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The use of cognitive task analysis to capture expert instruction in teaching mathematics
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The use of cognitive task analysis to capture expert instruction in teaching mathematics
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Content
Running head: THE USE OF COGNITIVE TASK ANALYSIS
THE USE OF COGNITIVE TASK ANALYSIS TO CAPTURE EXPERT
INSTRUCTION IN TEACHING MATHEMATICS
By
Acquillahs Muteti Mutie
A Dissertation Presented to the
FACULTY OF THE USC ROSSIER SCHOOL OF EDUCATION
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF EDUCATION
May 2015
Copyright 2015 Acquillahs Muteti Mutie
THE USE OF COGNITIVE TASK ANALYSIS
2
Dedication
This dissertation is dedicated to my best friend, my wife Esther. Thank you for your love
and support without which this endeavor would not have been successful. To my children,
Alfred, Allan, and Neema – you are amazing and I am blessed to be your dad. This one is for
you! You too can do it. To my mom Rebecca Nthambi, though you cannot read and write, your
love for education and encouragement has made me the man I am today. To my dad Benjamin
Mutie, you taught me hard work and persistence, thank you.
THE USE OF COGNITIVE TASK ANALYSIS
3
Acknowledgements
I would like to acknowledge a great professor, mentor and teacher, Dr. Kenneth A. Yates
for his time and dedication to this work on Cognitive Task Analysis (CTA). Dr. Yates, you truly
exemplify what great educators do, motivate! Thank you for the countless Sunday early morning
review sessions you provided me when you did not have to. Phone calls, text messages and “Join
me” rounds made all the difference; you went over and beyond the call of duty to guide this
study. You made me!
To Dr. Camille Ramos-Beal, my friend and biggest cheerleader at Palomares Academy,
thank you for accepting to be in my committee. Your encouragement and high fives made all the
difference. I am excited when I imagine what lies ahead for both of us. Thank you!
To Dr. Angela “Laila” Hasan, a mathematics teacher education expert, thank you for
supporting this study and being a member of my committee. Your contributions and insights
were priceless. I look forward to collaborations on ways of improving teacher education
programs that you are passionate about.
I am extremely grateful to the five mathematics teachers who participated in this study. I
sincerely appreciate your perseverance through the CTA interview process that for some of you
took up to three hours. That was a sacrifice on your part and for that I will always be indebted to
you. Through your commitment to student achievement, your participation in this study will
make a difference in the lives of many students.
To Megan McGuinness, Christine Gayle Corpus, Kari Cole, Judith Franco, Chad
Hammitt, Charlotte Ann Garcia, Milo Jury, Deidre Larson, Nicholas Lim and Douglas Weiland,
you are the best CTA thematic group ever. Thank you for being my partners. You are the best
team I could have hoped for. Together we were a great team! Megan, you have a great spirit and
THE USE OF COGNITIVE TASK ANALYSIS
4
you made our Saturday dissertation “writes” pleasant experiences! I will always remember you.
To my friend Catherine Kawaguchi, thank you. To the Tuesday 2012 cohort, thank you for your
friendship. We are the class of 2015! I am.
Thank you to my professors and lecturers at the University of Nairobi’s Chiromo Campus
for giving me great education. You gave me the start I enjoy today.
Thank you to all the teachers in my elementary school, Mii Primary School in Makueni
County, KENYA for teaching me how to read and write. Yes, you are my heroes!
To my wife, Esther, thank you for the sacrifices we both made. This was a challenging
journey for both of us. Thank you for your love, patience and enduring support.
To Neema, what a gift to my life! Thank you for being an awesome daughter.
To Allan and Alfred, you are awesome sons! I am so proud of you and thank you for your
unwavering love.
Lastly, to the USC Rossier School of Education Ed.D. program faculty and student
support team, you are amazing and student oriented. This was a rigorous program and with your
collective dedication to student success, I too completed this journey. USC, so proud of you!
Fight on!
THE USE OF COGNITIVE TASK ANALYSIS
5
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 ..................................................................................................... 13
Methodology of the Study ............................................................................................ 13
Definition of Terms ....................................................................................................... 14
Organization of the Study ............................................................................................. 15
CHAPTER TWO: LITERATURE REVIEW ................................................................... 17
United States Math Performance .................................................................................. 17
Job Market Consequences ............................................................................................. 19
Algebra as a Foundational Course ................................................................................ 20
Quadratic Equations ...................................................................................................... 22
Teachers' Knowledge in Teaching Mathematics .......................................................... 23
Expertise in Mathematics .......................................................................................... 25
Professional Development to Improve Teacher Learning ........................................ 26
Cognitive Task Analysis ........................................................................................... 27
Knowledge Types ......................................................................................................... 29
Declarative Knowledge ............................................................................................. 29
Procedural and Conditional Knowledge ................................................................... 30
Automaticity ................................................................................................................. 30
Expertise ....................................................................................................................... 31
Characteristics of Experts ......................................................................................... 31
Building Expertise .................................................................................................... 32
Consequences of Expertise ....................................................................................... 33
Expert Omissions ...................................................................................................... 33
Cognitive Task Analysis (CTA) ................................................................................... 35
Definition of CTA ..................................................................................................... 35
Brief History of CTA ................................................................................................ 35
Cognitive Task Analysis Methodology .................................................................... 36
Taxonomies of Knowledge Elicitation Techniques .................................................. 36
Pairing Knowledge Elicitation with Knowledge Analysis ....................................... 37
Effectiveness of CTA ................................................................................................ 38
Benefits of CTA for Instruction ................................................................................ 39
Summary ....................................................................................................................... 41
THE USE OF COGNITIVE TASK ANALYSIS
6
CHAPTER THREE: METHODOLOGY ......................................................................... 42
Task ............................................................................................................................... 42
Population and Sample ................................................................................................. 43
Data Collection and Instrumentation ............................................................................ 44
Step 1: Collecting preliminary knowledge ............................................................... 44
Step 2: Identifying Knowledge Representations ....................................................... 44
Stage 3: Applying Focused Knowledge Elicitation Methods ................................... 44
Step 4: Analyzing and Verifying Data Acquired ...................................................... 45
Step 5: Formatting Results for the Intended Application ......................................... 46
Data Analysis ............................................................................................................ 46
CHAPTER FOUR: RESULTS ......................................................................................... 49
Overview of Results ...................................................................................................... 49
Research Questions ....................................................................................................... 49
Question 1 ................................................................................................................. 49
Recalled action and decision steps. .......................................................................... 52
Question 2 ................................................................................................................. 60
CHAPTER FIVE: DISCUSSION ..................................................................................... 65
Overview of the Study .................................................................................................. 65
Process of Conducting Cognitive Task Analysis .......................................................... 66
Selection of Experts .................................................................................................. 66
Collection of Data ..................................................................................................... 68
Discussion of Findings .................................................................................................. 70
Differences in SMEs ................................................................................................. 71
Action Steps Verses Decision Steps ......................................................................... 73
Expert Review Of Draft Gold Standard Protocol ..................................................... 75
Limitations .................................................................................................................... 77
Confirmation Bias ..................................................................................................... 77
Internal Validity ........................................................................................................ 77
External Validity ....................................................................................................... 78
Implications ................................................................................................................... 78
Future Research ............................................................................................................ 80
Conclusion .................................................................................................................... 80
References ......................................................................................................................... 82
Appendix A ....................................................................................................................... 95
Appendix B ....................................................................................................................... 98
Appendix C ....................................................................................................................... 99
Appendix D ..................................................................................................................... 101
Appendix E ..................................................................................................................... 127
Appendix F...................................................................................................................... 151
Appendix G ..................................................................................................................... 181
Appendix H ..................................................................................................................... 183
THE USE OF COGNITIVE TASK ANALYSIS
7
List of Tables
Table 4.1: Example of Aggregating Action and Decision Steps for the PGSP 51
Table 4.2: Counts of Action and Decision Steps for Each SME 53
Table 4.3: Percentage of Action and Decision Steps for Each SME Compared to the
Total Number of action and decision steps 54
Table 4.4: Additional Action and Decision Steps After Round Two Interviews 56
Table 4.5: Count and Percentage of Action and Decision Steps that are Fully,
Substantially, Partially, or Not Aligned with the GSP 59
Table 4.6: Total Action and Decision Steps Omitted by SMEs When Compared
to the GSP 60
Table 4.7: Percentage of Total Action and Decision Steps Omitted by SMEs
When Compared to the GSP 61
THE USE OF COGNITIVE TASK ANALYSIS
8
List of Figures
Figure 3.1: Process of conducting CTA Semi-Structured Interviews and Aggregating
the GSP 47
Figure 4.1: Number of Action, Decision Steps and Total Action and Decision Steps for
SMEs A, B, C and D Captured Through CTA 55
Figure 4.2: Percentage of Action Steps of SMEs for Round One and Round Two
Interviews 57
Figure 4.3: Percentage of Decision Steps of SMEs for Round One and Round Two
Interviews 58
Figure 4.4: Percentage of Total Action and Decision Steps Omitted by SMEs
When Compared to the GSP 62
Figure 4.5: Percentage of Action Steps Omitted by the SMEs Compared to
the GSP 63
Figure 4.6: Percentage of Action Steps Omitted by the SMEs Compared
to the GSP 64
THE USE OF COGNITIVE TASK ANALYSIS
9
List of Abbreviations
CCSS: Common Core State Standards 79
CTA: Cognitive Task Analysis 10
EQ: Essential Question 137
GSP: Gold Standard Protocol 10
IRR: Inter-rater Reliability 46
K-12: Kindergarten Through Twelfth Grade 15
M: Mean 54
OECD: Organization for Economic Cooperation and Development 18
p: Probability Value 76
PGSP: Preliminary Gold Standard Protocol 51
PISA: Program in International Student Assessment 18
QE: Quadratic Equation 102
QF: Quadratic Formula 105
R1: Round 1 Interview 56
R2: Round 2 Interview 56
SD: Standard Deviation 54
SME: Subject Matter Expert 10
t: t-Statistic 76
TIMSS Trends in International Mathematics and Science Study 17
ZPP: Zero Product Property 136
THE USE OF COGNITIVE TASK ANALYSIS
10
Abstract
The purpose of this study was to apply Cognitive Task Analysis (CTA) methods to capture
expert mathematics instruction in solving quadratic equations. CTA seeks to elicit the highly
automated and often-unconscious knowledge experts use to solve difficult problems and perform
complex tasks. Students taking algebra find solving and understanding quadratic equations very
challenging yet quadratic equations are a major component of building mastery in algebra. Four
8
th
and 9
th
grade Algebra teachers, who were qualified as experts using both qualitative and
quantitative measures, were interviewed to capture the action and decision steps they use to teach
quadratic equations. The individual protocols were then aggregated as a gold standard protocol
(GSP) that was reviewed by a fifth senior SME for accuracy and consistency. Overall, there
were found to be seven main procedures for solving quadratic equations. However, there was
full alignment among the four experts on only seven percent of the action and decision steps,
suggesting that multiple experts should be used to capture complex procedures, such as teaching
algebra. Moreover, the experts omitted an average of 59.90% of the total action and decision
steps, thus supporting previous research finding that experts may omit up to 70% of the critical
information required to perform a complex task. The expert knowledge and skills captured may
be used to train student teachers in teacher prep-programs and also offer professional
development to Algebra teachers for teaching this highly complex subject.
THE USE OF COGNITIVE TASK ANALYSIS
11
CHAPTER ONE: OVERVIEW OF THE STUDY
Statement of the Problem
The percentage of high-achieving math students in the United States – and most of its
individual states – are below those of many of the world’s leading industrialized nations
(Hanushek, Peterson, & Woessmann, 2011). Hanushek, et al., (2011) observe that unless U.S.
schools find the tools to bring students up to the highest level of accomplishment, it places the
nation at risk in international economy of the 21
st
century. There is a need for the United States
to maintain its inventive advantage (Nord, Jenkins, Chan, & Kastberg, 2013). Both Hanushek et
al. (2011) and Nord et al. (2013) argue that to maintain our inventive advantage, our system of
education requires teaching that produces students with advanced mathematics and science skills.
Algebra is a foundation on which advanced mathematics, science, technology, and
engineering courses are built (Evan, Gray, & Olchefske, 2006). According to both Gamoran and
Hannigan (2000) and Musen (2010), success in Algebra opens doors to more advanced math, a
college preparatory high school curriculum, higher college going and graduation rates.
Therefore, Algebra is the gateway to success in career and college (Gamoran & Hannigan, 2000;
Musen, 2010) and success in algebra is equally linked to career and job readiness and higher
earnings once a student has joined the job market (Achieve 2008). The highest level of
mathematics reached in high school is a key marker in students’ success in college (Adelman,
2006; Evan et al. 2006; and Musen, 2010). In fact Evan et al. (2006) makes a vital observation in
that passing algebra no later than 9
th
grade significantly increases the chances of the student
graduating from high school, going to college, and graduating from college.
Algebra students find it challenging to solve and understand quadratic equations
(Vaiyavutjamai, Ellerton, & Clements, 2005). Seeing the significance of quadratic equations in
THE USE OF COGNITIVE TASK ANALYSIS
12
algebra and mathematics in general, Vaiyavutjamai et al. (2005) wonders why there has been so
little research into the teaching and learning of quadratic equations. In fact students that have not
mastered and understood how to solve quadratic equations, struggle in their later years in high
school because all the other math concepts build on quadratic equations (Chazan & Yerushalmy,
2003; Vaiyavutjamai et al., 2005). Students are taught procedures that provide them solutions
(Chazan & Yerushalmy, 2003) and the impression behind the curriculum is that if students have
mastered these procedures, then they will be able to apply them in the context of new problems
(Vaiyavutjamai et al., 2005).
This problem in student achievement in math is compounded by math teachers’ lack of
basic understanding of mathematical ideas and procedures of teaching mathematics (Ball, 2000).
Ball (2000) noted that teachers do not have the knowledge that matters for teaching and therefore
find teaching math difficult. This places a huge burden on Algebra students who are expected to
pass this class to have opportunities for higher levels of math in preparation for college and other
careers. In order to improve student achievement, schools must attend to the training of teachers
because student learning is enhanced by the efforts of teachers who are skillful at teaching it to
others (Ball, 2000; Darling-Hammond, 1999). Darling-Hammond (1999) observes that the
effects of well-prepared teachers on student achievement are stronger than the influences of
poverty, language barriers, and minority status. If we can capture expertise of teaching quadratic
equations, then we can improve instruction.
Cognitive task analysis (CTA) is a best practice for capturing expertise. CTA is a
methodology that has been used to capture the cognitive processes, decision-making, and
judgments that underlie expert behaviors (Yates, 2007). Although there are no published studies
that offer evidence on how many mathematics experts must be interviewed in order to capture
THE USE OF COGNITIVE TASK ANALYSIS
13
enough critical information to aid in teaching quadratic equations, Chao and Salvendy (1994)
concluded three experts were needed to acquire the optimal vital knowledge and skills needed to
solve a complex software-debugging task. As such, this study seeks to use CTA to capture three
to four math experts’ knowledge and skills for solving quadratic equations.
Purpose of the Study
The purpose of this study is to conduct a CTA with math teachers who have been
identified as experts, to capture the action and decision steps they use when teaching solving
quadratic equations to 8
th
and 9
th
grade students. The following questions will guide this study:
1. What are action and decision steps that expert math teachers recall when they
describe how to teach solving quadratic equations in Algebra?
2. What percentage of actions and/or decision steps, when compared to a gold
standard, do expert math teachers omit when they describe how to solve
quadratic equations in Algebra?
Methodology of the Study
The methodology of this study was to conduct a Cognitive Task Analysis to determine
the action and decision steps expert algebra teachers’ use while teaching solving quadratic
equations to 8
th
and 9
th
grade students. The teachers came from two unified school districts in
Southern California identified as experts through students’ achievement data on State
Standardized Tests and by their peers. Five SMEs were chosen, four to participate in semi-
structured interviews and the fifth to verify the data collected form the SMEs on the steps to
solve quadratic equations. The CTA followed a five-step process of:
1) A preliminary phase to build general familiarity mostly known as
“bootstrapping;”
THE USE OF COGNITIVE TASK ANALYSIS
14
2) The identification of declarative and procedural knowledge and any hierarchical
relationships in the application of these knowledge types;
3) Knowledge elicitation through semi-structured interviews;
4) Data analysis involving coding, inter-rater reliability, and individual SME
protocol verification; and
5) The development of a gold standard protocol that was used to analyze and
determine expert omissions and eventually to be used for training of novice
algebra teachers.
Definition of Terms
The following are the definition 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, Pokomy, Dibble, & Glaser, 1993;
Hatano & Inagaki, 2000)
Automaticity: An unconscious fluidity of task performance following sustained and
repeated execution results in automated mode of functioning (Anderson, 1996; Ericsson, 2004).
Automated knowledge: Knowledge about how to do something: operates outside of
conscious awareness due to repetition of task (Wheatley & Wegner, 2001)
Cognitive load: Simultaneous demands placed on working memory during information
processing that can present challenges to learners (Sweller, 1988).
Cognitive tasks: Tasks that require mental effort and engagement to perform (Clark &
Estes, 1996).
THE USE OF COGNITIVE TASK ANALYSIS
15
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,
1996; 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, Feldon, van Merriënboer, Yates, & Early, 2008).
Organization of the Study
Chapter Two of this study reviews the literature in two parts; the first part of the literature
review assesses the relevant literature associated to mathematics student performance and
achievement in K-12 in the United States while the second part concentrates on literature
THE USE OF COGNITIVE TASK ANALYSIS
16
relevant to Cognitive Task Analysis as a knowledge elicitation method for subject matter
expertise. Chapter Three addresses the methodology of this study and how the approach to the
research answers the research questions. Chapter Four analyses the collected data and results of
the study. This chapter also compares these results in relation to the research questions. Chapter
Five discusses the findings, the implications of the findings and CTA, limitations of the study,
and implications for future research.
THE USE OF COGNITIVE TASK ANALYSIS
17
CHAPTER TWO: LITERATURE REVIEW
United States Math Performance
Countries all over the world have been participating in common international assessments
of mathematics and science, the Trends in International Mathematics and Science Study
(TIMSS) and Program in International Student Assessment (PISA) (Hanushek & Woessmann,
2010). According to Bishop (1992) and Hanushek and Woessmann (2010) these assessments
provide countries with data that help them understand both the significance of low achievement
and its impact of skills on economic and social outcomes. The proportion of U.S. students
performing at proficient levels is lower than most of the world’s leading industrialized countries
(Fleischman, Hopstock, Pelczar, & Shelley, 2010; Hanushek et al., 2011; Nord, Jenkins, Chan, &
Kastberg, 2013Va; Perterson, Woessmann, Hanushek, & Lastra-Andon, 2011). Students in the
United States are not just underperforming because of the many English learners in United
States’ schools; “only 8% of white students in the U.S. class of 2009 scored at the advanced
level, a percentage that was less than the share of advanced students in 24 other countries
regardless of their ethnic background” (Hanushek, Peterson, & Woessmann, 2011, p.5).
Hanushek et al. (2011) lament that the inability of American schools to bring students up to the
advanced level of achievement in mathematics is much more deep-rooted. Within the 50 states,
student achievement at the advanced level in mathematics varies significantly, but all do poorly
when compared internationally (Hanushek et al., 2011). Therefore it is not a question of some
individual states performing at higher levels being offset by the low achievement of other states,
it is a question of the United States not preparing its students to learn and master the skills to
perform competitively amongst other developed countries.
THE USE OF COGNITIVE TASK ANALYSIS
18
In 2013, Nord et al. reported that 9 percent of 15-year-old students in the United States
scored at proficient level 5 or above in PISA assessment, which was lower than the Organization
for Economic Cooperation and Development (OECD) average of 13%. The percentage of 15-
year-old students that scored below the baseline proficient level 2 was reported at 26 percent,
which was higher than the OECD average of 23 percent (Nord et al., 2013). Hanushek et al.
found that the percentage of students in the U.S. Class of 2009 who were proficient in PISA’s
math assessment was well below that of most countries the U.S. normally compares itself. The
average math scores in 2012 in the U.S. were not significantly different from the average scores
in 2003, 2006 and 2009 (Nord et al., 2013).
With high unemployment in the United States, many are wondering whether our schools
are preparing students effectively for the job market of the 21
st
century (Peterson et al., 2011). As
President Barack Obama said in his 2011 State of the Union address, “We know what it takes to
compete for the jobs and industries of our time. We need to out-innovate, out-educate, and out-
build the rest of the world” (as cited in Peterson et al., 2011). In affirming the president’s view,
Peterson et al. (2011) observe:
The United States could enjoy a remarkable increment in its annual GDP growth per
capita by enhancing the math proficiency of U.S. students. Increasing the percentage of
proficient students to the levels attained in Canada and Korea would increase the annual
U.S. growth rate by 0.9 percentage points and 1.3 percentage points, respectively. Since
long-term average annual growth rates hover around 2 and 3 percentage points, that
increment would lift growth rates by between 30 and 50 percent (p. viii).
THE USE OF COGNITIVE TASK ANALYSIS
19
According to Peterson et al. (2011), when this is translated into dollar terms, these
percentage increases in the annual U.S. growth amount to nothing less than 75 trillion dollars
over a period of 80 years. Therefore those who say mathematics performance does not matter
are clearly wrong and furthermore there is strong evidence that mathematics competence in high
school is a major predictor of potential earnings and economic stability in the future than other
skills acquired in high school (Bishop, 1992; Hanushek et al., 2011).
Job Market Consequences
With the growing economy, the demands of college faculty and employers for graduates
with advanced math skills are increasing (Musen, 2010) and there is concern that future workers
will not have the necessary skills they need to succeed in the 21
st
century economy (Evan et al.
2006). Musen (2010) acknowledges that the United States has made “a significant shift from a
manufacturing- and agriculture- based economy to a knowledge- and service- based economy (p.
3). Consequently demand for highly qualified workers will continue to rise while at the same
time the high unemployment rate will likely continue because those seeking for work are not
qualified.
“Employers in manufacturing, high tech, health care, and other fields are struggling to
find employees with the skills necessary to function well and meet expectations” (Achieve, 2008,
p. 10) and this has long-term implications for the U.S. economy. According to Achieve (2008) a
labor market whose qualifications are not keeping up with the rest of the world impedes the
capacity for the U.S. to compete with other nations. Lack of skills has severe consequences for a
country’s overall growth and productivity (Hanushek et al., 2011; see also Achieve, 2008; Evan
et al., 2006; Musen, 2010). To this end, the United States must invest in its K-12 education
system by providing highly qualified teachers especially for algebra, which is the building block
THE USE OF COGNITIVE TASK ANALYSIS
20
for advanced mathematics, science, engineering, and technology. “Algebra is not simply a means
to an end; it is a gatekeeper” (Evan et al., 2006, p.9).
Algebra as a Foundational Course
This country needs to radically increase the percentage of students leaving high school
with skills that are competitive by increasing the number of students who achieve proficiency in
algebra in their middle school and early high school years (Evan et al., 2006). In fact, Evan et al.
notes, “… successfully passing algebra early in a student’s career – no later than 9
th
grade –
greatly improves the chances of the student graduating from high school, going to college, and
graduating from college” (p. 9). History tells us that algebra was never a regular course offered
in high school. It was not until the Massachusetts’ act of 1827 (the first high school law in
America) algebra was introduced as a compulsory course in high schools of every town in the
state with a population of more than 500 families (Overn, 1937). Overn (1937) noted that as high
schools became widespread in the country in the nineteenth century, algebra became a regular
course. As algebra became a regular course in high school curriculum, “… its position was
greatly strengthened by the fact that one college after another added elementary algebra to its
admission requirements” (Overn, 1937, p.374). Therefore algebra has been linked to college
entrance requirements for a long time and its importance cannot be overemphasized.
Many students are frustrated by algebra and see it as a monster that haunts and follows
them everywhere. The following quotation captures so well how algebra has been regarded for
many years:
If there is a heaven for school subjects, algebra will never go there. It is the one subject in
the curriculum that has kept children from finishing high school, from developing their
special interests and from enjoying much of their home study work. It has caused more
THE USE OF COGNITIVE TASK ANALYSIS
21
family rows, more tears, more heartaches, and more sleepless nights than any other
school subject.
Algebra is required in practically every course except those courses which
are frankly dumb-bell courses. It is a requirement for graduation; it is a requirement for
college entrance … (Anonymous editorial writer in a metropolitan newspaper as cited in
Reeve, 1936, p.2).
According to Musen (2010), students that are successful in algebra have the opportunity
to take more advanced math courses and college preparatory high school curriculum. The
academic strength of a high school student’s curriculum counts more than anything else in
providing impetus toward completing a bachelor’s degree (Adelman, 2006; Musen, 2010).
Moses and Cobb (2001) through their book, Radical Equations: Civil Rights from Mississippi to
the Algebra Project saw algebra not just as a gatekeeper but as a civil rights issue:
So algebra, once solely in place as a gatekeeper for higher math and the priesthood who
gained access to it, now is the gatekeeper for citizenship; and people who don’t have it
are like the people who couldn’t read and write in the industrial age. … [Algebra has
become] … a barrier to citizenship (p.14).
Moses and Cobb (2001) are reaffirming the belief that all students should learn algebra; making
math literacy and economic access a civil rights issue of our time. Schools have to commit to
every student to gain math literacy instead of weeding all but the brightest students out of
advanced math (Moses & Cobb, 2001).
Algebra matters (Rose & Betts, 2004). In other words, Rose and Betts (2004) believe that
“a curriculum that includes algebra is systematically related to higher earnings for graduates a
decade after graduation” (p.510). Several studies (Adelman, 2006; Evan et al., 2006; Gamoran &
THE USE OF COGNITIVE TASK ANALYSIS
22
Hannigan, 2000; Moses & Cobb, 2001; Musen, 2010; Smith, 1996) have all concluded that
algebra is the gateway to success in career and college. Vaiyavutjamai, Ellerton, and Clements
(2005) established that students taking algebra find solving and understanding quadratic
equations very challenging yet quadratic equations are a major component of building mastery in
algebra.
Quadratic Equations
Researchers that have studied the teaching and learning of algebra have established that
in order to have a rich understanding of the function concept that is the basis of the quadratic
equations one must know how to represent functions in different ways (Vaiyavutjamai, 2009). In
fact, Vaiyavutjamai (2009) continues, “… teachers often emphasize procedural skills more than
the links between representations” (p. 1) which does not build the conceptual understanding that
students need to solve quadratic equations. Chaysuwan’s (1996) study of 661 grade 9 students in
Bangkok reported that 70 percent of students’ responses to standard quadratic equations tasks
were incorrect immediately after participating in lessons on quadratic equations (as cited in
Vaiyavutjamai et al., 2005). In a study spanning three countries, Vaiyavutjamai et al. (2005)
established that more than 50 percent of students in Thailand and Brunei that were involved in
the study were confused with respect to the concept of a variable as it manifested itself in
quadratic equations. The United States, which was the third country in the study, 41 percent of
second year university students in the study were confused with the concept of two solutions and
that symbol means “the positive square root of.” Students find it difficult to understand and
solve quadratic equations (Eraslan, 2005; Vaiyavutjamai, 2009; Vaiyavutjamai et al., 2005).
Furthermore Zaslavsky’s (1997) study of 800 10
th
and 11
th
grade students in Israel from
eight high schools found that students could not differentiate between a quadratic function and a
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23
quadratic equation after they had completed the study of functions. Students treated a quadratic
function as though it was a quadratic equation. Zaslavsky (1997) concluded that the relation
between quadratic functions and quadratic equations seemed to hamper students’ understanding
of both quadratic functions and equations. In another study Vaiyavutjamai and Clements (2006)
revealed that many students in their study that got the correct solutions had grave misconceptions
about what quadratic equations were. “There answers were correct but, from a mathematical
point of view, they did not know what they were talking about” (p. 73). Therefore, with quadratic
equations remaining as important as they are, research is needed to inform teachers about how
students think and what needs to be done to assist teachers to improve their students’ concepts of
a variable in the context of quadratic equations (Vaiyavutjamai & Clements, 2006;
Vaiyavutjamai et al., 2005). Students must possess the knowledge of solving quadratic equations
to gain fluency in algebra. This requires direct and systematic instruction on the recognition of
and interaction with variables.
Teachers' Knowledge in Teaching Mathematics
While many factors contribute to a student’s academic success, access to teachers’ with
the knowledge to teach mathematics contributes to gains in students’ mathematics achievement
(Boston, 2012; Hill, Rowan, & Ball, 2005). According to Hill et al. (2005) teachers’ knowledge
for teaching mathematics positively predicted student gains in mathematics. In this study, what
was being measured was the relationship between the knowledge teachers were assumed to have
for teaching mathematics, not just computational aptitude or courses taken. Knowledge for
teaching mathematics goes beyond the courses math teachers take or basic mathematical skills
(Hill et al., 2005; see also Boston, 2012; Leinhardt, 1989). Shechtman, Roschelle, Haertel, and
Knudsen (2010) conducted a study in which they collected data in 125 seventh-grade and 56
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24
eighth-grade classrooms where they investigated the relationship between teachers’ mathematics
knowledge for teaching and student achievement among other variables. They found that
teachers’ mathematical knowledge for teaching was associated with student achievement in
mathematics.
Also teachers’ knowledge of mathematics demonstrated strong links with the
mathematical quality of instruction in their classroom (Hill, Blunk, Charalambous, Lewis,
Phelps, Sleep, & Ball, 2008; Shechtman et al., 2010) and in turn showed a positive association to
student achievement (Shechtman et al., 2010). Hill et al. (2008) further shows that there is a
significant relationship between teachers’ knowledge of mathematics, how they know it, and
what they do in the classroom in the context of instruction. “Effective teaching is teaching that
produces high levels of student performance skill” (Leinhardt, 1989, p. 52). Overall, these
studies show that teachers play an important role in student achievement and more so when the
teachers are knowledgeable in their content and the art of teaching. In fact, according to Darling
–Hammond (2007), teacher expertise is one of the primary factors and single most important
predictor influencing student achievement gains and therefore teachers who lack preparation in
either subject matter or teaching methods are significantly less effective in producing student
learning gains than those who are fully prepared and certified. The weight of substantial
evidence indicates that teachers who have had more preparation for teaching are more confident
and successful with students than those who have had little or none (Darling-Hammond, 2000;
Darling-Hammond, 2007). All teachers must be provided with stronger understanding of how
students learn and develop, how a variety of curricular and instructional strategies can address
their needs, and how changes in school and classroom practices can support student growth and
achievement (Darling-Hammond, 2007). Darling-Hammond (2004) observed that states and
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25
school districts that have focused on broader notions of accountability include investments in
teacher knowledge and skill, organization of schools to support teacher and student learning, and
systems of assessment that drive curriculum reform and teaching improvements.
Expertise in Mathematics
Borko and Livingston’s (1989) study found that “Expert teachers have larger, better-
integrated stores of facts, principles, and experiences upon which to draw as they engage in
planning, reflection, and other forms of pedagogical reasoning” (p. 475) while novice math
teachers seem not to have those same skills which are major components of learning to teach.
Like experts in other fields, expert teachers have well-formed schemas that provide an outline for
the reasonable analysis of information (Westerman, 1991; see also Borko & Livingston, 1989;
Livingston & Borko, 1989). Mathematics teacher education is mostly concerned with the content
knowledge required to teach mathematics (Liljedahl et al., 2009). Liljedahl et al. made this
observation:
Teacher education is a unique enterprise. The reason for this is that the what is also the
how. That is, what we teach is also how we teach. As such, pre-service teachers have a
unique experience. What they are learning is also how they are learning. Through their
experiences as student teachers they are both student and teacher, and through the
constant shifting between student and teacher they are given the opportunity to not only
acquire the knowledge that they will require to become effective teachers, but also are
given the opportunity to recast their initial (pre-conceived) beliefs about what it means to
be a teacher, what it means to teach, what it means to learn, and even what it means for
something to be mathematics. (p. 29)
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26
Ultimately, what is at stake here is the teacher they eventually become, what they will
teach and how they will teach mathematics especially quadratic equations to build long lasting
conceptual understanding of their students. Darling-Hammond (1999) observes that the effects of
well-prepared teachers on student achievement are stronger than the influences of poverty,
language barriers, and minority status. The U.S. school system must provide high quality
teaching and learning to all students. Schools that provide professional development to build the
capacity of its teachers are better equipped to meet the needs of diverse learners (Darling-
Hammond, 2004).
Professional Development to Improve Teacher Learning and Student Achievement
Professional development that focuses on concrete tasks of teaching is vital to enhancing
teacher expertise and improving instructional practices that ultimately support an increase in
student learning outcomes. Darling-Hammond and Richardson (2009) note that the most useful
professional development emphasizes active teaching, assessment, observation, and reflection
rather than abstract discussions. Effective professional development is intensive, ongoing, and
connected to the teaching practice; focuses on the teaching and learning of specific academic
content; is connected to other school initiatives; and builds strong working relationships among
teachers (Darling-Hammond, Wei, Andree, Richardson & Orphanos, 2009). While teachers
typically need substantial professional development in a given area (close to 50 hours) to
improve their skills and their students’ learning, most professional development opportunities in
the U.S. are much shorter (Darling-Hammond, et al., 2009). Penuel, Fishman, Yamaguchi and
Gallagher (2007) emphasize that professional development that is of longer duration and time
span is more likely to contain the kinds of learning opportunities necessary for teachers to
integrate new knowledge into practice. Garet, Porter, Desimone, Birman, and Yoon (2001) in
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27
their study: What makes professional development effective? Results from a national sample of
teachers, noted that professional development may help contribute to a shared professional
culture, in which teachers in a school or teachers who teach the same subject develop a common
understanding of instructional goals, methods, problems, and solutions. Their study also
indicated that professional development that focused on academic subject matter (content), gave
teachers opportunities for practical work (active learning), and was incorporated into the daily
life of the school (coherence), was more likely to produce enhanced knowledge and skills. In
another study of mathematics teaching in California based on data on teachers’ professional
development experiences and school-level data on student achievement on a mathematics test
administered statewide, Cohen and Hill (2000) found that controlling for the characteristics of
students enrolled, average mathematics achievement was higher in schools in which teachers had
participated in extensive professional development focusing on teaching specific mathematics
content, compared to the achievement in schools where teachers had not. Their study also found
that participation in professional development focusing on general pedagogy was not related to
student achievement. Darling-Hammond et al. (2009) adds that improving professional
development and collaborative learning opportunities for educators is a crucial step in
transforming schools and improving academic achievement for all students. Therefore, if we
capture expertise in teaching solving quadratic equations, then we can offer targeted professional
development that would assist in improving instruction.
Cognitive Task Analysis
Cognitive task analysis (CTA) refers to a set of methods for capturing expertise. CTA is a
methodology that has been used to capture the cognitive processes, decision-making, and
judgments that underlie expert behaviors (Yates, 2007). Although there are no published studies
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28
that offer evidence on how many mathematics experts must be interviewed in order to capture
enough critical information to aid in teaching quadratic equations, Chao and Salvendy (1994)
concluded three experts were needed to acquire the optimal vital knowledge and skills needed to
solve a complex software-debugging task. This study seeks to use CTA to capture math experts’
knowledge and skills for solving quadratic equations.
Experts are frequently called upon for their knowledge and skills to teach, to inform
curriculum content and instructional materials, and to mentor and coach others to perform
complex tasks and solve difficult problems. The purpose of education is to replicate knowledge.
According to Jackson (1985), traditions in education started with the expert and novice model in
which the objective was for the novice to imitate the expert. The expert knows what precedes
what in the range of steps and she devotedly follows such an order when deciding what her
student is to learn at any one time (Jackson, 1985). However, current research shows that experts
may omit up to 70 percent of the critical knowledge and skills novices need to replicate expert
performance (Feldon & Clark, 2006). In their study, Feldon and Clark found that self-reports by
experts were incomplete and inaccurate. In fact, as Jackson (1985) inquired, if teachers do not
cover it all, then what do they leave out? According to several studies (Clark, Feldon, van
Merriënboer, Yates, & Early, 2008; Feldon & Clark, 2006; see also Bedard & Chi 1992; Feldon,
2007), experts often omit critical knowledge and skills during self-report because they have
automated their knowledge and skills through repeated practice so that it becomes unconscious
and unavailable for recall.
Clark et al. (2008) noted that declarative knowledge is recalled from long-term memory
and is consciously available in working memory. Declarative knowledge alone is not sufficient
for performance. Procedural knowledge is required for skill performance and, as skills are
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29
continuously practiced, automaticity is attained (Clark et al., 2008). Since automated knowledge
is outside the consciousness of the expert, CTA has been shown to be an effective method for
capturing both the conscious and automated knowledge experts uses to perform complex skills
and solve difficult problems (Feldon & Clark, 2006). To further understand why CTA is
effective, the following sections examine the types of knowledge, the nature of automaticity, and
the characteristics of expertise.
Knowledge Types
The procedure of teaching, learning, and assessing skills requires the transmission of
knowledge from the expert (teacher) to the novice (student). The key to a learning experience
that is productive is the achievement of the vital components of knowledge: declarative (why and
what it is), procedural (when and how to do it) and the conditions under which to perform a
procedural task.
Declarative Knowledge
Declarative is knowledge that is controlled and can be changed abruptly in the working
memory (Clark & Estes, 1996). Declarative knowledge is explicit knowledge about facts, of why
or that something is. It is overt and comprised of information about “why, what and that.”
According to Anderson (1982) the declarative stage of acquiring automated procedural
knowledge is where the domain knowledge is directly embodied in procedures for performing
the skill although by itself declarative knowledge is insufficient to execute skilled performance.
When teaching new concepts and procedures to students such as teaching how to solve quadratic
equations, it is necessary to focus on the required declarative knowledge so they are able to
understand and comprehend the steps needed to solve the problem. Declarative knowledge,
knowing why and what something is, clears the path for and supports the attainment of the how
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30
and when something is to be performed as procedural knowledge (Anderson, 1982; Anderson &
Fincham, 1994; Anderson & Schunn, 2000; Clark & Estes, 1996).
Procedural and Conditional Knowledge
Declarative, procedural, and conditional knowledge are required for completing complex
tasks and are acquired as one transitions from novice to expert. Procedural knowledge is the
knowledge about “when and how” to perform a task (Ambrose, Bridges, DiPietro, Lovett, &
Norman, 2010; Anderson, 1982; Anderson & Krathwohl, 2001). According to Clark and Estes
(1996) procedural knowledge is difficult to learn and fast to execute. It requires practice and
feedback but once it becomes a high level of expertise or automated, it is very difficult to change
(Anderson, 1993; Clark & Estes, 1996). Procedural knowledge accounts for 70 to 90 percent of
the total knowledge adults have (Clark & Elen, 2006). Paris, Lipson and Wilson (1983) noted
that conditional knowledge is a form of procedural knowledge. Conditional knowledge is
knowing when; provides the circumstances or rational for various actions including value
judgments, and helps modulate procedural and declarative knowledge (Paris et al., 1983). With
repetition and practice, both declarative and procedural knowledge become stronger and
performance becomes more fluid, consistent, and automated.
Automaticity
Through repeated performance and deliberate practice of a task, declarative and
procedural knowledge becomes automated and unconscious in nature and the speed in
performing the task increases while the amount of active mental effort decreases (Feldon, 2007).
Clark and Elen (2006) emphasize that the automation process is advantageous to expertise as it
supports the capacity to respond to novel problems with speed, accuracy, and consistency within
an expert’s domain. Automated processes often initiate without prompting and once SMEs
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31
initiate, automated processes run to completion without being available for conscious monitoring
(Clark, 1999; Feldon, 2007). Automaticity frees up the working memory by unconsciously
processing and running procedures (Wheatley & Wegner, 2001) and with repeated practice,
cognitive tasks becomes fluid and automatic and SMEs are able to deploy strategies to solve
problems with ease (Clark, 1999). Automaticity enables SMEs to perform tasks requiring
declarative and procedural knowledge unconsciously freeing up working memory to address
novelty, however due to the unconscious nature of automaticity it is resistant to change (Clark,
2008c; Wheatley & Wegner, 2001) and difficult to modify, eliminate, or express to others using
concrete language and examples (Clark & Elen, 2006; Clark & Estes, 1996; Kirschner, Sweller,
& Clark, 2006).
Expertise
Characteristics of Experts
The characteristics of expertise include extensive and highly structured knowledge of the
domain, effective strategies for solving problems within the domain, and expanded working
memory that utilizes elaborated schemas to organize information effectively for rapid storage,
retrieval, and manipulation (Bedard & Chi, 1992; Ericsson & Lehmann, 1996; Feldon, 2007). In
fact, Chi (2006) defines an expert as a distinguished or brilliant journeyman, highly regarded by
peers, whose judgments are uncommonly accurate and reliable, whose performance shows
consummate skill and economy of effort, and who can deal with certain types of rare or tough
cases. Also, an expert is one who has special skills or knowledge derived from extensive
experience sub-domains. According to Bedard and Chi (2006), what sets the expert apart from
novice is that experts have developed schemas allowing them to efficiently organize information
so it is quickly and efficiently retrieved with minimal effort. An expert can see beyond function
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32
and simple schemas, they create mental models while novices are more literal and predictable.
According to Ericsson and Lehmann (1996) the ability of experts to exceed usual capacity
limitations is important because it demonstrates how acquired skills can supplant critical limits
within a specific type of activity. Extensive evidence indicates that experts are able to attend to
and process much more domain-relevant information in working memory that is possible for
novices (Ericsson & Lehmann, 1996; Feldon, 2007; see also Glaser & Chi, 1988). Therefore, as
will be described later in this review, by using CTA, the expertise of subject matter experts can
be captured and taught to novice learners to begin to build their own expertise.
Building Expertise
Expertise, by its nature, is acquired as a result of continuous and deliberate practice in
solving problems in a domain. Expert performance continues to improve as a function of more
experience, coupled with purposeful practice (Alexander, 2003; Ericsson, 2004; Ericsson &
Charness, 1994; see also Ericsson, Krampe, & Tesch-Römer, 1993); the challenge for aspiring
expert performers is to avoid the arrested development associated with automaticity and to
acquire cognitive skills to support their continued learning and improvement (Ericsson, 2004).
According to Ericsson (see also Ericsson & Charness, 1994), deliberate practice is therefore
designed to improve aspects of performance in a manner that assures that attained changes can
be successfully integrated into representative performance. However, Ericsson et al., (1993)
warn that although experts outperform novices, research has shown that expertise does not
transfer to domains unfamiliar to the expert. Thus, the domain-specific nature of expert’s
superior performance implies that acquired knowledge and skill are important to attainment of
expert performance. Once we conceive of expert performance as mediated by complex integrated
systems of representations for the execution, monitoring, planning, and analyses of performance,
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33
it becomes clear that its acquisition requires an orderly and deliberate approach (Ericsson, 2004).
Therefore, by engaging in purposeful practice and problem solving, a novice learner develops
over time (usually 10 years) more efficient schema, knowledge, skills and decision steps.
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,
negatively impacting instructional efficacy and leading to subsequent difficulties for learners
(Chi, 2006; Feldon, 2007). Feldon (2007) observes that automated procedures are deeply rooted
and not easy to change and therefore automaticity impairs the development of expertise. Experts
regularly cannot articulate their knowledge
because much of their knowledge is implied and their
overt intuitions can be flawed (Chi, 2006). Evidence (Feldon, 2007) suggests that routine
approaches to problems are goal-activated and significantly limits the solution search. This is
also made worse because experts tend overestimate their capabilities by being overly confident
(Chi, 2006) and therefore fail to articulate relevant cues seen in problem states (Feldon, 2007).
Feldon (2007) observed in his study that, extensive practice using procedures to solve problems
in a specific domain led experts to automate portions of their skills. Consequently, the most
frequently employed elements – presumably those of greatest utility within a domain of experts –
were the most difficult to articulate through recall. Therefore, the automaticity of experts impairs
their ability to consciously identify many of the decisions they make thereby omitting key details
and process information necessary to provide instruction on optimal performance.
Expert Omissions
Experts in an instructional role may unintentionally leave out information that students
must master when learning procedural skills. Recent research (Clark, Pugh, Yates, Inaba, Green,
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34
and Sullivan, 2011) has shown that when experts describe how they perform a difficult task, they
may unintentionally omit up to 70 percent of the critical information novices need to learn to
successfully perform the task. According to Clark (2008) this is a serious problem because it
forces novices to “fill in the blanks” using less efficient and error-prone trial-and-error methods.
Furthermore, as these errors are practiced over time, they become more difficult to “unlearn” and
to correct. There are two reasons for this problem (Clark & Estes, 1996). First, as SMEs gain
expertise, their skills become automated and the steps of the procedure blend together. Experts
perform tasks largely without conscious knowledge as a result of years of practice and
experience. This causes experts to omit critical steps when describing a procedure because this
information is no longer accessible to conscious processes (Clark & Elen, 2006). Secondly, many
SMEs are not able to share the complex thought processes of behavioral execution of skills. Even
experts who make an attempt to “think out aloud” during the process of complex problems often
omit essential information because their knowledge is automated (Clark & Elen, 2006; Clark &
Estes, 1996). Consequently, it is difficult to identify points during a procedure where an expert
makes decisions (Clark & Elen, 2006). Clark et al. (2011) further reported that when experts are
asked to describe a procedure, they rely on self-recall of specific skills but studies from the field
of cognitive psychology suggest that the use of standard self-report or interview protocols to
extract descriptions of events, decision making, and problem solving strategies can lead to
inaccurate or incomplete reports. In fact, experts often do not recognize these errors because of
the automated and unconscious nature of the knowledge described (Wheatley & Wegner, 2001).
Experts’ self-reports about their approaches to complex tasks have revealed that they leave out
up to 70 percent of critical information (Feldon & Clark, 2006).
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35
Cognitive Task Analysis (CTA)
Definition of CTA
Cognitive task analysis (CTA) is a group of “methods used for studying and describing
reasoning and knowledge” (Crandall, Klein, & Hoffmann, 2006, p. 9). CTA has evolved from
traditional task analysis methods, and is utilized in order to elicit and clarify expert knowledge
within a specific domain. According to Clark et al. (2008) CTA uses a variety of interview and
observation strategies to capture a description of the explicit and implicit knowledge that experts
use to perform complex tasks. Crandall et al. (2006) adds that CTA is a type of knowledge
elicitation, analysis of data, and representation of knowledge tool that captures expert knowledge
on the way the mind works. CTA is an extension of traditional task analysis that identifies the
knowledge, thought processes and goal structures that underlie observable task performance, as
well as overt and covert cognitive functions that form the integrated whole (Chipman, 2000;
Clark et al., 2008). Therefore, CTA yields information about the knowledge, thought processes,
and goal structures that underlie observable task performance; the explicit and implicit
knowledge that is explicated from the CTA can be used to teach, train, assess performance and
develop expert systems.
Brief History of CTA
The seeds of CTA were planted as far back as the 1800s and can be found throughout the
history of applied psychology, industrial engineering and human factors (Clark & Estes, 1996;
Militello & Hoffmann, 2008). And then as recently as the 1980s, CTA methods emerged as a
response to capturing cognitive processes as a result of workplace demands and have become
refined over the last 20 years due to the demand of modern technology (Militello & Hoffmann,
2008). CTA has been long in evolution and is related to many fields and is now one of the most
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36
successful methods of elicitation of expert knowledge that can be used today. Hoffmann and
Woods (2000) noted that CTA evolved from traditional task analysis and the study of cognitive
engineering, in order to aid in human performance, and improve skill in the use of new
technology. According to Annett (2000) the foundations for cognitive psychology took root in
the 1950s, and evolved into a definition of CTA in the 1970s with an emphasis on capturing
human expertise that was never captured with older methods of task analysis. Therefore, CTA is
the advanced task analysis system that fills in that gap.
Cognitive Task Analysis Methodology
A number of researchers have identified the stages through which a typical, ideal
cognitive task analysis would proceed. In Chipman, Schraagen and Shalin (2000) view, the ideal
model of cognitive task analysis is one that is not subject to resource restrictions, is typified by a
series of discrete steps. According to Chipman et al. (2000) and Clark, Feldon, van Merriënboer,
Yates, and Early (2008) these discrete stages are: (a) collect preliminary knowledge, (b) identify
knowledge representations, (c) apply knowledge elicitation techniques, (d) verify/analyze data
elicited, and (e) format results of the analysis as a basis for an expert system or expert cognitive
model. Although there are over 100 varieties of cognitive task analysis (Yates, 2007), in a
general sense, most varieties follow a five-stage process. Multiple authors have developed
taxonomies that categorize these techniques according to a number of criteria.
Taxonomies of Knowledge Elicitation Techniques
Knowledge elicitation is the process of extracting domain specific knowledge that
underlies human performance (Cooke, 1999). There are four categories of knowledge elicitation:
(a) observations, (b) interviews, (c) process tracing, and (d) conceptual methods (Cooke, 1999).
According to Cooke (1994, 1999), knowledge elicitation begins with observing task performance
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37
within the domain of interest and provides a general conceptualization of the domain and any
constraints or issues to be addressed in the later phases. While interviews are the most frequently
used elicitation method, process tracing is the most often used method to elicit procedural
information, such as conditional rules used in decision making (Cooke, 1999). Cooke (1994,
1999) added that conceptual methods elicit and represent conceptual structure in the form of
domain-related concepts and their interrelations. This method is mainly used to elicit knowledge
to improve user interface design, guide development of training programs, and understand
expert-novice differences. Wei and Salvendy (2004) identified formal models as a fourth family
of knowledge elicitation technique. Since these techniques are based on processes, such
taxonomies/typologies may make it difficult for analysts to choose an appropriate CTA
approach, especially when the desired result is a particular type of knowledge. In order to elicit
accurate and complete expert knowledge descriptions, Cooke (1994, 1999) proposed using
multiple knowledge elicitation techniques to capture rich representations of the task.
Pairing Knowledge Elicitation with Knowledge Analysis
Since the current classification schemes organize CTA methods by process rather than
the desired outcome or application, practitioners find it difficult to select an optimal method for
their specific purpose (or knowledge outcome). Therefore, Yates (2007) identified the most
frequently used CTA methods and the knowledge types associated with the respective methods
and outcomes (product approach versus the existing process approach). Although data analysis
and knowledge representation are considered as two distinct techniques of CTA, they are often
linked with elicitation methods (Yates, 2007). Additionally, since both share common
characteristics, data analysis and knowledge representation are often combined into a single
category in a classification scheme. Crandall, Klein, and Hoffman (2006) noted that many
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38
knowledge elicitation methods have analytical processes and representational formats embedded
within the method. Therefore, it may be more appropriate to examine CTA as a pairing of
knowledge elicitation with an analysis/representation technique (Yates, 2007). The results of
Yate’s (2007) study revealed that the most frequently used CTA method pairings included
standardized methods and informal methods and the application of these methods were
associated more with declarative knowledge than procedural knowledge. Also, this study found
that standardized methods (protocol analysis and conceptual methods) appeared to provide
greater consistency in the results than informal models (observations and interviews). CTA relies
on the use of both elicitation and analysis/representation methods. For efficiency and optimal use
of CTA, CTA methods need to be classified in terms of desired outcome rather than process.
Effectiveness of CTA
Cognitive task analysis has proven to be an effective method for capturing the explicit
observable behaviors, as well as the tacit, unobservable knowledge of experts. According to
Hoffman and Militello (2009) the use of CTA identifies the explicit and implicit knowledge of
experts to use for training; the knowledge elicited from experts includes domain content,
concepts and principles, experts’ schemas, reasoning and heuristics, mental models and sense
making. Data captured from CTA supports effective and efficient training and instructional
activities in complex systems. Particularly for domains that emphasize technical-functional
capabilities, such as engineers or military personnel, simply listing the action steps for a
particular procedure or task is not an adequate way to train (Means & Gott, 1988). Even if
context is captured, traditional methods such as asking experts to list steps or making
observations, do not accurately account for abstract knowledge in experts. Therefore, use of CTA
is useful for educators to identify the subtle skills, perceptual differences, and procedures that
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39
may be left out during instruction (Crandall et al., 2006). Compared to other strategies, Cognitive
Task Analysis is more effective at capturing the unconscious, complex cognitive action and
decision steps of experts.
Research has shown instruction using Cognitive Task Analysis is more cost effective and
efficient than other instructional models (Clark, Feldon, van Merriënboer, Yates, & Early, 2008).
This research found that in a number of studies reviewed, CTA-based instruction resulted in
higher achievement compared to non CTA-based instruction. Clark et al. (2008) noted the
importance of CTA was based on compelling evidence that experts are not fully aware of about
70% of their own actions, decisions and mental analysis of tasks and so are unable to explain
them fully even when they intend to support professional training of novices. Therefore,
according to Clark et al. (2008) CTA methods attempt to overcome this problem by employing
observational and interview strategies that allow knowledge elicitors to capture more accurate
and complete descriptions of how experts succeed at complex tasks and this in turn reduces the
total training days by nearly a half. Flynn’s (2012) research found that much of the literature
emphasized the degree to which the CTA methodology positively impacted costs and
efficiencies; however, more meaningful, was the degree to which the method elicited expert
knowledge that could be translated into guided instructional or Guided Experimental Learning
(Clark, 2004). The use of CTA in instruction and training has been proven to be positively
related to cost savings due to reduced training times with comparable learning outcomes (Clark,
2011).
Benefits of CTA for Instruction
Studies that have applied Cognitive Task Analysis to capture knowledge and deliver
instruction have uncovered several benefits and useful design strategies as compared to other
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40
forms of instruction. According to Hoffman and Militello (2009) CTA is able to identify the
explicit and implicit knowledge of experts to use for training and technology. The authors
indicate that the knowledge that can be elicited from experts includes domain content, concepts
and principles, experts’ schemas, reasoning and heuristics, mental models and sense making. In
fact these authors observed that data captured from CTA supported effective and efficient
training and instructional activities in complex systems. Others like Crandall et al. (2006) noted
that CTA could be used for training in a variety of ways, such as “cognitive training
requirements, scenario design, cognitive feedback, and on-the-job training” (p. 196). CTA has
proven to be an effective method for eliciting the nuances in expert knowledge, such as decision
points and perspectives, resulting in a variety of instructional strategies utilizing the outcomes of
CTA (Crandall et al., 2006; Hoffman & Militello, 2009; Means & Gott, 1998).
Studies across a variety of domains have explored the degree to which CTA-informed
instruction has influenced learning outcomes. In another review, Clark (2014) noted that CTA
results in nearly 30-45% learning performance increases as compared to instruction that is
informed by traditional observation or task analysis. The author further points out that there is
evidence pertaining to the front-end of training being advantageous for increasing learning and
reducing the number of mistakes made by recently graduated students within the field of
healthcare. Therefore, CTA-informed learners or employees may be considered better trained
and perhaps more appealing to employers throughout the medical field. In Crandall and Getchell-
Reiter’s (1993) study, their findings were favorable in supporting CTA as the most effective
method for capturing expertise. They discovered that the formal analysis helped generate more
instances from experts, which captured the subtle nuances of what was considered highly
subjective material. Other studies (see Schaafstal, Schraagen, & Marcel, 2000; Velmahos, et al.,
THE USE OF COGNITIVE TASK ANALYSIS
41
2004) associated to the task of troubleshooting were conducted in order to gain a sense of what
analyses surfaced the most accuracies and abilities in solving problems. The CTA method proved
to be effective because it gained expertise from both people who were from a theoretical
background as well as practitioners. Overall, CTA has been shown to be effective in capturing
expertise and informing instruction in a wide range of domains, including software development
(Schaafstal et al., 2000), military (Fackler, et al., 2009; Flynn, 2011), business sector (Klein,
2004; Mayer, 2011), and medical fields (Clark, 2014).
Summary
Cognitive Task Analysis is an interview and observation methodology that is used to
capture underlying cognitive procedures experts use to perform and solve complex problems.
When experts are asked to describe how to perform complex tasks, they unconsciously omit up
to 70% of action and decision steps novices need to successfully perform the complex task.
Teaching solving quadratic equations is a complex task which expert knowledge and skills is
required to meet the instructional needs of 8
th
and 9
th
grade students in K-12 who need to build
mastery of Algebra One because Algebra One is a building block for higher level mathematics in
high school. The level of mathematics students in high school attain in their senior year has a
correlation to whether they will earn a bachelor’s degree. As such, the task of solving quadratic
equations in algebra may gain from doing a CTA to inform teaching. Therefore, the purpose of
this study was to perform a CTA to examine the knowledge and skills that expert math teachers
use to teach solving quadratic equations in Algebra and the extent at which expert math teachers
omit the critical conceptual knowledge, action steps, and decision steps when describing
instruction for solving quadratic equations.
THE USE OF COGNITIVE TASK ANALYSIS
42
CHAPTER THREE: METHODOLOGY
The purpose of this study was to investigate the expertise of mathematics teachers that
teach algebra, specifically how to solve quadratic equations, to 8
th
and/or 9
th
grade students in K-
12. The study used Cognitive Task Analysis (Clark, Feldon, van Merriënboer, Yates, & Early,
2008; Cooke, 1999; see also Cooke, 1994; Wei & Salvendy, 2004) methods to capture the
knowledge of expert math teachers when they describe how they teach solving quadratic
equations. Clark et al. (2008) suggested four steps that could be used for conducting a Cognitive
Task Analysis procedure to elicit knowledge. These four steps were used for knowledge
elicitation in this study and they included, (a) collecting preliminary domain-specific knowledge,
(b) identifying the types of knowledge associated with the task, (c) applying the knowledge
elicitation technique in a semi-structured interview, and finally (d) verifying and analyzing the
results from the interviews.
The following were the research questions that guided this study:
1. What are action and decision steps that expert math teachers recall when they
describe how to teach solving quadratic equations in Algebra?
2. What percentage of actions and/or decision steps, when compared to a gold
standard, do expert math teachers omit when they describe how to solve
quadratic equations in Algebra One?
Task
The task for this study was to elicit the action and decision steps Algebra One teachers
can recall when describing how to teach solving quadratic equations to 8
th
and/or 9
th
grade
students. Students taking algebra find understanding and solving quadratic equations very
challenging yet quadratic equations are a major component of building mastery in algebra. To be
THE USE OF COGNITIVE TASK ANALYSIS
43
proficient in solving quadratic equations, students require both declarative (what and why to
perform the task) and procedural (when and how to perform the task) knowledge. Although
solving quadratic equations is common in algebra, it involves complex procedural steps, which
can result in getting an erroneous solution if the steps are inaccurately executed. As described in
Chapter Two, expert math teachers’ knowledge and skills may be automated and unconscious to
the extent that when they teach mathematics they may be providing incomplete or inaccurate
instruction that ultimately diminishes students’ achievement of the task.
Population and Sample
Yates, Sullivan, and Clark (2012) suggest that 3 to 4 experts are needed for CTA to
capture the optimal amount of significant information during a procedure. Therefore, this study’s
sample included four algebra teachers from two Southern California School Districts plus one
senior expert to review the final gold standard protocol. The researcher collaborated with the two
school districts superintendents and other district leaders to identify expert math teachers for
Algebra One. The teachers were selected based upon their expertise in teaching algebra to 8
th
and
9
th
grade students. Feldon (2007) describes the characteristics of expertise to include extensive
and highly structured knowledge of the domain, effective strategies for solving problems within
the domain, and expanded working memory that utilizes elaborated schemas to organize
information effectively for rapid storage, retrieval, and manipulation. According to Clark et al.,
(2008) a subject matter expert (SME) is a person with wide experience and is capable of
performing a range of tasks fast and successfully. For this study, expertise was shown by the
algebra teacher’s years of experience, teaching expertise, and more so performance of their
students in state standardized tests. The sample of expert math teachers was asked to voluntarily
participate in CTA guided semi-structured interviews to capture their subject matter expertise in
THE USE OF COGNITIVE TASK ANALYSIS
44
teaching solving quadratic equations for the purpose of aggregating a gold standard protocol.
Data Collection and Instrumentation
Using Clark et al. (2008) guidelines, CTA was conducted using the five common steps
that entailed: (a) collecting preliminary knowledge, (b) identifying knowledge representations,
(c) applying focused knowledge elicitation methods, (d) analyzing and verifying acquired data,
and (e) formatting results for the intended application. The steps are described below, as they
were used during each step of the CTA process.
Step 1: Collecting preliminary knowledge
In this preliminary stage, the researcher is a high school mathematics teacher and is
familiar with the task of solving quadratic equations. As part of this research, the researcher
identified the task sequences and procedures that became the focus of the CTA. While the
researcher did not need to become an SME, the researcher was familiar with the procedures and
steps of solving quadratic equations. This stage also included document analysis of books and
other resources describing this task. This orientation prepared the researcher for subsequent task
analysis activities. The information elicited during semi-structured interviews may be more
robust when analysts are already familiar with experts’ language.
Step 2: Identifying Knowledge Representations
The information collected in the first step was examined to identify and generate a
preliminary list of subtasks and types of knowledge required to solve quadratic equations. This
involved using flowcharts in order to provide a systematic way of organizing the information that
was elicited from the SMEs.
Stage 3: Applying Focused Knowledge Elicitation Methods
At this stage, semi-structured interviews following the protocol attached as Appendix A
THE USE OF COGNITIVE TASK ANALYSIS
45
were contacted to elicit information from the four SMEs. During the semi-structured CTA
interview, the SMEs were asked a series of questions that focused on the major tasks and
potential difficulties a student may encounter when solving quadratic equations. The action and
decision steps are considered the critical information novices need to perform the task. Action
steps begin with a verb and are statements about what a person should do, such as “When driving
out of a garage, close the garage door.” Decision steps contain two or more alternatives to
consider before taking an action, such as “When driving, IF you are backing up, THEN look
back and the rear view mirrors; IF it is not clear and safe, THEN wait; IF it is clear and safe,
THEN proceed to back up.”
These experts were asked to describe the action and decision steps they used to solve
quadratic equations. The preliminary semi-structured interview began with a clear description of
the CTA process by the researcher. The SMEs were asked to list or outline the performance
sequence of all key subtasks necessary to successfully solve quadratic equations. SMEs were
also asked to describe at least five problems that an expert should be able to solve if they have
mastered how to solve quadratic equations. These problems ranged from routine to highly
complex ones. The initial Round One interviews lasted approximately three hours, followed by
Round Two interviews that allowed the SMEs to review the individual draft protocol to add or
delete any unnecessary steps for this task. The Round Two interviews lasted approximately two
hours.
Step 4: Analyzing and Verifying Data Acquired
The information generated from the SMEs through the semi-structured interviews was
used to create a protocol for solving quadratic equations. However, before this protocol was
prepared, the researcher coded the data using domain specific codes and formatted the results for
THE USE OF COGNITIVE TASK ANALYSIS
46
verification, refinement, and revision by participating SMEs to ensure that the representations of
procedures were complete and accurate.
CTA coding scheme. After the four interviews were transcribed, a CTA coding scheme
was generated (Appendix B) that allowed the researcher to analyze the data from the semi-
structured interviews. The coding categories used were: objective, conditions/cues, main
procedures, action steps, decision steps, standards, equipment, reasons, new concepts and others
that were determined as the coding was in progress.
Inter-rater reliability (IRR). The researcher and another trained coder for inter-rater
reliability coded one of the coded transcripts. After the coding was completed, it was analyzed
and an inter-rater reliability was calculated as a percentage of agreement between the two coders.
Once there is an 85% or higher agreement in inter-rater reliability, then the coding process is
considered as reliable among the different coders (Hoffman, Crandall, & Shadbolt, 1998).
Crandall, et al., (2006) suggested that if the inter-rater reliability is less than 85%, then the
coding scheme may need to be further revised and refined.
Step 5: Formatting Results for the Intended Application
Data Analysis
Research Question 1. What are action and decision steps that expert math teachers recall
when they describe how to teach solving quadratic equations in Algebra?
Gold standard protocol (GSP). Each of the individual reviewed and corrected CTA
protocols was aggregated to develop one draft protocol. This protocol served as a preliminary
gold standard protocol (PGSP) and was given to another senior expert who was not part of the
initial CTA interviews to review and edit for completeness and accuracy to produce a corrected
and final GSP (Appendix D) that was used to generate the action and decision steps expert
THE USE OF COGNITIVE TASK ANALYSIS
47
mathematics teachers make to teach solving quadratic equations in Algebra One. See Appendix
C for a description of a complete procedure for creating a GSP. Figure 3.1 visually represents the
five-stage process.
Figure 3.1: Process of conducting CTA semi-structured interviews and aggregating the GSP
THE USE OF COGNITIVE TASK ANALYSIS
48
Research Question 2. What percentage of action and/or decision steps, when compared
to a gold standard, do expert math teachers omit when they describe how to solve quadratic
equations in Algebra One?
Microsoft Excel spreadsheet analysis. The gold standard protocol that was generated
from the four individual CTA protocols was transferred to a Microsoft Excel spreadsheet. This
GSP included the action and decision steps that were recorded for teaching solving quadratic
equations. Each of these steps that were included in the individual SME, a “1” was put in the
column under that SME, otherwise if the step was missing, then a “0” was put in that column
under the SME. Microsoft Excel spreadsheet formulas were used to add the number of “1’s” for
each SME and the number of “0’s,” which indicated the number of omissions recorded for each
SME. The analysis of this data allowed the researcher to calculate the percentage of omissions of
actions and decision steps compared to the gold standard protocol. The mean percentage of
omissions was then calculated and recorded to respond to Research Question 2.
THE USE OF COGNITIVE TASK ANALYSIS
49
CHAPTER FOUR: RESULTS
Overview of Results
This study examined the action and decision steps of five expert mathematics teachers
using a CTA knowledge elicitation procedure to capture their expertise in teaching solving
quadratic equations to 8
th
and 9
th
grade students in K-12. The data will be presented using
frequency counts, percentages and graphs in order to answer the two research questions.
Research Questions
Question 1
What are action and decision steps that expert math teachers recall when they describe
how to teach solving quadratic equations in Algebra One?
Inter-rater reliability. The researcher and one other knowledge analyst coded data from
one of the four interview transcripts for the purpose of knowledge type coding of action and
decision steps captured from the SME identified as SME C. After the researcher and the other
knowledge analyst coded SME C’s transcript independently, they got together and discussed
their respective coding and attempted to reconcile any differences. Tallying all the coded items
that were in agreement between the two coders and dividing this number with the total number of
coded items determined inter-rater reliability. The inter-rater reliability showed a consistent
interpretation of knowledge items for this SME with a 97.5% agreement across 16 distinct codes
as shown in Appendix B. This researcher coded the remaining three SME interview transcripts
before developing the initial individual protocol for solving quadratic equations for each SME.
Flowchart analysis. A flowchart was created from the interview data of SME C that is
shown as Appendix D. After the flowchart was created, the researcher analyzed it to ensure that
action and decision steps captured from the first interview were logical and could be executed
THE USE OF COGNITIVE TASK ANALYSIS
50
when teaching solving quadratic equations. Also the flowchart showed instances where some
decisions were being made without appropriate actions. These observations led the researcher to
ask further questions to the SMEs during Round Two of the knowledge elicitation interviews that
followed. On the agreed day for the review of the initial draft protocol and Round Two
interviews, the researcher emailed the draft protocol to the SMEs to download into their
computers. Each draft protocol steps where numbered and the transcript line number where that
information was extracted from were indicated at the end of each sentence for ease during the
review process. The SMEs were asked to use the Microsoft Word Track Changes feature on their
computer to track any changes made on the draft protocol. Each SME was asked to read their
draft protocol entirely before starting to make any changes to allow them to understand the
document and have a general view of what was captured from their transcript. The researcher
had at their disposal the original transcript that was used to generate the draft protocol. The
researcher asked the SME to add any new steps that may make the execution of the procedures
complete and/or delete any steps that may have been misleading and to give reasons.
Gold standard protocol. Once all the SMEs had reviewed their respective draft
protocols, the researcher generated a preliminary gold standard protocol (PGSP). The researcher
aggregated the data from the protocols generated from the four SMEs to solve quadratic
equations. An example of the process of aggregating the gold standard protocol (GSP) is shown
in Table 4.1.
THE USE OF COGNITIVE TASK ANALYSIS
51
Table 4.1
Example of aggregating action and decision steps for the preliminary gold standard protocol
(PGSP)
SME C – Action step:
• Sing to students a song on the quadratic formula: “x-equals negative b, plus or minus
square root of b squared minus 4ac, all over 2a” (P1)
SME D – Additions (in bold) to SME C’s Action step:
• Teach students a way to memorize the quadratic formula: Sing, “x-equals negative b,
plus or minus square root of b squared minus 4ac, all over 2a” Or say: “A negative boy
could not decide whether or not to go to a radical party. He decided to be square
and he missed out on 4 awesome chicks. The party was all over at 2 am.” (P1, P2)
SME A – Additions to SMEs C and D Action step
SME A – said: “Play the quadratic formula song video (You Tube) so students can memorize the
quadratic formula." Therefore the action step as aggregated from SMEs C and D remained as is.
• Teach students a way to memorize the quadratic formula: Sing: “x-equals negative b,
plus or minus square root of b squared minus 4ac, all over 2a” Or say: “A negative boy
could not decide whether or not to go to a radical party. He decided to be square
and he missed out on 4 awesome chicks. The party was all over at 2 am.” (P1, P2,
P3)
SME B – Additions (underlined) to PGSP (as Action step reads in final GSP)
SME B – said: “Play the quadratic formula song and make students sing along.
GSP Step: 3.1: Teach students a way to memorize the quadratic formula: GSP Step:
3.1.1: Sing: “x-equals negative b, plus or minus square root of b squared minus 4ac, all
over 2a.” Make students sing along (P4) or teach students to memorize the quadratic
formula using the phrase: “A negative boy could not decide whether or not to go to a
radical party. He decided to be square and he missed out on 4 awesome chicks. The
party was all over at 2 am.” (P1, P2, P3, P4)
Note: P1 – represents SME C’s contribution to the GSP, P2 – represents SME D’s contribution to the GSP, P3 –
represents SME A’s contribution to the GSP, and P4 – represents SME B’s contribution to the GSP.
Thereafter, the researcher set an appointment with a fifth senior SME to review the initial
gold standard protocol for accuracy, consistency, and completeness. The initial preliminary gold
standard protocol was not send to this SME prior to the meeting because the researcher wanted
the SME to read the protocol in his presence. The SME was asked to download the document
into her computer and to turn on the Microsoft Word Track Changes feature and then review the
protocol in its entirety. Any additions, deletions, modifications and re-arrangements of the main
THE USE OF COGNITIVE TASK ANALYSIS
52
procedures, action and decision steps from the initial gold standard were captured. After the
senior SME’s review, a final gold standard protocol was created, which is the response to
Research Question 1 and is attached as Appendix E. The gold standard protocol represents the
action and decision steps that expert math teachers use to teach solving quadratic equations.
Overall there were found to be seven main procedures for solving quadratic equations. These
seven procedures are:
1. Review linear equations to activate prior knowledge
2. Teach solving quadratic equations by factoring
3. Teach solving quadratic equations by using the quadratic formula
4. Teach solving quadratic equations by graphing
5. Teach solving quadratic equations by completing the square
6. Teach solving quadratic equations by the square root method
7. Teach application of these methods of solving quadratic equations to solving real-
life word problems
The disaggregated results are described in the following sections.
Recalled action and decision steps. In this study, action steps and decision steps
indicate those steps that provide “how-to” procedural information for solving quadratic
equations. Action steps refer to the steps that SMEs may be observed performing while decision
steps refer to steps that are unobservable cognitive decisions that may inform an action. Table
4.2 and Table 4.3 show the data results for action steps and decision steps in frequency counts
and percentages respectively.
THE USE OF COGNITIVE TASK ANALYSIS
53
Table 4.2
Counts of action and decision steps for each SME
SME Summary Statistics
A B C D Median M SD
Action steps 81 58 262 115 98 129 91.71
Decision steps 16 16 76 24 20 33 28.91
Total action and decision steps 97 74 338 139 106 156 122.49
There were a total of 402 action and decision steps on the GSP for solving quadratic
equations. The SME with the highest count of action and decision steps was SME C with a total
of 338 steps that accounted for 83.66% of the GSP action and decision steps for solving
quadratic equations. SME B had the lowest recorded action and decision steps with a total count
of 74 steps that represented 18.32% of the GSP action and decision steps. It should be noted that
total tally of the action and decision steps shown in Table 4.2 do not add up to 402 because the
action and decision steps elicited through CTA may not be distinctive to one SME, as a result the
action and decision steps in Figure 4.1 when aggregated do not equal the total number of action
and decision steps reported in the GSP because some action steps were described by more than
one SME. Therefore, these data confirm that SMEs will tend to provide the same action or
decision steps when describing how to perform a complex task through CTA techniques of
knowledge elicitation. Of the combined 402 action and decision steps recalled by the four SMEs,
there were 317 action steps and 85 decision steps for solving quadratic equations.
As shown in Table 4.3, individual SMEs recalled between 18.32% and 83.66% of action
and decision steps, a range of 65.34% and median of 29.21%. The variation between the SMEs
when describing the action and decision steps for solving quadratic equations was extremely
THE USE OF COGNITIVE TASK ANALYSIS
54
high with a standard deviation (SD) of 29.79% and a mean (M) of 40.10% of action and decision
steps described.
Table 4.3
Percentage of action steps and decision steps for each SME compared to the total number of
action and decision steps.
SME Summary statistics
A B C D Range Median M SD
Action
steps
25.47% 18.24% 82.39% 36.16% 64.15% 30.82% 40.57% 28.84%
Decision
steps
18.60% 18.60% 88.37% 27.91% 69.77% 23.26% 38.37% 33.62%
Total
action
and
decision
steps
24.01%
18.32%
83.66%
34.41%
65.34%
29.21%
40.10%
29.79%
SME C had the highest recall at 83.66% of the 402 action and decision steps captured in
the GSP, which was more than the sum of the other three SMEs. SME B had the lowest recall of
action and decision steps at 18.32%, which was a difference of 65.34%. All in all, the SMEs
recalled more action steps than decision steps. Figure 4.3 shows a graphical representation of the
number of action steps, decision steps and total action and decision steps captured from SMEs A,
B, C and D. These results will be discussed further in Chapter Five: Discussions.
THE USE OF COGNITIVE TASK ANALYSIS
55
Figure 4.1: Number of action steps, decision steps, and total action and decision steps for SMEs
A, B, C and D captured through CTA
Note. The graph represents the individual non-repeating SME action and decision steps captured from the
four SMEs. There were a total of 317 action steps and 85 decision steps captured in the gold standard
making a total of 402 steps for solving quadratic equations.
Action and decision steps captured in Round Two interviews. Following the initial
interview with each SME, an individual protocol of action and decision steps for each SME was
generated. The researcher conducted a round two interview with each SME to respond to
questions that arose while preparing the individual CTA protocol. This exercise gave the SMEs
an opportunity to make corrections to the individual protocol as much as they could recall the
necessary action and decision steps required to solve quadratic equations. In addition to these
THE USE OF COGNITIVE TASK ANALYSIS
56
SMEs, a fifth senior SME reviewed the final combined draft gold standard protocol and was also
given an opportunity to make changes that would make the protocol effective as a job-aid in
teaching solving quadratic equations. Table 4.4 shows both round 1 (R1) and round 2 (R2)
additional action steps and decision steps that were added during the Round Two interview
process.
Table 4.4
Additional action and decision Steps during Round two interviews
SME
A B C D Senior
SME
R1 R2 R1 R2 R1 R2 R1 R2
Action steps 44 37 42 16 234 28 110 5 0
Decision steps 4 12 6 10 74 2 21 3 0
Total action
and decision
steps
48
49
48
26
308
30
131
8
0
Round Two
Interviews
In
person
no
prior
email
In
person
no
prior
email
In
person
no
prior
email
In
person
no
prior
email
In
person
no
prior
email
Note: The senior SME did not participate in initial CTA semi-structured interviews but reviewed the preliminary
gold standard protocol for accuracy and completeness.
THE USE OF COGNITIVE TASK ANALYSIS
57
Figure 4.2: Percentage of action steps of SMEs for Round 1 and Round 2 interviews
All the four SMEs recalled extra action and decision steps at this stage. SME A and SME
B recalled more decision steps compared to their initial interview. SME A recalled 12 (75%) new
decision steps while SME B recalled an additional 10 (62.5%) decision steps as shown in Table
4.4. Equally important were the extra action steps recalled by both SME A and SME B, which
were 37 (45.7%) and 16 (27.6%) respectively as shown in both Table 4.4 and Figure 4.2.
THE USE OF COGNITIVE TASK ANALYSIS
58
Figure 4.3: Percentage of decision steps of SMEs for Round 1 and Round 2 interviews
It appeared that SME C and SME D recalled more during Round One interviews with
SME C adding only 28 (10.7%) new action steps and 2 (5.1%) new decision steps while SME D
added up to 4.4% of new recalled action steps and 12.5% new recalled decision steps as shown
in Figure 4.4 and Figure 4.5 respectively.
Alignment of SMEs in describing the same action and decision steps. The spreadsheet
analysis was also used to determine the number and percentage of action and decision steps
described by each SME that were fully aligned, substantially aligned, partially aligned, or not
THE USE OF COGNITIVE TASK ANALYSIS
59
aligned. For each action and decision step, if the step was only added by one SME, it was
identified as being “not aligned” then the number “1” was added in the alignment column. If an
action or decision step was described by two of the SMEs, then the number “2” was added in the
alignment column to indicate that the step was “partially aligned.” If an action or decision step
was described by three of the four SMEs, then the number “3” was added in the alignment
column indicating that the step was “substantially aligned” with the GSP. Finally, if an action or
decision step was described by all four SMEs the number “4” was added in the alignment
column to indicate the step was “fully aligned” with the GSP. Table 4.4 shows a summary of the
results of this analysis.
Table 4.5
Count and percentage of action and decision steps that are fully, substantially, partially, or not
aligned with the GSP
Count Percentage
Full Alignment 26 6.68%
Substantial Alignment 44 10.89%
Partial Alignment 81 20.05%
No Alignment 251 62.38%
Together the SMEs were “fully aligned” on 26 (6.68%), “substantially aligned” on 44
(10.89%), “partially aligned” on 81 (20.05%), and “not aligned” on 251 (62.38%) of the total
action and decision steps (402) on the GSP. The implications of these findings are discussed in
Chapter Five.
THE USE OF COGNITIVE TASK ANALYSIS
60
Question 2
What percentage of actions and/or decision steps, when compared to a gold standard, do
expert math teachers omit when they describe how to solve quadratic equations in Algebra One?
To answer Research Question 2, the number of steps omitted was aggregated using Microsoft
Excel to determine the extent expert mathematics teachers omit critical action and decision steps
required when describing instruction for solving quadratic equations. This data was analyzed and
is shown in Tables 4.6, Table 4.7 and Figure 4.4 that shows a graphical representation of the
same data. Action and decision steps that were included in the GSP but omitted by individual
SMEs were marked “0.” Using the Microsoft Excel, the omitted action and decision steps were
aggregated and summarized in Table 4.4 and Table 4.5 that show the frequency and percentage
omissions respectively for each SME.
Table 4.6
Total action and decision steps omitted by SMEs when compared to the GSP
SME Summary statistics
A B C D Median M SD
Action steps 237 260 56 203 220 189 91.706
Decision steps 70 70 10 62 66 53 28.914
Total action and
decision steps
307
330
66
265
286
242
120.488
THE USE OF COGNITIVE TASK ANALYSIS
61
Table 4.7
Percentage of total action and decision steps omitted by SMEs when compared to the GSP
SME Summary statistics
A B C D Range Median M SD
Action
steps
74.53% 81.76% 17.61% 63.84% 64.15% 69.19% 59.44% 28.84%
Decision
steps
81.40% 81.40% 11.63% 72.09% 69.77% 76.75% 61.63% 33.62%
Total
action and
decision
steps
75.99%
81.68%
16.34%
65.59%
65.34%
70.79%
59.90%
29.79%
The GSP had a total of 402 action and decision steps for solving quadratic equations. The
lowest percentage of action and decision steps omissions was 16.34% while the highest
percentage of omissions was 81.68% after Round Two interviews as shown in Figure 4.4. When
the SMEs described how to solve quadratic equations, the mean percentage of omissions of
actions steps was 59.44% with a standard deviation of 28.84% of 338 action steps recorded in the
GSP.
THE USE OF COGNITIVE TASK ANALYSIS
62
Figure 4.4: Percentage of total action and decision steps omitted by SMEs when compared to the
GSP
As shown in Figure 4.5, SME A and SME B omitted 74.53% and 81.76% respectively of
action steps required to successfully solve quadratic equations while SME D omitted 63.84% of
action steps.
THE USE OF COGNITIVE TASK ANALYSIS
63
Figure 4.5: Percentage of action steps omitted by the SMEs compared to the GSP
There were a total of 402 action and decision steps of which SME A and SME B omitted
75.99% and 81.68% respectively as indicated in Figure 4.4. SME D omitted up to 65.59% of the
402 action and decision steps shown in the GSP.
THE USE OF COGNITIVE TASK ANALYSIS
64
Figure 4.6: Percentage of decision steps omitted by the SMEs compared to the GSP
The mean percentage of omissions of decision steps was 61.63% with a standard
deviation of 33.62% of 86 decision steps recorded in the GSP. Overall, it was established that
expert mathematics teachers that participated in this study omitted up to an average of 59.90%
with a standard deviation of 29.79% of the total action and decision steps captured in the GSP
(Table 4.7).
Chapter Five will include an overview of the study, a discussion of the findings,
limitations of the study, implications, and future research.
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65
CHAPTER FIVE: DISCUSSION
Overview of the Study
The main purpose of this study was to use cognitive task analysis to capture the
knowledge and skills expert mathematics teachers of Algebra One use to teach solving quadratic
equations to 8
th
and 9
th
grade students in the K-12 education system. Also this study sought to
establish the percentage of critical information omitted when describing solving quadratic
equation procedures. While no formal hypotheses were stated, these two research questions
guided and informed this study. Research has shown that experts in an instructional role may
unintentionally leave out information that students must master when learning procedural skills.
Recent research (Clark, Pugh, Yates, Inaba, Green, and Sullivan, 2011) has shown that when
experts describe how they perform a difficult task, they may unintentionally omit up to 70
percent of the critical information novices need to learn to successfully perform the task.
According to Clark (2008) this is a serious problem because it forces novices to “fill in the
blanks” using less efficient and error-prone trial-and-error methods.
As new knowledge becomes automated and unconscious, experts are often unable to
completely and accurately recall the knowledge and skills that comprise their expertise,
negatively impacting instructional efficacy and leading to subsequent difficulties for learners
(Chi, 2006; Feldon, 2007). Feldon (2007) observes that automated procedures are deeply rooted
and not easy to change and therefore automaticity impairs the development of expertise. Experts
regularly cannot articulate their knowledge because much of their knowledge is implied and their
overt intuitions can be flawed (Chi, 2006). Therefore, in K-12 consultants who are experts in
their field and may be hampered by the effects of expertise and automaticity when recalling the
critical knowledge and skills often do professional development for teachers. While CTA has
THE USE OF COGNITIVE TASK ANALYSIS
66
been used successfully to capture explicit and implicit knowledge of experts in various domains
to use for training (Hoffman & Militello, 2009), this is the first study to use CTA to capture
expert mathematics teachers’ knowledge and skills to solve quadratic equations. This study may
form the basis for more research in K-12 education to provide effective professional
development modules to teachers and training institutions. Furthermore, the expert knowledge
and skills elicited through this study may provide clear guidelines in designing effective training
programs for mathematics teachers than those currently in use. Darling-Hammond et al. (2009)
notes that improving professional development and collaborative learning opportunities for
educators is a crucial step in transforming schools and improving academic achievement for all
students. Therefore, capturing expertise in teaching solving quadratic equations, can lead to
offering targeted professional development that would assist in improving instruction.
As such, this chapter discusses the process of conducting CTA, discussion of findings,
limitations of the study, its implications and suggestions for future research.
Process of Conducting Cognitive Task Analysis
Selection of Experts
Yates, Sullivan, and Clark (2012) and Crispen (2010) recommend that 3 to 4 experts are
needed for CTA to capture the optimal amount of significant information during a procedure.
Merriam (2009) notes, “If the purpose is to maximize information, the sampling is terminated
when no new information is forthcoming from new sampled units” (p. 80). And as Crispen
(2010) confirmed, the investment of resources into three to four experts yields a reliable amount
of expert knowledge to create a gold standard protocol. Further Merriam (2009) asserts that
selecting information-rich cases is important.
The selection of the participants to this study was a challenge to the researcher. This
THE USE OF COGNITIVE TASK ANALYSIS
67
researcher identified one unified school district in Southern California and approached its
superintendent for clearance to allow three of its 8
th
or 9
th
grade Algebra One teachers who have
shown high student achievement in state standardized tests over a period of five or more years.
Unfortunately it took longer than anticipated for these experts to be identified. Meanwhile, this
researcher approached a second unified school district that within a short time identified three
subject matter experts and the process of Round One interviews began. No sooner had the
researcher began interviewing experts from the second school district than the first school district
identified three experts for this study. That meant that this CTA study had six identified experts
though according to Crispen (2010), three to four SMEs provide the optimal level of action and
decision steps in a CTA gold standard protocol.
Since this study involved a small sample of mathematics teachers, selecting the sample
purposefully was necessary (Merriam, 2009; Patton, 1990) to allow the researcher to select
teachers that were highly qualified as determined by their training, student achievement scores
and experience. Feldon (2007) describes the characteristics of expertise to include extensive and
highly structured knowledge of the domain, effective strategies for solving problems within the
domain, and expanded working memory that utilizes elaborated schemas to organize information
effectively for rapid storage, retrieval, and manipulation. According to Clark et al., (2008) a
subject matter expert (SME) is a person with wide experience and is capable of performing a
range of tasks fast and successfully. For this study, expertise was shown by the algebra teachers’
years of experience, teaching expertise, and moreover, performance of their students in state
standardized tests. This was a critical criterion of selecting the experts.
Of the six SMEs identified, five were interviewed but only four experts’ interviews were
aggregated to generate the GSP, as the fifth expert was not able to verbally describe the action
THE USE OF COGNITIVE TASK ANALYSIS
68
and decision steps required to successfully solve quadratic equations. It may be that this expert’s
procedural knowledge was so automated that delineating the individual actions and decisions
became impossible, similar to the difficulty a person would have in describing how to drive a
car. The sixth SME was used to review the aggregated gold standard protocol for accuracy and
completeness.
Collection of Data
Multiple CTA methods. During the process of collecting data, the semi-structured
interview protocol described in Chapter Three was used for the first two experts, SME A and
SME B. This method of eliciting knowledge was followed strictly, however, the experts found it
difficult to recall critical action and decision steps for solving quadratic equations. When the
third and fourth SMEs (C and D) were interviewed, the researcher followed the same semi-
structured interview protocol but when requested, permitted the SMEs to write out the action and
decision steps as they verbalized them. This change in tact seemed to elicit significantly more
critical action and decision steps from these two SMEs (C and D) compared to those elicited
from SMEs A and B. In fact, SME C and SME D recalled more than double the number of action
and decision steps required to solve quadratic equations compared to both SME A and SME B
after they were allowed to think aloud as they wrote these steps. A possible explanation of these
results follows.
According to Yates (2007), there are over 100 types of CTA methods that have been
identified and classified. Yates suggests that since the current classification schemes organize
CTA methods by process rather than the desired outcome or application, practitioners may find it
difficult to select an optimal method for their specific purpose (or knowledge outcome). Crandall
et al. (2006) note that many knowledge elicitation methods have analytical processes and
THE USE OF COGNITIVE TASK ANALYSIS
69
representational formats embedded within the method. Therefore, Yates (2007) identified the
most frequently used CTA methods and the knowledge types associated with the respective
methods and outcomes (product approach versus the existing process approach) supporting Chao
and Salvendy’s (1994) conclusion that the percentage of accurately recalled decisions and
procedures varied by task type and by elicitation method used.
Considering the literature, it may be that the researcher inadvertently changed the
methods of elicitation for SME C and D. The method used in SME C and D process of collecting
data has been described in the literature as the Think Aloud protocol. Ericsson and Simon (1984)
describe this method as verbalizing a description of task performance while actually performing
the task or visualizing performing the task. As such, it may be that SME C and D recalled more
action and decision steps than the other SMEs because the Think Aloud method was more
appropriate for the task than the intended semi-structured interview method described in Chapter
Three. Further research is needed to examine these results.
Length of interviews. The CTA interviews took more time than it had been anticipated,
however, the additional time may have increased the number of action and decision steps
recalled by the experts. The initial round one interviews SMEs A and B took approximately two
hours, while SMEs C and D took on average three hours. These interviews were expected to take
about 90 minutes but the SMEs seemed to “get into it” and continued to give information that
sometimes may not have been relevant to the process of solving quadratic equations. SMEs C
and D interviews may have taken about one hour longer because these SMEs were given the
opportunity to freely talk and write out the steps without interruption from the interviewer. This,
in turn, may have helped to elicit more action and decision steps from SME C and SME D than
were elicited from SMEs A and B.
THE USE OF COGNITIVE TASK ANALYSIS
70
In sum, the length of the interviews in this study depended on the process of collecting
data. The process that involved the CTA semi-structured interview protocol took about two hours
while the process in which the SMEs were given the opportunity to write freely as they described
the action and decision steps took approximately one-hour more time. Further research is needed
in this area to examine the length of the interviews and their effect on the knowledge elicited.
Confirmation bias. The researcher was a high school mathematics teacher with 15 years
experience. To avoid confirmation bias, the researcher recorded and took notes on what the
SMEs were describing as the critical action and decision steps they make while solving quadratic
equations. According to Nickerson (1998) and Plous (1993), confirmation bias is the tendency to
search for, gather and/or interpret information to confirm one’s beliefs or hypothesis. The
researcher made sure by constantly reviewing the interview recording and transcript that the
protocol and the flowchart were based on what the SMEs said they do while teaching 8
th
and/or
9
th
grade students how to solve quadratic equations and not what the researcher thought should
have been done.
Discussion of Findings
While no formal hypotheses were developed for this study, the study was guided by two
research questions. The results from the data collection are discussed here below.
Research Question 1: What are action and decision steps that expert math teachers
recall when they describe how to teach solving quadratic equations in Algebra?
To answer this question, four subject matter experts described the action and decision
steps used to solve quadratic equations in algebra for 8
th
and 9
th
grade students in K-12 education
system. Out of all the procedural steps that were recalled, 78.7% were action steps, a
significantly higher number than decision steps, which were 21.3%.
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71
Differences in SMEs
In knowledge elicitation, experts are more likely to recall more action than decision steps
because decision steps are unobservable cognitions and are often more difficult to recall when
performing a task (Clark, 2014; Clark & Estes, 1996) as a consequence of automated expert
knowledge (see also Ericsson, 2004; Ericsson et al., 1993). In this study, the SMEs had
significance differences in the number of action steps and the number decision steps they
recalled (SME A = 97; SME B = 74; SME C = 338; SME D = 139). An examination of the
biographical differences among these SMEs may provide insight into these findings.
Biographical data. The five SMEs that participated in this study had varied
experiences and educational backgrounds. SME A had six years of experience teaching
mathematics the least experience of all the SMEs. SME B, on the other hand, was an expert with
16 years experience teaching Algebra One to 8
th
grade students. SME B became a teacher after
spending 23 years in an unrelated industry. The third expert, SME C had 11 years experience
teaching Algebra One and the fourth SME, SME D, had 11 years experience teaching Algebra
One to 8
th
grade students.
The expert with most experience, SME B recalled the least action and decision steps
which was in step with literature which indicates that as SMEs gain expertise, their skills become
automated and the steps of the procedure blend together. Experts perform tasks largely without
conscious knowledge as a result of years of practice and experience. This causes experts to omit
critical steps when describing a procedure because this information is no longer accessible to
conscious processes (Clark & Elen, 2006). Also SME B with six-year experience recalled less
action and decision steps compared to SME C and SME D who had 11 years of teaching Algebra
One. Based on previous studies (Canillas, 2010; Clark & Elen, 2006) and the concept of
THE USE OF COGNITIVE TASK ANALYSIS
72
automaticity in experts (Feldon, 2007), the SME with the least experience should recall the most
action and decision steps compared to the SMEs with more experience in the subject matter.
However, SME B with six years experience compared to SMEs C and D who had 11 years of
experience teaching algebra one recalled less action and decision steps. Therefore, these results
are inconclusive and do not support previous studies. It may also be that the knowledge
elicitation method, which changed from semi-structured interview for SMEs A and B to a think
aloud method for SMEs C and D and discussed in the next section might have influenced the
results.
Interview methods. It was intended that the semi-structured CTA interview
protocol described in Chapter Three be used to elicit expert knowledge from the four experts.
During the semi-structured CTA interview process, the SMEs were asked a series of questions
that focused on the major tasks when solving quadratic equations. The first two SMEs (A and B)
interview protocol was adhered to without deviation, however, SMEs C and D had more
difficulty responding to the semi-structured interview questions and thus were allowed and
encouraged to think aloud and even write what they were thinking as they responded to the semi-
structured CTA interview questions. The results show that SME C and D recalled significantly
more action and decision steps compared to the first two experts interviewed after the method of
eliciting knowledge was changed. These data support the conclusions of Yates (2007) and Chao
and Salvendy (1994) that different knowledge elicitation methods may be more appropriate for
specific tasks and knowledge types than other methods. The differences in the number of action
and decision steps recalled by the experts can be further examined by comparing the number of
action steps recalled versus the number of decision steps recalled by the experts.
THE USE OF COGNITIVE TASK ANALYSIS
73
Action Steps Verses Decision Steps
On average the four SMEs recalled more action steps than decision steps. These SMEs
recalled an average of 317 (78.7%) action steps compared to an average of 85 (21.3%) decision
steps. These findings confirm Canillas’ (2010) findings that SMEs are consistently able to
describe more knowledge steps on “how” to do a performance task, than knowledge of “when”
to do the task in a decision step. Canillas (2010) found that experts described 75.8% action steps
compared to 24.2% decision steps in describing the critical information required for the
placement of a central venous catheter (CVC). Due to automaticity, experts perform tasks largely
without conscious knowledge as a result of years of practice and experience. This phenomenon
causes experts to omit critical steps when describing a procedure because this information is no
longer accessible to conscious processes (Clark & Elen, 2006). Secondly, many SMEs are not
able to share the complex thought processes of behavioral execution of skills. Even experts who
make an attempt to “think aloud” during the process of complex problems often omit essential
information because their knowledge is automated (Clark & Elen, 2006; Clark & Estes, 1996).
Consequently, it is difficult to identify points during a procedure where an expert makes
decisions (Clark & Elen, 2006) and as such they are not able to describe these decision steps and
procedures.
SME C was the most efficient expert in recalling 83.66% of action and decision steps
aggregated in the GSP while the other three SMEs, A, B and D recalled an average of 24.58%
action and decision steps as enumerated in the GSP. The reason SME C may have recalled more
action and decision steps than the other SMEs may have been because of how the interview was
done. This researcher did not use the CTA semi-structured interview protocol alone like it was
done for SMEs A and B. Several elicitation methods were used together to maximize the
THE USE OF COGNITIVE TASK ANALYSIS
74
knowledge recall for both SME C and D. These elicitation methods were semi-structured
interview paired with both Diagram Drawing and Think Aloud (Clark & Estes, 1996; Yates,
2007). According to Yates and Feldon (2011), these methods result in both declarative and
procedural knowledge. Therefore SME C made more contributions to the GSP than three other
SMEs combined. During the Round One interview, SME D found it very hard to describe the
action and decision steps required to solve quadratic equations. This researcher then asked SME
D to “think aloud” and write these steps on paper as he verbalized what he was writing so that
the recording device could pick his voice to transcribe later. When this option was offered to
SME D, the SME was able to articulate and describe action and decision steps for solving
quadratic equations with ease. This essence of performing the task of solving quadratic equations
instead of describing the task allowed SME D to recall more action and decision procedural steps
required to solve quadratic equations though it was only 34.41% of the steps in the GSP. Clark
and Estes (1996) notes that differences between the various CTA approaches tend to be based
more on the specific nature of the types of tasks being analyzed and the eventual use of the
information being collected. The first two SMEs (A and B) found it challenging to describe these
steps because their knowledge was automated and they were not given the option to Think Aloud
during the initial interview. In fact, SME B kept on stating “let the kids play around” but never
articulated what “playing around” was in the context of solving quadratic equations.
Follow up interviews. During the Round Two interviews, all the SMEs recalled more
action and decision steps as shown in Table 4.4 and this may have been attributed to two things:
(1) the SMEs were able to recall more action and decision steps having known what the task was
about after the first interview, and/or (2) this researcher gained CTA interview skills to ask the
right probing questions and allowed all the SMEs to Think Aloud and utilize the Diagram
THE USE OF COGNITIVE TASK ANALYSIS
75
Drawing method of eliciting critical knowledge. At this stage SME A and SME B added
significantly more action and decision steps during Round Two interviews after they were
encouraged to verbalize their thinking and also to write the processes step by step as they
remembered them. The implication of this process was that the knowledge analyst must have a
variety of tools for interview, to match up the methods with the kinds of knowledge being sought
(Chao & Salvendy, 1994).
Expert Review of Draft Gold Standard Protocol
After this researcher aggregated the four individual protocols generated from the SMEs, a
fifth senior SME was asked to review the preliminary gold standard protocol for accuracy and
completeness. The fifth SME did not add any action or decision steps to the existing preliminary
gold standard protocol. The fifth SME noted that the preliminary gold standard protocol
represented the complete process for teaching how to solve quadratic equations and therefore did
not contribute to the PGSP. During the review of the preliminary gold standard protocol by the
fifth senior SME, this researcher noted that the senior SME was distracted because at the same
time the SME was supervising students doing club activities. What this researcher found very
useful was that during the Round Two interviews each of the four SMEs read their individual
protocols and clarified all the questions this researcher generated during the process of preparing
the individual protocols. This was important to capture all the knowledge and skills these experts
could recall in response to Research Question 1 as shown in the GSP (Appendix E).
Research Question 2: What percentage of actions and/or decision steps, when compared
to a gold standard, do expert math teachers omit when they describe how to solve quadratic
equations in Algebra?
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76
Expert knowledge omissions. Recent research (Clark et al., 2011) has shown that when
experts describe how they perform a difficult task, they may unintentionally omit up to 70
percent of the critical information novices need to learn to successfully perform the task. As new
knowledge becomes automated and unconscious, experts are often unable to completely and
accurately recall the knowledge and skills that comprise their expertise, negatively impacting
instructional efficacy and leading to subsequent difficulties for learners (Chi, 2006; Feldon,
2007). As such, in this study when compared to the GSP, on average 59.90% of action and
decision steps were omitted by SMEs with a standard deviation of 29.79% when describing the
overall steps needed to successfully solve quadratic equations. Individually, SME A and SME B
had the most omissions of 75.99% and 81.68%, which was much higher than the literature
suggests that expert may omit up to 70% of the critical information novices need to successfully
perform the task. SME D omitted 65.59% compared to the gold standard protocol while SME C
had the least omissions at 16.34% of action and decision steps enumerated in the GSP for solving
quadratic equations.
On further analysis of these data, with n = 4 participants, sample mean of omissions =
59.90%, and standard deviation (SD) = 29.79% and with the assumption that the population
mean of omissions of experts is 70% based on literature, the calculated two-sided statistic is t = -
0.6780 with a p = 0.5464 which is substantial evidence against an alpha level, 𝛼= 0.05,
showing this could not have happened by chance and therefore provides further evidence that
experts may omit up to 70% of critical information novices need to successfully perform the
task. Therefore, these results confirm that experts do omit critical information when describing
how to solve quadratic equations.
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77
Limitations
Though the study’s findings are consistent with results from other CTA studies related to
capturing expert knowledge in the form of action and decision steps and expert knowledge
omissions, there were several study limitations.
Confirmation Bias
The first limitation of this study is that the researcher is a high school mathematics
teacher with extensive knowledge and experience in teaching solving quadratic equations to high
school students for 15 years. This background and experience put the researcher to be attentive
of biases that may crop up while conducting the CTA interviews. The researcher had to
constantly control his facial outlook and emotions while listening to responses by the SMEs.
According to Clark (2014), when a knowledge analyst has experience in a performance task, the
analyst tends to change the information captured from SMEs to suit the analyst’s knowledge and
expectations. This knowledge analyst did not need to participate in any bootstrapping
procedures, where the analyst should read materials to gain a general familiarity with job or task
and knowledge of the specialized vocabulary (Crandall et al., 2006; Schraagen et al., 2000)
because this analyst had experience and knowledge of solving quadratic equations. Therefore,
extra effort was needed by the researcher to not put his preexisting knowledge and experiences in
solving quadratic equations onto the data collected. The potential for bias can never be
completely eliminated.
Internal Validity
The second limitation of this study is the validation of these results against what the
SMEs do in practice as they teach solving quadratic equations to 8
th
and 9
th
grade students in the
K-12 education system. According to Merriam (2009) “internal validity deals with the question
THE USE OF COGNITIVE TASK ANALYSIS
78
of how research findings match reality” (p. 213). In other words, validity must be assessed in
terms of something other than reality itself. In order to validate the gold standard protocol
developed from the CTA interviews, there would need to be a study on the effectiveness of CTA
based instruction. The validity test will be to observe the teachers whose data produced the GSP
and see how many of the actions and decisions steps s/he actually performs in reality.
External Validity
The final limitation of the present study is it may not be generalizable because of the
small sample size (n = 4) and that the participants were limited to two neighboring school
districts in Southern California. Merriam (2009) observes that the question of generalizability
has plagued qualitative researchers for some time. In other words, Merriam believes that part of
the difficulty lies in thinking of generalizability in the same way as do researchers using
experimental designs. However, the generalizability of the use of CTA for this task could be
measured as more teachers use it with successful results in student achievement. Also, future
research may replicate this study that will result in increasing the sample size and therefore
reduce external validity.
Implications
Darling-Hammond et al. (2009) assert that improving professional development and
collaborative learning opportunities for educators is a crucial step in transforming schools and
improving academic achievement for all students. Therefore, the declarative and procedural
knowledge captured from the CTA study when applied to training and instruction may increase
novice performance and decreases the amount of time and resources needed for training. In fact
according various studies (Embry, 2010; Zepeda-McZeal, 2014; Canillas, 2010), CTA has been
shown to be an effective methodology of capturing expert knowledge needed for the
THE USE OF COGNITIVE TASK ANALYSIS
79
performance of complex tasks. As such, in most schools and school districts expert consultants
may omit up to 70% of the critical action and decision steps when they conduct professional
development; training based on the results of CTA may prove to be advantageous (Clark et al.,
2011; Clark et al., 2008).
The current study supports the use of CTA research to capture expert knowledge and
skills in complex instructional tasks, such as solving quadratic equations in Algebra One not just
for training and instruction but to a major extend improve student achievement in algebra which
is a gateway to success in career and college (Gamoran & Hannigan, 2000; Moses & Cobb,
2001; Smith, 1996). Most importantly, although none of the SMEs admitted it, they did not seem
to know how to describe how to solve real-life word problems that involve quadratic equations.
As this is a requirement for the new Common Core State Standards’ (CCSS) performance tasks
in mathematics, omissions such as these may have an implication in student performance. CTA
may be an efficient method of capturing these skills for Common Core professional
development.
Clark (2011) notes that the use of CTA in instruction and training has been proven to be
positively related to cost savings due to reduced training times with comparable learning
outcomes. Further Clark (2011) maintains that CTA training results in 50% learning gains and
with reduced training times and cost savings, the implication is that school districts will spend
less resources in training and achieve more in well prepared teachers which will be expected to
translate to improved performance for students in the classroom.
Overall, CTA training and instruction has been shown to significantly improve
performance, including patent examiners finish 75% faster, six months vs. two years (Clark,
2011), surgical residents finish 25% faster, learn 40% more and important mistakes are reduced
THE USE OF COGNITIVE TASK ANALYSIS
80
by 50% (Velmahos et al., 2004), a meta-analysis of 34 studies averaged 47% performance
increase (Lee, 2004) and another meta-analysis of more than 100 studies averaged 25% learning
increase (Tofel-Grehl & Feldon, 2013). Therefore, K-12 professional development that includes
CTA training and instruction would benefit teachers by offering a targeted professional
development that would aid in learning that in turn may lead to improved instruction.
Future Research
A search of studies in this field of research did not result in any studies in the area of
solving quadratic equations in algebra using cognitive task analysis. Therefore as a result of this
current study, future studies may consider using the gold standard protocol generated by the
research and implement a randomized experimental study with mathematics teachers teaching
solving quadratic equations to 8
th
and 9
th
grade students. This study would involve a control
group of teachers who would teach solving quadratic equations using the current traditional
method and an experimental group in which the teachers would use the gold standard protocol.
These two groups would be compared using a two-sample t-test of differences to see whether
there is a significant difference in performance between the control group and the experimental
group of students taught using the GSP. Longitudinal research may also benefit this body of
research to determine short- and long-term learning gains in solving quadratic equations. During
this research study, it was also established that the experts did not articulate clearly how real-life
word problems are introduced while teaching quadratic equations and this would be one area that
may warrant future research using CTA methods.
Conclusion
The purpose of this study was to add to the body of knowledge on the benefits of CTA
for capturing critical information experts use when solving challenging problems and performing
THE USE OF COGNITIVE TASK ANALYSIS
81
complex tasks and the omissions experts make when describing their knowledge and skills. The
complex task of capturing the expert information that mathematics teachers use when teaching
8
th
and 9
th
grade students how to solve quadratic equations in Algebra is the first of its kind in K-
12, however there are other similar studies that explore the knowledge and skills captured and
omitted by experts through CTA methods. Expert mathematics teachers in this study omitted up
to an average of 60%, which was statistically not different from 70% and therefore this study
gives evidence in support of earlier studies that experts omit up to 70% of the critical action and
decision steps needed to successfully solve complex tasks when describing how to solve
quadratic equations. CTA methods were shown in this research to be effective in capturing the
unconscious, automated knowledge of expert mathematics teachers when they perform the
complex task of solving quadratic equations. The expert knowledge captured and aggregated into
a gold standard protocol in this study may be used to train teachers in teacher education
programs and in professional development in schools and school districts to assist in cutting
down costs and improving student achievement in Algebra One which is a building block for
upper level mathematics courses in high school. Darling-Hammond (1999) observed that the
effects of well-prepared teachers on student achievement are stronger than the influences of
poverty, language barriers, and minority status. Therefore capturing expertise of teaching
solving quadratic equations can improve Algebra One instruction that may lead to higher student
achievement.
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82
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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):
Performance Objective:
Ask: “What is the objective of solving quadratic equations? What action verbs should be used?”
Step 1:
Objective: Capture a complete list of student learning outcomes for teaching solving quadratic
equations.
(a) Ask the SME to list student outcomes when these tasks are complete. Ask them to make
the list as complete as possible.
(b) How are the students assessed on these outcomes?
Step 2:
Objective: Provide practice exercises that are authentic to the teaching context in which the
tasks are performed.
(a) Ask the SME to list all the contexts in which these tasks are performed (e.g. using the
quadratic formula, completing the square, graphing, factorization, or real-life problem
type)
(b) Ask the SME how the tasks would change for each method of solving quadratic
equations.
Step 3:
Objective: Identify main steps or stages to accomplish the task.
(a) Ask SME the key steps or stages required to accomplish the task.
(b) Ask SME to arrange the list of main steps (procedures) 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
(b) Ask: “Please describe how you would accomplish this task step-by-step, so a novice
teacher could perform it.”
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(c) 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.
(d) If the 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 novice teachers 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.
i) Ask the SME about Cues and Conditions
Ask: “For this task, what must happen before someone starts the task? What prior
knowledge, order, or other initiating event must happen? Who decides?”
ii) 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
novice teacher?
Concepts – terms mentioned by the SME that may be new to the novice
Ask for a definition and at least one example
Processes – How something works
If the trainee is teaching solving quadratic equations, then ask the SME to “Please describe how
and at what stage quadratic equations fit in teaching algebra – in words that novices will
understand what procedures are taught first? Think of it as a flowchart.”
Ask: “Must novices know these procedures 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.
a) Ask the SME about Equipment and Materials
Ask: “What equipment if any and materials are required to succeed at this task in
routine situations? Where are they located? How are they accessed?”
b) Performance Standard
Ask: “How do we know the objective has been met? What are the criteria, such as
time, efficiency, quality indicators (if any)?”
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c) Sensory experiences required for task
Ask: “Must novices see, hear, or touch 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 touch something (such as a calculator)?”
Step 6:
Objective: Identify problems that can be solved by using the procedure
(a) Ask the SME to describe at least one routine problem and two or three complex problems
that the novice should be able to solve if they can perform each of the tasks on the list
you just made.
Ask: “Of the tasks we just discussed, describe at least one simple or routine problem and
two to three complex problems that the novice should be able to solve IF they learn to
perform the task.”
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Appendix B
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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
THE USE OF COGNITIVE TASK ANALYSIS
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protocol” (PGSP).
7. Verify the PGSP by either:
a) Asking a senior SME, who has not been interviewed for a CTA, to review 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:
a. 1. Attempt to resolve the differences by communicating
with the SMEs, or
b. 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).
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Appendix D
SME C Initial Interview Flowchart for solving Quadratic Equations
Procedure 1
Yes
No
Start of
Procedure 1
Review/show:
• Multiplication and division
of rationals
• Factor whole numbers
• Factor linear expressions
• Show two examples
• Guided practice
Check for
understanding
(CFU):
Are students
proficient?
Activate
prior
knowledge
Introduce
solving
quadratic
equations
using
Draw a thinking
map
End of Procedure 1
Legend
Start/End
Preparation
Action or
Process
Input/
Output
Decision
Off-page
connector
Flow line
Connector:
Flow
continues
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Procedure 2
Start Procedure 2
Define factor
CFU
Are students
proficient with
factoring
numbers?
Teach how to use sum
and product tables to
find factors of QE
Guided practice:
Give students one or two
numbers to find factors on
whiteboards
Yes
No
Teach how to write quadratic
equations (QE) in standard
form:
Procedure 2
continued on
p.103
Show students
one or two
more examples
Teach how to factor
numbers with a couple of
examples
THE USE OF COGNITIVE TASK ANALYSIS
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Guided practice:
Give students one or two
quadratic expressions to
practice
Procedure 2
continued on
p.104
CFU
Are students
proficient using
sum and product
tables to find
factors of QE?
Show students
one or two more
examples
Show students how to
factor quadratic
expressions:
Show students two more
examples
No
Yes
CFU
Are students’
proficient
factoring
expressions?
Show students two more
examples step-by-step
No
Yes
Procedure 2
continued from
p.102
Guided practice:
Give students a QE to find two
numbers that give the sum of
the middle term and product of
the constant term
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Yes
Use zero product property to
solve for x:
If AB = 0, then A = 0 or B = 0
Guided Practice:
Give students two problems to
practice and walk around the room to
monitor progress
CFU
Have students
mastered the
factoring method
to solve QE?
Show students how to
solve real-life
problems, for example,
area problems
End of
Procedure 2
Do more examples
for students, showing
step-by-step
No
Procedure 2
continued from
p.103
Show students how to solve a
QE: using
the factorization method
(x+2)(x+3) = 0
Show students a couple more
examples
THE USE OF COGNITIVE TASK ANALYSIS
105
Procedure 3
Start procedure 3
Write the quadratic
formula (QF)
and sing a song on the
QF
Remind students to write
QEs in standard form
before
using the QF to solve for x
CFU
Are students
proficient
writing QEs in
standard form?
Show students one or
two examples on how
to write QE in
standard form
Guided practice: Give
students one or two
QEs to write in std.
form on their
whiteboards
Make 3-
columns in
notebook
No
Yes
Write essential question on
the left-hand column and
the steps of using QF on
the right-hand column
Procedure 3
continued on
p.106
THE USE OF COGNITIVE TASK ANALYSIS
106
Step 1
Write QE in standard form
followed by
underneath it
Procedure 3
continued from
p.105
Step 2
Label the values of a, b,
and c
Step 3
Write the QF
as you
sing along
Below the QF in Step 3:
Write
and substitute the values of a, b, and
c from Step 2
Show students a QE to
solve using the
quadratic formula
Is the QE
written in
standard
form?
Yes
No
Procedure 3
continued on
p.107
Solve
for x
THE USE OF COGNITIVE TASK ANALYSIS
107
Show
students
examples
using the
steps
provided
Procedure 3
continued from
p.106
Write a QE on the board
to solve using the QF. For
example, solve for x:
Step 1:
Write QE in standard form
Subtract 15 from both sides of
the equal sign:
3x
2
+ 4x – 15 = 15 – 15
3x
2
+ 4x – 15 = 0
Is QE
written in
standard
form?
Step 2:
Label the values of a = 3,
b = 4 and c = -15
Step 3:
Write the QF as you sing along:
“x equals negative b, plus or minus square root of b
squared minus 4ac, all over 2a”
Procedure 3
continued on
p.108
No
Yes
THE USE OF COGNITIVE TASK ANALYSIS
108
If b
2
– 4ac > 0, then
QE has two
solutions and its
parabola intersects
the x-axis twice.
Procedure 3
continued on
p.109
Below the QF in Step 3:
Write
substitute the values of a = 3, b =
4, and c = -15 from Step 2
Substitute the value of b = 4 into the first
Parenthesis, - ( ); substitute the value of b
again into the second parenthesis, the
Exponent part ( )
2
; substitute the values of
a = 3 and c = -15 into the third and fourth
Parenthesis respectively, - 4( )( ) and
circle -4ac; finally substitute a = 3 in the
Parenthesis in the denominator 2 ( )
Procedure 3
continued from
p.107
Tell students:
“b
2
– 4ac” is the
discriminant and it
determines the number of
solutions the QE has.
If b
2
– 4ac < 0, then
the QE has no real
solutions and its
parabola does not
intersect the x-axis
twice.
If b
2
– 4ac = 0, then
QE has one real
solution and its
parabola intersects
the x-axis once.
Show students b
2
– 4ac is part of the QF.
Point at the discriminant in the formula
discriminant
THE USE OF COGNITIVE TASK ANALYSIS
109
Procedure 3
continued on
p.110
Simplify:
, take the square root of 196 (discriminant)
Solutions:
The solutions are also called x-intercepts, roots or zeros of
the QE
Procedure 3
continued from
p.108
Show students a couple
more examples of solving
QE using the QF
CFU
Are students
proficient in
using the QF
to solve QE?
Guided practice:
Give students one problem
at a time to do and walk
around the classroom to
see what they are doing
and who needs help
Show students one
more example
No
Yes
THE USE OF COGNITIVE TASK ANALYSIS
110
Procedure 3
continued from
p.109
Use the QF to solve any QE for
its roots, which are also called
its solution, its zeros, or its x-
intercepts
Then introduce students to real-
life application problems like
vertical motion problems
Show students how to use the QF to
solve real-life application problems
End of Procedure 3
THE USE OF COGNITIVE TASK ANALYSIS
111
Procedure 4
Start Procedure 4
Graphing and
solving QE
should not be
taught in
isolation
Post a x-y coordinate
plane (graph) on the
whiteboard
Write the QE in standard
form:
Identify and write a the
coefficient of x
2
, b the
coefficient of x and c,
the constant
Tell students: If a is
positive, then the
parabola (graph) opens
up otherwise it opens
down.
Find the axis of symmetry:
, this is an x-value
Relate it to the QF:
Procedure 4
continued on p.112
THE USE OF COGNITIVE TASK ANALYSIS
112
Procedure 4 continued
from p.111
Substitute the values of a
and b into the axis of
symmetry equation
to find the axis of symmetry
Draw a dotted line through
the axis of symmetry on the
x-y coordinate plane on
the board.
Find the vertex by
substituting the x-value of
the axis of symmetry into
the original equation:
to find
the y-value of the vertex
Write the vertex in
the form (x, y) and
plot the point on the
x-y coordinate plane
on the board
Tell students: “The
vertex is where the
parabola opens from”
Procedure 4
continued on p.113
THE USE OF COGNITIVE TASK ANALYSIS
113
Procedure 4
continued from p.112
Draw a two-column t-table of values
(712-713) and put the vertex
coordinates at the center of the t-table
Choose two x-values to the left and
right of the x-value of the vertex (axis
of symmetry) and substitute them into
the original QE to find the
corresponding y-values
x y
Plot the points on the x-y
coordinate and connect
the points to plot the
graph with a smooth
curve
Label on the graph the
vertex, y-intercept, axis
of symmetry, and
direction of opening, up
or down
Procedure 4
continued on p.114
THE USE OF COGNITIVE TASK ANALYSIS
114
CFU:
Are students
proficient with
the graphing
QEs?
Guided practice:
Give students one QE at a
time to graph in pairs. Walk
around the room to check
for understanding.
Procedure 4
Continued on p.113
Show students one
more example step-
by-step
End of Procedure 4
No
Yes
THE USE OF COGNITIVE TASK ANALYSIS
115
Procedure 5
Start Procedure 5
Divide notebook into
three columns.
Label the steps on the
right-hand side, do your
work in the middle, and
write the essential
question on the left-hand
side.
Start with QE with the
coefficient of x
2
, a = 1:
Write the steps for
solving QE by
completing the
square
Step 1:
Is the QE
in standard
form?
Write the
equation in
standard form.
Can the
equation be
factored at
this stage?
Solve the QE
using the
factorization
method
(Procedure 2)
Step 2:
Isolate the constant
on the opposite side
of the equal sign
No
Yes
No
Yes
Procedure 5
continued on
p.116
See Procedure 2
on p.101
THE USE OF COGNITIVE TASK ANALYSIS
116
Procedure 5
continued
from p.115
Step 3:
Sing: “half of b squared,
add it to both sides”
Tell students:
Take add it to both sides
Show students an example:
Write x
2
+ 10x = 0, b = 10
Take half of b squared;
add it to both
sides:
Factorize the left-hand side
of:
Do students
know how to
factorize?
Go to
Procedure 2 on
p.101
Factorize:
Factors of x
2
are x and x and
factors of 25 are 5 and 5
Procedure 5
continued on
p.117
No
Yes
THE USE OF COGNITIVE TASK ANALYSIS
117
Procedure 5
continued from
p.116
Take the square root of
both sides of the equal
sign:
Solve for x by isolating the x
by itself on one side of the
equation, subtract 5 from
both sides
x = 5 – 5 or -5 – 5
Solution: x = 0 or x = 10
Show students two
more examples.
Guided practice:
Give students one QE at
a time to solve by
completing the square.
Walk around the
classroom to check for
understanding
CFU:
Are students proficient
solving QE with a
coefficient of x
2
, a = 1
using the completing the
square procedure?
Procedure 5
continued on
p.118
No
Yes
THE USE OF COGNITIVE TASK ANALYSIS
118
Procedure 5
continued from
p.117
Introduce a QE with a
coefficient of x
2
,
Is the QE
in
standard
form?
Write the
equation in
standard form
Divide both sides of
the equal sign by a,
Can the
equation be
factored at
this stage?
Solve the QE using
the factorization
method
(Procedure 2)
See Procedure 2
on p.102
Procedure 5
continued on
p.119
No
Yes
No
Yes
Is the
coefficient of
x, even?
Solve the QE using
the quadratic formula
(Procedure 3)
See Procedure 3
on p. 105
No
Yes
THE USE OF COGNITIVE TASK ANALYSIS
119
Procedure 5
continued from
p.118
Isolate the constant
on the opposite side
of the equal sign
See Step 3 on
p.116-117
End of Procedure 5
THE USE OF COGNITIVE TASK ANALYSIS
120
Procedure 6
Start Procedure 6
Is the QE of the form
Or
?
That is, is it missing
the b-term?
Use the square root
procedure to solve the
QE
Is the QE
factorable?
No Solve the QE
using the
factoring
procedure
Yes
See procedure 2
on p.102
Is the
coefficient
of x
2
,
?
No
Solve the QE
using the QF
procedure
See procedure 3
on p.105
Yes
Is the
coefficien
t of x, b
even?
No
No
Solve the QE
using the QF
procedure
See procedure 3
on p.105
Solve the QE using
completing the
square procedure
See procedure 5
on p.115
Yes
Yes
Write an example on the board
of perfect squares, x
2
= 9 and
show students how to solve it.
Take the square root
of both sides,
The solution is
x = -3 or x = 3
OR
Procedure 6
continued on
p.121
THE USE OF COGNITIVE TASK ANALYSIS
121
Procedure 6
continued from
p.120
Use factorization of sum
and difference of products
of perfect squares
to solve the QE
Show students an example
of sum and difference of
perfect squares.
Factorize and solve
Use the zero product property,
If AB = 0, then A = 0 or B = 0
to solve for x.
Solve:
If (x - 4)(x + 4) = 0,
then x – 4 = 0 or
x + 4 = 0
The solution is: x
= 4 or x = -4
Show another example
without a perfect square,
Solve by
adding 8 to both
sides to get:
Procedure 6
continued on
p.122
THE USE OF COGNITIVE TASK ANALYSIS
122
Show students one
or two QE using
the square root
method.
Procedure 6
continued from
p.121
Take the square root of
both sides of x
2
= 8:
=
Write:
or
and these
are the solutions,
roots, zeros or x-
intercepts of the QE,
x
2
– 8 = 0.
Guided practice:
Give students one QE at a time to
solve by taking the square root
procedure. Walk around the
classroom to check for
understanding
CFU:
Are students proficient
solving QE using the
square root procedure?
Check for understanding
with students writing on
their whiteboards and
raising them up for the
teacher to see.
Procedure 6
continued on
p.123
Yes
No
THE USE OF COGNITIVE TASK ANALYSIS
123
Procedure 6
continued from
p.122
Proceed to the
application of all these
procedures to solve
real-life problems that
involve QE
End of Procedure 6
THE USE OF COGNITIVE TASK ANALYSIS
124
Procedure 7
Start of Procedure 7
Write a practice real-
life problem on the
board or give students a
copy of the problem
Does the QE
extracted from
the problem
have a b-term,
the coefficient of
x?
Make notes on the 3-
column table on all
discussions as you
show students how to
solve the problem
Use the 3-column
table on the board
as a graphic
organizer
Solve the QE using
the square root
procedure
See procedure 6 on
p.120
Procedure 7
continued on
p.125
No
Yes
THE USE OF COGNITIVE TASK ANALYSIS
125
See procedure 2
on p.102
Procedure 7
continued from
p.124
Is the
coefficient
of x
2
,
𝒂≠𝟏?
Use the QF to solve the
QE generated from the
real-life problem.
Yes
See procedure 3
on p.105
Is b the
coefficient
of x even?
No
Use the QF to solve the
QE generated from the
real-life problem.
No
Is the QE
factorable
?
Yes
See procedure 3
on p.105
Use the factorization
procedure to solve the
problem
Use completing the
square procedure to
solve the problem.
No Yes
See procedure 5
on p.115
Procedure 7
continued on
p.126
THE USE OF COGNITIVE TASK ANALYSIS
126
Procedure 7
continued from
p.125
Take about two to
three days
working with
students
Assess students why they
choose the procedure they want
to solve the QE
Can
students
solve basic
quadratic
equations?
Introduce practical real-life problems,
like finding the width of a parabolic
disc, problems involving projectiles
that ask for the maximum height and
how long it will take to reach the
ground.
Demonstrate how to draw a
picture that shows the parabolic
path of the projectile. If given an
application problem (word
problem), then draw a picture to
represent the story
Draw a picture back to the
graph
Yes
No
Review with
students the
procedures for
solving QE
and check for
understanding
Relate back to previous
lessons because QE are
not an isolated unit.
End of Procedure 7
THE USE OF COGNITIVE TASK ANALYSIS
127
Appendix E
Solving Quadratic Equations – Gold Standard Protocol
Task: To Teach Solving Quadratic Equations in Algebra 1
Objective: Students understand how quadratic equations connect to real-life situations and that
there are multiple ways to solve a quadratic equation
Main Procedures:
1. Review Linear Equations to Activate prior knowledge
2. Teach solving quadratic equations by Factoring.
3. Teach solving quadratic equations by using the quadratic formula.
4. Teach solving quadratic equations by graphing
5. Teach solving quadratic equations by completing the square.
6. Teach solving quadratic equations by the Square root method.
7. Teach Application of these methods of solving quadratic equations to solving real-life
problems.
Procedure 1: Review linear equation to Activate prior knowledge
1.1 Give students an overview of the unit of solving quadratic equations.
1.1.1 Tell students: “I’m going to teach you how to solve quadratic equations”
1.1.2 Tell students: “There are multiple methods for solving quadratic equations:
factorization, graphing, the quadratic formula, the square root method, and
completing the square
1.1.3 Tell students: “Some of these methods will work better for some of the quadratic
equations. Some quadratic equations are perfect square binomials and therefore
will be easy to recognize that it can be factored and solved, while other quadratic
equations may be missing a “b” value and so it will be useful to use the square
root method. Other quadratic equations may be factorable but have many factors
to try and so quadratic formula will be quickest”.
1.2 Draw a full page size Tree map – see Figure E1
1.2.1 Relate solving quadratic equations to solving linear equations
1.2.1.1 Reason: Get students to see there are many ways to solve a quadratic equation
by always referring back to the tree map and these procedures are related to
solving linear equations
THE USE OF COGNITIVE TASK ANALYSIS
128
Procedures for Solving
Quadratic Equations
Factoring
(And use zero
product property
(zpp))
Quadratic
formula
Graphing Completing the
Square
Square Root
QE in standard
form must be
factorable
Example:
x
2
+ 7x + 10 = 0
sum product
(x + 5)(x + 2) = 0
x+5=0 x+2=0
zpp
x = -5 and x = -2
these are zeros,
roots, solutions of
the QE
Non-example:
x
2
+ 1x + 5 = 0
sum product
There are no
factors of 5 that
have a sum of 1
Use with QE in
standard form:
ax
2
+ bx + c = 0
The quadratic
formula will work
to solve any
quadratic equation.
Standard form
(Brace Map)
• Opening up
or down
depends on
a.
• c-value is the
y-intercept
• Axis of
symmetry,
• Axis of
symmetry
cuts parabola
into 2
symmetrical
pieces
• Vertex
!
!!
!!
,𝑓!
!!
!!
!!
substitute the
x-value of
the axis of
symmetry
into the
function to
find the y-
value of the
vertex.
• The vertex is
a minimum
or a
maximum
• Other points:
select and
substitute an
x to the left
and right of
the vertex
• Isolate c
• Song/dance
• Use when b-
value is even
• Use only
when b-
value is 0
(b = 0)
• ax
2
= c
Figure E1: Tree map
THE USE OF COGNITIVE TASK ANALYSIS
129
1.3 Put patterns on the screen over the computer or write them on the board, see Figure E2
Display Patterns (P3)
How many squares will be in the 5
th
pattern?
1
st
2
nd
3
rd
4
th
5
th
Position
of
squares
Number
of
squares
1
st
diff
2
nd
diff
0 0
1 1 1
2 3 2 1
3 6 3 1
4 10 4 1
5 ?
Therefore, there are:
y = ½ (5
2
) + ½ (5)
y = 12.5 + 2.5
y = 15
There are 15 squares in the 5
th
pattern.
Show students how to find the constants a, b, and c:
The second difference is a constant, which indicates the pattern is
quadratic.
Find a by dividing the common difference of 1 by 2: 𝑎=
1
2
! , b = ?
and c = 0, y-intercept
Use the 4
th
pattern with 10 squares: (4, 10) ; y = ax
2
+ bx + c
Substitute for x and y into the quadratic equation
𝑦= 𝑎𝑥
!
+𝑏𝑥+𝑐
10 = ½ (4
2
) + b(4) + 0
10 = 8 + 4b
2 = 4b, b = ½
Therefore the quadratic equation is y = ½ x
2
+ ½ x
Check: Use the 3
rd
pattern, (3, 6)
6 = ½ (3
2
) + ½ (3)
6 = 4.5 + 1.5
6 = 6, therefore this quadratic equation checks for this pattern.
Figure E2: Patterns
THE USE OF COGNITIVE TASK ANALYSIS
130
1.3.1 Show students how to solve linear equations step-by-step
1.3.2 Draw a two-column table on the board
1.3.3 Show students how to generate a table of values from patterns or linear equations
(see Fig. 2)
1.3.4 Plot the points on an x-y coordinate plane drawn on the white board
1.3.5 Examine and compare linear equations by changing the slope (m) and/or changing
the y-intercept (b)
1.3.6 IF students have access to graphing calculators or graphing software, THEN give
opportunities to check the graph they have made the linear equations from the
table of values
1.3.7 IF you show students a problem or two, THEN give them a few to try individually
to check for understanding
1.3.8 IF students are not proficient in solving linear equations, THEN reteach the
concept
1.3.9 Ask students randomly to come to the board to show that they have successfully
completed the problem
1.3.10 Assess students by walking around the room to get a visual of what they are doing
and that they are communicating using math language in their groups
1.3.11 IF you see that students are showing that they can do it on the board and you are
walking around the room and making sure that students are understanding from
what you can see, THEN continue to progress with the lesson of activating prior
knowledge
1.4 Review with students how to multiply and divide rational numbers
1.4.1 Give examples: ; ; ; .
1.4.1.1 Reason: The intention is to remind students about the rules for multiplying
integers since factoring quadratics assumes students can factor constant values
1.4.2 Define a factor to students: “Factors are numbers you can multiply together to get
another number”
1.5 Review with students how to factor a whole number
1.5.1 Use factor trees (see Figure E3)
1.5.2 Show students the factors of 6: 1 and 6, and 2 and 3 are factors because the
product of each pair is 6.
1.5.3 Give students a number to factorize, for example: factorize 18
1.5.3.1 Show students the factors of 18 are: 1 and 18, 2 and 9, 3 and 6, -1 and -18, -2
and -9, and -3 and -6
1.5.3.2 Show students that the product of these factors is 18
1.5.4 Remind students the rules for multiplying integers
1.5.4.1 Multiply a positive number by a positive number the product is another
positive number; multiply a negative number by another negative number the
product is positive while the product of a positive number by a negative
number is a negative number
1.5.5 IF you are factorizing a positive number, THEN get two positive factors or two
negative factors
1.5.6 IF you are factorizing a negative number, THEN get one positive factor and one
negative factor. Be sure that the sign of the greater factor matches the sign of the
middle term
12 4 3 = ∗ 10 5 2 − = ∗ − 10 5 2 − = − ∗ 6 2 3 = − ∗ −
THE USE OF COGNITIVE TASK ANALYSIS
131
1.6 Review with students how to multiply polynomials
1.6.1 Teach students exponent rules
1.6.1.1 Remind students, for example that times equals
1.6.1.2 Factorize or into factors: x
3
= (x)(x)(x) and x
2
= (x)(x)
1.6.1.3 Show students that on the whiteboard
1.6.1.3.1 Reason: To know the difference between addition and multiplication
when factorizing (breaking down) polynomials into factors
1.7 Review with students how to factor linear expressions
1.7.1 Show students how to factor an expression like 3x + 12. Tell students: “The terms
3x and 12 have a common factor of 3 because 3 can divide both 3x and 12.
Factorize 3 and the new expression is 3(x + 4)”
1.7.2 Give students another linear expression to practice factorizing
1.7.3 IF students are not proficient factoring linear expressions, THEN show more
examples like in step 1.7.1
1.7.4 IF students are proficient factoring linear expressions, THEN introduce solving
quadratic equations by factoring
Procedure 2: Teach solving quadratic equations by factoring
2.1 Remind students what a “factor” is
2.1.1 Define a factor to students again (line 1.4.2): “Factors are numbers you can
multiply together to get another number”
2.1.2 Explain (step 1.5) what you will do when you factor a certain problem. When
“factoring” we are showing students another way to write a product—as a
multiplication problem. Sometimes the factored form will look like an expanded
version of the original problem.
2.1.3 Give students a few factor tree problems to practice, for example: find the factors
24 (Fig. 3)
2.1.3.1 Factor 24: 24 = (2)(12) or (3)(8) or (4)(6). At this point, label (2)(12) as the
“factored form” of 24, Figure E3.
2.1.3.2 Give students a number to factor individually
2.1.3.3 IF some students are not proficient in multiplication, then assign
multiplication flash cards for homework practice, and encourage those
students to use multiplication charts when factorizing
2.1.3.4 IF students are not proficient in factorizing, THEN the teacher does one more
as students follow along
x x
2
x
3
x
2
x
2
, 2 x x x and x x x = • = +
24 24 24
2 12 3 8 4 6
Figure E3: Factor trees
THE USE OF COGNITIVE TASK ANALYSIS
132
2.1.3.5 IF students are proficient with factoring numbers, THEN teach students how
to write a quadratic expression in standard form, like
2.2 Give an example of the standard form: and a non-example: .
Tell students: “All the terms should be on one side of the equal sign
2.3 Show students coefficients of the quadratic equation. For example, the coefficients of
are 1, 5, and 6”
2.4 Teach students how to use distributive property
2.4.1 Write an example of two binomials on the board, for example (x + 2) and (x + 1)
2.4.2 Show students how to use algebra tiles (see Figure E5)
2.4.3 Find the area of the product of these binomials: x
2
+ 2x + x + 2
2.4.4 Write area x
2
+ 3x + 2 by looking at tiles
2.4.5 Tell students: “Factoring is how we undo the distributive property (product of
binomials) for example, getting the binomials that gave the product 3x
2
+ 4x – 15”
2.5 Show students how to factorize x
2
+ 5x + 6
2.6 Show students a sum and product table
2.6.1 Show students how to use a sum and product table to find factors
2.6.1.1 Tell students: “Draw a sum and product table on your whiteboard”
2.6.1.2 Tell students: “Two numbers have a sum of 5 and a product of 6. With your
partner, figure out which numbers they are”
2.6.1.3 Tell students to do the problem on their whiteboards and hold them up
2.6.1.4 Scan across the room as students raise their whiteboards checking for
understanding
2.6.1.5 IF students have not mastered the use of sum and product tables to factorize,
THEN show students another example. Like two numbers have a sum of 8 and
a product of 15, show them how to find these two numbers
2.6.1.6 Repeat this procedure with different problems until students are proficient
2.6.1.7 IF students have mastered the use of sum and product tables, THEN show
students how to factor the original problem
2.6.1.8 Circle the term 6 which is the constant in and write the word
constant above the 6. IF you circle the constant, THEN write the word product
underneath it
2.6.1.8.1 Show students some numbers that give a product of 6, write these
numbers in the sum and product table. Some pairs are 1 and 6, and 2
and 3
2.6.1.9 Circle the x-term, 5x and then write sum underneath it
2.6.1.9.1 Use the pairs of numbers in the sum and product table to determine
which pair, 1 and 6 or 2 and 3 adds up to 5
2.6.1.9.2 Choose 2 and 3
2.6.1.10 Circle the x
2
-term. Tell students: “ x times x is x
2
and therefore x and x are
the factors of x
2
”
2.7 Teach students how to use algebra tiles, the X-BOX (Diamond Method), and the
Parenthesis methods to factorize
2.7.1 Use algebra tiles to factor quadratic equations (see Figure E5)
2.7.2 Introduce quadratic equations that can be factorized using algebra tiles
6 5
2
+ + x x
0 6 5
2
= + + x x x x 5 6
2
− = +
0 6 5
2
= + + x x
6 5
2
+ + x x
6 5
2
+ + x x
THE USE OF COGNITIVE TASK ANALYSIS
133
2.7.2.1 Ask students: IF I have x
2
+ 3x + 2, THEN what tiles would I need? (Figure
E4)
2.7.2.2 Ask students to gather tiles
2.7.2.3 Tell students that x
2
+ 3x + 2 represents an area
2.7.2.4 Arrange tiles in a rectangle, to find factors by looking at length and width of
the rectangle
2.7.2.5 Tell students: “The factors of x
2
+ 3x + 2 are (x + 2) and (x + 1)” (step 2.7.2.4)
2.7.2.6 IF students are not yet comfortable with factoring using algebra tiles, THEN
reteach the concept as in step 2.7.2.4
2.7.3 If students are comfortable with factoring using algebra tiles, then introduce the
X-BOX (Diamond) method for factoring so that students may have an alternative
way of factorizing. For example, factorize x
2
+ 3x + 2 using the X-BOX method,
see below
Product
a.c
2
2 1
3
b
sum
Factors: (x+2) and (x+1)
x
2
x x
1
1
x
Figure E4: Algebra tiles
x
2
x
x
1 1
x
x + 2
x
+
1
Figure E5: Factorization using tiles
THE USE OF COGNITIVE TASK ANALYSIS
134
2.7.4 IF students are not proficient factorizing using the X-BOX method (step 2.7.3),
THEN reteach
2.7.5 IF students are proficient factorizing using the X-BOX method (step 2.7.3),
THEN proceed to teach students the Parenthesis Method
2.8 Show students how to factorize a quadratic expression
using the Parenthesis
Method
2.8.1 Put two sets of parentheses, at the top for the binomials that we are trying
to factor this problem into Then, at the bottom of the parentheses, write the
problem:
2.8.2 Tell students
: “We are going to find the factors of this problem,
2.8.3 Tell students: “Since factors of the first term are x and x, put each on a
different parenthesis , since x times x is x
2
, followed by 2 in the first
parenthesis and 3 on the second parenthesis ”
2.8.4 IF we factor , THEN the product of these parts in the parentheses
have to match with the original expression,
2.8.5 Give students another problem to factorize
2.8.6 IF students are not proficient with factoring , THEN show students
how to do it
2.8.6.1 Use the sum and product table. Two numbers have a sum of 11 and a product
of 30.
2.8.6.2 Repeat the same process as in 2.3.1.1 through 2.3.1.4 to factorize
2.8.7 IF students are proficient with factoring expressions, THEN introduce them to
solving quadratic equations using the factorization method
2.8.7.1 Show students how to solve by finding the value of x that
satisfies this equation.
2.8.7.2 Factorize the left hand side of the equation
2.8.7.3 Put below the original quadratic equation
2.8.7.4 Use the zero product property, IF AB = 0, THEN A = 0 or B = 0
2.8.7.5 IF , THEN (x + 2) = 0 or (x + 3) = 0
2.8.7.6 Solve: x + 2 = 0, subtract 2 from both sides of the equation: x+2-2 = 0-2
therefore x = -2. And x + 3 = 0, subtract 3 from both sides of the equation:
x+3-3=0-3 therefore x = -3.
2.8.7.7 Solution is x = -2 or -3. Solution of a quadratic equation is also called the x-
intercepts, zeros, and roots.
2.8.8 Give students another example to practice:
2.8.9 Use the sum and product table to factorize the left hand side of the quadratic
equation
2.8.9.1 Factorize to (x +5)(x + 6) = 0
2.8.9.2 Use the zero product property, IF AB = 0, THEN A=0 or B=0
2.8.9.3 Solve: IF (x + 5)(x +6) = 0, THEN (x + 5) = 0 or (x + 6) = 0
6 5
2
+ + x x
( )( )
6 5
2
+ + x x
" 6 5
2
+ + x x
2
x
( )( ) + + x x
( )( ) 3 2 + + x x
6 5
2
+ + x x
( )( ) 3 2 + + x x 6 5
2
+ + x x
30 11
2
+ + x x
30 11
2
+ + x x
30 11
2
+ + x x
0 6 5
2
= + + x x
0 6 5
2
= + + x x
( )( ) 0 3 2 = + + x x
( )( ) 0 3 2 = + + x x
0 30 11
2
= + + x x
0 30 11
2
= + + x x
THE USE OF COGNITIVE TASK ANALYSIS
135
2.8.9.4 IF x + 5 = 0, THEN x = -5 and IF x + 6 = 0, THEN x = -6
2.8.9.5 Solution is x = -5 or -6. Tell students: “The solution of a quadratic equation is
also called x-intercepts, zeros, and roots.
2.9 Give students 3-5 question assessment (open ended)
2.9.1 IF students have mastered factoring when the leading coefficient is 1, THEN
move on to factoring where the leading coefficient is other than 1. IF not, THEN
reteach.
2.10 Write a quadratic equation like
on the board
2.10.1
Identify a = 6, b = 1, and c = -1
2.10.2
Use the X-Box (Diamond) method, for example (Step 2.10.3)
2.10.3
Make a big cross (X) underneath the equation. Inside the top of the X, write the
product of a and c and inside the bottom of the X, write b (the sum) –
(see below).
2.10.4
Find factors with a product of -6 and a sum of 1, as shown
2.10.5
Tell students: “There are many pairs that will give a product of -6 but they are not
all going to give a sum of the middle coefficient, 1”
2.10.6
Show students the factors of -6 that give a sum of 1 are: 3 and -2
2.10.7 Write the expanded form of the quadratic equation:
2.10.8
Factor by grouping
2.10.9
Find common factors:
2.10.10
Check by multiplying using the box shown below
2.10.11Write out
to confirm the product is the original quadratic
equation
6x
2
+x−1=0
(a.c=−6)
6x
2
+3x−2x−1=0
3x(2x+1)−1(2x+1)=0
(2x+1)(3x−1)
6x
2
+3x−2x−1=0
6x
2
+x−1=0
Product
a.c
-6
3 -2
1
b
sum
6x
2
+3x – 2x – 1 = 0
a b b c
Factor by grouping: 3x(2x+1) -1(2x+1) = 0
Find common factors: (2x + 1)(3x – 1) = 0
Multiply 2x +1
3x 6x
2
3x
-1 -2x -1
THE USE OF COGNITIVE TASK ANALYSIS
136
2.10.12
Solve the quadratic equation for x using the zero product property (ZPP):
. If (2x + 1)(3x – 1) = 0, THEN 2x + 1 = 0 or 3x – 1 = 0
2.10.13Solve 2x + 1 = 0 to get x = −1/2 and 3x – 1 = 0 to get x = 1/3
2.11 IF teacher does one example on the board, THEN give students one problem to
try. Walk around the room to monitor what students are doing to check for understanding
2.12 IF students have not mastered factoring, THEN reteach the concept
2.13 IF students have mastered factoring, THEN give proceed to give assessment
2.14 Give students 5-10 question assessment (open ended) on factoring.
2.14.1 IF students have mastered factoring, THEN move on to solving quadratic
equations using the quadratic formula. IF NOT, THEN reteach.
Procedure 3: Teach solving quadratic equations by using the quadratic formula,
3.1 Teach students a way to memorize the quadratic formula
3.1.1 Sing: “x equals negative b, plus or minus square root of b squared minus 4ac, all
over 2a.” Make students sing along, or teach students to memorize the quadratic
formula using this phrase: “A negative boy could not decide whether or not to go
to a radical party. He decided to be square and he missed out on 4 awesome
chicks. The party was all over at 2 am.”
3.1.2 Tell students: “The quadratic formula is a “catchall” for solving quadratic
equations, it works every time”
3.1.3 Tell students: “When in doubt while solving quadratic equations, revert back to
the quadratic formula, that is the reason for singing the song every day, multiple
times during the period while using the quadratic formula”
3.2 Write the quadratic equation in standard form, before using the
quadratic formula
3.2.1 IF the equation is not in standard form, THEN the equation may be misleading
because either the value of a, b, or c may not be correct
3.3 Show students how to write a quadratic equation in standard form
3.3.1 Write an example on the board that has the x-term on the other side,
3.3.2 Show students that the standard form would be 0 5 3 2
2
= + − x x or
3.3.3 Choose which side of the equal sign to take all the terms to get the equation to
standard form and pay attention to the sign change
3.3.4 IF there are terms on both sides of the equal sign, THEN the signs will be
different when all the terms a collected on the same side
3.4 Tell students: ‘Make three-columns in your notebook’ see Table 1
(2x+1)(3x−1)=0
a
ac b b
x
2
4
2
− ± −
=
0
2
= + + c bx ax
x x 3 5 2
2
= +
5 3 2 0
2
− + − = x x
THE USE OF COGNITIVE TASK ANALYSIS
137
3.4.1 Write the essential question (EQ): “How is the quadratic formula used to solve a
quadratic equation” on the left-hand side
3.4.2 Write the steps of using the quadratic formula on the right-hand side [see Table
E1)
3.4.3 Do the steps along with the students
3.4.3.1 Step 1: Write: Standard form
3.4.3.2 Write: or
underneath “Standard form”
3.4.3.3 Step 2: Label a, b, and c. To the left of a, write “opening”
3.4.3.3.1 Reason: So students know when they graph it, a is going tell them the
direction the graph will open, either “up” or “down”
3.4.3.4 IF a equals a negative number, THEN the graph opens down
3.4.3.5 IF a equals a positive number, THEN graph opens up
0
2
= + + c bx ax y c bx ax = + +
2
Essential Question (EQ): How is the quadratic formula used to solve a quadratic equation?
Steps
Solve: 2x
2
+ 3x – 5 = 0 2x
2
+ 3x – 5 = 0 Step 1: Standard form
ax
2
+ bx + c = 0
a = 2 (opening-up), b = 3; c = -5 (y-intercept) Step 2: Label a, b, and c
Step 3: Write quadratic formula
Step 4: Substitute values of a, b,
and c into the quadratic formula
Step 5: PEMM (4 steps in one)
Step 6: Add and simplify the
discriminant
Step 7: Take the square root the
discriminant
x = 10/4 or x = −4/4
Step 8: Simplify and write: zeros,
roots, solutions or x-intercept
Table E1: Three-column table
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138
3.4.3.6 Write y-intercept next to c
3.4.3.6.1 Reason: So that students know this is not in isolation
3.4.3.7 Step 3: Write the quadratic formula, , as you sing along
with students the quadratic formula song: “x equals negative b, plus or minus
square root of b squared minus 4ac, all over 2a”
3.4.3.7.1 Do not substitute the values of a, b, and c into the formula before
writing the formula out
3.4.3.7.2 Write below the quadratic formula. Sing the
remix together with students: “x equals negative parenthesis plus or
minus square root of parenthesis squared minus four parenthesis
parenthesis all over two parenthesis” as you write this:
3.4.3.7.3 Get students in the habit of writing the formula every time
3.4.3.7.4 IF students substitute directly without writing the formula first, THEN
they do not get points for it
3.4.3.7.4.1 Reason: Because students must use parenthesis when substituting
into the formula
3.4.3.8 Tell students: “To substitute is to replace”
3.4.4 Show students how to do the four steps in one (see Table E1)
3.4.4.1 Do order of operations, do PEMM
3.4.4.2 Substitute the value of b into the first Parenthesis, negative ( )
3.4.4.3 Substitute the value of b again into the Exponent part ( )
2
and circle the b-
squared part
3.4.4.4 Substitute the values of a and c into third and fourth parenthesis respectively,
negative 4( )( ) and circle -4ac followed by putting M over it to indicate
Multiplication will take place
3.4.4.5 Circle the denominator, 2a, substitute the value of a into the parenthesis 2( )
and put M over it to indicate Multiplication will take place
3.4.4.6 Circle what is under the square root sign
3.4.4.7 IF you are going to add the numbers b
2
and 4ac, THEN put A for addition
above it and add the quantities
3.4.4.8 IF you are going to subtract the quantities b
2
and 4ac, THEN put S for
subtraction above it and subtract the quantities
3.4.4.9 Write D for Division but wait on D
3.4.4.10 Take the square root of the quantity b
2
– 4ac
3.4.4.11 Divide the numerator by the denominator and then write x equals the result
of the division
3.4.4.11.1 Reason: Because we are going through PEMMDAS, the order of
operations
3.4.4.12 Circle using a red marker on the board and write the step being done
3.4.4.13 Draw a box around x equals, see below
a
ac b b
x
2
4
2
− ± −
=
x=
−()± ()
2
−4()()
2()
( ) ( ) ( )( )
( ) 2
4
2
− ± −
= x
x =
THE USE OF COGNITIVE TASK ANALYSIS
139
3.4.4.14 IF x has two solutions, THEN the graph intercepts the x-axis twice
3.4.4.15 IF x has only one solution, THEN the graph touches the x-axis once
3.4.4.16 IF x has no solution, THEN the graph does not touch the x-axis
3.4.5 Show an example:2x
2
−3x−5=0
3.4.6 Substitute the values of a, b, and c from the standard quadratic equation,
into the formula with the parenthesis in place of a, b, and c:
( ) ( ) ( )( )
( ) 2
4
2
− ± −
= x
3.4.6.1 Substitute the value of b from the standard quadratic equation,
2x
2
−3x−5=0 into the first and second parenthesis,
x=
− −3
( )
± −3
( )
2
−4
( )( )
2
( )
the negative sign is still outside the parenthesis
3.4.6.2 IF b was negative, THEN substitute it together with its sign
3.4.6.3 Tell students: “Pay attention to the ”
3.4.6.3.1 IF b is a negative number, THEN negative times negative is positive.
3.4.6.3.2 IF b is a positive number, THEN positive times positive is positive
3.4.6.4 IF any number is squared, THEN the product is always positive (P1)
3.4.6.5 Substitute values of a, the c and a again into: x=
− −3
( )
± −3
( )
2
−4 2
( )
−5
( )
2 2
( )
3.4.6.6 Simplify: x=
3± 9+40
4
=
x=
3± 49
4
which becomes
x=
3±7
4
3.4.6.7 Tell students: “The plus or minus 7 means we have two solutions for this
quadratic equation
3.4.6.8 Solution: x = 10/4 or x = −4/4 which are simplified to x = 2.5 or -1. These
are also called roots, x-intercepts or zeros of the quadratic equation
3.4.7 See Table 1 (step 3.4)
3.4.7.1 Tell students: “ is the discriminant and it helps determine the number
of solutions of a quadratic equation”
3.4.7.2 IF the discriminant is positive, THEN the parabola intercepts the x-axis twice
3.4.7.2.1 Draw the graph to show students the parabola intercepts the x-axis
twice
3.4.7.3 IF the discriminant is zero, THEN the parabola touches the x-axis once and
turns around
3.4.7.3.1 Draw the graph to show students the parabola touches the x-axis once
3.4.7.4 IF the discriminant is negative, THEN the parabola does not touch the x-axis
3.4.7.4.1 Draw the graph to show students the parabola does not intersect the x-
axis
3.4.7.5 IF students get the solution, THEN they have to write all the names every
time: x-intercept(s), solution(s), zero(s) and root(s) of the quadratic equation
3.4.7.6 Tell students: “Write x-intercepts, solutions, zeros and roots under the answer
on every quadratic equation problem you solve”
0
2
= + + c bx ax
( )
2
ac b 4
2
−
THE USE OF COGNITIVE TASK ANALYSIS
140
3.5 Show students b
2
– 4ac is the discriminant and is part of the quadratic formula
3.5.1 Point at the discriminant:
3.6 Use the quadratic formula to solve any quadratic equation for its roots, its solution, and
its zeros
3.7 IF students are able to solve a quadratic equation for its roots, solutions, or its zeros,
THEN they can solve real-life application problems like vertical motion problems
(Appendix D)
3.8 IF students are not comfortable using the quadratic formula, THEN show them two more
examples and give them a few problems to practice (guided practice)
3.9 Give students 3-5 problems to solve using the quadratic formula.
3.9.1 IF students have mastered solving using the quadratic formula, THEN begin
teaching graphing. IF not, THEN reteach.
3.9.2 IF students are proficient using the quadratic formula, THEN proceed with the
lesson to show students the next procedure for solving quadratic equations
Procedure 4: Teach solving quadratic equations by graphing
4.1 Post an x-y coordinate plane on the whiteboard throughout the unit of quadratic equations
(see below)
4.2 Solve all quadratic equations next to the graph so that students make connections and also
see multiple representations
4.3 Solve while relating back to the graph because students have a hard time connecting
different representations
4.4 Do not teach graphing of quadratic equations and solving quadratic equations using other
procedures in isolation
4.4.1 Go back and forth between various methods of solving quadratic equations
and their graphs
4.4.2 Relate the x-intercepts of the graph of a quadratic equation to its solutions
after solving using any of the other procedures
a
ac b b
x
2
4
2
− ± −
=
Discriminant: 2 roots, 1 root or
no root
THE USE OF COGNITIVE TASK ANALYSIS
141
4.5 Start with a quadratic equation in standard form
and reflects on the y-
axis
4.6 Identify the parts that are obvious based on the equation
4.6.1 Identify a the coefficient of x
2
, b the coefficient of x and c, the constant
4.6.2 IF a is positive, THEN the graph (parabola) will open up
4.6.3 IF a is negative, THEN the graph (parabola) will open down
4.6.4 Write c, the y-intercept
4.7 Teach how to find the axis of symmetry
4.7.1 Find axis of symmetry and relate to the quadratic formula,
4.7.2 Break apart the quadratic equation and show students
that is the axis of symmetry
4.7.3 Write and carefully explain to students that this does not represent x-
intercepts
4.7.4 Tell students: “This is an x-value and it is where on the x-axis the axis of
symmetry cuts through”
4.7.5 IF the quadratic equation has a middle term bx, THEN the parabola will not
reflect over the y-axis because x=
−b
2a
≠0 . The axis of symmetry will not be
on the y-axis
4.7.6 Substitute the values of a and b into the axis of symmetry equation: x=
−b
2a
to
find the axis of symmetry
4.7.7 IF you fold a parabola in half through the axis of symmetry, THEN there are
two identical parts
4.7.8 IF there is a y-value to the left of the axis of symmetry, THEN there is an
equivalent y-value same distance from the axis of symmetry on the right of
the axis of symmetry
4.7.9 IF you have the y-intercept on one side of the axis of symmetry, THEN there
is another point at the same height on the other side of the parabola
4.8 Teach students how to find the vertex
4.8.1 Use the axis of symmetry, to find the x-value of the vertex
4.8.2 Substitute the x-value of the vertex into to find the y-value
of the vertex
4.8.3 Write the vertex in the form of (x, y) coordinate point
4.8.4 Point to students that the vertex is the highest or lowest point of the parabola
4.9 Teach Graphing procedure
4.9.1 Draw an x-y coordinate plane
y c bx ax = + +
2
a
ac b b
x
2
4
2
− ± −
=
a
ac b b
x
2
4
2
− ± −
=
a
b
x
2
−
=
a
b
x
2
−
=
a
b
x
2
−
=
y c bx ax = + +
2
THE USE OF COGNITIVE TASK ANALYSIS
142
4.9.2 Draw a dotted line through the axis of symmetry found in step 4.7.6
4.9.3 Draw an x-y table of values. An example is here below when the x-value of
the vertex is 0
4.9.4 Put the vertex coordinates at the center of the table
4.9.5 Choose an x-value to the left or to the right of the vertex
4.9.6 Show students the mirror point(s)
4.9.7 Plot the vertex as found in step 4.8.3
4.9.8 IF you find the vertex, THEN get 2 or 3 points on one side of the axis of
symmetry
4.9.9 IF you have 2 or 3 points on one side of the axis of symmetry, THEN you
will get 2 or 3 points on the opposite side of the axis of symmetry
4.9.10 IF you have the axis of symmetry (x-value), THEN choose two x-values to
the left or right of the axis of symmetry to substitute into the original
equation to find the corresponding y-values
4.9.11 IF the parabola opens up, THEN the vertex is a minimum
4.9.12 IF the parabola opens down, THEN the vertex is a maximum
4.9.13 Tell students: “The vertex is where our parabola opens from”
4.9.14 Remind students the graph opens upwards or opens downwards depending on
the a-value
4.10 Connect the points to plot the graph with a smooth curve
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143
4.11 Label on the graph the vertex, y-intercept, axis of symmetry and direction of
opening, up or down
4.12 Label the x-intercepts if they exist
4.13 Show an example, x
2
+ 4x – 12 = 0
4.13.1 Draw a two-column t-table of values (step 4.9.3)
4.13.2 Identify the coefficients: a = 1, b = 4, and c = -12
4.13.3 Find the axis of symmetry,
( )
( )
2
2
4
1 2
4
2
− =
−
=
−
=
−
=
a
b
x
4.13.4 Put the axis of symmetry, x = -2 in the middle of the t-table and then choose
integers on either side of -2 that are equidistant from the axis of symmetry
4.13.5 Draw the axis of symmetry x = -2
4.13.6 Choose two x-values less than -2 and two x-values greater than -2: -4, -3, -2, -
1, and 0 (step 3.1.5)
4.13.7 Substitute these x-values into the quadratic equation, x
2
+ 4x – 12 = 0 to find
the corresponding y-values (step 4.8.2)
4.13.8 Plot the pairs of points on x-y coordinate plane
4.13.9 Ask aloud: “How many times does the graph of x
2
+ 4x – 12 = 0 cross the x-
axis?”
4.13.10 Show students the x-intercepts, which are also the solutions of the quadratic
equation
4.14 IF the graph of a quadratic equation intercepts the x-axis twice, THEN the
quadratic equation has two real solutions
4.15 IF the graph of a quadratic equation touches the x-axis once, THEN the quadratic
equation has one real solution
4.16 IF the graph of a quadratic equation does not touch the x-axis, THEN the
quadratic equation has no real solution
4.17 Check for understanding by giving students two quadratic equations to graph
4.18 IF students are not proficient with graphing, THEN show them the process with
two more examples
4.19 IF students are proficient with graphing quadratic equations, THEN proceed to the
next procedure for solving quadratic equations
4.20 Give students a 5 question graphing assessment
4.21 IF students have mastered graphing, THEN move on to completing the Square.
IF NOT, then reteach.
Procedure 5: Teach solving quadratic equations by Completing the square
5.1 Complete the square of a quadratic equation with a leading coefficient of 1
5.1.1 Divide your notepaper or notebook into three columns
5.1.2 Label the steps on the right hand side of your notepaper or notebook
5.1.3 Do your work in the middle of your paper
5.1.4 Write the essential question on the left hand side of your paper: Essential
Question, “How is completing the square used to solve a quadratic
equation?”
5.2 Write steps for completing the square
5.2.1 Step 1 – If the equation is not in standard form, THEN re-arrange the
terms in standard form
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144
5.2.1.1 Reason: Because it gives students consistency and therefore write
the equation in standard form
5.2.2 IF the equation can be factored at this point, THEN tell students to solve
by factorization
5.2.3 Step 2 – Pull the constant
5.2.3.1 Isolate the constant on the opposite side
5.2.3.2 IF the constant is already isolated, THEN skip step 2
5.2.4 Work on either side of the equal sign
5.2.4.1 Reason: Because students should feel constrained to have
everything on the left
5.2.5 Sing: “half of b squared, add it to both sides” (while drumming)
5.2.6 Take and add it to both sides
5.2.7 Sing to students again: “half of b squared, add it to both sides” (while
drumming)
5.2.7.1 Tell students: “Let us sing, “half of b squared, add it to both sides”
(while drumming)
5.2.7.2 Sing together: “half of b squared, add it to both sides” (while
drumming)
5.2.7.3 IF teacher sings, THEN teacher asks students to sing with her
5.2.7.4 IF students sing, THEN show them how to do it
5.2.8 Tell students: “We are taking half of b”
5.2.8.1 Show students what half of something means, say half of $4 is $2,
half of $12 is $6
5.2.8.2 Practice with students: half of 6, half of 10 …
5.2.8.3 Check for understanding with students writing the answers on their
individual whiteboards and lifting them up to show the teacher
5.3 Start with an expression with a coefficient of 1 for x
2
, x
2
+ bx to complete the square
5.3.1 Give an example x
2
+ 6x, start with an even b-term
5.3.2 Complete the square by dividing 6 by 2 to get 3 and then square 3 to get 9:
x
2
+ 6x + 9 = (x + 3)
2
5.3.3 Show how to complete the square using algebra tiles
5.3.4 Explain to students that 9 6 6
2 2
+ + ≠ + x x x x but just showing the process
of completing the square
5.3.5 Show students more completing the square: x
2
+ 4x, to complete the
square, add the square of
2
4
, which is 2
2
= 4 and the expression becomes
x
2
+ 4x + 4. Therefore add 4 squares to complete the square
5.3.6 Show students another example: x
2
+ 8x, to complete the square, add the
square of
2
8
, which is 4
2
= 16 and the expression becomes x
2
+ 8x + 16.
Therefore add 16 squares to complete the square.
5.4 Introduce students to a quadratic equation to solve using by the completing the square
procedure
5.5 Show students how to write the quadratic equation in the form x
2
+ bx = c
2
2
⎟
⎠
⎞
⎜
⎝
⎛b
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145
5.5.1 Give students an example, like
5.5.2 Teacher says: “b = 10, take half of 10”
5.5.3 Teacher says: “IF I say half of b, THEN you say the answer”
5.5.4 Teacher says: “IF I say half of 10, THEN you say 5!”
5.5.5 Teacher says: IF I say 5 squared, THEN you say 25!”
5.5.6 Teacher says: “IF I say add it to both sides, THEN you add it to both
sides”
5.5.7 Add 25 to both sides of the equation:
5.5.8 Remind students the song: “half of b squared, add it to both sides”
5.5.9 Tell students: “We squared it, so the title of completing the square. We are
making it squared so that we can write it as a quantity squared”
5.5.10 Tell students: “x was squared to get x-squared and 5 was squared to get
25”
5.5.11 Take square root to undo squares
5.5.12 Factorize the left hand side: factors of x
2
are x and x and factors of 25 are 5
and 5. So,
5.5.13 IF you take the square root of one side, THEN you must take the square
root of the other side
5.5.14 Take the square root of both sides:
5.5.15 Solve for x from
5.5.15.1 Tell students: “Let’s look at our essential question: How do
we use completing the square to solve quadratic equations?”
5.5.16 Circle x:
5.5.17 Isolate x by itself
5.5.18 Subtract 5 from both sides:
5.5.19 Box the answer: x = 0 or -10 and write solutions, roots, x-intercepts and
zeros of the quadratic equation
5.6 Show students another example following the steps shown on step 5.5
5.7 Check for understanding by giving students one problem at time to do on their
whiteboards in pairs
5.8 IF students are not proficient solving quadratic equations with a coefficient of 1 for x
2
by completing the square procedure, THEN show them one more example.
5.9 IF students are proficient solving quadratic equations with a coefficient of 1 for x
2
by
completing the square procedure, THEN introduce an equation with an a-value
greater than 1
5.10 Introduce an equation with an a-value greater than 1
0 10
2
= + x x
25 25 10
2
= + + x x
( ) 25 5
25 25 10
2
2
= +
= + +
x
x x
( )
5 5
25 5
2
± = +
± = +
x
x
5 5 ± = + x
5 5 ± = + x
10 0
5 5 5 5
5 5
5 5 5 5
− =
− − − + =
− ± =
− ± = − +
or x
or x
x x
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146
5.10.1 Step 1 – IF the equation is not in standard form, THEN re-arrange the
terms in standard form [see step 5.2]
5.10.2 IF the a-value is not equal to one, THEN divide both sides by a
5.10.3 Tell students: “It is going to be a challenge because you may start dealing
with a b-value that is a fraction or an odd number”
5.10.4 Repeat steps 5.2.1 through 5.2.7
5.11 Teach students easy ways to remember the steps. Singing seems to work all the
time
5.12 Teach students how to use algebra tiles to complete the square – see Figure E7
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Completing the square using Algebra tiles
Students use tiles to model:
Find the missing tile: 𝑥
!
+2𝑥+𝐶
Find tiles Arrange the tiles to form a square
Determine tiles needed to complete the square
Students explore patterns given the following table:
𝑥
!
+2𝑥+𝐶 𝑥+1 𝑥+1 𝑥+1
!
:𝐶= 1
𝑥
!
+4𝑥+𝐶 𝑥+2 𝑥+2 𝑥+2
!
:𝐶= 4
𝑥
!
+6𝑥+𝐶 ? ?
After students have progressed through enough examples with algebra tiles, they should be able to verbalize that the pattern
developing is taking the b-term and dividing it by 2, then squaring that quantity to get the missing number C to complete the
square. Once students understand 𝐶=
!
!
!
, then introduce other equations such as 𝑥
!
−8𝑥= 9 and complete the square to
solve for x.
Figure E7: Completing the square
x
2
x
2
x x
x
x
x
2
x
1
x
2
x
x
1
Missing tile
𝑥
!
+2𝑥+1= (𝑥+1)(𝑥+1)
= (𝑥+1)
!
Therefore, C = 1
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5.13 IF it is about solving quadratic equations, THEN the quadratic formula is the
fallback method, it works for every quadratic equation
5.14 Give students 3 problems to solve by completing the square, and 1 problem that is
already solved but solution steps are out of order and students must order the steps
correctly, and the proof of the quadratic formula by completing the square with steps
out of order where students must correctly order the steps of the proof.
5.15 IF students have mastered completing the square, THEN go on to solving
quadratics using square roots. IF NOT, THEN reteach.
Procedure 6: Solving quadratics using the square root method
6.1 IF the equation is of the form , THEN the square root method
is appropriate
6.2 Show students how to solve x
2
= 9
6.3 Take the square root of both sides, , which gives .
6.4 Write x = -3 or 3 which are the solutions, x-intercepts, zeros and also the roots of the
quadratic equation x
2
= 9.
6.5 Show another example without a perfect square, solve
6.5.01 Add 8 to both sides, to isolate x
2
: x
2
– 8 + 8 = 0 + 8
6.5.02 Simplify: x
2
= 8
6.5.03 Take the square root of both sides:
6.5.04 Write and these are the roots, zeros, solutions or x-
intercepts of this quadratic equation x
2
– 8 = 0.
6.6 Show another example: x
2
– 16 = 0
6.6.01 Use factorization and find the sum and difference of products
6.6.01.1 Use zero product property, IF AB = 0, THEN A = 0
or B = 0
6.6.01.2 IF x
2
– 16 = 0, THEN (x – 4)(x + 4) = 0
6.6.01.3 Solve: IF (x – 4)(x + 4) = 0, THEN x – 4 = 0 or x +
4 = 0
6.6.01.4 Simplify: IF x – 4 = 0 or x + 4 = 0, THEN x = 4 or x
= -4
6.6.02 Use square roots to solve: x
2
– 16 = 0
6.6.02.1 Add 16 to both sides of the equal sign
6.6.02.2 Solve x
2
= 16
6.6.02.3 Take the square root of both sides
6.6.02.4 Write , which is the same solution.
6.7 Remind students: “Those problems that they did in factoring are very similar to our
square roots problems”
6.8 IF a quadratic equation is missing the b-term (middle term), THEN solve using the
square root procedure
0
2 2
= − = c ax or c ax
9
2
± = x 3 ± = x
0 8
2
= − x
2 2
2 4
2 4
8
2
± =
± =
• ± =
± =
x
x
x
x
2 2 2 2 − = or x
4 ± = x
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6.9 IF students are not proficient using the square root procedure, THEN show them two
more examples
6.10 Check for understanding with students writing on their whiteboards and showing
the teacher their solution
6.11 IF students are proficient with solving quadratic equations with the square root
procedure, THEN proceed to the application of all these procedures that students have
been learning
6.12 Give students a 5 question assessment (open ended quadratic equation problems
to solve)
6.13 IF students have mastered completing the square, THEN go to application. IF
NOT, reteach.
Procedure 7: Teach Application of these methods of solving quadratic equations to solve
real-life problems
7.1 Apply knowledge of solving quadratic equation to real-life problems
7.2 IF students have been taught the skill base for solving quadratic equations, THEN spend
two to three days where students practice in class real-life problems
7.3 Write (Project) a practice problem on the board
7.4 Choose the best procedure for solving the quadratic problem
7.5 Use the three-column table on the whiteboard as a graphic organizer
7.6 Make notes on it on all discussions as you solve the problem
7.7 Give students a couple of minutes to solve the problem
7.7.1 Ask students aloud: “Which way did you solve it?”
7.7.2 Go back to the three-column table and make notes
7.8 Ask students: “Was is it most convenient to use factoring?” “Why?”
7.9 Ask students: “Can I use square roots?” “Why?” (DOK 3)
7.9.1 IF a student solved it really quickly, THEN ask, “What did you do? Which
method did you use?”
7.9.1.1 IF a student took a little bit longer, THEN ask, “What did you do? Which
method did you use?”
7.9.1.2 Go back to the graphic organizer and edit it with specifics of what you did
7.10 IF the quadratic equation does not have a b-term, THEN use the square roots
method”
7.10.1 Reason: Because it is the easiest. It is the quickest
7.11 IF the coefficient of x
2
is greater than 1 or not factorable, THEN use the quadratic
formula
7.11.1 Reason: Because you do not have to list all possible different factors and then
finally find the problem is not factorable while the quadratic formula always
works
7.12 Take about two to three days doing this with students. Practicing to build their
confidence
7.13 Assess students on their skills
7.13.1 Assess students on solving quadratic equations using a method/procedure of their
choice.
7.13.2 Give students two problems, where they have to use a specific method to solve
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7.14 IF students can solve basic quadratic equations, THEN introduce practical real-
life problems.
7.15 IF students can solve a quadratic equation, THEN ask them to do different things,
like find the width of a parabolic disc
7.16 Give students problems that involve projectiles
7.17 IF a rocket is launched, THEN what would be the maximum height, or how long
will it take to reach the ground?
7.17.1 Make connection: maximum height of rocket corresponds to vertex of a parabola;
time rocket takes to reach the ground is the difference between the x-intercepts of
the parabola. These are practical applications of solving quadratic equations
7.18 Demonstrate drawing a picture that shows the parabolic path of the projectile
7.18.1 IF given an application problem (word problem), THEN draw a picture to
represent the story
7.18.2 Draw a picture back to the graph
7.18.3 Tell students: “This is the skill you learned to do, when we were graphing”
7.19 Draw a picture always
7.20 Relate back to previous lessons because quadratic equations are not an isolated
unit
7.21 Make students understand why they are solving quadratic equations
7.21.1 Make connections for students! For example: When solving a problem involving
the jumping path of a kangaroo, impose a graph that shows the horizontal time
and vertical distance by labeling the x- and y-axis, Draw in the axis of symmetry
and vertex reminding students that the kangaroo reached a specific height (y) after
so many seconds (x). The students can see on the graph that the maximum height
reached by the kangaroo is at the vertex. Talk about the height the kangaroo
started at (was there a y-intercept other than 0?) and then where he ended up.
7.22 Remind students a, b, and c in the quadratic equation, always relates back to the
graph or real-life application problem
7.23 Ask students aloud: “What does it mean to solve for x?”
7.23.1 Ask students: “What is the significance of finding x on the graph? What does it
mean?”
7.23.2 Know solving for x is finding solutions, the roots, the zeros and they are also the
x-intercepts of the graph of the quadratic equation
7.24 Give students a UNIT assessment.
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Appendix F
Data Analysis Spreadsheet
Gold Standard Protocol Procedures: Action and Decision Steps
Type SME Steps Alignment
Final Gold Standard Protocol Data
Analysis
A(P3) B(P4) C(P1) D(P2) A D
Procedure 1. Review linear
equation to Activate prior knowledge
34 8
1 A 1.1 Give students an overview of
the unit of solving quadratic equations.
(P1)
0 0 1 0 1
2 A 1.1.1 Tell students: “I’m going to
teach you how to solve quadratic
equations” (P1)
0 0 1 0 1
3 A 1.1.2 Tell students: “There are
multiple methods for solving quadratic
equations: factorization, graphing, the
quadratic formula, the square root
method, and completing the square
(P1)
0 0 1 0 1
4 A 1.1.3 Tell students: “Some of these
methods will work better for some of
the quadratic equations. Some
quadratic equations are perfect square
binomials and therefore will be easy to
recognize that it can be factored and
solved, while other quadratic equations
may be missing a “b” value and so it
will be useful to use the square root
method. Other quadratic equations
may be factorable but
have many factors to try and so
quadratic formula will be quickest”.
(P1)
0 0 1 0 1
5 A 1.2 Draw a full page size Tree map
(P1)
0 0 1 0 1
6 A 1.2.1 Relate solving quadratic
equations to solving linear equations
(P1)
0 0 1 0 1
7 A 1.2.1.1 Reason: Get students to see
there are many ways to solve a
quadratic equation by always referring
back to the tree map and these
procedures are related to solving linear
0 0 1 0 1
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152
equations (P1)
8 A 1.3 Put linear patterns and linear
equations of the form y = mx + b on the
screen over the computer or write them
on the board (P2, P3, P4)
1 1 0 1 3
9 A 1.3.1 Show students how to solve
linear equations step-by-step (P2)
0 0 0 1 1
10 A 1.3.2 Draw a two-column table on
the board (P3, P4)
1 1 0 0 2
11 A 1.3.3 Show students how to
generate a table of values from patterns
or linear equations (see Appendix E –
P3) (P3)
1 0 0 0 1
12 A 1.3.4 Plot the points on an x-y
coordinate plane drawn on the white
board (P4)
0 1 0 0 1
13 A 1.3.5 Play around with the linear
equations by changing the slope (m)
and/or changing the y-intercept (b) (P4)
0 1 0 0 1
14 D 1.3.6 IF students have access to
graphing calculators or graphing
software, THEN give opportunities to
check the graph they have made
the linear equations from the table of
values (P3)
1 0 0 0 1
15 D 1.3.7 IF you show students a
problem or two, THEN give them a
few to try individually to check for
understanding (P2)
0 0 0 1 1
16 D 1.3.8 IF students are not proficient
in solving linear equations, THEN
reteach the concept (P2)
0 0 0 1 1
17 A 1.3.9 Ask students randomly to
come to the board to show that they
have successfully completed the
problem (P2)
0 0 0 1 1
18 A 1.3.10 Assess students by walking
around the room to get a visual of what
they are doing and that they are
communicating using math language in
their groups (P2)
0 0 0 1 1
19 D 1.3.11 IF you see that students are
showing that they can do it on the
board and you are walking around the
room and making sure that students are
understanding from what you can see,
0 0 0 1 1
THE USE OF COGNITIVE TASK ANALYSIS
153
THEN continue to progress with the
lesson of activating prior knowledge
(P2)
20 A 1.4 Review with students how to
multiply and divide rational numbers
(P1)
0 0 1 0 1
21 A 1.4.1 Give examples: 3*4 = 12; -
2*5 = -10; 2*-5 = -10; and (-2)*(-3) =
6 (P1)
0 0 1 0 1
1.4.1.1 Reason: The intention is to
remind students about the rules for
multiplying integers since factoring
quadratics assumes students can factor
constant values (P1)
22 A 1.4.2 Define a factor to students:
“Factors are numbers you can multiply
together to get another number” (P1).
0 0 1 0 1
23 A 1.5 Review with students how to
factor a whole number (P1)
0 0 1 0 1
24 A 1.5.1 Use factor trees (P1) 0 0 1 0 1
25 A 1.5.2 Show students the factors of 6:
1 and 6, and 2 and 3 are factors
because the product of each pair is 6
(P1).
0 0 1 0 1
26 A 1.5.3 Give students a number to
factorize, for example: factorize 18
(P1)
0 0 1 0 1
27 A 1.5.3.1 Show students the factors of 18
are: 1 and 18, 2 and 9, 3 and 6, -1 and -
18, -2 and -9, and -3 and -6 (P1)
0 0 1 0 1
28 A 1.5.3.2 Show students that the product
of these factors is 18 (P1)
0 0 1 0 1
29 A 1.5.4 Remind students the rules for
multiplying integers (P1)
0 0 1 0 1
30 A 1.5.4.1 Multiply a positive number by a
positive number the product is another
positive number; multiply a negative
number by another negative number
the product is positive while the
product of a positive number by a
negative number is a negative number
(P1)
0 0 1 0 1
31 D 1.5.5 IF you are factorizing a
positive number, THEN get two
positive factors or two negative factors
(P1)
0 0 1 0 1
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154
32 D 1.5.6 IF you are factorizing a
negative number, THEN get one
positive factor and one negative factor.
Be sure that the sign of the
greater factor matches the sign of the
middle term (P1)
0 0 1 0 1
33 A 1.6 Review with students how to
multiply polynomials (P1)
0 0 1 0 1
34 A 1.6.1 Teach students exponent rules
(P1)
0 0 1 0 1
35 A 1.6.1.1 Remind students, for example
that x times x equals x
2
(P1)
0 0 1 0 1
36 A 1.6.1.2 Factorize x
3
or x
2
into factors:
x
3
= (x)(x)(x) and x
2
= (x)(x) (P1)
0 0 1 0 1
37 A 1.6.1.3 Show students that x + x = 2x
and (x)(x) = x
2
on the whiteboard (P1)
0 0 1 0 1
1.6.1.3.1 Reason: To know the
difference between addition and
multiplication when factorizing
(breaking down) polynomials
into factors (P1)
38 A 1.7 Review with students how to
factor linear expressions (P1)
0 0 1 0 1
39 A 1.7 Review with students how to
factor linear expressions (P1)
0 0 1 0 1
40 A 1.7.2 Give students another linear
expression to practice factorizing (P1)
0 0 1 0 1
41 D 1.7.3 IF students are not proficient
factoring linear expressions, THEN
show more examples like in step 1.7.1
(P1)
0 0 1 0 1
42 D 1.7.4 IF students are proficient
factoring linear expressions, THEN
introduce solving quadratic equations
by factoring (P1)
0 0 1 0 1
Procedure 2: Teach solving
quadratic equations by factoring
73 17
43 A 2.1 Remind students what a “factor” is
(P1)
0 0 1 0 1
44 A 2.1.1 Define a factor to students
again (line 1.4.2): “Factors are numbers
you can multiply together to get
another number” (P1)
0 0 1 0 1
45 A 2.1.2 Explain (step 1.5) what you
will do when you factor a certain
1 0 1 0 1
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155
problem. When “factoring” we are
showing students another way to write
a product—as a multiplication
problem. Sometimes the factored form
will look like an expanded version of
the original problem. (P1, P3)
46 A 2.1.3 Give students a few factor tree
problems to practice, for example: find
the factors 24 (P1)
0 0 1 0 1
47 A 2.1.3.1 Factor 24: 24 = (2)(12) or (3)(8)
or (4)(6). At this point, label (2)(12) as
the “factored form” of 24 (P1)
0 0 1 0 1
48 A 2.1.3.2 Give students a number to
factor individually (P1)
0 0 1 0 1
49 D 2.1.3.3 IF some students are not
proficient in multiplication, then assign
multiplication flash cards for
homework practice, and
encourage those students to use
multiplication charts when factorizing
(P1)
0 0 1 0 1
50 D 2.1.3.4 IF students are not proficient in
factorizing, THEN the teacher does one
more as students follow along (P1)
0 0 1 0 1
51 D 2.1.3.5 IF students are proficient with
factoring numbers, THEN teach
students how to write a quadratic
expression in standard form, like x
2
+
5x + 6 (P1)
0 0 1 0 1
52 A 2.2 Give an example of the standard
form: x
2
+ 5x + 6 = 0 and a non-
example: x
2
+ 6 = -5x. Tell students, "
All the terms should be on one side of
the equal sign (P1, P2, P4)
0 1 1 1 3
53 A 2.3 Show students coefficients of the
quadratic equation. For example, the
coefficients of x
2
+ 5x + 6 = 0 are 1, 5,
and 6 (P2)
0 0 0 1 1
54 A 2.4 Teach students how to use
distributive property (P2)
0 0 0 1 1
55 A 2.4.1 Write an example of two
binomials on the board, for example (x
+ 2) and (x + 1) (P2)
0 0 0 1 1
56 A 2.4.2 Show students how to use
algebra tiles (P2, P3)
1 0 0 1 2
57 A 2.4.3 Find the area of the product of 1 0 0 1 2
THE USE OF COGNITIVE TASK ANALYSIS
156
these binomials: x
2
+ 2x + x + 2 (P2,
P3)
58 A 2.4.4 Write area x
2
+ 3x + 2 by
looking at tiles (P3)
1 0 0 0 1
59 A 2.4.5 Tell students: “Factoring is
how we undo the distributive property
(product of binomials) for example,
getting the binomials that gave the
product 3x
2
+ 4x – 15” (P2, P3)
1 0 0 1 2
60 A 2.5 Show students how to factorize x
2
+ 5x + 6 (P1, P2, P3, P4)
1 1 1 1 4
61 A 2.6 Show students a sum and product
table (P1)
0 0 1 0 1
62 A 2.6.1 Show students how to use a
sum and product table to find factors
(P1)
0 0 1 0 1
63 A 2.6.1.1 Tell students: “Draw a sum and
product table on your whiteboard” (P1)
0 0 1 0 1
64 A 2.6.1.2 Tell students: “Two numbers
have a sum of 5 and a product of 6.
With your partner, figure out which
numbers they are” (P1)
0 0 1 0 1
65 A 2.6.1.3 Tell students to do the problem
on their whiteboards and hold them up
(P1)
0 0 1 0 1
66 A 2.6.1.4 Scan across the room as
students raise their whiteboards
checking for understanding (P1)
0 0 1 0 1
67 D 2.6.1.5 IF students have not mastered
the use of sum and product tables to
factorize, THEN show students another
example. Like two numbers have a sum
of 8 and a product of 15, show them
how to find these two numbers (P1).
0 0 1 0 1
68 A 2.2.1.6 Repeat this procedure with
different problems until students are
proficient (P1)
0 0 1 0 1
69 D 2.2.1.7 IF students have mastered the
use of sum and product tables, THEN
show students how to factor the
original problem x
2
+ 5x + 6 (P1, P2)
0 0 1 1 2
70 A 2.2.1.8 Circle the term 6 which is the
constant in x
2
+ 5x + 6 and write the
word constant above the 6. IF you
circle the constant, THEN write the
word product underneath it (P1)
0 0 1 0 1
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157
71 A 2.2.1.8.1 Show students some
numbers that give a product of 6, write
these numbers in the sum and product
table. Some pairs are 1 and 6, and 2
and 3 (P1)
0 0 1 0 1
72 A 2.2.1.9 Circle the x-term, 5x and then
write sum underneath it (P1)
0 0 1 0 1
73 A 2.2.1.9.1 Use the pairs of numbers
in the sum and product table to
determine which pair, 1 and 6 or 2 and
3 adds up to 5 (P1)
0 0 1 0 1
74 A 2.2.1.9.2 Choose 2 and 3 (P1) 0 0 1 0 1
75 A 2.2.1.10 Circle the x
2
-term. Tell
students: “ x times x is x
2
and therefore
x and x are the factors of x
2
” (P1)
0 0 1 0 1
76 A 2.7 Teach students how to use algebra
tiles, the X-BOX, and the Parenthesis
methods to factorize (P1, P2)
0 0 1 0 1
77 A 2.7.1 Use algebra tiles to factor
quadratic equations (P3)
1 0 0 0 1
78 A 2.7.2 Introduce quadratic equations
that can be factorized using algebra
tiles (P3)
1 0 0 0 1
79 A 2.7.2.1 Ask students: IF I have x
2
+ 3x
+ 2, THEN what tiles would I need?
(P3)
1 0 0 0 1
80 A 2.7.2.2 Ask students to gather tiles (P3) 1 0 0 0 1
81 A 2.7.2.3 Tell students that x
2
+ 3x + 2
represents an area (P1, P3)
1 0 1 0 2
82 A 2.7.2.4 Arrange tiles in a rectangle, to
find factors by looking at length and
width of the rectangle (P3)
1 0 0 0 1
83 A 2.7.2.5 Tell students: “The factors of x
2
+ 3x + 2 are (x + 2) and (x + 1)” (step
2.7.2.4) (P3, P4)
1 1 0 0 2
84 D 2.7.2.6 IF students are not yet
comfortable with factoring using
algebra tiles, THEN reteach the
concept as in step 2.7.2.4 (P3, P4)
1 1 0 0 2
85 D 2.7.3 If students are comfortable
with factoring using algebra tiles, then
introduce the X-BOX method for
factoring so that students may have an
alternative way of factorizing. For
example, factorize x
2
+ 3x + 2 using
the X-BOX method, see below (P3)
1 0 0 0 1
THE USE OF COGNITIVE TASK ANALYSIS
158
86 D 2.7.5 IF students are proficient
factorizing using the X-BOX method
(step 2.7.3), THEN proceed to teach
students the Parenthesis Method (P1)
0 0 1 0 1
87 A 2.8 Show students how to factorize a
quadratic expression using the
Parenthesis Method (P1)
0 0 1 0 1
89 A 2.8.1 Put two sets of parentheses,
( )( ) at the top for the binomials that
we are trying to factor this problem
into Then, at the bottom of the
parentheses, write the problem: (P1)
0 0 1 0 1
90 A 2.8.2 Tell students: “We are going
to find the factors of this problem, (P1)
0 0 1 0 1
91 A 2.8.3 Tell students: “Since factors of
the first term are x and x, put each on a
different parenthesis, since x times x is
x
2
, followed by 2 in the first
parenthesis and 3 on the second
parenthesis ” (P1, P3)
1 0 1 0 2
92 D 2.8.4 IF we factor, THEN the
product of these parts in the
parentheses have to match with the
original expression, (P1)
0 0 1 0 1
93 A 2.8.5 Give students another problem
to factorize (P1)
0 0 1 0 1
94 D 2.8.6 IF students are not proficient
with factoring, THEN show students
how to do it (P1)
0 0 1 0 1
95 A 2.8.6.1 Use the sum and product table.
Two numbers have a sum of 11 and a
product of 30. (P1)
0 0 1 0 1
96 A 2.8.6.2 Repeat the same process as in
2.3.1.1 through 2.3.1.4 to factorize (P1)
0 0 1 0 1
97 D 2.8.7 IF students are proficient with
factoring expressions, THEN introduce
them to solving quadratic equations
using the factorization method (P1)
0 0 1 0 1
98 A 2.8.7.1 Show students how to solve by
finding the value of x that satisfies this
equation. (P1)
0 0 1 0 1
99 A 2.8.7.2 Factorize the left hand side of
the equation (P1, P2, P4)
0 1 1 1 3
100 A 2.8.7.3 Put below the original quadratic
equation (P1)
0 0 1 0 1
101 A 2.8.7.4 Use the zero product property, 1 1 1 1 4
THE USE OF COGNITIVE TASK ANALYSIS
159
IF AB = 0, THEN A = 0 or B = 0 (P1,
P2, P3, P4)
102 D 2.8.7.5 IF , THEN (x + 2) = 0 or (x + 3)
= 0 (P1, P2, P4)
0 1 1 1 3
103 A 2.8.7.6 Solve: x + 2 = 0, subtract 2
from both sides of the equation: x + 2 -
2 = 0-2 therefore x = -2. And x + 3 = 0,
subtract 3 from both sides of the
equation: x + 3 -3 = 0 - 3 therefore x =
-3. (P1, P2, P4)
0 1 1 1 3
104 A 2.8.7.7 Solution is x = -2 or -3.
Solution of a quadratic equation is also
called the x-intercepts, zeros, and roots.
(P1)
0 0 1 0 1
105 A 2.8.8 Give students another example
to practice: (P1)
0 0 1 0 1
106 A 2.8.9 Use the sum and product table
to factorize the left hand side of the
quadratic equation (P1)
0 0 1 0 1
107 A 2.8.9.1 Factorize to (x +5)(x + 6) = 0
(P1)
0 0 1 0 1
108 A 2.8.9.2 Use the zero product property,
IF AB = 0, THEN A=0 or B=0 (P1, P2)
0 0 1 1 2
109 A 2.8.9.3 Solve: IF (x + 5)(x +6) = 0,
THEN (x + 5) = 0 or (x + 6) = 0 (P1,
P2)
0 0 1 1 2
110 A 2.8.9.4 IF x + 5 = 0, THEN x = -5 and
IF x + 6 = 0, THEN x = -6 (P1, P2)
0 0 1 1 2
111 A 2.8.9.5 Solution is x = -5 or -6. Tell
students: “The solution of a quadratic
equation is also called x-intercepts,
zeros, and roots. (P1)
0 0 1 0 1
112 A 2.9 Give students 3-5 question
assessment (open ended) (P1)
0 0 1 0 1
113 D 2.9.1 IF students have mastered
factoring when the leading coefficient
is 1, THEN move on to factoring where
the leading coefficient is other than 1.
IF not, THEN reteach. (P1, P2)
0 0 1 1 2
114 A 2.10 Write a quadratic equation like on
the board (P1, P2, P3)
1 0 1 1 3
115 A 2.10.1 Identify a = 6, b = 1, and c =
-1 (P1, P2, P3)
1 0 1 1 3
116 A 2.10.2 Use the X-Box method, for
example (Step 2.10.3) (P2, P3)
1 0 0 1 2
117 A 2.10.3 Make a big cross (X) 1 0 1 1 3
THE USE OF COGNITIVE TASK ANALYSIS
160
underneath the equation. Inside the top
of the X, write the product of a and c
and inside the bottom of the X, write b
(the sum) (P1, P2, P3)
118 A 2.10.4 Find factors with a product of
-6 and a sum of 1, as shown (P1, P2,
P3)
1 0 1 1 3
119 A 2.10.5 Tell students: “There are
many pairs that will give a product of -
6 but they are not all going to give a
sum of the middle coefficient, 1” (P2)
0 0 0 1 1
120 A 2.10.6 Show students the factors of
-6 that give a sum of 1 are: 3 and -2
(P1, P2, P3)
1 0 1 1 3
121 A 2.10.7 Write the expanded form of
the quadratic equation: (P3)
1 0 0 0 1
122 A 2.10.8 Factor by grouping (P3) 1 0 0 0 1
123 A 2.10.9 Find common factors: (P1,
P2, P3)
1 0 1 1 3
124 A 2.10.10 Check by multiplying using
the box (P3)
1 0 0 0 1
125 A 2.10.11 Write out to confirm the
product is the original quadratic
equation (P3)
1 0 0 0 1
126 A 2.10.12 Solve the quadratic equation
for x using the zero product property
(ZPP): If (2x + 1)(3x – 1) = 0, THEN
2x + 1 = 0 or 3x – 1 = 0 (P2, P3)
1 0 0 1 2
127 A 2.10.13 Solve 2x + 1 = 0 to get x = -
1/2 and 3x – 1 = 0 to get x = 1/3 (P3)
1 0 0 0 1
128 A 2.11 IF teacher does one example on
the board, THEN give students one
problem to try. Walk around the room
to monitor what students are doing to
check for understanding (P1, P2, P3)
1 0 1 1 3
129 D 2.12 IF students have not mastered
factoring, THEN reteach the concept
(P1, P2)
0 0 1 1 2
130 D 2.13 IF students have mastered
factoring, THEN give proceed to give
assessment (P1)
0 0 1 0 1
131 A 2.14 Give students 5-10 question
assessment (open ended) on factoring.
(P1)
0 0 1 0 1
132 D 2.14.1 IF students have mastered
factoring, THEN move on to solving
0 1 1 1 3
THE USE OF COGNITIVE TASK ANALYSIS
161
quadratic equations using the quadratic
formula. IF NOT, THEN reteach. (P1,
P2, P4)
Procedure 3: Teach solving
quadratic equations by using the
quadratic formula
57 18
133 A 3.1 Sing to students a song on the
quadratic formula (P1, P2, P3, P4)
1 1 1 1 4
134 A 3.1.1 Sing: “x equals negative b,
plus or minus square root of b squared
minus 4ac, all over 2a”; Or teach
students to memorize the quadratic
formula using this phrase: “A negative
boy could not decide whether or not to
go to a radical party. He decided to be
square and he missed out on 4
awesome chicks. The party was all
over at 2 am.” (P1, P2, P3, P4)
1 1 1 1 4
135 A 3.1.2 Tell students: “The quadratic
formula is a “catchall” for solving
quadratic equations, it works every
time” (P1, P4)
0 1 1 0 2
136 A 3.1.3 Tell students: “When in doubt
while solving quadratic equations,
revert back to the quadratic formula,
that is the reason for singing the song
every day, multiple times during the
period while using the quadratic
formula” (P1, P3, P4)
1 1 1 0 3
137 A 3.2 Write the quadratic equation in
standard form, before using the
quadratic formula (P1)
0 0 1 0 1
138 D 3.2.1 IF the equation is not in
standard form, THEN the equation may
be misleading because either the value
of a, b, or c may not be correct (P1, P2)
0 0 1 1 2
139 A 3.3 Show students how to write a
quadratic equation in standard form
(P1, P2)
0 0 1 1 2
140 A 3.3.1 Write an example on the board
that has the x-term on the other side,
(P1)
0 0 1 0 1
141 A 3.3.2 Show students that the
standard form would be or (P1, P2)
0 0 1 1 2
142 A 3.3.3 Choose which side of the 0 0 1 0 1
THE USE OF COGNITIVE TASK ANALYSIS
162
equal sign to take all the terms to get
the equation to standard form and pay
attention to the sign change (P1)
143 D 3.3.4 IF there are terms on both
sides of the equal sign, THEN the signs
will be different when all the terms a
collected on the same side (P1)
0 0 1 0 1
144 A 3.4 Tell students: ‘Make three-columns
in your notebook’ (P1)
0 0 1 0 1
145 A 3.4.1 Write the essential question
(EQ): “How is the quadratic formula
used to solve a quadratic equation” on
the left-hand side (P1).
0 0 1 0 1
146 A 3.4.2 Write the steps of using the
quadratic formula on the right-hand
side (P1)
0 0 1 0 1
147 A 3.4.3 Do the steps along with the
students (P1)
0 0 1 0 1
148 A 3.4.3.1 Step 1: Write: Standard form
(P1, P2, P3)
1 0 1 1 3
149 A 3.4.3.2 Write: ax
2
+ bx + c = 0 or ax
2
+
bx + c = y underneath “Standard form”
(P1)
0 0 1 0 1
150 A 3.4.3.3 Step 2: Label a, b, and c. To the
left of a, write “opening” (P1)
0 0 1 0 1
3.4.3.3.1 Reason: So students know
when they graph it, a is going tell them
the direction the graph will open, either
“up” or “down” (P1)
151 D 3.4.3.4 IF a equals a negative number,
THEN the graph opens down (P1)
0 0 1 0 1
152 D 3.4.3.5 IF a equals a positive number,
THEN graph opens up (P1)
0 0 1 0 1
153 A 3.4.3.6 Write y-intercept next to c (P1) 0 0 1 0 1
3.4.3.6.1 Reason: So that students
know this is not in isolation (P1)
154 A 3.4.3.7 Write the quadratic formula, as
you sing along with students the
quadratic formula song: “x equals
negative b, plus or minus square root of
b squared minus 4ac, all over 2a” (P1,
P2)
0 0 1 1 2
155 A 3.4.3.7.1 Do not substitute the
values of a, b, and c into the formula
before writing the formula out (P1, P2)
0 0 1 1 2
156 A 3.4.3.7.2 Write the quadratic 0 0 1 1 2
THE USE OF COGNITIVE TASK ANALYSIS
163
formula with parenthesis below the
quadratic formula. Sing the remix
together with students: “x equals
negative parenthesis plus or minus
square root of parenthesis squared
minus four parenthesis parenthesis all
over two parenthesis” as you write it
(P1, P2)
157 A 3.4.3.7.3 Get students in the habit of
writing the formula every time (P1, P2)
0 0 1 1 2
158 D 3.4.3.7.4 IF students substitute
directly without writing the formula
first, THEN they do not get points for it
(P1)
0 0 1 0 1
3.4.3.7.4.1 Reason: Because students
must use parenthesis when substituting
into the formula (P1)
159 A 3.4.3.8 Tell students: “To substitute is
to replace” (P1)
0 0 1 0 1
160 A 3.4.4 Show students how to do the
four steps in one (P1)
0 0 1 0 1
161 A 3.4.4.1 Do order of operations, do
PEMM (P1)
0 0 1 0 1
162 A 3.4.4.2 Substitute the value of b into
the first Parenthesis, negative ( ) (P1)
0 0 1 0 1
163 A 3.4.4.3 Substitute the value of b again
into the Exponent part ( )
2
and circle
the b-squared part (P1)
0 0 1 0 1
164 A 3.4.4.4 Substitute the values of a and c
into third and fourth parenthesis
respectively, negative 4( )( ) and circle
-4ac followed by putting M over it to
indicate Multiplication will take place
(P1)
0 0 1 0 1
165 A 3.4.4.5 Circle the denominator, 2a,
substitute the value of a into the
parenthesis 2( ) and put M over it to
indicate Multiplication will take place
(P1)
0 0 1 0 1
166 A 3.4.4.6 Circle what is under the square
root sign (P1)
0 0 1 0 1
167 D 3.4.4.7 IF you are going to add the
numbers b2 and 4ac, THEN put A for
addition above it and add the quantities
(P1)
0 0 1 0 1
168 D 3.4.4.8 IF you are going to subtract the 0 0 1 0 1
THE USE OF COGNITIVE TASK ANALYSIS
164
quantities b2 and 4ac, THEN put S for
subtraction above it and subtract the
quantities (P1)
169 A 3.4.4.9 Write D for Division but wait
on D (P1)
0 0 1 0 1
170 A 3.4.4.10 Take the square root of the
quantity b2 – 4ac (P1)
0 0 1 0 1
171 A 3.4.4.11 Divide the numerator by the
denominator and then write x equals
the result of the division (P1)
0 0 1 0 1
3.4.4.11.1 Reason: Because we are
going through PEMMDAS, the order
of operations (P1)
172 A 3.4.4.12 Circle using a red marker
on the board and write the step being
done (P1)
0 0 1 0 1
173 A 3.4.4.12 Circle using a red marker on
the board and write the step being done
(P1)
0 0 1 0 1
174 A 3.4.4.13 Draw a box around x equals
(P1)
0 0 1 0 1
175 D 3.4.4.14 IF x has two solutions, THEN
the graph intercepts the x-axis twice
(P1)
0 0 1 0 1
176 D 3.4.4.15 IF x has only one solution,
THEN the graph touches the x-axis
once (P1)
0 0 1 0 1
177 D 3.4.4.16 IF x has no solution, THEN
the graph does not touch the x-axis (P1)
0 0 1 0 1
178 A 3.4.5 Show an example: (P1, P2) 0 0 1 1 2
179 A 3.4.6 Substitute the values of a, b, and
c from the standard quadratic equation,
into the formula with the parenthesis in
place of a, b, and c (P1, P2, P4)
0 1 1 1 1
180 A 3.4.6.1 Substitute the value of b from
the standard quadratic equation, 2x
2
-
3x - 5 = 0 into the first and second
parenthesis (P1, P2)
0 0 1 1 2
181 D 3.4.6.2 IF b was negative, THEN
substitute it together with its sign (P1)
0 0 1 0 1
182 A 3.4.6.3 Tell students: “Pay attention to
the ( )
2
" (P1)
0 0 1 0 1
183 D 3.4.6.3.1 IF b is a negative number,
THEN negative times negative is
positive. (P1)
0 0 1 0 1
184 D 3.4.6.3.2 IF b is a positive number, 0 0 1 0 1
THE USE OF COGNITIVE TASK ANALYSIS
165
THEN positive times positive is
positive (P1)
185 D 3.4.6.4 IF any number is squared,
THEN the product is always positive
(P1)
0 0 1 0 1
186 A 3.4.6.5 Substitute values of a, the c and
a again into the remaining parenthesis
(P1)
0 0 1 0 1
187 A 3.4.6.6 Simplify (P1, P2) 0 0 1 1 2
188 A 3.1.1.1 Tell students: “The plus or
minus 7 means we have two solutions
for this quadratic equation (P2)
0 0 0 1 1
189 A 3.1.1.2 Solution: x = 10/4 or x = -4/4
which are simplified to x = 2.5 or -1.
These are also called roots, x-intercepts
or zeros of the quadratic equation (P1,
P2)
0 0 1 1 2
3.4.7 See Table 1 (P1)
190 A 3.4.7.1 Tell students: "b
2
- 4ac is the
discriminant and it helps determine the
number of solutions of a quadratic
equation” (P1, P2, P3, P4)
1 1 1 1 4
191 D 3.4.7.2 IF the discriminant is positive,
THEN the parabola intercepts the x-
axis twice (P1, P2, P3, P4)
1 1 1 1 4
192 A 3.4.7.2.1 Draw the graph to show
students the parabola intercepts the x-
axis twice (P1, P4)
0 1 1 0 2
193 D 3.4.7.3 IF the discriminant is zero,
THEN the parabola touches the x-axis
once and turns around (P1, P2, P3, P4)
1 1 1 1 4
194 A 3.4.7.3.1 Draw the graph to show
students the parabola touches the x-axis
once (P1, P4)
0 1 1 0 2
195 D 3.4.7.4 IF the discriminant is negative,
THEN the parabola does not touch the
x-axis (P1, P2, P3, P4)
1 1 1 1 4
196 A 3.4.7.4.1 Draw the graph to show
students the parabola does not intersect
the x-axis (P1, P4)
0 1 1 0 2
197 D 3.4.7.5 IF students get the solution,
THEN they have to write all the names
every time: x-intercept(s), solution(s),
zero(s) and root(s) of the quadratic
equation (P1)
0 0 1 0 1
198 A 3.5 Show students b
2
- 4ac is the 0 0 1 0 1
THE USE OF COGNITIVE TASK ANALYSIS
166
discriminant and is part of the
quadratic formula (P1)
199 A 3.4.7.5 IF students get the solution,
THEN they have to write all the names
every time: x-intercept(s), solution(s),
zero(s) and root(s) of the quadratic
equation (P1)
0 0 1 0 1
200 A 3.5.1 Point at the discriminant (P1) 0 0 1 0 1
201 A 3.6 Use the quadratic formula to solve
any quadratic equation for its roots, its
solution, and its zeros (P1, P2)
0 0 1 1 2
202 D 3.7 IF students are able to solve a
quadratic equation for its roots,
solutions, or its zeros, THEN they can
solve real-life application problems
like vertical motion problems (P1)
0 0 1 0 1
203 D 3.8 IF students are not comfortable
using the quadratic formula, THEN
show them two more examples and
give them a few problems to practice
(guided practice) (P1, P3)
1 0 1 0 2
204 A 3.9 Give students 3-5 problems to solve
using the quadratic formula. (P1)
0 0 1 0 1
205 D 3.9.1 IF students have mastered
solving using the quadratic formula,
THEN begin teaching graphing. IF
not, THEN reteach. (P1, P3)
1 0 1 0 2
206 D 3.9.2 IF students are proficient using
the quadratic formula, THEN proceed
with the lesson to show students the
next procedure for solving quadratic
equations (P1)
0 0 1 0 1
Procedure 4: Teach solving
quadratic equations by graphing
47 17
207 A 4.1 Post an x-y coordinate plane on the
whiteboard throughout the unit of
quadratic equations (P1)
0 0 1 0 1
208 A 4.2 Solve all quadratic equations next
to the graph so that students make
connections and also see multiple
representations (P1)
0 0 1 0 1
209 A 4.3 Solve while relating back to the
graph because students have a hard
time connecting different
representations (P1)
0 0 1 0 1
THE USE OF COGNITIVE TASK ANALYSIS
167
210 A 4.4 Do not teach graphing of quadratic
equations and solving quadratic
equations using other procedures in
isolation (P1)
0 0 1 0 1
211 A 4.4.1 Go back and forth between
various methods of solving quadratic
equations and their graphs (P1)
0 0 1 0 1
212 A 4.4.2 Relate the x-intercepts of the
graph of a quadratic equation to its
solutions after solving using any of the
other procedures (P1)
0 0 1 0 1
213 A 4.5 Start with a quadratic equation in
standard form and reflects on the y-axis
(P1, P2, P3, P4)
1 1 1 1 4
214 A 4.6 Identify the parts that are obvious
based on the equation (P1, P3, P4)
1 1 1 0 3
215 A 4.6.1 Identify a the coefficient of x
2
, b
the coefficient of x and c, the constant
(P1, P3, P4)
1 1 1 0 3
216 D 4.6.2 IF a is positive, THEN the graph
(parabola) will open up (P1)
0 0 1 0 1
217 D IF a is negative, THEN the graph
(parabola) will open down (P1)
0 0 1 0 1
218 A 4.6.4 Write c, the y-intercept (P1, P4) 0 1 1 0 2
219 A 4.7 Teach how to find the axis of
symmetry (P1, P2)
0 0 1 1 2
220 A 4.7.1 Find axis of symmetry and relate
to the quadratic formula (P1)
0 0 1 0 1
221 A 4.7.2 Break apart the quadratic
formula and show students the axis of
symmetry (P1, P2, P4)
0 1 1 1 3
222 A 4.7.3 Write the axis of symmetry
formula and carefully explain to
students that this does not represent x-
intercepts (P1)
0 0 1 0 1
223 A 4.7.4 Tell students: “This is an x-value
and it is where on the x-axis the axis of
symmetry cuts through” (P1)
0 0 1 0 1
224 D 4.7.5 IF the quadratic equation has a
middle term bx, THEN the parabola
will not reflect over the y-axis (P2)
0 0 0 1 1
225 A 4.7.6 Substitute the values of a and b
into the axis of symmetry equation to
find the axis of symmetry (P1, P3, P4)
1 1 1 0 3
226 D 4.7.7 IF you fold a parabola in half
through the axis of symmetry, THEN
0 1 1 0 2
THE USE OF COGNITIVE TASK ANALYSIS
168
there are two identical parts (P1, P4)
227 D 4.7.8 IF there is a y-value to left of the
axis of symmetry, THEN there is an
equivalent y-value same distance from
the axis of symmetry on the right of the
axis of symmetry (P1)
0 0 1 0 1
228 D 4.7.9 IF you have the y-intercept on
one side of the axis of symmetry,
THEN there is another point at the
same height on the other side of the
parabola (P1)
0 0 1 0 1
229 A 4.8 Teach students how to find the
vertex (P1, P2, P3)
1 0 1 1 3
230 A 4.8.1 Use the axis of symmetry to find
the x-value of the vertex (P1, P2, P3)
1 0 1 1 3
231 A 4.8.2 Substitute the x-value of the
vertex into ax
2
+ bx + c = y to find the
y-value of the vertex (P1, P2, P3, P4)
1 1 1 1 4
232 A 4.8.3 Write the vertex in the form of (x,
y) coordinate point (P1, P2, P3, P4)
1 1 1 1 4
233 A 4.8.4 Point to students that the vertex is
the highest or lowest point of the
parabola (P3, P4)
1 1 0 0 2
234 A 4.9 Teach Graphing procedure (P1, P2,
P3, P4)
1 1 1 1 4
235 A 4.9.1 Draw an x-y coordinate plane (P1,
P2, P3, P4)
1 1 1 1 4
236 A 4.9.2 Draw a dotted line through the
axis of symmetry found in step 4.7.6
(P1, P2, P4)
0 1 1 1 3
237 A 4.9.3 Draw an x-y table of values. An
example is here below when the x-
value of the vertex is 0 (P1, P2, P3, P4)
1 1 1 1 4
238 A 4.9.4 Put the vertex coordinates at the
center of the table (P1, P2, P3, P4)
1 1 1 1 4
239 A 4.9.5 Choose an x-value to the left or to
the right of the vertex (P1, P3, P4)
1 1 1 0 3
240 A 4.9.6 Show students the mirror point(s)
(P1, P4)
0 1 1 0 2
241 A 4.9.7 Plot the vertex as found in step
4.8.3 (P1, P3, P4)
1 1 1 0 3
242 D 4.9.8 IF you find the vertex, THEN get
2 or 3 points on one side of the axis of
symmetry (P1, P3, P4)
1 1 1 0 3
243 D 4.9.9 IF you have 2 or 3 points on one
side of the axis of symmetry, THEN
1 1 1 0 3
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169
you will get 2 or 3 points on the
opposite side of the axis of symmetry
(P1, P3, P4)
244 D 4.9.10 IF you have the axis of
symmetry (x-value), THEN choose two
x-values to the left or right of the axis
of symmetry to substitute into the
original equation to find the
corresponding y-values (P1, P2, P3,
P4)
1 1 1 1 4
245 D 4.9.11 IF the parabola opens up, THEN
the vertex is a minimum (P1)
0 0 1 0 1
246 D 4.9.12 IF the parabola opens down,
THEN the vertex is a maximum (P1)
0 0 1 0 1
247 A 4.9.13 Tell students: “The vertex is
where our parabola opens from” (P1)
0 0 1 0 1
248 A 4.9.14 Remind students the graph
opens upwards or opens downwards
depending on the a-value (P1)
0 0 1 0 1
249 A 4.10 Connect the points to plot the
graph with a smooth curve (P1, P3, P4)
1 1 1 0 3
250 A 4.11 Label on the graph the vertex, y-
intercept, axis of symmetry and
direction of opening, up or down (P1)
0 0 1 0 1
251 A 4.12 Label the x-intercepts if they exist
(P3)
1 0 0 0 1
252 A 4.13 Show an example, x
2
+ 4x – 12 =
0 (P2)
0 0 0 1 1
253 A 4.13.1 Draw a two-column t-table of
values (step 4.9.3) (P2)
0 0 0 1 1
254 A 4.13.2 Identify the coefficients: a = 1, b
= 4, and c = -12 (P2)
0 0 0 1 1
255 A 4.13.3 Find the axis of symmetry (P2) 0 0 0 1 1
256 A 4.13.4 Put the axis of symmetry, x = -2
in the middle of the t-table and then
choose integers on either side of -2 that
are equidistant from the axis of
symmetry (P2)
0 0 0 1 1
257 A 4.13.5 Draw the axis of symmetry x = -
2 (P2)
0 0 0 1 1
258 A 4.13.6 Choose two x-values less than -2
and two x-values greater than -2: -4, -3,
-2, -1, and 0 (step 3.1.5) (P2)
0 0 0 1 1
259 A 4.13.7 Substitute these x-values into the
quadratic equation, x
2
+ 4x – 12 = 0 to
find the corresponding y-values (step
0 0 0 1 1
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170
4.8.2) (P2)
260 A 4.13.8 Plot the pairs of points on x-y
coordinate plane (P2)
0 0 0 1 1
261 A 4.13.9 Ask aloud: “How many times
does the graph of x
2
+ 4x – 12 = 0 cross
the x-axis?” (P2)
0 0 0 1 1
262 A 4.13.10 Show students the x-intercepts,
which are also the solutions of the
quadratic equation (P2)
0 0 0 1 1
263 D 4.14 IF the graph of a quadratic
equation intercepts the x-axis twice,
THEN the quadratic equation has two
real solutions (P2)
0 0 0 1 1
264 D 4.15 IF the graph of a quadratic
equation touches the x-axis once,
THEN the quadratic equation has one
real solution (P2)
0 0 0 1 1
265 D 4.16 IF the graph of a quadratic
equation does not touch the x-axis,
THEN the quadratic equation has no
real solution (P2)
0 0 0 1 1
266 A 4.17 Check for understanding by giving
students two quadratic equations to
graph (P1, P4)
0 1 1 0 2
267 D 4.18 IF students are not proficient with
graphing, THEN show them the
process with two more examples (P1,
P4)
0 1 1 0 2
268 D 4.19 IF students are proficient with
graphing quadratic equations, THEN
proceed to the next procedure for
solving quadratic equations (P1, P4)
0 1 1 0 2
269 A 4.20 Give students a 5 question
graphing assessment (P1, P4)
0 1 1 0 2
270 D 4.21 IF students have mastered
graphing, THEN move on to
Completing the Square. IF NOT, then
reteach. (P1, P4)
0 1 1 0 2
Procedure 5: Completing the square 57 10
271 A 5.1 Complete the square of a quadratic
equation with a leading coefficient of 1
(P1)
0 0 1 0 1
272 A 5.1.1 Divide your notepaper or
notebook into three columns (P1)
0 0 1 0 1
273 A 5.1.2 Label the steps on the right 0 0 1 0 1
THE USE OF COGNITIVE TASK ANALYSIS
171
hand side of your notepaper or
notebook (P1)
274 A 5.1.3 Do your work in the middle of
your paper (P1)
0 0 1 0 1
275 A 5.1.4 Write the essential question on
the left hand side of your paper:
Essential Question, “How is
completing the square used to solve a
quadratic equation?” (P1)
0 0 1 0 1
276 A 5.2 Write steps for completing the
square (P1)
0 0 1 0 1
277 A 5.2.1 Step 1 – If the equation is not in
standard form, THEN re-arrange the
terms in standard form (P1)
0 0 1 0 1
278 A 5.2.1.1 Reason: Because it gives
students consistency and therefore
write the equation in standard form
(P1)
0 0 1 0 1
279 D 5.2.2 IF the equation can be factored
at this point, THEN tell students to
solve by factorization (P1)
0 0 1 0 1
280 A 5.2.3 Step 2 – Pull the constant (P1) 0 0 1 0 1
281 A 5.2.3.1 Isolate the constant on the
opposite side (P1)
0 0 1 0 1
282 D 5.2.3.2 IF the constant is already
isolated, THEN skip step 2 (P1)
0 0 1 0 1
283 A 5.2.4 Work on either side of the equal
sign (P1)
0 0 1 0 1
5.2.4.1 Reason: Because students
should feel constrained to have
everything on the left (P1)
284 A 5.2.5 Sing: “half of b squared, add it
to both sides” (while drumming) (P1)
0 0 1 0 1
285 A 5.2.6 Take and add it to both sides
(P1)
0 0 1 0 1
286 A 5.2.7 Sing to students again: “half of
b squared, add it to both sides” (while
drumming) (P1)
0 0 1 0 1
287 A 5.2.7.1 Tell students: “Let us sing, “half
of b squared, add it to both sides”
(while drumming) (P1)
0 0 1 0 1
288 A 5.2.7.2 Sing together: “half of b
squared, add it to both sides” (while
drumming) (P1)
0 0 1 0 1
289 D 5.2.7.3 IF teacher sings, THEN teacher
asks students to sing with her (P1)
0 0 1 0 1
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172
290 D 5.2.7.4 IF students sing, THEN show
them how to do it (P1)
0 0 1 0 1
291 A 5.2.8 Tell students: “We are taking
half of b” (P1)
0 0 1 0 1
292 A 5.2.8.1 Show students what half of
something means, say half of $4 is $2,
half of $12 is $6 (P1)
0 0 1 0 1
293 A 5.2.8.2 Practice with students: half of 6,
half of 10 … (P1)
0 0 1 0 1
294 A 5.2.8.3 Check for understanding with
students writing the answers on their
individual whiteboards and lifting them
up to show the teacher (P1)
0 0 1 0 1
295 A 5.3 Start with an expression with a
coefficient of 1 for x
2
, x
2
+ bx to
complete the square (P2)
0 0 0 1 1
296 A 5.3.1 Give an example x
2
+ 6x, start
with an even b-term (P2, P3)
1 0 0 1 2
297 A 5.3.2 Complete the square by
dividing 6 by 2 to get 3 and then square
3 to get 9: x
2
+ 6x + 9 = (x + 3)
2
(P2,
P3, P4)
1 1 0 1 3
298 A 5.3.3 Show how to complete the
square using algebra tiles (P2, P3, P4)
1 1 0 1 3
299 A 5.3.4 Explain to students the process
of completing the square (P2, P3)
1 0 0 1 2
300 A 5.3.5 Show students more completing
the square: x
2
+ 4x, to complete the
square, add the square of , which is 2
2
= 4 and the expression becomes x
2
+ 4x
+ 4. Therefore add 4 squares to
complete the square (P2, P3, P4)
1 1 0 1 3
301 A 5.3.6 Show students another example:
x
2
+ 8x, to complete the square, add the
square of, which is 4
2
= 16 and the
expression becomes x
2
+ 8x + 16.
Therefore add 16 squares to complete
the square. (P2, P3, P4)
1 1 0 1 3
302 A 5.4 Introduce students to a quadratic
equation to solve using by the
completing the square procedure (P1,
P2, P3)
1 0 1 1 3
303 A 5.5 Show students how to write the
quadratic equation in the form x
2
+ bx
= c (P2, P3)
1 0 0 1 2
304 A 5.5.1 Give students an example, like 1 0 1 1 3
THE USE OF COGNITIVE TASK ANALYSIS
173
(P1, P2, P3)
305 A 5.5.2 Teacher says: “b = 10, take half
of 10” (P1, P2, P3)
1 0 1 1 3
306 A 5.5.3 Teacher says: “IF I say half of b,
THEN you say the answer” (P1)
0 0 1 0 1
307 A 5.5.4 Teacher says: “IF I say half of
10, THEN you say 5!” (P1)
0 0 1 0 1
308 A 5.5.5 Teacher says: IF I say 5 squared,
THEN you say 25!” (P1)
0 0 1 0 1
309 A 5.5.6 Teacher says: “IF I say add it to
both sides, THEN you add it to both
sides” (P1)
0 0 1 0 1
310 A 5.5.7 Add 25 to both sides of the
equation (P1, P2, P3)
1 0 1 1 3
311 A 5.5.8 Remind students the song: “half
of b squared, add it to both sides” (P1)
0 0 1 0 1
312 A 5.5.9 Tell students: “We squared it,
so the title of completing the square.
We are making it squared so that we
can write it as a quantity squared” (P1)
0 0 1 0 1
313 A 5.5.10 Tell students: “x was squared
to get x-squared and 5 was squared to
get 25” (P1, P2)
0 0 1 1 2
314 A 5.5.11 Take square root to undo
squares (P1, P2, P3, P4)
1 1 1 1 4
315 A 5.5.12 Factorize the left hand side:
factors of x
2
are x and x and factors of
25 are 5 and 5 (P1, P2, P3, P4)
1 1 1 1 4
316 A 5.5.13 IF you take the square root of
one side, THEN you must take the
square root of the other side (P1, P2,
P3, P4)
1 1 1 1 4
317 A 5.5.14 Take the square root of both
sides (P1, P2, P3, P4)
1 1 1 1 4
318 A 5.5.15 Solve for x (P1, P2, P3, P4) 1 1 1 1 4
319 A 5.5.15.1 Tell students: “Let’s look at
our essential question: How do we use
completing the square to solve
quadratic equations?” (P1)
0 0 1 0 1
320 A 5.5.16 Circle x (P1, P2, P3, P4) 1 1 1 1 4
321 A 5.5.17 Isolate x by itself (P1, P2, P3,
P4)
1 1 1 1 4
322 A 5.5.18 Subtract 5 from both sides
(P1, P2, P3, P4)
1 1 1 1 4
323 A 5.5.19 Box the answer: x = 0 or -10
and write solutions, roots, x-intercepts
1 1 1 1 4
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174
and zeros of the quadratic equation (P1,
P2, P3, P4)
324 A 5.6 Show students another example
following the steps shown on step 5.5
(P1)
0 0 1 0 1
325 A 5.7 Check for understanding by giving
students one problem at time to do on
their whiteboards in pairs (P1, P2, P3,
P4)
1 1 1 1 4
326 D 5.8 IF students are not proficient
solving quadratic equations with a
coefficient of 1 for x
2
by completing
the square procedure, THEN show
them one more example (P1, P2, P3).
1 0 1 1 3
327 D 5.9 IF students are proficient solving
quadratic equations with a coefficient
of 1 for x2 by completing the square
procedure, THEN introduce an
equation with an a-value greater than 1
(P1, P2, P3, P4)
1 1 1 1 4
328 A 5.10 Introduce an equation with an a-
value greater than 1 (P1)
0 0 1 0 1
329 D 5.10.1 Step 1 – IF the equation is not in
standard form, THEN re-arrange the
terms in standard form [see step 5.2]
(P1)
0 0 1 0 1
330 D 5.10.2 IF the a-value is not equal to
one, THEN divide both sides by a (P1)
0 0 1 0 1
331 A 5.10.3 Tell students: “It is going to be a
challenge because you may start
dealing with a b-value that is a fraction
or an odd number” (P1)
0 0 1 0 1
332 A 5.10.4 Repeat steps 5.2.1 through 5.2.7
(P1)
0 0 1 0 1
333 A 5.11 Teach students easy ways to
remember the steps. Singing seems to
work all the time (P1)
0 0 1 0 1
334 A 5.12 Teach students how to use algebra
tiles to complete the square (P3)
1 0 0 0 1
335 D 5.13 IF it is about solving quadratic
equations, THEN the quadratic formula
is the fallback method, it works for
every quadratic equation (P1)
0 0 1 0 1
336 A 5.14 Give students 3 problems to solve
by completing the square, and 1
problem that is already solved but
0 0 1 1 2
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175
solution steps are out of order and
students must order the steps correctly,
and the proof of the quadratic formula
by completing the square with steps out
of order where students must correctly
order the steps of the proof. (P1, P2)
337 D 5.3 IF students have mastered
completing the square, THEN go on to
solving quadratics using square roots.
IF NOT, THEN reteach. (P1, P2)
0 0 1 1 2
Procedure 6: Solving quadratics
using the square root method
21 6
338 D 6.1 IF the equation is of the form ax
2
=
c or ax
2
- c = 0, THEN the square root
method is appropriate (P1, P2)
0 0 1 1 2
339 A 6.2 Show students how to solve x
2
= 9
(P1, P2)
0 0 1 1 2
340 A 6.3 Take the square root of both sides
(P1, P2)
0 0 1 1 2
341 A 6.4 Write x = -3 or 3 which are the
solutions, x-intercepts, zeros and also
the roots of the quadratic equation x
2
=
9. (P1)
0 0 1 0 1
342 A 6.5 Show another example without a
perfect square, solve (P1, P2)
0 0 1 1 2
343 A 6.5.1 Add 8 to both sides, to isolate
x
2
: x
2
– 8 + 8 = 0 + 8 (P1, P2)
0 0 1 1 2
344 A 6.5.2 Simplify: x
2
= 8 (P1, P2) 0 0 1 1 2
345 A 6.5.3 Take the square root of both sides
(P1, P2)
0 0 1 1 2
346 A 6.5.4 Write the answers and these are
the roots, zeros, solutions or x-
intercepts of this quadratic equation x
2
– 8 = 0. (P1, P2)
0 0 1 1 2
347 A 6.6 Show another example: x
2
– 16 = 0
(P1, P2)
0 0 1 1 2
348 A 6.6.1 Use factorization and find the
sum and difference of products (P1 P2)
0 0 1 1 2
349 A 6.6.2 Use zero product property, IF
AB = 0, THEN A = 0 or B = 0 (P1, P2)
0 0 1 1 2
350 D 6.6.3 IF x
2
– 16 = 0, THEN (x – 4)(x
+ 4) = 0 (P1, P2)
0 0 1 1 2
351 A 6.6.4 Solve: IF (x – 4)(x + 4) = 0,
THEN x – 4 = 0 or x + 4 = 0 (P1, P2)
0 0 1 1 2
352 A 6.6.5 Simplify: IF x – 4 = 0 or x + 4 = 0 0 1 1 2
THE USE OF COGNITIVE TASK ANALYSIS
176
0, THEN x = 4 or x = -4 (P1, P2)
353 A 6.6.6 Use square roots to solve: x
2
– 16
= 0 (P1, P2)
0 0 1 1 2
354 A 6.6.7 Add 16 to both sides of the equal
sign (P1, P2)
0 0 1 1 2
355 A 6.6.8 Solve x
2
= 16 (P1, P2) 0 0 1 1 2
356 A 6.6.9 Take the square root of both sides
(P1, P2)
0 0 1 1 2
357 A 6.6.10 Write x = 4 or x = -4, which is
the same solution. (P1, P2)
0 0 1 1 2
358 A 6.2 Remind students: “Those problems
that they did in factoring are very
similar to our square roots problems”
(P1)
0 0 1 0 1
359 D 6.3 IF a quadratic equation is missing
the b-term (middle term), THEN solve
using the square root procedure (P1,
P2)
0 0 1 1 2
360 D 6.4 IF students are not proficient using
the square root procedure, THEN show
them two more examples (P1, P2)
0 0 1 1 2
361 A 6.5 Check for understanding with
students writing on their whiteboards
and showing the teacher their solution
(P1, P2)
0 0 1 1 2
362 D 6.6 IF students are proficient with
solving quadratic equations with the
square root procedure, THEN proceed
to the application of all these
procedures that students have been
learning (P1, P2)
0 0 1 1 2
363 A 6.7 Give students a 5 question
assessment (open ended quadratic
equation problems to solve) (P1)
0 0 1 0 1
364 D 6.8 IF students have mastered
completing the square, THEN go to
application. IF NOT, reteach. (P1)
0 0 1 0 1
Procedure 7: Teach Application of
these methods of solving quadratic
equations to solve real-life problem
30 9
365 A 7.1 Apply knowledge of solving
quadratic equation to real-life problems
(P1, P3, P4)
1 1 1 0 3
366 D 7.2 IF students have been taught the
skill base for solving quadratic (P1)
0 0 1 0 1
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177
367 A 7.3 Write (Project) a practice problem
on the board (P1, P4)
0 1 1 0 2
368 A 7.4 Choose the best procedure for
solving the quadratic problem (P1, P4)
0 1 1 0 2
369 A 7.5 Use the three-column table on the
whiteboard as a graphic organizer (P1)
0 0 1 0 1
370 A 7.6 Make notes on it on all discussions
as you solve the problem (P1)
0 0 1 0 1
371 A 7.7 Give students a couple of minutes
to solve the problem (P1)
0 0 1 0 1
372 A 7.7.1 Ask students aloud: “Which way
did you solve it?” (P1)
0 0 1 0 1
373 A 7.7.2 Go back to the three-column table
and make notes (P1)
0 0 1 0 1
374 A 7.8 Ask students: “When is it most
convenient to use factoring?” (P1)
0 0 1 0 1
375 A 7.9 Ask students: “When can I use
square roots?” (P1)
0 0 1 0 1
376 D 7.9.1 IF a student solved it really
quickly, THEN ask, “What did you do?
Which method did you use?” (P1)
0 0 1 0 1
377 D 7.9.1.1 IF a student took a little bit
longer, THEN ask, “What did you do?
Which method did you use?” (P1)
0 0 1 0 1
378 A 7.9.1.2 Go back to the graphic
organizer and edit it with specifics of
what you did (P1)
0 0 1 0 1
379 D 7.10 IF the quadratic equation does not
have a b-term, THEN use the square
roots method” (P1)
0 0 1 0 1
7.10.1 Reason: Because it is the
easiest. It is the quickest (P1)
380 D 7.11 IF the coefficient of x2 is greater
than 1 or not factorable, THEN use the
quadratic formula (P1)
0 0 1 0 1
7.11.1 Reason: Because you do not
have to list all possible different factors
and then finally find the problem is not
factorable while the quadratic formula
always works (P1)
381 A 7.12 Take about two to three days
doing this with students. Practicing to
build their confidence (P1)
0 0 1 0 1
382 A 7.13 Assess students on their skills
(P1)
0 0 1 0 1
383 A 7.13.1 Assess students where they 0 0 1 0 1
THE USE OF COGNITIVE TASK ANALYSIS
178
choose any method they want to solve
(P1)
384 A 7.13.2 Give students two problems,
where they have to use a specific
method to solve (P1)
0 0 1 0 1
385 D 7.14 IF students can solve basic
quadratic equations, THEN introduce
practical real-life problems (P1)
0 0 1 0 1
386 D 7.15 IF students can solve a quadratic
equation, THEN ask them to do
different things, like find the width of a
parabolic disc (P1, P3, P4)
0 0 1 0 1
387 A 7.16 Give students problems that
involve projectiles (P1, P3, P4)
1 1 1 0 3
388 D 7.17 IF a rocket is launched, THEN
what would be the maximum height, or
how long will it take to reach the
ground? (P1, P3, P4)
1 1 1 0 3
389 A 7.17.1 Make connection: maximum
height of rocket corresponds to vertex
of a parabola; time rocket takes to
reach the ground is the difference
between the x-intercepts of the
parabola. These are practical
applications of solving quadratic
equations (P1, P3, P4)
1 1 1 0 3
390 A 7.18 Demonstrate drawing a picture
that shows the parabolic path of the
projectile (P1, P3, P4
1 1 1 0 3
391 D 7.18.1 IF given an application problem
(word problem), THEN draw a picture
to represent the story (P1, P3, P4)
1 1 1 0 3
392 A 7.18.2 Draw a picture back to the
graph (P1)
0 0 1 0 1
393 A 7.18.3 Tell students: “This is the skill
you learned to do, when we were
graphing” (P1)
0 0 1 0 1
394 A 7.19 Draw a picture always (P1) 0 0 1 0 1
395 A 7.20 Relate back to previous lessons
because quadratic equations are not an
isolated unit (P1)
0 0 1 0 1
396 A 7.21 Make students understand why
they are solving quadratic equations
(P1)
0 0 1 0 1
397 A 7.21.1 Make connections for students!
For example: When solving a problem
1 1 1 0 3
THE USE OF COGNITIVE TASK ANALYSIS
179
involving the jumping path of a
kangaroo, impose a graph that shows
the horizontal time and vertical
distance by labeling the x- and y-axis.
Draw in the axis of symmetry and
vertex reminding students that the
kangaroo reached a specific height (y)
after so many seconds (x). The
students can see on the graph that the
maximum height reached by the
kangaroo is at the vertex. Talk about
the height the kangaroo started at (was
there a y-intercept other than 0?) and
then where he ended up. (P1, P3, P4)
398 A 7.22 Remind students a, b, and c in the
quadratic equation, always relates back
to the graph or real-life application
problem (P1)
0 0 1 0 1
399 A 7.23 Ask students aloud again: “What
does it mean to solve for x?” (P1)
0 0 1 0 1
400 A 7.23.1 Ask students: “What is the
significance of finding x on the graph?
What does it mean?” (P1)
0 0 1 0 1
401 A 7.23.2 Tell students that solving for x
is finding solutions, the roots, the zeros
and they are also the x-intercepts of the
graph of the quadratic equation (P1)
0 0 1 0 1
402 A 7.24 Give students a UNIT assessment.
(P1)
0 0 1 0 1
SME
A B C D
Action Steps 81 58 262 115
Decision Steps 16 16 76 24
Total Action and Decision Steps 97 74 338 139
SME
A B C D
Total Action and Decision Steps 24.01% 18.32% 83.66% 34.41%
Action Steps 25.47% 18.24% 82.39% 36.16%
Decision Steps 18.60% 18.60% 88.37% 27.91%
THE USE OF COGNITIVE TASK ANALYSIS
180
SME
A B C D
Total Action and Decision Steps Omitted 307 330 66 265
Action Steps Omitted 237 260 56 203
Decision Steps Omitted 70 70 10 62
SME
A B C D
Total Action and Decision Steps Omitted 75.99% 81.68% 16.34% 65.59%
Action Steps Omitted 74.53% 81.76% 17.61% 63.84%
Decision Steps Omitted 81.40% 81.40% 11.63% 72.09%
Full Alignment 26 6.68%
Substantial Alignment 44 10.89%
Partial Alignment 81 20.05%
No Alignment 251 62.38%
Total Action and Decision Steps 402 100.00%
THE USE OF COGNITIVE TASK ANALYSIS
181
Appendix G
Information Sheet
University of Southern California
Rossier School of Education
3470 Trousdale Parkway
Los Angeles, CA 90089
INFORMATION/FACTS SHEET FOR NON-MEDICAL
RESEARCH
USING COGNITIVE TASK ANALYSIS TO CAPTURE EXPERT INSTRUCTION
IN ALGEBRA FOR 8
th
and 9
th
GRADE HIGH SCHOOL STUDENTS
You are invited to participate in a research study conducted by Acquillahs Muteti Mutie,
a doctoral candidate at Rossier School of Education at the University of Southern California,
because you are a teacher identified as highly knowledgeable in Algebra instruction. Your
participation is voluntary. You should read the information below, and ask questions about
anything you do not understand or that is unclear to you, before deciding whether to participate.
PURPOSE OF THE STUDY
The purpose of the study is to use Cognitive Task Analysis methods to capture the knowledge of
expert Algebra teachers as they implement research-based instructional practices in solving
quadratic equations for 8
th
and 9
th
grade students. You are being asked to participate in this
study because you have been identified as highly knowledgeable in algebra instruction for this
student population. The information gathered will be used to help better understand quadratic
equations instruction in Algebra. Your participation in the study will aid in capturing the
implicit and non-observable decisions, judgments, analyses, and other cognitive processes used
during quadratic equations instruction.
PARTICIPANT INVOLVEMENT
There is only one way to participate in this inquiry through participating in an interview with a
follow-up round two interview. The paper survey will be distributed to teachers and is
anticipated to take no more than 20 minutes to complete. After you take the survey you can also
choose to participate in the interview with follow-up interview by emailing the Principal
Investigator, Acquillahs Mutie, at mutie@usc.edu. Completing the survey is a requirement for
participation in the interview process. The interview should take about 90 minutes to complete.
Your participation is voluntary and if you choose not to participate no penalty will occur. You
may choose not to participate at any time. Your identity as a participant will be de-identified and
will remain confidential at all times during and after the inquiry project.
THE USE OF COGNITIVE TASK ANALYSIS
182
PAYMENT/COMPENSATION FOR PARTICIPATION
No payment will be offered. Participation will aid in capturing the knowledge of expert Algebra
teachers, through Cognitive Task Analysis, as they implement research-based strategies for
solving quadratic equations instructional practices for 8
th
and 9
th
grade Algebra students.
CONFIDENTIALITY
Any identifiable information obtained in connection with this study will be de-identified and
remain confidential. Your responses will be coded with a false name (pseudonym) and
maintained separately. The data will be stored on a password-protected computer in the
researcher’s office for three years after the study has been completed and then destroyed.
The members of the research team, and the University of Southern California’s Human
Subjects Protection Program (HSPP) may access the data. The HSPP reviews and monitors
research studies to protect the rights and welfare of research subjects.
When the results of the research are published or discussed in conferences, no identifiable
information will be used.
INVESTIGATOR CONTACT INFORMATION
If you have any questions or concerns about the study, please feel free to contact the Principal
Investigator, Acquillahs Mutie, by email at mutie@usc.edu
IRB CONTACT INFORMATION
University Park Institutional Review Board (UPIRB), 3720 South Flower Street #301, Los
Angeles, CA 90089-0702, (213) 821-5272 or upirb@usc.edu
THE USE OF COGNITIVE TASK ANALYSIS
183
Appendix H
Interview Letter to Participants
Dear Teachers:
My name is Acquillahs Muteti Mutie, and I am a doctoral candidate in the Rossier School of
Education at University of Southern California. I am conducting a study as part of my doctoral
dissertation that focuses on capturing the expertise of Algebra teachers that teach solving
quadratic equations to 8
th
and/or 9
th
grade students in the K-12 education system.
The method used in this study is Cognitive Task Analysis (CTA). CTA is a knowledge elicitation
and analysis technique that involves interviewing subject matter experts (SME) to identify the
tacit action and decision steps experts use when performing a complex cognitive task.
You are invited to participate in this study and the information gathered will help in developing
strategies to support instruction for solving quadratic equations. The interview should not take
more than 90 minutes to complete. Participation in this study is voluntary. Your identity as a
participant will be de-identified and will remain confidential at all times during and after the
study.
If you have questions or would like to participate, please contact me at mutie@usc.edu
Thank you for your participation,
Acquillahs
Abstract (if available)
Abstract
The purpose of this study was to apply Cognitive Task Analysis (CTA) methods to capture expert mathematics instruction in solving quadratic equations. CTA seeks to elicit the highly automated and often-unconscious knowledge experts use to solve difficult problems and perform complex tasks. Students taking algebra find solving and understanding quadratic equations very challenging yet quadratic equations are a major component of building mastery in algebra. Four 8th and 9th grade Algebra teachers, who were qualified as experts using both qualitative and quantitative measures, were interviewed to capture the action and decision steps they use to teach quadratic equations. The individual protocols were then aggregated as a gold standard protocol (GSP) that was reviewed by a fifth senior SME for accuracy and consistency. Overall, there were found to be seven main procedures for solving quadratic equations. However, there was full alignment among the four experts on only seven percent of the action and decision steps, suggesting that multiple experts should be used to capture complex procedures, such as teaching algebra. Moreover, the experts omitted an average of 59.90% of the total action and decision steps, thus supporting previous research finding that experts may omit up to 70% of the critical information required to perform a complex task. The expert knowledge and skills captured may be used to train student teachers in teacher prep-programs and also offer professional development to Algebra teachers for teaching this highly complex subject.
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Asset Metadata
Creator
Mutie, Acquillahs Muteti
(author)
Core Title
The use of cognitive task analysis to capture expert instruction in teaching mathematics
School
Rossier School of Education
Degree
Doctor of Education
Degree Program
Education (Leadership)
Publication Date
01/15/2015
Defense Date
11/24/2014
Publisher
University of Southern California
(original),
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(digital)
Tag
cognitive task analysis,declarative knowledge,expert,OAI-PMH Harvest,procedural knowledge,subject matter experts
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Language
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Advisor
Yates, Kenneth A. (
committee chair
), Hasan, Angela Laila (
committee member
), Ramos-Beal, Camille (
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)
Creator Email
mutetiam@gmail.com,mutie@usc.edu
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Tags
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