1.1.3.3.1 IF student has
correct answer, THEN go to step
1.1.3.4 (A)
1 0 0 1
10 D
1.1.3.3.2 IF student has
incorrect answer, THEN call on
another student whose hand isn’t
raised, until a student gives the
correct answer (A)
1 0 0 1
11 A
1.1.3.4 Check for
understanding with discussion (A)
1 0 0 1
12 A
1.1.4 Model how to divide
those two boxes by 3, the other
multiplicand, so that there are 3
columns (A)
1 0 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 171
13 A
1.1.4.1 Demonstrate how
to count the number of boxes
created by the 2 rows and 3
columns, which is 6 (A)
1 0 0 1
14 A
1.1.4.2 Reinforce the point of the
exercise that 2 rows by 3 columns
is a graphical representation of
2x3=6 (A)
1 0 0 1
15 A
1.2 Repeat above sequence
with a new multiplication problem
with unequal multiplicands;
students independently create area
models corresponding to that
problem (A) 1 0 0 1
16 A
1.3 Wander around and check
visually (A)
1 0 0 1
17 A
1.4 Ask students,
following several additional
problems area models with
unequal multiplicands, “What
shape have we been creating?” (A)
1 0 0 1
18 D
1.4.1 IF students demonstrate
understanding, THEN step 1.5 (A)
1 0 0 1
19 D
1.4.2 IF students are
unclear, THEN give them a
worksheet for homework (A)
1 0 0 1
20 A
1.5 Reinforce that all
examples have been rectangles (A)
1 0 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 172
21 A
1.6 Ask, “What does it
mean to be a rectangle?” (A)
1 0 0 1
22 A
1.7 Ask, “How do you
know a rectangle when you see
one?” (A)
1 0 0 1
23 A
1.8 Challenge students to
create a definition of a rectangle (2
pairs of parallel sides, the
opposites being equal in length,
and 4 right angles) (A)
1 0 0 1
24 A
1.9 Reinforce that what
you have built is a rectangle (A)
1 0 0 1
25 A
1.10 Introduce several
multiplication problems with 2
equal multiplicands (A)
1 0 0 1
26 A
1.11 Ask students to create
one area model with unequal
multiplicands and then one area
model with equal multiplicands
(A)
1 0 0 1
27 A
1.12 Check visually (A)
1 0 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 173
28 A
1.12.1 Build a connection
by asking, “What plane figure is
created when all sides are equal?”
(A)
1 0 0 1
29 A
1.12.2 Ask, after student
response of “square” is elicited,
“Is a square not also a rectangle?”
(A)
1 0 0 1
30 A
1.12.3 Reinforce concept
that a square is a rectangle with 4
equal sides (A)
1 0 0 1
31 A
1.12.4Display visual on
board during discussion. (A)
1 0 0 1
32 A
1.12.5 Display all squares
whose sides range in length from 1
to 10 units. (A)
1 0 0 1
33 A
1.12.6 Discuss that
squares are a subset of rectangles
(A)
1 0 0 1
34 A
1.12.7 Discuss here that
all squares are rectangles (A)
1 0 0 1
35 D
1.12.7.1 IF students are
struggling to conceptualize these
understandings, problems. THEN
give students a worksheet (A)
1 0 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 174
36 D
1.12.7.2 IF students
understand concepts, THEN
step1.13
1 0 0 1
37 A
1.13 Introduce a new
problem with unequal
multiplicands (e.g. 3x4) (A)
1 0 0 1
38 A
1.13.1 Ask students to
compare their models with a
neighbor (A)
1 0 0 1
39 A
1.13.2 Ask students to
decide if their area model looks
like their neighbor’s (A)
1 0 0 1
40 A
1.13.2.1 Showcase
students whose models differ (e.g.
3x4 area models versus 4x3 area
models) (A)
1 0 0 1
41 A
1.13.2.2 Circulate and ask
students with different models to
draw their models on board and
then have a discussion (A)
1 0 0 1
42 A
1.13.2.3 Ask, “If our
multiplication problem is 3x4,
how many boxes do we expect in
our area model,” the answer, of
course to which is 12 (A)
1 0 0 1
43 A
1.13.2.4 Challenge
students to pair with someone else
with a differently shaped area
model and establish that both have
the same number of boxes (A)
1 0 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 175
44 A
1.13.2.5 Explain that 3x4
and 4x3 area models, although
they are shaped differently, yield
the same number of boxes (A)
1 0 0 1
45 A
1.13.2.5.1 Equate this with
the notion of groups: a 3x4 area
model is the same multiplication
problem as a 4x3 area model (A)
1 0 0 1
46 A
1.13.2.5.2 Establish a
connection to the understanding
that “3 groups of 4 is the same
idea as 4 groups of 3” and explain
that this relationship between
multiplicands is known as the
commutative property of
multiplication (A)
1 0 0 1
Procedure 2: Review
concepts of division
42 10
47 A
2.1 Explain that division is
the inverse of multiplication so we
have to work backwards in
applying multiplication to division
(A)
1 0 0 1
48 A
2.1.1 Explain this involves
changing the product of our
multiplication problem so that it is
now the dividend of our division
problem (A)
1 0 0 1
49 A
2.1.2 Demonstrate that
2x3=6, and working backward,
this is related to the equation
6÷3=2 (A)
1 0 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 176
50 A
2.1.3 Pose division
problem with a quotient greater
than one, such as 12÷2 (A, C)
1 0 1 2
51 D
2.1.3.1 IF students derive
correct answer, 6, THEN step
2.1.4 (A)
1 0 0 1
52 D
2.1.3.2 IF students seem
unclear, THEN step 2.1 (A)
1 0 0 1
53 A
2.1.4 Ask students, “What
is the meaning of this problem?”
(A, C)
1 0 1 2
54 A
2.1.5 Direct students to
discuss this with a neighbor (C)
0 0 1 1
55 A
2.1.6 Check for
understanding by calling on
students randomly (A, C)
1 0 1 2
56 D
2.1.6.1.1 IF student
explains that she decided how
many times 2 fits into 12, THEN
go to step 2.1.8 (A)
1 0 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 177
57 D
2.1.6.1.2 IF student says
she broke the dividend into the
number of groups indicated by the
divisor and determined how many
were in each group THEN step
2.1.6.2 (A)
1 0 0 1
58 A
2.1.6.2 Realize that this is
a pattern of student thought that
needs to be rewired (A)
1 0 0 1
59 A
2.1.6.3 Explain that
because multiplication is
commutative there are two ways to
think about division (A)
1 0 0 1
60 A
2.1.6.3.1 Explain that one
way is to break the dividend into
the number of groups indicated by
the divisor and then determine the
size of each group (A)
1 0 0 1
61 A
2.1.6.3.2 Explain that this
is a method that we are going to
avoid (A)
1 0 0 1
62 A
2.1.6.3.3 Explain that the
other method is to determine how
many times the divisor fits into the
dividend, just as we do in long
division and then show an
example on board (A, C)
1 0 1 2
63 A
2.1.6.3.4 Establish the
expectation that this is the way
students need to begin to think
about division (A, C)
1 0 1 2
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 178
64 A
2.1.7 Pose new problem,
such as 12÷2 (A, C)
1 0 1 2
65 A
2.1.7.1 Ask students to
explain what the meaning of this
problem is (A,C)
1 0 1 2
66 D
2.1.7.1.1 IF student says
she broke the dividend into the
number of groups indicated by the
divisor and determined how many
were in each group THEN step
2.1.6.2 (A)
1 0 0 1
67 D
2.1.7.1.2 IF student says
she determined how many times
the divisor fit into the dividend,
THEN step 2.1.8 (A)
1 0 0 1
68 A
2.1.8 Ask students to
formulate a real-life example of
the problem (C)
0 0 1 1
69 A
2.1.9 Direct students to
discuss this with a partner (C)
0 0 1 1
70 A
2.1.10 Check for
understanding by calling on
students randomly (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 179
71 A
2.1.11 Highlight correct
examples, such as, “If Juan has
$12, how many $2 bills would be
needed to match that amount?” (C)
0 0 1 1
72 A
2.2 Continue to reinforce
that the goal of division is not to
determine how many equal groups
are indicated by the divisor, but
rather, to determine how many
times the divisor fits into the
dividend, as in long division (A,
C)
1 0 1 2
73 A
2.3 Conclude lesson with a
similar problem, to which students
must provide a written answer (A,
C)
1 0 1 2
74 A
2.4 Review all answers
(A, C)
1 0 1 2
75 D
2.4.1 IF more than 3
answers indicate
misunderstanding, THEN begin
next class session with a review of
the concept of the meaning of
division (A, C) 1 0 1 2
76 D
2.4.2 IF only 2 or 3
answers indicate
misunderstanding, THEN sit with
these students during next class
session and provide remediation
(A, C) 1 0 1 2
77 A
2.5 Pose division problem
with a quotient less than one, such
as 1÷2 (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 180
78 A
2.5.1 Ask students, “What
is the meaning of this problem?”
(A,C)
1 0 1 2
79 A
2.5.2 Direct students to
discuss this with a neighbor (C)
0 0 1 1
80 A
2.5.3 Check for
understanding by calling on
students randomly (C)
0 0 1 1
81 A
2.5.4 Entertain a
discussion centering on the
concept that the problem involves
determining how many groups of
2 are in 1 (A, C)
1 0 1 2
82 A
2.5.5 Ask students to
formulate a real-life example of
the problem (C)
0 0 1 1
83 A
2.5.6 Direct students to
discuss this with a partner (C)
0 0 1 1
84 A
2.5.7 Check for
understanding by calling on
students randomly (C)
0 0 1 1
85 A
2.5.8 Highlight correct
examples, such as, “If Juan has 1
bagel, how many groups of 2
bagels are in that 1 bagel?” (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 181
86 A
2.6 Pose another similar
problem, such as 1÷3 (C)
0 0 1 1
87 A
2.6.1 Ask students, “What
is the meaning of this problem?”
(A, C)
1 0 1 2
88 A
2.6.2 Direct students to
discuss this with a neighbor (C)
0 0 1 1
89 A
2.6.3 Check for
understanding by calling on
students randomly (C)
0 0 1 1
90 A
2.6.4 Entertain a
discussion centering on the
concept that the problem involves
determining how many groups of
3 are in 1 (A, C)
1 0 1 2
91 A
2.6.5 Ask students to
formulate a real-life example of
the problem (C)
0 0 1 1
92 A
2.6.6 Direct students to
discuss this with a partner (C)
0 0 1 1
93 A
2.6.7 Check for
understanding by calling on
students randomly (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 182
94 A
2.6.8 Highlight correct
examples, such as, “If Rachel has
1 bagel, how many groups of 3
bagels are in that 1 bagel (C)
0 0 1 1
95 A
2.7 Conclude lesson with a
similar problem, to which students
must provide a written answer (C)
0 0 1 1
96 A
2.8 Review all answers
(C)
0 0 1 1
97 D
2.8.1 IF a majority of
answers indicate
misunderstanding, THEN begin
next class session with a review of
the concept of the meaning of
division (C) 0 0 1 1
98 D
2.8.2 IF only 2 or 3
answers indicate
misunderstanding, THEN sit with
these students during next class
session and provide remediation
(C) 0 0 1 1
Procedure 3: Teach
operations with integers
33 8
99 A
3.1 Teach integer
operations using a number line (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 183
100 A
3.1.1 Demonstrate
addition of a negative: 4 + -2
involves starting at 4 and going
back 2 on number line to 2 (B)
0 1 0 1
101 A
3.1.2 Demonstrate
addition of a positive: -2 + 7
involves starting at -2 and going
forward 7 to 5 (B)
0 1 0 1
102 A
3.1.3 Demonstrate
subtraction of a positive: -4 – 2
involves starting at -4 and going
back 2 to -6 (B)
0 1 0 1
103 A
3.1.4 Demonstrate
subtraction of a negative: -4 – (-2)
is the opposite of -4 -2, so we go
forward 2 to -2 (B)
0 1 0 1
104 A
3.1.5 Assign similar
problems for homework (B)
0 1 0 1
105 A
3.1.6 Visually inspect one
or two key problems on following
day using clipboard (B)
0 1 0 1
106 D
3.1.6.1 IF student
demonstrates understanding,
THEN go to step 3.2 (B)
0 1 0 1
107 D
3.1.6.2 IF student’s
answers do not demonstrate
understanding, THEN check in
with student before end of class to
provide brief review (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 184
108 A
3.2 Teach integer
operations using manipulatives (B)
0 1 0 1
109 A
3.2.1 Distribute integer
chips (B)
0 1 0 1
110 A
3.2.2 Explain that the
black side indicates positive and
the red side indicates negative (B)
0 1 0 1
111 A
3.2.3 Demonstrate
addition (B)
0 1 0 1
112 A
3.2.3.1 Show that -4 + 2
can be represented as four red
chips and two black chips (B)
0 1 0 1
113 A
3.2.3.2 Match one red chip
and one black chip to indicate
zero, based on the inverse property
of addition (B)
0 1 0 1
114 A
3.2.3.3 Repeat for the
other black chip to yield a total of
two zeroes, which yields a sum of
-2 (B)
0 1 0 1
115 A
3.2.4 Demonstrate
subtraction (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 185
116 A
3.2.4.1 Show that -4 – 2
can be represented as four red
chips (B)
0 1 0 1
117 A
3.2.4.2 Explain that it’s
impossible to match negatives and
positives because you only have 4
black chips (B)
0 1 0 1
118 A
3.2.4.3 Explain that
identity property of addition says
that adding zero to a number
results in no change to that number
(B)
0 1 0 1
119 A
3.2.4.4 Demonstrate
placing two zero pairs (one black
and one red chip) next to the four
red chips (B)
0 1 0 1
120 A
3.2.4.5 Demonstrate that
you can now subtract the two
positive (black) chips and the
resulting value is -6 (B)
0 1 0 1
121 A
3.2.5 Assign similar
problems for homework (B)
0 1 0 1
122 A
3.2.6 Visually inspect one
or two key problems on following
day using clipboard (B)
0 1 0 1
123 D
3.2.6.1 IF student
demonstrates understanding,
THEN go to step 3.2.7 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 186
124 D
3.2.6.2 IF student’s
answers do not demonstrate
understanding, THEN check in
with student before end of class to
provide brief review (B)
0 1 0 1
125 A
3.2.7 Demonstrate
multiplication (B)
0 1 0 1
126 A
3.2.7.1 Show that 3 x -2
can be represented as three groups
of two red chips, which is -6 (B)
0 1 0 1
127 A
3.2.7.2 Show that -3 x -2
would be the opposite (additive
inverse) of 3 x -2 (B)
0 1 0 1
128 A
3.2.7.3 Demonstrate that
we can derive the opposite of 3 x -
2 by simply flipping the integer
chips over, yielding 6 positives,
which signifies that -3 x -2 equals
positive 6 (B)
0 1 0 1
129 A
3.2.8 Demonstrate
division (B)
0 1 0 1
130 A
3.2.8.1 Show that -6 ÷ 2
can be represented as six red chips
(B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 187
131 A
3.2.8.2 Ask, “How many
groups of positive 2 can I make
out of these 6 negatives?” (B)
0 1 0 1
132 A
3.2.8.3 Explain that the
answer is -3 (B)
0 1 0 1
133 A
3.2.8.4 Show next that -6
÷ -2 would be the opposite
(additive inverse) of -6 ÷ 2 by
flipping the six red tiles (-6) over
and then showing that the answer
is 3 positive groups (B)
0 1 0 1
134 A
3.2.8.5 Assign similar
problems for homework (B)
0 1 0 1
135 A
3.2.8.6 Visually inspect
one or two key problems on
following day using clipboard (B)
0 1 0 1
136 D
3.2.8.6.1 IF student
demonstrates understanding,
THEN go to step 4 (B)
0 1 0 1
137 D
3.2.8.6.2 IF student’s
answers do not demonstrate
understanding, THEN check in
with student before end of class to
check for understanding (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 188
138 D
3.2.8.6.2.1 IF student
demonstrates understanding,
THEN go to step 3 (B)
0 1 0 1
139 D
3.2.8.6.2.2 IF student does
not demonstrate understanding,
direct her to come to class during
Muir Time and provide
remediation (B)
0 1 0 1
Procedure 4: Teach
number domains
31 20
140 A
4.1 Explain that before we
can begin operations with
fractions, we need to understand
the different types of numbers (B)
0 1 0 1
141 A
4.2 Review that whole
numbers are 0,1,2,3,4…to infinity
(B)
0 1 0 1
142 A
4.3 Review that integers
are whole numbers and their
opposites, such as 1 and -1, 2 and -
2, etc. (B)
0 1 0 1
143 A
4.4 Teach that in between
the whole numbers and integers
are other numbers (B)
0 1 0 1
144 A
4.5 Explain that these are
called rational numbers, defined as
any number that can be written as
a fraction (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 189
145 A
4.6 Explain that whole
numbers and integers are also
rational numbers, because they can
also be written as fractions (B)
0 1 0 1
146 A
4.7 Explain that decimals
are also rational numbers, because
they can be written as fractions (B)
0 1 0 1
147 A
4.71 Display the example
of 0.7 (B)
0 1 0 1
148 A
4.72 Ask students to write
on their whiteboards the name of
the place the 7 occupies (B)
0 1 0 1
149 D
4.7.2.1 IF student
understands it is in the tenths
place, THEN go to step 4.7.3 (B)
0 1 0 1
150 D
4.7.2.2 If student does not
understand it is in the tenths place,
check in with student before end
of class to provide brief review (B)
0 1 0 1
151 A
4.73 Ask students to write
how 0.7 is read on their
whiteboards (B)
0 1 0 1
152 D
4.7.3.1 IF student
understands it is read as “seven
tenths”, THEN go to step 4.7.4 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 190
153 D
4.7.3.2 IF student does not
understand it is read as “seven
tenths”, then check in with student
before end of class to provide brief
review (B)
0 1 0 1
154 A
4.7.4 Ask students to write
0.7 as a fraction on their
whiteboards (B)
0 1 0 1
155 D
4.7.4.1 IF student
understands it is written as 7/10
THEN go to step 4.7.5 (B)
0 1 0 1
156 D
4.7.4.2 IF student does not
understand it is written as 7/10
THEN check in with student
before end of class and provide
brief review (B)
0 1 0 1
157 A
4.7.5 Ask students to write
0.03 as a fraction on their
whiteboards (B)
0 1 0 1
158 D
4.7.5.1 IF student
understands it is written as 3/100
then go to step 4.7.6 (B)
0 1 0 1
159 D
4.7.5.2 IF student does not
understand it is written as 3/100
THEN check in with her before
end of class and provide brief
review (B)
0 1 0 1
160 A
4.7.6 Ask students to write
0.004 as a fraction on their
whiteboards (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 191
161 D
4.7.6.1 IF student
understands it is written as 4/1000
THEN go to step 4.7.7 (B)
0 1 0 1
162 D
4.7.6.2 IF student does not
understand it is written as 4/1000
THEN check in with her before
end of class and provide brief
review (B)
0 1 0 1
163 A
4.7.7 Review that the
place of the number on the right in
any decimal determines what the
denominator will be when it is
converted to a fraction (B)
0 1 0 1
164 A
4.8 Review procedure for
converting a fraction to a decimal
(B)
0 1 0 1
165 A
4.8.1 Review that we need
a power of 10 in the denominator
in order to convert many fractions
to a decimal (B)
0 1 0 1
166 A
4.8.2 Demonstrate that a
fraction such as 3/5 needs to be
converted to an equivalent fraction
with a power of 10, in this case 10,
in the denominator (B)
0 1 0 1
167 A
4.8.3 Demonstrate
multiplying 3/5 x 2/2 to yield 6/10
(B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 192
168 A
4.8.4 Demonstrate that this
converts to the decimal 0.6 (B)
0 1 0 1
169 A
4.8.5 Assign similar
problems (B)
0 1 0 1
170 A
4.8.6 Visually check for
understanding (B)
0 1 0 1
171 D
4.8.6.1 IF students
demonstrate understanding, THEN
go to step 4.8.7 (B)
0 1 0 1
172 D
4.8.6.2 IF student does not
demonstrate understanding, THEN
check in with her before end of
class to provide brief review (B)
0 1 0 1
173 A
4.8.7 Review that some
fractions, such as 1/16 , have
denominators that cannot be
converted to a power of 10 (B)
0 1 0 1
174 A
4.8.8 Review that with
fractions such as these, we can
divide the numerator by the
denominator, such that 1÷ 16 =
.0625 (B)
0 1 0 1
175 A
4.8.9 Assign similar
problems (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 193
176 D
4.8.9.1 IF student
demonstrates understanding,
THEN go to step 4.8.10 (B)
0 1 0 1
177 D
4.8.9.2 IF student does not
demonstrate understanding, THEN
check in with student before end
of class to provide brief review (B)
0 1 0 1
178 A
4.8.10 Review that some
fractions, such as 1/3 yield non-
terminating, repeating decimals
(B)
0 1 0 1
179 A
4.8.11 Demonstrate that,
again, we can divide the numerator
by the denominator, such that 1 ÷
3 = 0.3333 ?, which repeats and
does not terminate, as indicated by
the superscript (B)
0 1 0 1
180 A
4.8.12 Assign similar
problems (B)
0 1 0 1
181 D
4.8.12.1 IF student
demonstrates understanding,
THEN go to step 4.8.13 (B)
0 1 0 1
182 D
4.8.12.2 IF student does
not demonstrate understanding,
THEN check in with her before
end of class and provide brief
review (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 194
183 A
4.8.13 Assign homework
problems for decimal/fraction
concepts (B)
0 1 0 1
184 A
4.8.14 Visually inspect
one or two key problems on
following day using clipboard (B)
0 1 0 1
185 D
4.8.14.1 IF student
demonstrates understanding,
THEN go to step 4.9 (B)
0 1 0 1
186 D
4.8.14.2 IF student does
not demonstrate understanding,
THEN check in with her before
end of class to check for
understanding (B)
0 1 0 1
187 D
4.8.14.2.1 IF student
demonstrates understanding,
THEN go to step 4.9 (B)
0 1 0 1
188 D
4.8.14.2.2 IF student does
not demonstrate understanding,
THEN direct her to come to class
during Muir Time and provide
remediation (B)
0 1 0 1
189 A
4.9 Explain that not all
rational numbers can be integers
or whole numbers (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 195
190 A
4.10 Explain that a
number that cannot be written as a
fraction (a non-repeating, non-
terminating decimal, such as
square root of 12, is an irrational
number (B)
0 1 0 1
Procedure 5: Review
representations of fractions
22 14
191 A
5.1 Explain that, although
this review involves fractions as
parts of a whole, later we will be
dealing with fraction concepts
involving ratios and proportions
(B)
0 1 0 1
192 A
5.2 Remind students that
the concept of fractions as parts of
a whole requires us to split up that
whole (B)
0 1 0 1
193 A
5.3 Distribute fraction
circle manipulatives (B)
0 1 0 1
194 A
5.4 Ask students to
represent 3/8 (B)
0 1 0 1
195 A
5.4.1 Visually check for
understanding (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 196
196 D
5.4.1.1 IF student
demonstrates understanding,
THEN go to step 5.5 (B)
0 1 0 1
197 D
5.4.1.2 IF student does not
demonstrate understanding, THEN
check in with student before end
of class to provide brief review (B)
0 1 0 1
198 A
5.5 Ask students to
represent 1/2 (B)
0 1 0 1
199 A
5.5.1 Visually check for
understanding (B)
0 1 0 1
200 D
5.5.1.1 IF student
demonstrates understanding,
THEN go to step 5.6 (B)
0 1 0 1
201 D
5.5.1.2 IF student does not
demonstrate understanding, THEN
check in with student before end
of class to provide brief review (B)
0 1 0 1
202 A
5.6 Ask students to show
another way to make 1/2 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 197
203 A
5.7 Explain to students
that representing 1/2 by its
equivalents, such as 2/4 or 3/6,
exemplifies the identity property
of multiplication, which states that
multiplying any number by a form
of 1 yields that same number (B)
0 1 0 1
204 A
5.8 Ask students, “How do
you write an equivalent fraction
for 1/2 ?” (B)
0 1 0 1
205 A
5.9 Remind students that
you do that by multiplying by a
form of 1, such as 2/2, resulting in
the equivalent fraction, 2/4 (B)
0 1 0 1
206 A
5.10 Assign similar
problems (B)
0 1 0 1
207 A
5.10.1 Visually check for
understanding (B)
0 1 0 1
208 D
5.10.1.1 IF student
demonstrates understanding,
THEN go to step 5.11 (B)
0 1 0 1
209 D
5.10.1.2 IF student does
not demonstrate understanding,
THEN check in with student
before end of class to provide brief
review (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 198
210 A
5.11 Distribute fraction
strip manipulatives (B)
0 1 0 1
211 A
5.12 Ask students, in
groups, or with partners to
represent 1/2 (B)
0 1 0 1
212 A
5.12.1 Visually check for
understanding (B)
0 1 0 1
213 D
5.12.1.1 IF student
demonstrates understanding,
THEN go to step 5.13 (B)
0 1 0 1
214 D
5.12.1.2 IF student does
not demonstrate understanding,
THEN check in with student
before end of class to provide brief
review (B)
0 1 0 1
215 A
5.13 Ask students to
demonstrate as many ways as
possible to represent 1/2 (B)
0 1 0 1
216 A
5.13.1 Visually check for
understanding (B)
0 1 0 1
217 D
5.13.1.1 IF student
demonstrates understanding,
THEN go to step 5.14 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 199
218 D
5.13.1.2 IF student does
not demonstrate understanding,
THEN check in with student
before end of class to provide brief
review (B)
0 1 0 1
219 A
5.14 Remind students that
dividing any number by a form of
1 is identical to the identity
property of multiplication: the
result in both cases is the original
number (B)
0 1 0 1
220 A
5.15 Distribute fraction
circle manipulatives (B)
0 1 0 1
221 A
5.16 Ask students to
represent a fraction such as 4/8 in
simplest terms (B)
0 1 0 1
222 A
5.17 Visually check for
understanding (B)
0 1 0 1
223 D
5.17.1 IF student
demonstrates understanding,
THEN go to step 6 (B)
0 1 0 1
224 D
5.17.2 IF student does not
demonstrate understanding, THEN
check in with student before end
of class to check for understanding
(B)
0 1 0 1
225 D
5.17.2.1 IF student
demonstrates understanding of all
elements of step 5, THEN go to
step 6 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 200
226 D
5.17.2.2 IF student does
not demonstrate understanding of
all elements of step 4, THEN
direct her to come to class during
Muir time and provide remediation
(B) 0 1 0 1
Procedure 6: Review
addition and subtraction of
fractions and mixed numbers
103 40
227 A
6.1 Review addition and
subtraction with like denominators
(A, B)
1 1 0 2
228 A
6.2 Distribute
manipulative fraction strips and
circles (A, B)
1 1 0 2
229 A
6.2.1 Ask students to
model a problem such as 1/2 + 1/2
with strips and circles (B)
0 1 0 1
230 A
6.2.2 Ask students to
demonstrate that the answer, two
halves, is visually equivalent to
one whole (B)
0 1 0 1
231 A
6.2.3 Ask students to
divide by a fraction equal to 1,
such as 2/2 , to also yield one
whole (A, B)
1 1 0 2
232 A
6.2.4 Ask students to
model a problem such as 3/8 + 1/8
(B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 201
233 A
6.2.5 Ask students to
demonstrate that the answer, four
eighths, is visually equivalent to
1/2 (B)
0 1 0 1
234 A
6.2.6 Ask students to
divide by a fraction equal to 1,
such as 4/4 , to also yield 1/2 (A,
B)
1 1 0 2
235 D
6.2.6.1 IF students
demonstrate understanding, then
go to step 6.2.7 (B)
0 1 0 1
236 D
6.2.6.2 IF student appears
unclear, then check in with her
before end of class to provide brief
review (B)
0 1 0 1
237 A
6.2.7 Ask students to
model a problem, such as 5/8 - 1/8
(B)
0 1 0 1
238 A
6.2.8 Ask students to
demonstrate that the answer is
visually equivalent to 1/2 (B)
0 1 0 1
239 A
6.2.9 Ask students to
divide by a fraction equal to 1,
such as 4/4 , to also yield 1/2 (A,
B)
1 1 0 2
240 D
6.2.9.1 IF students
demonstrate understanding, then
go to step 6.3 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 202
241 D
6.2.9.2 IF student appears
unclear, THEN check in with her
before end of class to provide brief
review (B)
0 1 0 1
242 A
6.3 Review algorithm for
adding and subtracting fractions
(A, B)
1 1 0 2
243 A
6.3.1 Ask students to add
two fractions with like
denominators, such as 1/4 + 1/4
(A, B)
1 1 0 2
244 A
6.3.2 Visually check that
students are adding numerators to
yield 2/4 (A, B)
1 1 0 2
245 A
6.3.3 Ask students to
divide by a fraction equal to 1,
such as 2/2 , to put in simplest
terms, 1/2 (A, B)
1 1 0 2
246 A
6.3.4 Ask students to
subtract two fractions with like
denominators, such as 3/6 - 1/6
(B)
0 1 0 1
247 A
6.3.5 Ask students to
divide by a fraction equal to 1,
such as 2/2 , to put in simplest
terms, 1/3 (A, B)
1 1 0 2
248 A
6.3.5.1 Assign similar
problems for homework (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 203
249 A
6.3.5.2 Visually inspect
one or two key problems on
following day using clipboard (B)
0 1 0 1
250 D
6.3.5.2.1 IF student
demonstrates understanding,
THEN go to step 6.4 (B)
0 1 0 1
251 D
6.3.5.2.2 IF student’s
answers do not demonstrate
understanding, THEN check in
with student before end of class to
check for understanding (B)
0 1 0 1
252 D
6.3.5.2.2.1 IF student
demonstrates understanding,
THEN go to step 6.4 (B)
0 1 0 1
253 D
6.3.5.2.2.2 IF student does
not demonstrate understanding,
direct her to come to class during
Muir Time and provide
remediation (B)
0 1 0 1
254 A
6.4 Review addition and
subtraction of fractions with unlike
denominators (A, B, C)
1 1 1 3
255 A
6.4.1 Demonstrate adding
two fractions with unlike
denominators, such as 1/2 + 1/4
(A, B, C)
1 1 1 3
256 A
6.4.2 Review that we can’t
add them unless the denominators
are the same (A, B, C)
1 1 1 3
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 204
257 A
6.4.3 Review that the
identify property of multiplication
means that we can multiply 1/2 by
a form of one, 2/2 to yield a
fraction with a denominator of 4,
2/4 (A, B)
1 1 0 2
258 A
6.4.4 Review that we can
now add 2/4 + 1/4 = 3/4 (A, B, C)
1 1 1 3
259 A
6.4.5 Assign similar
problems (A, B, C)
1 1 1 3
260 D
6.4.5.1 IF students
demonstrate understanding, then
go to step 6.4.6 (B)
0 1 0 1
261 D
6.4.5.2 IF student appears
unclear, then check in with her
before end of class to provide brief
review (B)
0 1 0 1
262 A
6.4.6 Demonstrate
subtracting two fractions with
unlike denominators, such as 1/2 -
1/4 (B)
0 1 0 1
263 A
6.4.7 Review that the
identify property of multiplication
means that we can multiply 1/2 by
a form of one, 2/2 to yield a
fraction with a denominator of 4,
2/4 (A,B)
1 1 0 2
264 A
6.4.8 Review that we can
now subtract 2/4 - 1/4 = 1/4 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 205
265 A
6.4.9 Assign similar
problems (B)
0 1 0 1
266 D
6.4.9.1 IF students
demonstrate understanding, then
go to step 6.4.10 (B)
0 1 0 1
267 D
6.4.9.2 IF student appears
unclear, then check in with her
before end of class to provide brief
review (B)
0 1 0 1
268 A
6.4.10 Assign addition and
subtraction of fractions with unlike
denominators for homework (B)
0 1 0 1
269 A
6.4.11 Visually inspect
one or two key problems on
following day using clipboard (B)
0 1 0 1
270 D
6.4.11.1 IF student
demonstrates understanding,
THEN go to step 6.5 (B)
0 1 0 1
271 D
6.4.11.2 IF student’s
answers do not demonstrate
understanding, THEN check in
with student before end of class to
check for understanding (B)
0 1 0 1
272 D
6.4.11.2.1 IF student
demonstrates understanding,
THEN go to step 6.5 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 206
273 D
6.4.11.2.2 IF student does
not demonstrate understanding,
direct her to come to class during
Muir Time and provide
remediation (B)
0 1 0 1
274 A
6.5 Review addition and
subtraction of mixed numbers with
like denominators (B)
0 1 0 1
275 A
6.5.1 Demonstrate 21/2 +
41/2 using hand-drawn circles to
represent the wholes and fractions
(B)
0 1 0 1
276 A
6.5.2 Review that the two
fractions add to one whole, which
we must add to the other six
wholes, to yield 7 (B)
0 1 0 1
277 A
6.5.3 Demonstrate 13/5 +
24/5 using hand-drawn circles (B)
0 1 0 1
278 A
6.5.4 Review that this
yields 3 wholes and seven-fifths
(B)
0 1 0 1
279 A
6.5.5 Review through use
of fraction strips that 7/5 is the
same as one whole and 2/5 (B)
0 1 0 1
280 A
6.5.6 Demonstrate that we
now add the 1 whole to the other 3
whole to yield 4 whole, and then
add 2/5 , resulting in 42/5 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 207
281 A
6.5.7 Demonstrate with
fraction strips the corollary of this
concept: 12/5 can be converted to
the improper fraction, 7/5 (B)
0 1 0 1
282 A
6.5.8 Demonstrate 61/8 -
45/8 using hand-drawn circles (B)
0 1 0 1
283 A
6.5.9 Review that one
whole can be represented by any
number over itself, such as 6/6 or
7/7 or 8/8 (B)
0 1 0 1
284 A
6.5.10 Review that we
need to borrow one whole from
the 6, which we can represent as
8/8 (B)
0 1 0 1
285 A
6.5.11 Review that we
must add the 8/8 we borrowed to
the 1/8 to yield 9/8 (B)
0 1 0 1
286 A
6.5.12 Demonstrate that
we can now subtract to yield 14/8
(B)
0 1 0 1
287 A
6.5.13 Review that we
need to simplify the 4/8 by
dividing by a form of one, 4/4 ,
yielding 11/2 (B)
0 1 0 1
288 A
6.5.14 Assign similar
problems (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 208
289 A
6.5.15 Visually check for
understanding (B)
0 1 0 1
290 D
6.5.15.1 IF students
demonstrate understanding, then
go to step 6.5.16 (B)
0 1 0 1
291 D
6.5.15.2 IF student
appears unclear, then check in
with her before end of class to
provide brief review (B)
0 1 0 1
292 A
6.5.16 Assign addition and
subtraction of mixed numbers with
like denominators for homework
(B)
0 1 0 1
293 A
6.5.17 Visually inspect
one or two key problems on
following day using clipboard (B)
0 1 0 1
294 D
6.5.17.1 IF student
demonstrates understanding,
THEN go to step 6.6 (B)
0 1 0 1
295 D
6.5.17.2 IF student’s
answers do not demonstrate
understanding, THEN check in
with student before end of class to
check for understanding (B)
0 1 0 1
296 D
6.5.17.2.1 IF student
demonstrates understanding,
THEN go to step 6.6 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 209
297 D
6.5.17.2.2 IF student does
not demonstrate understanding,
direct her to come to class during
Muir Time and provide
remediation (B)
0 1 0 1
298 A
6.6 Review addition and
subtraction of mixed numbers with
unlike denominators (B)
0 1 0 1
299 A
6.6.1 Demonstrate 11/2 +
21/4 using hand-drawn circles (B)
0 1 0 1
300 A
6.6.2 Review that we
cannot add fractions unless we
have the same denominators (A,
B, C)
1 1 1 3
301 A
6.6.3 Review that we can
multiply by a form of one, 2/2 , to
yield 12/4 (B)
0 1 0 1
302 A
6.6.4 Demonstrate adding
12/4 + 21/4 to yield 33/4 (B)
0 1 0 1
303 A
6.6.5 Demonstrate
subtracting 31/2 - 11/4 using hand-
drawn circles (B)
0 1 0 1
304 A
6.6.6 Review that we
cannot subtract fractions unless we
have the same denominators (B)
0 1 0 1
305 A
6.6.7 Review that we can
multiply by a form of one, 2/2 , to
yield 32/4 (A, B)
1 1 0 2
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 210
306 A
6.6.8 Demonstrate
subtracting 32/4 - 11/4 to yield
21/4 (B)
0 1 0 1
307 A
6.6.9 Assign similar
problems (B)
0 1 0 1
308 A
6.6.10 Visually check for
understanding (B)
0 1 0 1
309 A
6.6.10.1 IF students
demonstrate understanding, then
go to step 6.6.11 (B)
0 1 0 1
310 D
6.6.10.2 IF student
appears unclear, then check in
with her before end of class to
provide brief review (B)
0 1 0 1
311 D
6.6.11 Assign addition and
subtraction of mixed numbers with
unlike denominators for
homework (B)
0 1 0 1
312 A
6.6.12 Visually inspect
one or two key problems on
following day using clipboard (B)
0 1 0 1
313 D
6.6.12.1 IF student
demonstrates understanding,
THEN go to step 6.7 (B)
0 1 0 1
314 D
6.6.12.2 IF student’s
answers do not demonstrate
understanding, THEN check in
with student before end of class to
check for understanding (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 211
315 D
6.6.12.2.1 IF student
demonstrates understanding,
THEN go to step 6.7 (B)
0 1 0 1
316 D
6.6.12.2.2 IF student does
not demonstrate understanding,
direct her to come to class during
Muir Time and provide
remediation (B)
0 1 0 1
317 A
6.7 Review that because
rational numbers can be written as
both fractions and decimals, we
can add and subtract fractions and
mixed numbers by converting
them to decimals (B)
0 1 0 1
318 A
6.7.1 Demonstrate this for
fraction addition using an example
such as 1/4 + 1/4 = 1/2 (B)
0 1 0 1
319 A
6.7.2 Review that 1/4
needs to be converted to a fraction
with a denominator that is a power
of 10 (B)
0 1 0 1
320 A
6.7.3 Demonstrate that
multiplying 1/4 by 25/25 yields
25/100 , which in decimal form is
.25 (B)
0 1 0 1
321 A
6.7.4 Review that to add
.25 + .25 we must line up the
decimals and places, then add,
yielding a sum of .5 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 212
322 A
6.7.5 Demonstrate that .5
is the same as five tenths, or 5/10 ,
which can be reduced to simplest
terms as 1/2 (B)
0 1 0 1
323 A
6.7.6 Demonstrate that we
can derive the same answer, .5, by
converting our answer in 1/4 + 1/4
= 1/2 (B)
0 1 0 1
324 A
6.7.7 Demonstrate that if
we convert 1/2 to a fraction with a
power of 10 in the denominator,
we get 5/10 (B)
0 1 0 1
325 A
6.7.8 Review that 5/10 is
the same as .5 (B)
0 1 0 1
326 A
6.7.9 Reinforce idea that
regardless of whether we add
rational numbers as fractions or
decimals, the answer is the same
(B)
0 1 0 1
327 A
6.7.10 Assign similar
problems(B)
0 1 0 1
328 D
6.7.10.1 IF students
demonstrates understanding,
THEN go to step 6.7.11 (B)
0 1 0 1
329 D
6.7.10.2 IF student does
not demonstrate understanding,
THEN check in with her before
end of class and provide brief
review (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 213
330 A
6.7.11 Demonstrate same
concept for subtraction using an
example such as 3/4 - 1/4 = 2/4 ,
or 1/2 (B)
0 1 0 1
331 A
6.7.12 Demonstrate that
multiplying 3/4 and 1/4 by 25/25
yields 75/100 and 25/100 , which
in decimal form are .75 and .25
(B)
0 1 0 1
332 A
6.7.13 Review that to
subtract .75 - .25 we must line up
the decimals and places, then
subtract, yielding a difference of .5
(B)
0 1 0 1
333 A
6.7.14 Demonstrate that .5
is the same as five tenths, or 5/10 ,
which can be reduced to simplest
terms as 1/2 (B)
0 1 0 1
334 A
6.7.15 Demonstrate that
we can derive the same answer, .5,
by converting our answer in 3/4 -
1/4 = 1/2 (B)
0 1 0 1
335 A
6.7.16 Demonstrate that if
we convert 1/2 to a fraction with a
power of 10 in the denominator,
we get 5/10 (B)
0 1 0 1
336 A
6.7.17 Review that 5/10 is
the same as .5 (B)
0 1 0 1
337 A
6.7.18 Reinforce idea that
regardless of whether we subtract
rational numbers as fractions or
decimals, the answer is the same
(B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 214
338 A
6.7.19 Assign similar
problems (B)
0 1 0 1
339 D
6.7.19.1 IF students
demonstrates understanding,
THEN go to step 6.7.20 (B)
0 1 0 1
340 D
6.7.19.2 IF student does
not demonstrate understanding,
THEN check in with her before
end of class and provide brief
review (B)
0 1 0 1
341 A
6.7.20 Demonstrate same
concept for addition of mixed
numbers using an example such as
11/4 + 11/4 = 22/4 , or 21/2 (B)
0 1 0 1
342 A
6.7.21 Review that 1/4
needs to be converted to a fraction
with a denominator that is a power
of 10 (B)
0 1 0 1
343 A
6.7.22 Demonstrate that
multiplying 1/4 by 25/25 yields
25/100 , which in decimal form is
.25 (B)
0 1 0 1
344 A
6.7.23 Review that to add
1.25 + 1.25 we must line up the
decimals and places, then add,
yielding a sum of 2.5 (B)
0 1 0 1
345 A
6.7.24 Demonstrate that
2.5 is the same as two and five
tenths, or 25/10 , which can be
reduced to simplest terms as 21/2
(B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 215
346 A
6.7.25 Demonstrate that
we can derive the same answer,
2.5, by converting our answer in
11/4 + 11/4 = 21/2 (B)
0 1 0 1
347 A
6.7.26 Demonstrate that if
we convert 21/2 to a fraction with
a power of 10 in the denominator,
we get 25/10 (B)
0 1 0 1
348 A
6.7.27 Review that 25/10
is the same as 2.5 (B)
0 1 0 1
349 A
6.7.28 Reinforce idea that
regardless of whether we add
rational numbers as fractions or
decimals, the answer is the same
(B)
0 1 0 1
350 A
6.7.29 Assign similar
problems (B)
0 1 0 1
351 D
6.7.29.1 IF students
demonstrates understanding,
THEN go to step 6.7.30 (B)
0 1 0 1
352 D
6.7.29.2 IF student does
not demonstrate understanding,
THEN check in with her before
end of class and provide brief
review(B)
0 1 0 1
353 A
6.7.30 Demonstrate same
concept for subtraction of mixed
numbers using an example such as
23/4 - 11/4 = 12/4 , or 11/2 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 216
354 A
6.7.31 Demonstrate that
multiplying 3/4 and 1/4 by 25/25
yields 75/100 and 25/100 , which
in decimal form are .75 and .25
(B)
0 1 0 1
355 A
6.7.32 Review that to
subtract 2.75 - 1.25 we must line
up the decimals and places, then
subtract, yielding a difference of
1.5 (B)
0 1 0 1
356 A
6.7.33 Demonstrate that
1.5 is the same as one and five
tenths, or 15/10 , which can be
reduced to simplest terms as 11/2
(B)
0 1 0 1
357 A
6.7.34 Demonstrate that
we can derive the same answer,
1.5, by converting our answer in
23/4 - 11/4 = 11/2 (B)
0 1 0 1
358 A
6.7.35 Demonstrate that if
we convert 11/2 to a fraction with
a power of 10 in the denominator,
we get 15/10 (B)
0 1 0 1
359 A
6.7.36 Review that 15/10
is the same as 1.5 (B)
0 1 0 1
360 A
6.7.37 Reinforce idea that
regardless of whether we subtract
rational numbers as fractions or
decimals, the answer is the same
(B)
0 1 0 1
361 A
6.7.38 Assign similar
problems (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 217
362 D
6.7.38.1 IF students
demonstrates understanding,
THEN go to step 6.7.39 (B)
0 1 0 1
363 D
6.7.38.2 IF student does
not demonstrate understanding,
THEN check in with her before
end of class and provide brief
review (B)
0 1 0 1
364 A
6.7.39 Assign homework
problems for decimal/fraction
equivalence (B)
0 1 0 1
365 A
6.7.40 Visually inspect
one or two key problems on
following day using clipboard (B)
0 1 0 1
366 D
6.7.40.1 IF student
demonstrates understanding,
THEN go to step 7 (B)
0 1 0 1
367 D
6.7.40.2 IF student does
not demonstrate understanding,
THEN check in with her before
end of class to check for
understanding (B)
0 1 0 1
368 D
6.7.40.2.1 IF student
demonstrates understanding,
THEN go to step 7 (B)
0 1 0 1
369 D
6.7.40.2.2 IF student does
not demonstrate understanding,
THEN direct her to come to class
during Muir Time and provide
remediation (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 218
Procedure 7: Review
multiplication of fractions and
mixed numbers
24 12
370 A
7.1 Review multiplication
of fractions (B)
0 1 0 1
371 A
7.1.1 Demonstrate
multiplying a fraction by a whole
number, such as 1/3 x 2 (B)
0 1 0 1
372 A
7.1.2 Demonstrate we can
approach this through repeated
addition (B)
0 1 0 1
373 A
7.1.3 Review that 1/3 x 2
means two one-thirds, or 1/3 + 1/3
= 2/3 (B)
0 1 0 1
374 A
7.1.4 Demonstrate that we
can also convert the whole
number, 2, to a fraction by giving
the 2 a denominator, 1, yielding
2/1 (B)
0 1 0 1
375 A
7.1.5 Demonstrate that 1/3
x 2 is actually 1/3 x 2/1 , or 2/3
(B)
0 1 0 1
376 A
7.1.6 Review that in
multiplication of fractions, the
algorithm requires that we
multiply the numerators by the
numerators and the denominators
by the denominators (B)
0 1 0 1
377 A
7.1.7 Extend this
understanding to multiplication of
fractions by fractions (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 219
378 A
7.1.8 Demonstrate that a
problem such as 1/3 x 1/2 also
involves multiplying the
numerators by the numerators and
the denominators by the
denominators, yielding 1/6 (B)
0 1 0 1
379 A
7.1.9 Assign similar
problems (B)
0 1 0 1
380 A
7.1.10 Visually check for
understanding (B)
0 1 0 1
381 D
7.1.10.1 IF students
demonstrate understanding, then
go to step 7.1.11 (B)
0 1 0 1
382 D
7.1.10.2 IF student
appears unclear, then check in
with her before end of class to
provide brief review (B)
0 1 0 1
383 A
7.1.11 Assign
multiplication of fractions for
homework (B)
0 1 0 1
384 A
7.1.12 Visually inspect
one or two key problems on
following day using clipboard (B)
0 1 0 1
385 D
7.1.12.1 IF student
demonstrates understanding,
THEN go to step 7.1.13 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 220
386 D
7.1.12.2 IF student’s
answers do not demonstrate
understanding, THEN check in
with student before end of class to
check for understanding (B)
0 1 0 1
387 D
7.1.12.2.1 IF student
demonstrates understanding,
THEN go to step 7.1.13 (B)
0 1 0 1
388 D
7.1.12.2.2 IF student does
not demonstrate understanding,
direct her to come to class during
Muir Time and provide
remediation (B)
0 1 0 1
389 A
7.1.13 Demonstrate
multiplying a mixed number by a
mixed number, such as 21/4 x 21/6
(B)
0 1 0 1
390 A
7.1.14 Review that we
must convert each mixed number
into an improper fraction before
we can multiply (B)
0 1 0 1
391 A
7.1.15 Review that 21/4
can be represented as 9/4 and 21/6
can be represented as 13/6 (B)
0 1 0 1
392 A
7.1.16 Demonstrate that
we multiply the numerators by the
numerators and the denominators
by the denominators, yielding
117/24 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 221
393 A
7.1.17 Review that we
need to simplify, yielding 4 whole
and 21 twenty-fourths (B)
0 1 0 1
394 A
7.1.18 Review that 21/24
can be simplified by dividing by a
form of one, 3/3 , yielding 7/8 (B)
0 1 0 1
395 A
7.1.19 Display completed
product, 47/8 (B)
0 1 0 1
396 A
7.1.20 Assign similar
problems (B)
0 1 0 1
397 A
7.1.21 Visually check for
understanding (B)
0 1 0 1
398 D
7.1.21.1 IF students
demonstrate understanding, then
go to step 7.1.22 (B)
0 1 0 1
399 D
7.1.21.2 IF student
appears unclear, then check in
with her before end of class to
provide brief review (B)
0 1 0 1
400 A
7.1.22 Assign
multiplication of mixed numbers
for homework (B)
0 1 0 1
401 A
7.1.23 Visually inspect
one or two key problems on
following day using clipboard (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 222
402 D
7.1.23.1 IF student
demonstrates understanding,
THEN go to step 8 (B)
0 1 0 1
403 D
7.1.23.2 IF student’s
answers do not demonstrate
understanding, THEN check in
with student before end of class to
check for understanding (B)
0 1 0 1
404 D
7.1.23.2.1 IF student
demonstrates understanding,
THEN go to step 8 (B)
0 1 0 1
405 D
7.1.23.2.2 IF student does
not demonstrate understanding,
direct her to come to class during
Muir Time and provide
remediation (B)
0 1 0 1
Procedure 8: Teach
division of whole numbers by
fractions
53 6
406 A
8.1 Hold up a whole piece
of paper and pose question that
represents the problem 1÷ 1/2,
such as “If this piece of paper
represents a whole cake, how can I
divide it by one-halves?” (B, C)
0 1 1 2
407 A
8.2 Direct students to
discuss the question (B, C)
0 1 1 2
408 A
8.3 Circulate to check for
understanding (B, C)
0 1 1 2
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 223
409 A
8.4 Call on volunteers who
were able to understand that the
solution is two halves (B, C)
0 1 1 2
410 A
8.5 Distribute fraction
strips (C)
0 0 1 1
411 A
8.6 Relate the scenario, “If
I have one piece of paper, how
many halves are there in that one
piece?” (C)
0 0 1 1
412 A
8.6.1 Ask students to work
with a partner to model scenario
with manipulatives (C)
0 0 1 1
413 A
8.6.2 Circulate among
students to determine whether they
are creating concrete
representations of 1 ÷ 1/2 = 2 (C)
0 0 1 1
414 A
8.6.3 Check for
understanding by calling on
students randomly (C)
0 0 1 1
415 A
8.6.4 Validate correct
answers (C)
0 0 1 1
416 A
8.6.5 Demonstrate a
correct representation on
whiteboard using magnetic
fraction strips (A, C)
1 0 1 2
417 A
8.6.6 Ask students in pairs
to formulate a real-life example of
the problem, 1 ÷ 1/2 = 2 (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 224
418 A
8.6.7 Circulate among
students to check for
understanding (C)
0 0 1 1
419 A
8.6.8 Call on volunteers
who were able to formulate correct
examples, such as, “If I have a
whole pizza, how many halves are
contained therein?” (C)
0 0 1 1
420 A
8.6.9 Ask students to work
with a partner to explain what the
problem 1 ÷ 1/2 = 2 means (C)
0 0 1 1
421 A
8.6.10 Circulate to check
for understanding (C)
0 0 1 1
422 A
8.6.11 Call on volunteers
who were able to articulate that the
problem means, “How many one-
halves are contained in one whole”
(A, B, C)
1 1 1 3
423 A
8.7 Direct students to
work with a partner to create a
real-life example of the problem,
1÷ 1/4 (C)
0 0 1 1
424 A
8.8 Circulate among
students to check for
understanding (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 225
425 A
8.9 Call on volunteers who
were able to articulate examples,
such as, “If I have one dollar, how
many quarters are contained in
that dollar?” (C)
0 0 1 1
426 A
8.10 Ask students to work
with a partner to explain what the
problem 1÷ 1/4 means (C)
0 0 1 1
427 A
8.11 Circulate to check for
understanding (C)
0 0 1 1
428 A
8.12 Call on volunteers
who were able to articulate that the
problem means, “How many one-
fourths are contained in one
whole” (A, B, C)
1 1 1 3
429 A
8.12.1 Ask students to
work with a partner to model
scenario with manipulatives (C)
0 0 1 1
430 A
8.12.2 Circulate among
students to determine whether they
are creating concrete
representations of 1 ÷ 1/4 = 4 (C)
0 0 1 1
431 A
8.12.3 Check for
understanding by calling on
students randomly (C)
0 0 1 1
432 A
8.12.4 Validate correct
answers (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 226
433 A
8.12.5 Demonstrate a
correct representation on
whiteboard using magnetic
fraction strips (C)
0 0 1 1
434 D
8.12.5.1 teacher
observation indicates student
understands how to create concrete
representations of problems
involving whole numbers divided
by fractions, THEN step 8.13 (C)
0 0 1 1
435 D
8.12.5.2 IF teacher
observation indicates student does
not understand, THEN provide
remediation with similar examples
after class (C)
0 0 1 1
436 A
8.13 Distribute math paper
and pencilsC)
0 0 1 1
437 A
8.14 Relate the scenario,
“If I have one pizza and want to
share it among 3 people, how
much will each person get?” (C)
0 0 1 1
438 A
8.14.1 Direct students to
work with a partner to draw either
a number line or area model
representation of the scenario (C)
0 0 1 1
439 A
8.14.2 Circulate among
students to determine whether they
are creating pictorial
representations of 1 ÷ 1/3 = 3 (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 227
440 A
8.14.3 Ask students to
share their drawings with other
neighbors (C)
0 0 1 1
441 A
8.14.4 Check for
understanding by calling on
students randomly, asking them to
describe their drawings (C)
0 0 1 1
442 A
8.14.5 Validate correct
answers that represent 1 ÷ 1/3 = 3
(C)
0 0 1 1
443 A
8.14.6 Demonstrate a
correct representation on
whiteboard by drawing both an
area model and a number line
depicting 1 ÷ 1/3 = 3 (C)
0 0 1 1
444 A
8.15 Relate next scenario,
“If I have one bagel and want to
share it among 4 people, how
much will each person get?” (C)
0 0 1 1
445 A
8.15.1 Direct students to
work with a partner to draw either
a number line or area model
representation of the scenario (C)
0 0 1 1
446 A
8.15.2 Circulate among
students to determine whether they
are creating pictorial
representations of 1 ÷ 1/4 = 4 (C)
0 0 1 1
447 A
8.15.3 Ask students to
share their drawings with other
neighbors (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 228
448 A
8.15.4 Check for
understanding by calling on
students randomly, asking them to
describe their drawings (C)
0 0 1 1
449 A
8.15.5 Validate correct
answers that represent 1 ÷ 1/4 = 4
(C)
0 0 1 1
450 A
8.15.6 Demonstrate a
correct representation on
whiteboard by drawing both an
area model and a number line
depicting 1 ÷ 1/4 = 4 (C)
0 0 1 1
451 D
8.15.6.1 IF teacher
observation indicates student
understands how to create pictorial
representations of problems
involving whole numbers divided
by fractions, THEN step 8.16 (C)
0 0 1 1
452 D
8.15.6.2 IF teacher
observation indicates student does
not understand, THEN provide
remediation after class (C)
0 0 1 1
453 A
8.16 Ask students to
consider their answers to previous
problems, such as 1 ÷ 1/2 = 2 , 1 ÷
1/3 = 3 , and 1 ÷ 1/4 = 4 (C)
0 0 1 1
454 A
8.16.1 Ask students to pair
with a partner (C)
0 0 1 1
455 A
8.16.2 Ask students to
look for patterns in these problems
(A, C)
1 0 1 2
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 229
456 A
8.16.3 Ask students to
formulate a procedural rule that
can be followed (A, C)
1 0 1 2
457 A
8.16.4 Circulate among
students to check for
understanding (A, C)
1 0 1 2
458 A
8.16.5 Ask students that were
able to formulate a rule to share it
with the class (A, C)
1 0 1 2
459 A
8.16.6 Validate that the
rule is to multiply the whole
number by the reciprocal of the
divisor (A, B, C)
1 1 1 3
460 A
8.16.7 Assign similar
problems (A, C)
1 0 1 2
461 A
8.16.8 Direct students to
solve using the procedural rule and
to write an explanation of the
procedure they used (A, C)
1 0 1 2
462 A
8.16.9 Circulate
among students, and check written
explanations for understanding (A,
C)
1 0 1 2
463 D
8.16.9.1 IF student
demonstrates understanding,
THEN step 9 (A, C)
1 0 1 2
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 230
464 D
8.16.9.2 IF student does
not demonstrate understanding,
THEN provide remediation after
class (A, C)
1 0 1 2
Procedure 9: Teach
division of fractions by whole
numbers
45 6
465 A
9.1 Display an apple
that is cut into two pieces, asking,
“If I want to share one of these
halves of an apple between two
people, how much will each
person get?” (C)
0 0 1 1
466 A
9.2 Call on student
volunteers (C)
0 0 1 1
467 A
9.3 Confirm correct
answers by demonstrating that the
answer is one-fourth of the whole
apple for each person (C)
0 0 1 1
468 A
9.4 Distribute
fraction strips (C)
0 0 1 1
469 A
9.5 Relate the same scenario,
“If I have one half an apple and
want to share it between two
people, how much will each
person get?” (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 231
470 A
9.5.1 Ask students to work
with a partner to model scenario
with manipulatives (C)
0 0 1 1
471 A
9.5.2 Circulate among
students to determine whether they
are creating concrete
representations of 1/2 ÷ 2 = 1/4
(C)
0 0 1 1
472 A
9.5.3 Check for
understanding by calling on
students randomly (C)
0 0 1 1
473 A
9.5.4 Validate correct
answers (C)
0 0 1 1
474 A
9.5.5 Demonstrate a
correct representation on
whiteboard using magnetic
fraction strips (A, C)
0 0 1 1
475 A
9.6 Relate next scenario, “If I
have one half an apple and want to
share it among three people, how
much will each person get?” (C)
0 0 1 1
476 A
9.7 Display an apple that
is cut into two pieces, asking, “If I
want to share one of these halves
of an apple among three people,
how much will each person get?”
(C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 232
477 A
9.8 Call on student
volunteers (C)
0 0 1 1
478 A
9.9 Confirm correct answers by
demonstrating that the answer is
one-sixth of the whole apple for
each person (A, C)
1 0 1 2
479 A
9.9.1 Ask students to work with
a partner to model scenario with
manipulatives (C)
0 0 1 1
480 A
9.9.2 Circulate among
students to determine whether they
are creating concrete
representations of 1/2 ÷ 3 = 1/6
(C)
0 0 1 1
481 A
9.9.3 Check for understanding
by calling on students randomly
(C)
0 0 1 1
482 A
9.9.4 Validate correct answers
(C)
0 0 1 1
483 A
9.9.5 Demonstrate a correct
representation on whiteboard
using magnetic fraction strips (A,
C)
1 0 1 2
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 233
484 D
9.9.5.1 IF teacher observation
indicates student understands how
to create concrete representations
of problems involving fractions
divided by whole numbers, THEN
step 9.10 (C)
0 0 1 1
485 D
9.9.5.2 IF teacher observation
indicates student does not
understand, THEN provide
remediation after class (C)
0 0 1 1
486 A
9.10 Distribute math paper and
pencils (C)
0 0 1 1
487 A
9.11 Relate the scenario, “If I
have one third of a candy bar and
want to share it between 2 people,
how much will each person get?”
(C)
0 0 1 1
488 A
9.12 Draw both an area
model and a number line
representation on whiteboard of
problem (C)
0 0 1 1
489 A
9.13 Demonstrate
how both of these representations
are similar to the previous fraction
strip representation (C)
0 0 1 1
490 A
9.13.1 Direct
students to work with a partner to
draw either a number line or area
model representation of the
scenario (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 234
491 A
9.13.2 Circulate
among students to determine
whether they are creating pictorial
representations of 1/3 ÷ 2 (C)
0 0 1 1
492 A
9.13.3 Ask students
to share their drawings with other
neighbors (C)
0 0 1 1
493 A
9.13.4 Check for
understanding by calling on
students randomly, asking them to
describe their drawings (C)
0 0 1 1
494 A
9.13.5 Validate correct
answers that represent 1/3 ÷ 2 =
1/6 (C)
0 0 1 1
495 A
9.14 Relate next scenario,
“If I have one fourth of a bagel
and want to share it between 2
people, how much will each
person get?” (C)
0 0 1 1
496 A
9.14.1 Direct students to
work with a partner to draw either
a number line or area model
representation of the scenario (C)
0 0 1 1
497 A
9.14.2 Circulate among
students to determine whether they
are creating pictorial
representations of 1/4 ÷ 2 (C)
0 0 1 1
498 A
9.14.3 Ask students to share
their drawings with other
neighbors (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 235
499 A
9.14.4 Check for
understanding by calling on
students randomly, asking them to
describe their drawings (C)
0 0 1 1
500 A
9.14.5 Validate correct
answers that represent 1/4 ÷ 2 =
1/8 (C)
0 0 1 1
501 A
9.14.6 Demonstrate a
correct representation on
whiteboard by drawing both an
area model and a number line
depicting 1/4 ÷ 2 = 1/8 (C)
0 0 1 1
502 D
9.14.6.1 IF teacher
observation indicates student
understands how to create pictorial
representations of problems
involving fractions divided by
whole numbers, THEN step 9.15
(C)
0 0 1 1
503 D
9.14.6.2 IF teacher
observation indicates student does
not understand, THEN provide
remediation after class (C)
0 0 1 1
504 A
9.15 Ask students to
consider their answers to previous
problems, such as 1/2 ÷ 2 = 1/4 ,
1/2 ÷ 3 = 1/6 , and 1/4 ÷ 2 = 1/8
(C)
0 0 1 1
505 A
9.15.1 Ask students to pair
with a partner (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 236
506 A
9.15.2 Ask students to
look for patterns in the previous
problems (A, C)
1 0 1 2
507 A
9.15.3 Ask students to
formulate a procedural rule that
can be followed (A,C)
1 0 1 2
508 A
9.15.4 Circulate among
students to check for
understanding (A, C)
1 0 1 2
509 A
9.15.5 Ask students that
were able to formulate a rule to
share it with the class (A, C)
1 0 1 2
510 A
9.15.6 Validate that the rule is
to multiply the fraction by the
reciprocal of the whole number
(A, B, C)
1 1 1 3
511 A
9.15.7 Assign similar
problems, such as 1/5 ÷ 2 (A,C)
1 0 1 2
512 A
9.15.8 Direct students to solve
using fraction strips, then a
pictorial representation, and then
to validate using the procedural
rule, and include a written
description of the written
description (A, C) 1 0 1 2
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 237
3513 A
9.15.9 Circulate among
students and check strips, pictures,
and written descriptions for
understanding (A, C)
1 0 1 2
514 D
9.15.9.1 IF student
demonstrates understanding,
THEN step 10 (A, C)
1 0 1 2
515 D
9.15.9.2 IF student does not
demonstrate understanding, THEN
provide remediation after class (A,
C)
1 0 1 2
Procedure 10: Teach division
of fractions by fractions
59 10
516 A
10.1 Teach concrete
representation (C)
0 0 1 1
517 A
10.2 Distribute fractions strips
(C)
0 0 1 1
518 A
10.2.1 Pose problem such as
3/4÷ 1/4 (C)
0 0 1 1
519 A
20.2.2 Remind students that the
problem involves determining how
many fourths are in three fourths
(A, B, C)
1 1 1 3
520 A
10.2.3 Use magnetic fraction
strips on whiteboard to
demonstrate that the answer is 3:
there are 3 one-fourths in three-
fourths (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 238
521 A
10.2.4 Direct students to model
the same problem at their desks
(C)
0 0 1 1
522 A
10.2.5 Circulate among
students to check for
understanding (C)
0 0 1 1
523 A
10.2.6 Pose similar
problem, such as 1/2 ÷ 1/4 (C)
0 0 1 1
524 A
10.2.7 Direct students to model
problem with fraction strips or
linking cubes (C)
0 0 1 1
525 A
10.2.8 Circulate among
students to check for
understanding (C)
0 0 1 1
526 A
10.2.9 Demonstrate on
board that the answer is 2: there
are two one-fourths in one half (C)
0 0 1 1
527 A
10.2.10 Pose similar problems (C)
0 0 1 1
528 A
10.2.11 Circulate among
students to check for
understanding (C) 0 0 1 1
529 D
10.2.11.1 IF student
demonstrates understanding,
THEN step 10.3 (C) 0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 239
530 D
10.2.11.2 IF students does
not demonstrate understanding,
THEN provide remediation after
class (C)
0 0 1 1
531 A
10.3 Teach pictorial
representation (B, C)
0 1 1 2
532 A
10.3.1 Pose problem, such as
3/4÷ 1/4 (B, C)
0 1 1 2
533 A
10.3.2 Remind students that the
problem involves determining how
many fourths are in three fourths
(A, B, C)
1 1 1 3
534 A
10.3.3 Draw an area
model and a number line on
whiteboard to demonstrate that
there are three one-fourths in three
fourths (B, C)
0 1 1 2
535 A
10.3.4 Direct students to
model the same problem at their
desks (B, C)
0 1 1 2
536 A
10.3.5 Circulate among
students to check for
understanding (B, C) 0 1 1 2
537 A
10.3.6 Direct students to
validate the answer using the
procedural rule (C)
0 0 1 1
538 A
10.3.7 Circulate among
students to check for
understanding (B, C) 0 1 1 2
539 A
10.3.8 Call on volunteers
who were able to multiply three-
fourths by the reciprocal of one-
fourth, 4, to yield 3 (B, C)
0 1 1 2
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 240
540 A
10.3.9 Pose similar problem,
such as 1/2 ÷ 1/4 (C)
0 0 1 1
541 A
10.3.10 Direct students to
model problem with fraction
strips, and then area models or
number lines, and then validate
with the procedural rule (C)
0 0 1 1
542 A
10.3.11 Circulate among
students to check for
understanding (C)
0 0 1 1
543 A
10.3.12 Demonstrate on board
that there are two fourths in one
half (C)
0 0 1 1
544 A
10.3.13 Pose similar
problems (C)
0 0 1 1
545 A
10.3.14 Circulate among
students to check for
understanding (C)
0 0 1 1
546 D
10.3.14.1 IF student
demonstrates understanding,
THEN step 10.4 (C)
0 0 1 1
547 D
10.3.14.2 IF students
does not demonstrate
understanding, THEN provide
remediation after class (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 241
548 A
10.4 Teach complex
fraction approach (B)
0 1 0 1
549 A
10.4.1 Remind students
that division can be represented
through fractions: the dividend can
be represented as numerator, and
the divisor as denominator (B)
0 1 0 1
550 A
10.4.2 Demonstrate
that 3/4 ÷ 1/4 can be represented
as a complex fraction, in the form
(3/4)/(1/4) (B)
0 1 0 1
551 A
10.4.3 Remind students that
anything divided by one is itself
(B)
0 1 0 1
552 A
10.4.4 Remind students
that the identity property states
that multiplying a value by 1 does
not change that value (B)
0 1 0 1
553 A
10.4.5 Demonstrate
that multiplying (3/4)/(1/4) x
(4/1)/(4/1) is both multiplying by 1
and also going to create a 1 in the
denominator, based on the inverse
property of multiplication (B)
0 1 0 1
554 A
10.4.6 Draw the
outline of a “1” around (4/1)/(4/1)
(B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 242
555 A
10.4.7 Remind students
that because the denominator is 1,
we are left with 3/4 x 4/1 (B)
0 1 0 1
556 A
10.4.8 Remind
students that the algorithm for
multiplying fractions, by which we
multiply the numerators by
numerators and denominators by
denominators, means that the
product is going to be 12/4 (B)
0 1 0 1
557 A
10.4.9 Remind students to
reduce by dividing by a form of 1,
in this case 4/4 (B)
0 1 0 1
558 A
10.4.10 Explain that the answer
is 3, which is the same answer
derived through the pictorial
method (B)
0 1 0 1
559 A
10.4.11 Assign similar
problems asking students to solve
using complex fractions (B)
0 1 0 1
560 A
10.4.12 Visually check for
understanding (B)
0 1 0 1
561 D
10.4.12.1 IF students
demonstrate understanding, then
go to step 10.4.13 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 243
562 D
10.4.12.2 IF student appears
unclear, then check in with her
before end of class to provide brief
review (B)
0 1 0 1
563 A
10.4.13 Review that converting
a fraction division problem into a
complex fraction results in
multiplying the first fraction by
the reciprocal of the second
fraction, a fact we also discovered
when we divided whole numbers
by fractions and fractions by
whole numbers using
manipulatives and pictorial
representations (A, B, C)
1 1 1 3
564 A
10.4.14 Administer
assessment in which students must
solve similar problems and also
write an explanation of why they
can derive the same answer by
simply multiplying by the
reciprocal (B)
0 1 0 1
565 D
10.4.14.1 IF student
answers demonstrate
understanding, THEN go to step
10.5 (B)
0 1 0 1
566 D
10.4.14.2 IF student does
not demonstrate understanding,
direct her to come to class during
Muir Time and provide
remediation (B)
0 1 0 1
567 A
10.5 Teach alternate
method for dividing fractions by
fractions, which involves
converting the fractions to
decimals, and then dividing the
decimals (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 244
568 A
10.5.1 Demonstrate
conceptual nature of process with
an example using currency (B)
0 1 0 1
569 A
10.5.1.1 Demonstrate
using the example 1/2 ÷ 1/4 (B)
0 1 0 1
570 A
10.5.1.2 Review that we
must first convert both 1/2 and 1/4
to equivalent fractions with
denominators that are powers of
10 or 100 (B)
0 1 0 1
571 A
10.5.1.3 Demonstrate
multiplying 1/2 and 1/4 by 5/5
and 25/25 respectively, to yield
5/10 and 25/100 (B)
0 1 0 1
572 A
10.5.1.4 Review that these
new fractions convert to decimals
of .50 and .25 (B)
0 1 0 1
573 A
10.5.1.5 Connect these two
values to the concept of money by
explaining that .50 can be viewed
as fifty cents and .25 can be
viewed as twenty five cents (B)
0 1 0 1
574 A
10.5.1.6 Demonstrate that
.50 ÷ .25 is equivalent to asking,
“How many quarters make up fifty
cents?” (B)
0 1 0 1
575 A
10.5.1.7 Demonstrate that the
answer is “2” (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 245
576 A
10.5.2 Demonstrate
procedural process (B)
0 1 0 1
577 A
10.5.2.1 Review that 1/2 ÷
1/4 can be represented by decimals
as .50 ÷ .25 (B)
0 1 0 1
578 A
10.5.2.2 Review that we can
represent this in long division as
.25)(.50) ? (B)
0 1 0 1
579 A
10.5.2.3 Review that we must
shift both the decimal point in the
divisor and the decimal point in
the dividend two places to the
right before we can divide,
yielding 50 divided by 25 (B)
0 1 0 1
580 A
10.5.2.4 Demonstrate that the
quotient becomes 2, the same
answer we obtained in the money
example (B)
0 1 0 1
581 A
10.5.2.5 Assign similar
problems (B)
0 1 0 1
582 A
10.5.2.6 Visually check
for understanding (B)
0 1 0 1
583 D
10.5.2.6.1 IF student
demonstrates understanding,
THEN step 11 (B)
0 1 0 1
584 D
10.5.2.6.2 IF student does not
demonstrate understanding, THEN
direct her to come to class during
Muir Time and provide
remediation (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 246
Procedure 11: Teach
division of mixed numbers by
mixed numbers
23 8 0
585 A
11.1 Pose problem such as 21/2
÷ 11/4 (A, B, C)
1 1 1 3
586 A
11.2 Direct students to
create a concrete representation
with fraction strips (C)
0 0 1 1
587 A
11.2.1 Circulate among
students to check for
understanding (C)
0 0 1 1
588 A
11.2.2 Ask students with
correct answers to recreate their
representations on whiteboard
with magnetic fraction strips (C)
0 0 1 1
589 A
11.2.3 Validate that
correct answer is 2 (C)
0 0 1 1
590 A
11.2.4 Assign similar problems
(C)
0 0 1 1
591 D
11.2.4.1 IF student
demonstrates understanding,
THEN step 11.2.5 (C)
0 0 1 1
592 D
11.2.4.2 IF student does
not demonstrate understanding,
THEN provide remediation after
class (C)
0 0 1 1
593 A
11.2.5 Direct students to
create a pictorial representation of
same problem (B, C)
0 1 1 2
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 247
594 A
11.2.6 Circulate among
students to check for
understanding (B, C)
0 1 1 2
595 A
11.2.7 Ask students with correct
answers to recreate their area
models, circle models, or number
lines on whiteboard (B, C)
0 1 1 2
596 A
11.2.8 Validate that correct
drawings depict 21/2 ÷ 11/4 = 2
(B, C)
0 1 1 2
597 A
11.2.9 Assign similar problems
(B, C)
0 1 1 2
598 D
11.2.9.1 IF student
demonstrates understanding,
THEN step 11.3 (B, C)
0 1 1 2
599 D
11.2.9.2 IF student does
not demonstrate understanding,
THEN provide remediation after
class (B, C)
0 1 1 2
600 A
11.3 Direct students to solve
problem using the complex
fraction method (B)
0 1 0 1
601 A
11.3.1 Remind students that we
need to convert each term to an
improper fraction, yielding 5/2 ÷
5/4 (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 248
602 A
11.3.2 Demonstrate that
we can convert this to a complex
fraction, (5/2)/(5/4) (B)
0 1 0 1
603 A
11.3.3 Remind students
that we can multiply by (4/5)/(4/5)
to change the denominator to 1 (B)
0 1 0 1
604 A
11.3.4 Remind
students that converting to a
complex fraction and then
converting the denominator to 1 is
same as multiplying by the
reciprocal (B)
0 1 0 1
605 A
11.3.5 Assign similar
problems (B)
0 1 0 1
606 A
11.3.6 Visually check for
understanding (B)
0 1 0 1
607 D
11.3.6.1 IF students
demonstrate understanding, then
go to step 11.4 (B)
0 1 0 1
608 D
11.3.6.2 IF student does not
demonstrate understanding, direct
her to come to class during Muir
Time and provide remediation (B)
0 1 0 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 249
609 A
11.4 Direct students to
solve problem using the
procedural method (A, B, C)
1 1 1 3
610 A
11.4.1 Circulate among
students to check for
understanding (A, B, C)
1 1 1 3
611 A
11.4.2 Ask students with
correct answers to recreate their
algorithms on whiteboard (A, B,
C)
1 1 1 3
612 A
11.4.3 Validate that
correct procedure involves
converting 21/2 to 5/2 and 11/4 to
5/4 , and then multiplying 5/2 by
the reciprocal of 5/4 to yield 5/2 x
4/5 = 20/10 or 2 (A, B, C)
1 1 1 3
613 A
11.4.4 Assign similar problems
(A, B, C)
1 1 1 3
614 D
11.4.4.1 IF student
demonstrates understanding,
THEN step 12 (A, B, C)
1 1 1 3
615 D
11.4.4.2 IF student does not
demonstrate understanding, THEN
provide remediation after class (A,
B, C)
1 1 1 3
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 250
Procedure 12: Teach
division of fraction word
problems
15 2
616 A
12.1 Pose problem such as, “If a
recipe calls for 1/2 cup of flour for
one batch of bread, and I have 3/4
cup of flour, how many batches of
the recipe can I make?” (C)
0 0 1 1
617 A
12.1.1 Direct students to model
the problem with manipulatives
(C)
0 0 1 1
618 A
12.1.2 Circulate among
students to check for
understanding (C)
0 0 1 1
619 A
12.1.3 Ask students with
correct representations to recreate
their representations on
whiteboard with magnetic fraction
strips (C)
0 0 1 1
620 A
12.1.4 Validate correct concrete
representation of 3/4 ÷ 1/2 (C)
0 0 1 1
621 A
12.1.5 Direct students to model
the problem pictorially (C)
0 0 1 1
622 A
12.1.6 Circulate among
students to check for
understanding (C) 0 0 1 1
623 A
12.1.7 Ask students with
correct representations to recreate
their representations on
whiteboard with number lines or
area models (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 251
624 A
12.1.8 Validate correct pictorial
representation of 3/4 ÷ 1/2 (C)
0 0 1 1
625 A
12.1.9 Direct students to solve
the problem with the previously
discovered procedure (C)
0 0 1 1
626 A
12.1.10 Circulate among
students to check for
understanding (C)
0 0 1 1
627 A
12.1.11 Ask students with
correct procedure to write their
number sentences on the board (C)
0 0 1 1
628 A
12.1.12 Validate correct
number sentence is 3/4 ÷ 1/2 =
3/4 x 2/1 = 6/4 = 11/2 (C)
0 0 1 1
629 A
12.1.13 Assign similar
problems, asking students to create
both a concrete and pictorial
representation, and to solve using
the number sentence procedure (C)
0 0 1 1
630 A
12.1.14 Circulate among
students to check for
understanding (C)
0 0 1 1
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 252
631 D
12.1.14.1 IF student
demonstrates understanding,
THEN end task (C)
0 0 1 1
632 D
12.1.14.2 IF student does
not demonstrate understanding,
THEN provide remediation after
class (C)
0 0 1 1
632 Total Action and Decision Steps 130 385 227 490 142
490 Action Steps 110 279 199
142 Decision Steps 20 106 28
Total Action and Decision Steps 20.57% 60.92% 35.92%
Action Steps 22.45% 56.94% 40.61%
Decision Steps 14.08% 74.65% 19.72%
Action and Decision Steps
Omitted 502 247 405
Action Steps Omitted 380 211 291
Decision Steps Omitted 122 36 114
Action and Decision Steps
Omitted 79.43% 39.08% 64.08%
Action Steps Omitted 77.55% 43.06% 59.39%
Decision Steps Omitted 85.92% 25.35% 80.28%
Average Captured Omitted
Total Action and Decision Steps 39.14% 60.86%
Action Steps 40.00% 60.00%
Decision Steps 36.15% 63.85%
Highly Aligned 21 3.32%
Partially Aligned 68 10.76%
COGNITIVE TASK ANALYSIS AND DIVISION OF FRACTIONS 253
Slightly Aligned 543 85.92%
632 100.00%
Abstract (if available)
Abstract
This study applies Cognitive Task Analysis (CTA), a method for eliciting the automated, unconscious knowledge and skills of experts, to capture expertise in teaching K-12 mathematics. The purpose of this study was to conduct a CTA with middle school teachers who have been identified as experts, to capture the knowledge and skills they use when providing instruction in the division of fractions by fractions. Also, this study investigated whether experts’ knowledge omissions in this field would conform to those in other fields, which can approach 70%. CTA methods in this study included semi-structured interviews with three middle school mathematics teachers. The study’s findings indicated that these experts recalled, on average, 39.14% of the action and decision steps compared to the gold standard protocol, while omitting, on average, 60.86% of such steps. The study’s implications are that the degree of omissions among expert middle school teachers are similar to those of experts in other fields. Additionally, the greater degree of knowledge capture provided by the use of multiple experts, compared to that for a single expert, indicates that the use of CTA for the development of teacher preparation and professional development programs shows promise when compared to current models, which rely primarily on individual experts.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Using cognitive task analysis to capture expert reading instruction in informational text for students with mild to moderate learning disabilities
PDF
Using individual cognitive task analysis to capture expert writing instruction in expository writing for secondary students
PDF
Using incremental cognitive task analysis to capture expert instruction in expository writing for secondary students
PDF
Using cognitive task analysis to capture how expert principals conduct informal classroom walk-throughs and provide feedback to teachers
PDF
Using cognitive task analysis for capturing expert instruction of food safety training for novice employees
PDF
The use of cognitive task analysis to capture expert instruction in teaching mathematics
PDF
Using cognitive task analysis to capture palliative care physicians' expertise in in-patient shared decision making
PDF
The use of cognitive task analysis to capture expert patient care handoff to the post anesthesia care unit
PDF
Identifying the point of diminishing marginal utility for cognitive task analysis surgical subject matter expert interviews
PDF
The use of cognitive task analysis for the postanesthesia patient care handoff in the intensive care unit
PDF
The effect of cognitive task analysis based instruction on surgical skills expertise and performance
PDF
Towards a taxonomy of cognitive task analysis methods: a search for cognition and task analysis interactions
PDF
The use of cognitive task analysis for identifying the critical information omitted when experts describe surgical procedures
PDF
The use of cognitive task analysis to capture exterptise for tracheal extubation training in anesthesiology
PDF
Using cognitive task analysis to determine the percentage of critical information that experts omit when describing a surgical procedure
PDF
Using cognitive task analysis to capture how expert anesthesia providers conduct an intraoperative patient care handoff
PDF
The use of cognitive task analysis to determine surgical expert's awareness of critical decisions required for a surgical procedure
PDF
Cognitive task analysis for instruction in single-injection ultrasound-guided regional anesthesia
PDF
Employing cognitive task analysis supported instruction to increase medical student and surgical resident performance and self-efficacy
PDF
The use of cognitive task analysis to investigate how many experts must be interviewed to acquire the critical information needed to perform a central venous catheter placement
Asset Metadata
Creator
Wieland, Douglas C.
(author)
Core Title
Using cognitive task analysis to capture expert instruction in division of fractions
School
Rossier School of Education
Degree
Doctor of Education
Degree Program
Education
Publication Date
04/02/2015
Defense Date
01/26/2015
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
cognitive task analysis,division of fractions,K-12 instruction,mathematics,OAI-PMH Harvest
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Yates, Kenneth A. (
committee chair
), Kaplan, Sandra N. (
committee member
), Samkian, Artineh (
committee member
)
Creator Email
d-wieland@sbcglobal.net,dwieland@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-542305
Unique identifier
UC11298628
Identifier
etd-WielandDou-3244.pdf (filename),usctheses-c3-542305 (legacy record id)
Legacy Identifier
etd-WielandDou-3244.pdf
Dmrecord
542305
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Wieland, Douglas C.
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
cognitive task analysis
division of fractions
K-12 instruction