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Developing mathematical reasoning: a comparative study using student and teacher-centered pedagogies

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Developing mathematical reasoning: a comparative study using student and teacher-centered pedagogies

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Content Running head: DEVELOPING MATHEMATICAL REASONING 1
DEVELOPING MATHEMATICAL REASONING: A COMPARATIVE STUDY USING
STUDENT AND TEACHER-CENTERED PEDAGOGIES

by

Luciano Cid

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

December 2014

Copyright 2014 Luciano Cid
DEVELOPING MATHEMATICAL REASONING 2
Epigraph
“It is rarely, on the frontier of knowledge or elsewhere, that new facts are ‘discovered’ in
the sense of being encountered, as Newton suggested, in the form of islands of truth in an
uncharted sea of ignorance…Discovery, like surprise, favors the well-prepared mind.”
(Bruner, 1979, p. 82).
DEVELOPING MATHEMATICAL REASONING 3
Dedication
Whatever accolades I will be offered as a result of achieving my Doctorate, much like the
praises that have been received by anyone who has been fortunate enough to get to their own
mountain top, are only being attained because of the enormous amount of assistance and
sacrifice certain individuals have put forth. Consequently, my need to be selfish has time and
time again been matched by these individual’s constant selflessness. Many a days have I turned
their needs into means for my egoistical ends. Therefore, with this dedication, I would like to do
something more than simply state that this work has been accomplished for someone else – as if
I had written this work all by myself and was mine to give alone. Instead, I would like to take
this time to recognize the other contributors to this work, my lovely wife and two children. I
truly believe that my Doctorate has been your Doctorate, that my achievements are your
achievements, and that every sentence that I have written or statistical analysis that I have
performed has been performed by all of us. I could not have done any of it without your support
and effort. Matias, you were born a week before I began the program and, as a result, your entire
life has been characterized by sacrifice. Your willingness to allow daddy to read and type in his
computer, instead of playing swords, talking about pirates, or putting you to bed has shown an
immense sense of maturity. I am sorry I have taken so much time away from our relationship;
for the rest of my life I will try to make that up to you – let’s go surfing now. Italia, your smiles
have often allowed me to continue working, knowing that you loved daddy unconditionally even
if he had to spend more time with a laptop than with you. Janelle (my lovely wife), what can I
say; you are the best model of Jesus I have ever known – and this has been especially true during
the last four years. Your willingness to go to an extremely cold state, live in a tiny room for nine
months, give up our own space, love our children enough to fulfill the emotional needs that are
DEVELOPING MATHEMATICAL REASONING 4
often required of two parents, all while you found enough love to encourage, support, and care
for my needs can only be paralleled by the love of God. I thank you all from the bottom of my
heart and confidently state that this doctorate is your doctorate as much as it is mine. So, to
whom should we dedicate?
DEVELOPING MATHEMATICAL REASONING 5
Acknowledgements
“All I ever wanted really, and continue to want out of life, is to give 100 percent
to whatever I'm doing and to be committed to whatever I'm doing and then let the
results speak for themselves. Also to never take myself or people for granted and
always be thankful and grateful to the people who helped me” (Jackie Joyner-
Kersee)
I try to live by such words. Consequently, throughout my life I have worked very hard to
accomplish all my endeavors in a timely and successful manner. Still, I have not been as good at
the second part of the quote, giving thanks to those who have helped me. To amend that fault of
mine, I write the following words of acknowledgement:
 To my parents who sacrificed so much to give their kids a better life in America; I thank
you and hope that this work can stand as proof that your sacrifices have not been in vain.
I could not have done any of this without your love, your sacrifices, and your strength.
Mom, I remember all the times you forced me to get up early in the morning to help me
finish my homework, what I learned then gave me the diligence I have required to finish
my doctorate.
 To Marianne Smith, Erica Howell, and Jeff Sapp thank you for being so instrumental in
my professional life. Marianne the monetary and professional support that I received
through your grant gave me the ability to grow as a student, a teacher, and an individual.
Erica your friendship, professional advice, and assistance have been critical to where I
find myself today. Jeff the level of professionalism and effortless kindness that you show
are still in my mind the perfect models for which I constantly aspire.
DEVELOPING MATHEMATICAL REASONING 6
 To my committee members Dr. Kaplan, Dr. Gallagher, and Dr. Hasan, thank you for the
guidance and wisdom you have provided throughout my dissertation. You were always
positive, patient, and kind with me, which tremendously ameliorated the difficult
experience of writing such a lengthy and complex work.
 To the principals, teachers, and students who offered me not only their time, but also their
brain capacity in order for me to be collect the data that was analyzed for this work.
 To my friends and colleagues, Rena and Michael, our car rides to USC were filled with
encouragements and discussions that helped assuaged the difficulties that came from
trying to be a full time dad, student, and teacher for three years.
 Last, but certainly not least, to my in-laws who allowed my family to reside at their home.
Your love, assistance, kindness, and unending support for my entire family, as well as me,
have been an integral part of my ability to accomplish this doctorate.
Thank you all so much; I love you; I appreciate you; and, I promise you that I will continue
to work hard throughout my career to honor the sacrifices you have made in order for me to
reach my goals.

DEVELOPING MATHEMATICAL REASONING 7
Table of Contents
Epigraph 2
Dedication 3
Acknowledgements 5
List of Tables 9
List of Figures 10
Abstract 11
Chapter One: Introduction 12
Statement of the Problem 12
Purpose of the Study 14
Significance of the study 17
Research Questions 19
Research Hypotheses 19
Methodology 20
Assumptions 22
Limitations 22
Delimitations 23
Definitions 24
Chapter Two: Literature Review 30
Structure of the Literature Review 30
Introduction 30
What Contemporary Literature Says About Pedagogy 30
Recent Events Affecting the Discourse 32
The Economic Connection 39
Recent Shifts in Educational Policy 43
How has Mathematics been Taught During Past and Present Times? 45
The Recent Past 55
What Pedagogical Differences Exist between EDI and CGI? 66
How Do American Students Perform in Mathematical Reasoning? 72
What Does Research Say about How Students Like to be Taught? 79
Summary 85
Chapter Three: Methodology 88
Introduction 88
Research questions 89
Research Hypotheses 89
Nature of the Study 90
Subjects 93
Connecting With the School-Sites 93
Controlling for Confounding Variables 96
The Data Collection Process 99
Instruments 99
The Test 100
The Survey 104
Research procedure 106
Summary 110
DEVELOPING MATHEMATICAL REASONING 8
Chapter Four: Results 112
Statement of the Problem 112
Research Questions 113
Demographics 114
Research Question #1 115
Research Question #2 120
Research Question #3 124
Research Question #4 128
Summary 135
Chapter 5: Discussion 137
Introduction 137
Research Questions 138
Summary of the Findings 138
Key Findings 139
Research Question # 1 140
Research Question # 2 143
Research Question #3 146
Research Question #4 147
Limitations of the Study 148
Recommendations for Research and Practice 150
References 156
Appendix A 178
Appendix B 184
Appendix C 185
Appendix D 187
Appendix E 188
Appendix F 192
Appendix G 196
Appendix H 197

DEVELOPING MATHEMATICAL REASONING 9
List of Tables
Table 1: Test Items Between-Groups Comparison 98

Table 2: Research Questions with Corresponding Measurement Instruments and Supporting
Literature 100

Table 3: 12-Question Test Comparison Using Results from the Pilot Study and
the Actual Study 103

Table 4: Demographics of Each School Site 114

Table 5: Independent Samples T-test Using All the Items Included in the 12-Question
Test (Assessment of Subjects’ Overall Mathematical Performance) 120

Table 6: Independent Samples T-test Using Only the TIMSS Items
(Assessment of Subjects’ Mathematical Reasoning) 124

Table 7: Independent Samples T-test Using Data from the Survey’s Total Score
(Assessment of Subjects’ Mathematical Perception) 128

Table 8: Independent Samples T-test using the Survey’s Reflect Square-Root
Transformation Data ( Assessment of Subjects’ Mathematical Perception) 128

Table 9: Descriptive Statistics for the Linear Composite Curriculum/Perception
Using TIMSS Total Score as the Dependent Variable 129

Table 10: Normality Test Results Using the Linear Composite Curriculum/Perception
to Analyze the Subjects’ Ability to Reason Mathematically 132

Table 11: Two-way ANOVA results for Linear Composite Curriculum/Perception 132

DEVELOPING MATHEMATICAL REASONING 10
List of Figures
Figure 1: Model of the causal-comparative approached used in the study 91

Figure 2: Box plots of the 12-question test categorized using the curriculum type that was
employed to instruct subjects 117

Figure 3: Histograms and Q-Q plots for the 12-question test categorized using the curriculum
employed to instruct the subjects 119

Figure 4: Box plots using only the TIMSS items in the 12-question test designed to assess
mathematical reasoning 121

Figure 5: Histograms and Q-Q plots for TIMSS items total scores 123

Figure 6: Histograms, Q-Q plots, and Box plots displaying the reflect square-root transformation
scores of the survey designed to assess students’ mathematical perceptions 127

Figure 7: Box plots of the linear composite perception and curriculum used as the independent
variable of the dependent variable TIMSS total scores 131

Figure 8: Clustered bars of the two-way ANOVA results categorized by both the curriculum
experienced and perception designation given to the students 133

Figure 9: Graphs depicting the mean score attained in mathematical reasoning based on students’
mathematical perceptions and curriculum experienced 134

DEVELOPING MATHEMATICAL REASONING 11
Abstract
Focusing on the discrepancies that exist between student and teacher-centered pedagogies,
this study analyzed the ability of two professional development (PD) models, Explicit Direct
Instruction (EDI) and Cognitively Guided Instruction (CGI]), to increase the mathematical
performance, the mathematical reasoning, and the mathematical perception of elementary
students. A causal-comparative ex post facto model of investigation was employed to carry out
the study; consequently, purposeful sampling was used to identify all the participating subjects
(N=81). Three independent-samples T-tests and one two-way ANOVA were employed to
analyze whether differences existed between the groups regarding the following four criteria: (1)
overall mathematical performance, (2) ability to reason mathematically, (3) perceptions of math
and mathematics instruction, and (4) ability to reason mathematically when curriculum type and
mathematical perception are combined into one linear-composite independent variable. Results
showed a statistically significant difference in the overall mathematical performance and
mathematical reasoning between the two groups. Conversely, mathematical perception, as well
as the linear composite generated by combining perceptions and type of pedagogy, did not
display any statistical significance between the two groups. Nevertheless, interesting patterns
did arise once the results of the linear composite were segregated based upon perception
designations (i.e., low, medium, high, and very high) and type of curriculum (CGI+EDI or EDI).
Both research and practical implications are discussed given the results.
DEVELOPING MATHEMATICAL REASONING 12
CHAPTER ONE: INTRODUCTION
Statement of the Problem
The enduring discourse that ensued between supporters of teacher-centered pedagogies,
such as direct instruction and explicit direct instruction, and those who uphold a more student-
centered approach, such as activity based curriculum, problem-based learning, inquiry based
learning, continues. In fact, individuals on both sides of the argument can point to empirical
studies, quasi-experiments, meta-analysis, and/or action research projects to support their
positions (Adelson, 2004; Begley, 2004; Chall, 2000; Hmelo-Sliver, Duncan, & Chinn, 2007;
Klahr, Triona, & Williams, 2007; Kuhn, 2007; Ruby, 2001; Schimidt, Loyens, van Gog, & Pass,
2007; Schmidt, Molen, te Winkel, & Wijnen, 2009; Strauss, 2004; Tweed, 2004). Unfortunately,
this theoretical tug of war often left practitioners at the mercy of whatever popular hypothesis
and/or policies were instituted at the time (Campbell, 2001; Cunningham & Duffy, 1996;
Nichols, 2009; Piaget, 1973 ; Schoen, Fey, Hirsch, Coxford, 1999; Wilson, 2013; Wolfe, 1998),
a fact that generated a number of negative outcomes (Cohen, Moffit, Goldin, 2007; Smith, 2000;
Valli & Buese, 2007).
However, educators and, in turn, their students deserve more than the ability to blindly
follow a pendulum of strategies and methodologies that swings back and forth based on non-
instructional factors. Instead, teachers should be presented with an array of pedagogical options
that allow them to successfully instruct their students, what Hiebert, Gallimore, and Stigler
(2002) identified as the knowledge base of the teaching profession. In short, our educational
system needs a new model of practice, a model characterized by a more complex and inclusive
view of teaching rather than the dichotomic approach that has existed for so long. A model that
will allow teachers to know when it is best to employ teacher-centered rather than student-
DEVELOPING MATHEMATICAL REASONING 13
centered pedagogies, when it is best to drill and kill rather than to play games, when it is best to
place students in groups rather than having them work individually, or when it is best to direct
rather than guide their students’ learning. The ultimate purpose of this model, therefore, should
not be to affirm one pedagogy over all others, as many studies have done, but to determine which
pedagogical approach is best when developing a specific cognitive skill within one specific
content domain. That is, similarly to the way a handyman selects the tools required to
accomplish a specific task, so should a teacher choose which pedagogical approach must be
employed in order to successfully attain a lesson’s objective.
In order for practitioners to know when to employ a specific pedagogy, educational
researchers must alter the way they carry out their work. Pedagogical investigations will have to
become even more focused than they are today. That is, future studies will have to examine the
effects that a specific pedagogical technique can have upon a specific content and/or cognitive
domain. Such an increased focus should generate a model that tests the educational worth of all
pedagogies and theoretical views.
However, this approach is not what is used by researchers. Contemporary educational
research is characterized by contradictory theoretical views. Hence, to generate a new model of
pedagogical research, investigators must shift their own epistemological and exploratory
paradigms - a shift which a few have already begun.
With their revision of Blooms taxonomy, Anderson et al. (2001) demonstrated the need
to employ different instructional approaches based on the objective of the lesson. They
concluded that the memorization and recall of facts, the analysis and interpretation of
information, and the development of critical thinking skills should not be taught or assessed
using the same types of instructional methods. Consequently, they maintained that teachers
DEVELOPING MATHEMATICAL REASONING 14
should begin their pedagogical planning by identify the objective of the lesson and only then
choose the instructional and/or pedagogical approaches that they will employ.
The present study was designed to carry out such an investigation. That is, it identified
two diverse pedagogical approaches, Cognitively Guided Instruction and Explicit Direct
Instruction, currently employed by different districts in order to examine which is best suited to
developing one specific cognitive domain reasoning, within one specific content domain math),
a pedagogical combination in which American students have traditionally not performed well
(Cavanagh & Robelen, 2004).
Purpose of the Study
As previously seen, educational research is often characterized by oppositional views.
Examples of such pedagogical arguments can be seen in the debates of Carbo and Chall
regarding reading instruction (Turner, 1989) or in the disputes between direct instruction and
meaningful interactions in the field of second language acquisition (Goldenberg, 2008; Nichols,
2009). This work focuses on a similar pedagogical debate, the difference between teacher- and
student-centered pedagogies by comparing two diverse instructional approaches: Cognitively
Guided Instruction (CGI) and Explicit Direct Instruction (EDI).
Both of these instructional approaches were chosen based on the distinct philosophical,
theoretical, and pedagogical foundations that characterize them. Cognitively Guided Instruction,
for example, is a pedagogical approach that employs many of the features promoted by the
constructivist model of teaching, such as exploration, communication, group work, problem
solving. Conversely, the direct instructional model, exhibited by EDI, states that students learn
best when the teacher (a) maintains control of the lesson (Kirschner, Sweller, & Clark, 2006;
Mayer, 2004), (b) guides students’ learning through corrective measures, and (c) constantly
DEVELOPING MATHEMATICAL REASONING 15
checks for understanding – all principles arising from the behaviorist and cognitivist traditions of
learning (Scheurman, 1998).
Each of these approaches has supporters. For instance, multiple studies demonstrated the
effectiveness of direct instruction, especially regarding mathematics (Hattie, 2008), low-
performing students (Meyer 1984; Meyer, Gersten, & Gutkin, 1983, Baker, Gersten, & Lee,
2002), and/or novice learners (Kirschner et al., 2006; Klahr & Nigam, 2004). Kirschner et al.
(2006) state that minimally guided instructional approaches, such as problem, activity, or
inquiry-based learning, fail to account for a person’s cognitive architecture. Accordingly,
Kirschner and his colleagues maintain that constructivist pedagogies are less effective than the
combination of direct instruction and worked examples, which tend to alleviate the cognitive
load of the learner. Additionally, Kirschner et al. stated that problem-based learning is only
advantageous for individuals who have already acquired the necessary domain-expertise to
benefit from self-guidance, a premise that has been concurred with by Alfieri, Brooks, Aldirch,
and Tenenbaum’s (2011) work.
Conversely, constructivist methodologies have also proven their effectiveness. The
supporters of such methodologies argue that students must have the opportunity to build their
own knowledge by participating in thought provoking situations (Schmidt, Molen, te Winkel, &
Wijnen, 2009) - what Piaget identified as self-guided learning (1952, 1965, 1980). For example,
Carpenter, Fennema, Franke, Levi, and Empson (1999) have extensively studied and written
about the efficacy of Cognitively Guided Instruction (CGI). CGI is a problem-based approach to
teaching mathematics in which students develop mathematical conceptual understanding via
experimentation and reasoning. Their research showed that students in CGI classes “[have]
DEVELOPING MATHEMATICAL REASONING 16
significantly higher levels of achievement in problem solving… [and] number facts” (Carpenter,
et al., 1999, p. 109).
Not all researchers have opted to support one of these two theoretical camps. Some have
found common ground between these diverse perspectives (see Schmidt, Loyens, van Gog, and
Pass, 2007; Khun, 2007). Khun (2007), for example, stated that there is a “need to contemplate
instructional methods within the broader context of instructional goals” (p.112), meaning that, in
order to determine what pedagogy is most appropriate for a specific lesson, researchers and
practitioners should look towards the objective of the lesson, rather than pre-determined
epistemological perspectives. For instance, if a lesson requires a student to develop automaticity
rather than ingenuity, a teacher should employ repetitive methodology rather than a problem-
based exercise. Conversely, if the lesson calls for the students to develop creative and evaluative
skills, repeating and copying what the teacher modeled, as Engelman (1980) argued, might not
be the most appropriate way to increase such skills. Under such conditions, the teacher should
carry out a lesson that demands students to design, hypothesize, and/or construct original
information.
Others proposed a middle ground as well. In his article for the Mathematics Education
Research Journal, Lesh (2000) explained the necessity for schools to present their students with
a good foundation of knowledge, while also giving them the opportunity to meaningfully interact
with that knowledge. Galloway and Lasley II (2010) also presented a similar point of view.
They mentioned that “good teachers are not exclusively direct instruction or inquiry oriented;
they understand the needs of students in ways that cause them to structure instruction to [their]
unique learning needs …” (p.274). Furthermore, Cooper-Twamley and Null (2009) wrote about
the pedagogical and theoretical value of both Thorndike (a behaviorist) and Brooks (a
DEVELOPING MATHEMATICAL REASONING 17
constructivist) when instructing mathematics. And, Khun (2007) pointed to the work of Schwartz
and Bransford (1998) and Schwartz and Martin (2004) as examples of such a middle approach.
Similar to such views, this study assumed that no single educational methodology serves
to adequately instruct all students in all cognitive and/or content domains. Therefore, as advised
by Khun (2007), this study consisted of an empirical investigation to determine which
methodological approach, Explicit Direct Instruction or Cognitive Guided Instruction, is best
suited to increasing the mathematical reasoning of elementary school students.
Significance of the study
Ever since the enactment of NCLB, California schools have measured their success based
on reports provided by the Federal and State Departments of Education. These reports are
compiled using summative high-stakes multiple choice tests developed from standards that, for
the most part, required students to memorize rather than analyze information (Porter, Mc Maken,
Hwang, & Yang, 2011). As a result, for the past twelve years, many teachers have chosen to
ignore higher order thinking skills and, instead, focused on the retention and recall of factual and
procedural knowledge. Conversely, the new standards instituted by the Common Core will force
educators to look beyond their regular pedagogies in order to increase their students’ higher
order thinking skills (Porter et al., 2011), skills that American students lack (IEA’s Trends in
International Mathematics and Science Study – TIMSS 2011).
America’s need to connect specific cognitive and knowledge dimensions to appropriate
pedagogies is evident from the 2011 TIMSS (Trends in International Mathematics and Science
Studies) math test. The results show that, although American 4
th
and 8
th
grade students
performed fairly well when all three cognitive sub-skills, knowing, applying, and reasoning,
were scored, they did not do as well when these were segregated. That is, when each cognitive
DEVELOPING MATHEMATICAL REASONING 18
domain was evaluated individually, American students struggled most to solve items that asked
them to reason (IEA’s Trends in International Mathematics and Science Study – TIMSS 2011).
In fact, when these items were isolated, American 4
th
grade students placed 14
th
amongst 57
participating educational systems; whereas, 8
th
grade students placed 13
th
out of 56 educational
systems - both of the positions were lower than the 12
th
and 7
th
places American students attained
during the 2007 TIMSS.
The significance of these results becomes more critical when the scores are compared to
those of other industrialized countries. In that case, American students were at the bottom when
compared to most of their counterparts. This gives weight to the idea that the United States needs
identify and implement methodological approaches that can successfully prepare students to
reason mathematically, a skill that will be in great demand during the 21
st
century (Galloway &
Lasly, 2010; Peters, 2010; Putnam, Lampert, & Peterson, 1990).
Consequently, this study was designed to assist in the process of such a resolution. It uses
the taxonomies developed by Bloom and Krathwohl (1956) and Anderson and Krathwohl (2001)
as the theoretical framework through which to propose that all pedagogical approaches, such as
student- and teacher-centered, can be either beneficial or obstructive depending on the objective
of the lesson. Therefore, by following the tenets of such a dynamic taxonomy, the study sought
to construct an exploratory model that sets all theoretical and methodological perspectives within
the same playing field. To carry this out appropriately, this study adopted a causal comparative
model of investigation in which both perspectives, Cognitively Guided Instruction and Direct
Instruction, were tested in terms of serving to develop the mathematical reasoning skills of
elementary students – a skill set which, as previously presented, is problematic for many
American students.
DEVELOPING MATHEMATICAL REASONING 19
In closing, the study aimed to allow researchers to use the same research model presented
herein and, as a result, continue to increase the knowledge base of the field. By doing this,
practitioners may gain explicit knowledge to argue against the dichotomic models that provide
the inertia to swing the educational pendulum back and forth.
Research Questions
1. What are the mathematical performance differences that are exhibited by students who
were instructed using an amalgamation of student- and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
2. What are the mathematical reasoning differences that are exhibited by students who were
instructed using an amalgamation of student and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
3. What are the mathematical perception differences that are held by students who were
instructed using an amalgamation of student and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
4. What are the mathematical reasoning differences that are exhibited by students when
curriculum type and mathematical perception are combined in order to generate a linear
composite?
Research Hypotheses
 Research question 1 maintains a non-directional hypothesis: A difference exists between
the mathematical performance ( higher order thinking skills) of students taught using
DEVELOPING MATHEMATICAL REASONING 20
student-centered pedagogies ( Cognitive Guided Instruction) and that of students taught
using teacher-centered pedagogies (Data WORKS’ Explicit Direct Instruction model).
 Research question 2 maintains a non-directional hypothesis: A difference exists between
the mathematical reasoning abilities (higher order thinking skills) of students taught using
student-centered pedagogies (Cognitive Guided Instruction) and those of students taught
using teacher-centered pedagogies (DataWORKS’ Explicit Direct Instruction model).
 Research question 3 maintains a non-directional hypothesis: A difference exists between
the mathematical perceptions of students taught using constructivist student-centered
pedagogies (Cognitive Guided Instruction) and those of students taught using behaviorist
teacher-centered pedagogies (DataWORKS’ Explicit Direct Instruction model).
 Research question 4 maintains a non-directional hypothesis: The interaction created
between the type of curriculum to which students have been exposed and the perception
of mathematics that they maintain makes a difference in the students’ ability to reason
mathematically.
Methodology
Because this was a causal-comparative study, it was necessary to find at least two
different school sites, one that had employed Cognitively Guided Instruction and another that
only used Explicit Direct Instruction. Consequently, purposeful sampling was used to gather the
participants. Furthermore, due to the fact that the study focused on the development of
mathematical reasoning, it was important to develop a testing instrument that could assess that
particular cognitive function. A twelve-question test was developed for that purpose (Appendix
A). The test was constructed by combining previously released items from two well established
standardized tests, the California Standardized Test (CST) and the Trends in International
DEVELOPING MATHEMATICAL REASONING 21
Mathematics and Science Study (TIMSS). The final version of the test included six TIMSS
questions, which were specifically chosen to evaluate mathematical reasoning, and six CST
questions, which were specifically chosen based on the students’ familiarity with them. Three
independent-samples T-test and one two-way Analyses of Variance (ANOVA) were employed to
compare the means of the groups.
The second measuring tool that was specifically designed for this study was a survey
(Appendix B). The questions in the survey were modeled after those used in the 2011 TIMSS
examination. A total of nine-questions compose the entire survey, eight of which used a 4-point
Likert scale and one that followed a short-answer response criterion. The short-answer item was
included in order to provide the students with a chance to describe a more comprehensive
recollection of their prior mathematics experience. Out of the eight Likert-type questions three
of them were reverse coded, three were designed to evaluate a student’s personal feelings
regarding mathematics, two were intended to evaluate how students felt about word problems,
two were written to investigate how they perceived other students view mathematics, and one
was devised to explore how students felt about the pedagogies their specific school-site
employed to instruct mathematics. In essence, every question in the survey was designed to
analyze the students’ personal perceptions of mathematics at their school, which ,given Deci’s
(1980) work regarding self-determination theory and Deci and Ryan’s (1985) and Ryan and
Deci’s (2000) work regarding Cognitive Evaluation Theory, was seen as a critical factor of the
study. Similar to the twelve question test, the survey’s data was also analyzed using independent
samples T-tests and two-way ANOVAs.

DEVELOPING MATHEMATICAL REASONING 22
Assumptions
One of the assumptions of the study is the belief that no single pedagogical model is
entirely suited to providing students with all the cognitive, content, or motivational requirements
they will need to succeed in the 21
st
century. The second assumption of the study is the idea that
certain pedagogies, such as student-centered pedagogies can generate a greater appreciation of a
subject and, in turn, increase the students’ capacity to perform within that subject. The study’s
third assumption is grounded on the idea that students who experience less structured pedagogies
tend to develop a greater capacity for problem solving and creative thinking – two factors that
can increase one’s ability to reason mathematically. Another assumption is the belief that a
hybrid model of instruction, one that employs both teacher and student-centered pedagogical
approaches, will lead to more comprehensive knowledge. The last assumption extends from the
study’s inability to control what has already occurred in the classrooms of the participating
teachers. In other words, the study assumes that the teachers have been faithful in implementing
the tenets of their respective methodologies (EDI or CGI+EDI).
Limitations
Given that the design of the study is a causal comparative model that employs two very
specific teaching methodologies, the generalization of this study will be minimal. The only way
to generalize the outcome would be to carry out multiple experimental research studies, in
several schools, using a variety of students from different age groups, taught by teachers who
have not only been adequately trained in their respective methodologies, but have also proven to
faithfully and accurately employ such approaches in their work. Only after carrying out such a
large-scale comparison would the results of study be appropriately generalizable. That is, in the
absence of such an extensive comparison, the outcomes attained in this study could be due, in
DEVELOPING MATHEMATICAL REASONING 23
part, to a number of other confounding variables, such as the environmental and social factors
that typify each site, each teacher’s faithfulness to the methodology and pedagogy being
implemented, the aptitudes and attitudes that are exhibited by each class, teacher, or
administrator.
Delimitations
The major delimitation of this study is its inability to randomly place subjects in the
control and experimental groups. In fact, because this was a comparative study, no such
demarcations existed amongst the subjects. Instead, each school, one employing Explicit Direct
Instruction and one employing Cognitively Guided Instruction, provided the study with the
specific subjects needed to carry out the comparison.
A second delimitation is the training and years of expertise that the teachers of each
school had with their respective methodologies. For instance, the teachers who represented the
CGI group had only had a few years of experience with their approach whereas the teachers
representing the EDI group had lose to a decade of experience using their particular pedagogy.
A third delimitation was the population of each school. Although both schools were
considered Title I, based on the percentage of students who qualify for free or reduce lunch, only
one of the schools had also earned the privilege of being a GATE (Gifted and Talented
Education) clustered elementary. Consequently, part of the student population at one of the
schools, the site representing Cognitively Guided Instruction, was characterized by students who
qualified as being gifted and talented based on their superior intellectual ability.
The last delimitation had to do with a recent change in leadership that one of the sites underwent
during the time of the study. Such changes in leadership could ultimately affect the way teachers
carry out their work, especially regarding past methodological choices instituted by a different
DEVELOPING MATHEMATICAL REASONING 24
principal no longer at the school. In short, a change in leadership might mean a change in
pedagogical approaches, resulting in a shift away from what the study set out to examine.
Definitions
Direct Instruction
An “explicit or direct approach to teaching that consists of effective instructional design,
effective presentation techniques, and a logical organization of instruction” (Przychodin,
Marchand-Martella, Martella, Azim, 2004, p. 64). “Direct Instruction involves teaching rules,
concepts, principles, and problem-solving strategies in an explicit fashion” (Baker, Gersten, &
Lee, 2002).
Constructivism
“A view of learning that considers the learner as a responsible, active agent in his/her
knowledge acquisition process” (Loyens & Gijbels, 2008, p. 352). “Constructivism is a theory of
learning in which learners build knowledge in their working memory by engaging in appropriate
cognitive processing of mental representations during learning” (D'Angelo, Touchman, Clark,
O'Donnell, Mayer, Dean, Hmelo-Silver, 2009, discovery learning, para. 2).
Student-centered learning
A form of instruction that is “less authoritarian, less concerned with the past and ‘training
the mind,’ and more focused on individual needs, contemporary relevance, and preparing
students for a changing future…School is not seen as an institution that controls and
directs youth, or works to preserve and transmit the core culture, but as an institution that
works with youth to improve society or help students realize their individuality” (Sadker
& Zittleman, 2010a).

DEVELOPING MATHEMATICAL REASONING 25
Teacher-centered Pedagogy
“Traditionally, teacher-centered philosophies emphasize the importance of transferring
knowledge, information, and skills from the older (presumably wiser) generation to the younger
one. The teacher's role is to instill respect for authority, perseverance, duty, consideration, and
practicality” (Sadker & Zittleman, 2010b).
Problem-based Learning
“Problem-based Learning (PBL) is an approach to instruction that situates learning in
guided experience solving complex problems, such as medical diagnosis, planning
instruction, or designing a playground. Developed initially for use in medical schools it
has expanded to other settings such as teacher education, business, engineering, and K-12
instruction… students are actively engaged in learning content, strategies, and self-
directed learning skills through collaboratively solving problems, reflecting on their
experiences, and engaging in self-directed inquiry. The role of the teacher is to facilitate
the students' learning by providing opportunities for learners to engage in constructive
processing. The students take responsibility for their own learning and for the collective
progress of their collaborative group” (D'Angelo, Touchman, Clark, O'Donnell, Mayer,
Dean, Hmelo-Silver, 2009, problem base learning, para.1 and 2).
Discovery Learning
“Is an instructional method in which students are free to work in a learning environment
with little or no guidance. For example, discovery learning is the method of instruction
when students are given a math problem and asked to come up with a solution on their
own, when students are given a scientific problem and allowed to conduct experiments,
or when students are allowed to learn how a computer program works by typing
DEVELOPING MATHEMATICAL REASONING 26
commands and seeing what happens on a computer screen” (D'Angelo, Touchman, Clark,
O'Donnell, Mayer, Dean, Hmelo-Silver, 2009, discovery learning, para. 1).
Inquiry Based Learning
In “inquiry learning, students play the role of scientists…Their tasks include formulating
questions, designing informative investigations, analyzing patterns, drawing inferences,
accessing evidence in responding to questions, formulating explanations from evidence,
connecting explanations to knowledge, and communicating and justifying claims and
explanations” (D'Angelo, Touchman, Clark, O'Donnell, Mayer, Dean, Hmelo-Silver,
2009, inquiry based learning, para. 1).
Cognitively Guided Instruction
“A research-based model of children’s thinking that teachers can use to interpret,
transform, and reframe their informal or spontaneous knowledge about student’s
mathematical thinking” (Carpenter, Fennema, & Franke, 1996, p. 1) It maintains that
“children enter school with a great deal of informal or intuitive knowledge of
mathematics that can serve as the basis for developing much of the formal mathematics
of the primary school curriculum” (p.1)
Explicit Direct Instruction
“Explicit Direct Instruction, usually shortened to EDI, is a strategic collection of
instructional practices combined together to help teachers design and deliver well-crafted
lessons that explicitly teach content, especially grade-level content, to all students. EDI is
based on teacher-centered, direct instruction philosophy. EDI is an approach that
encompasses [the] goal of improving learning for all students and especially for low-
performing students” (DataWORKS, n.d.c, Explicit Direct Instruction, para. 1).
DEVELOPING MATHEMATICAL REASONING 27
Mathematical reasoning
“The attainment of abilities to construct mathematical conjectures, develop and evaluate
mathematical arguments, and select and use various types of representations. To help
students meet the standards, [teachers must] emphasize the importance of mathematical
discourse in the classroom. Students not only should discuss their reasoning on a regular
basis with the teacher and with one another but also should explain the basis for their
mathematical reasoning, both in writing and in their mathematical discourse” (Kramarski
& Mevarech, 2003, p.2).
Mathematical perceptions
A student’s evaluation of (1) how much s/he is interested in mathematics, (2) how much
s/he enjoys mathematics, (3) how he/she interprets the mathematical enjoyment and interest of
others, and (4) how much he/she and others enjoy solving mathematical word problems.
Cognitive skills
This term was generated from Anderson et al.’s (2001) work. That is, it is meant to
encompass all skills included within the taxonomy’s cognitive process dimension, as well as
some which are not.
"The cognitive process dimension contains six categories: Remember, Understand, Apply,
Analyze, Evaluate, and Create. The continuum underlying the cognitive process
dimension is assumed to be cognitive complexity; that is, Understand is believed to be
more cognitively complex than Remember, Apply is believe to be more cognitively
complex than Understand, and so on” (p. 5).

DEVELOPING MATHEMATICAL REASONING 28
Knowledge types
This term was generated from Anderson et al.’s (2001) work as well. That is, it is meant
to encompass all the knowledge types that are included within the taxonomy’s knowledge
dimension:
“The knowledge dimension contains four categories: Factual, Conceptual, Procedural,
and Metacognitive. These categories are assumed to lie along a continuum from concrete
(Factual) to abstract (Metacognitive). The Conceptual and Procedural categories
overlap in terms of abstractness, with some procedural knowledge being more concrete
than the most abstract conceptual knowledge” (p.5).,
Bloom’s Taxonomy
“Bloom identified six levels within the cognitive domain, from the simple recall or
recognition of facts, as the lowest level, through increasingly more complex and abstract mental
levels, to the highest order which is classified as evaluation” (Instructional Technology Program
- Graduate School of Education - George Mason University, n.d., Bloom's Taxonomy of
Cognitive Domain, para. 1) .
Lower Order Thinking Skills
Lower order thinking demands only routine or mechanical application of previously
acquired information, such as listing information previously memorized and inserting numbers
into previously learned formulas (Lewis & Smith, 1993, citing Newman, 1990).
Higher Order Thinking Skills
“Higher order thinking skills include critical, logical, reflective, metacognitive, and
creative thinking. They are activated when individuals encounter unfamiliar problems,
uncertainties, questions, or dilemmas. Successful applications of the skills result in
DEVELOPING MATHEMATICAL REASONING 29
explanations, decisions, performances, and products that are valid within the context of
available knowledge and experience and that promote continued growth in these and
other intellectual skills. Higher order thinking skills are grounded in lower order skills
such as discriminations, simple application and analysis, and cognitive strategies and are
linked to prior knowledge of subject matter content. Appropriate teaching strategies and
learning environments facilitate their growth as do student persistence, self-monitoring,
and open-minded, flexible attitudes” (King, Goodson, Rohani, n.d.)
Unfamiliar (non-routine) Problems
Problems in which the “problem solver does not initially know how to go about solving.
For example, the following problem is non-routine for most high-school students: ‘If the
area covered by water lilies in a lake doubles every 24 hours, and the entire lake is
covered in 60 days, how long does it take to cover half the lake?’ Robert Sternberg and
Janet Davidson (1995) refer to this kind of problem as an insight problem because
problem solvers need to invent a solution method” (Mayer & Wittrock, 2009, types of
problems, para. 2).
Familiar (routine) Problems
Problems in which the “problem solver knows how to go about solving... For example,
two-column multiplication problems, such as 25 x 12 = ___, are routine for most high school
students because they know the procedure” (Mayer & Wittrock, 2009, types of problems, para.
2).
DEVELOPING MATHEMATICAL REASONING 30
CHAPTER TWO: LITERATURE REVIEW
Structure of the Literature Review
In order to appreciate the importance of this study, one must first be presented with, and
understand, the events that have led to its inception. To accomplish this, the following literature
review is presented. The review employs a thematic organization to answer five major questions.
(1)How has mathematical pedagogy been thought of during ancient and contemporary times? (2)
What are the pedagogical differences between Explicit Direct Instruction and Cognitively
Guided Instruction? (3) How do American students perform with regards to mathematical
reasoning? (4) What does research say about how students like to be taught? And, (5) where do
we go from here? Prior to delving into these questions, the review first presents an introductory
section explaining the ways in which contemporary literature framed the discourse of student-
versus teacher-centered pedagogies. It will discuss how different policies led to the current
academic situation as well as the need for a more precise model of pedagogical research and
practice.
Introduction
What Contemporary Literature Says About Pedagogy
The theoretical and pedagogical discourses that have occurred between the supporters of
teacher-centered pedagogies (Kirschner, Sweller, Clark, 2006; Meyer, 1984, Adams &
Engelmann, 1996) and those who support more student-centered approaches (Loyens & Gijbels,
2008; Schmidt, Loyens, van Gog, & Paas, 2007; Schmidt, van der Molen, Wilco, te Winkel, &
Wijnen, 2009;) still persist. The discussion, however, recently focused on the amount of
cognitive load that learners must contend with during instruction. For instance, several
supporters of teacher-centered methodologies claimed that constructivist approaches such as
DEVELOPING MATHEMATICAL REASONING 31
inquiry, discovery, or problem-based learning, ask learners to process a debilitating amount of
information which inhibits their ability to learn (Tuovinen & Sweller, 1999; Kirschner, Sweller,
& Clark, 2006). Consequently, models such as Information Processing Theory (IPT) or
Cognitive Load Theory (CLT), two teacher-centered approaches that spring from the cognitive
tradition of psychology, maintain that the amount of information embedded within a lesson
should be considered the fundamental pedagogical question. That is, because information
overload can cause negative learning outcomes, especially when instructing complex material,
teachers should design lessons that take into consideration what research discovered about
human cognitive architecture (Kirschner, Sweller, Clark, 2006).
An example of the success of this pedagogical approach can be found in Vogel-Walcutt,
Gebrim, Carper, Nicholson’s (2011) work. They demonstrated that, by following the premises of
Cognitive Load Theory (CLT), they could teach complex skills more efficiently and effectively
to military personnel. Still, it is important to note that the Vogel-Walcutt’s study was not
exceptionally conclusive regarding every aspect of learning. In fact, the data showed mixed
results when CLT was compared to student-centered methodologies regarding the “acquisition of
procedural, declarative, and conceptual knowledge, as well as, decision-making skills” (p.133),
skills which have become pertinent to 21
st
century students (Peters, 2010; Putnam, Lampert, &
Peterson, 1990). Moreover, when such inconclusiveness is coupled with the findings of Schmidt,
van der Molen, Wilco, te Winkel, & Wijnen (2009), who demonstrated an increase in the
interpersonal and practical skills of medical students instructed using student-centered
pedagogies ( problem-based learning), the superiority of teacher-centered methodologies begins
to falter. Michael (2006), for example, wrote a meta-analysis of the effectiveness of active
learning within physiological university classrooms.
DEVELOPING MATHEMATICAL REASONING 32
This paper, therefore, argues that Vogel-Walcutt et al.’s (2011) results are a microcosmic
example of the larger macrocosmic reality of American schools. As a result, it presents a new
research model to restructure the dichotomic impasse that characterized the educational
landscapes of the 20
th
and 21
st
century. This review, therefore, presents evidence in support of
both pedagogical approaches as well as the amalgamation of each, student-, teacher-, and student
plus teacher-centered instruction. However, the affirmation of these three instructional
approaches leads to two questions: (1) when should a teacher choose one pedagogy over another?
And, (2) how should a teacher choose one pedagogy over another?
The answers to these two questions are what should lead educational researchers to
understand the necessity for a newer model of investigation constructed upon specificity rather
than generality. That is, researchers need to investigate the ability that particular instructional
approaches can offer regarding specific content domains, knowledge types, and cognitive sub-
skills. Using this type of framework, the current study examined which pedagogical approach,
EDI or CGI+EDI, is most suitable at developing the mathematical reasoning of elementary
school students.
Recent Events Affecting the Discourse
To understand why this framework should be considered a change away from the current
paradigm, one must look at the recent history of American education. Between 2000 and 2010,
American education became engulfed by the need for accountability (Linn, Baker, & Betebenner,
2002). This need was nationally institutionalized by one of most ample educational policies the
country had ever experienced.
By signing the No Child Left Behind Act (NCLB) into law, George W. Bush initiated a
new era in American education characterized by high-stakes tests, teacher accountability
DEVELOPING MATHEMATICAL REASONING 33
measures, and greater federal control. To guarantee the success of this new policy, the federal
government instituted a top-down system of control that used the results of standardized tests to
monitor the Adequate Yearly Progress (AYP) of schools. This pressure to perform compelled
many school-districts, school-sites, and individual educators to affirm teacher-centered
pedagogical models, such as Engelmann’s (1980) Direct Instruction, as favorable approaches by
which to control and manage what occurred within their classrooms. As a result, this period
became a time in which lessons were less cognitively demanding (Au, 2007). A fact that became
especially true for schools impacted by a high number of students identified to be of low socio-
economic status, who were part of a minority groups, or who were English Language Learners
(Ellis, 2008; Valli & Buese, 2007).
Ultimately, the new policy accomplished what it was intended to achieve; it forced school
districts to focus on every student’s needs. In addition, the law also succeeded in a second way;
it forced districts to find ways by which to increase their teacher’s scores. That is, in order to
meet their targets, schools and school-districts looked for and trained teachers in pedagogical and
methodological approaches known to increase the standardized test scores of students, regardless
of the overall educational cost that would entail.
The academic results brought forth by NCLB continue to be disputed. Depending on which
content domain, grade level, demographic variable, or examination instrument is employed,
NCLB has both positive and negative results. Dee and Jacob’s (2010), for example, reported an
increase in math scores, but not in reading scores whereas Azzam (2007), using research
conducted by the Center for Educational Policy (CEP) mentions that both reading and math
scores improved since the law was enacted. In fact, the CEP (2007) reported that their data
affirmed the following: (1) most states with comparable data have exhibited an uptrend in scores,
DEVELOPING MATHEMATICAL REASONING 34
(2) the math scores’ of elementary schools have improved greatly since the inception of NCLB,
(3) a greater number of high schools seem to be improving their math and reading scores than
those which are not, and (4) the achievement gaps that have historically divided minority and
low socioeconomic status (SES) students from their counterparts seem to be narrowing.
Still, the Center for Education Policy (CEP) also stated how difficult it is to determine the
connection between these positive trends and NCLB. According to the CEP, this is due to the
fact that, since the enactment of NCLB, state governments, school districts, and individual
schools have implemented a number of policies in order to raise the achievement of their
students, some that have been in direct response to NCLB and some that have not.
Others also questioned the connection that exists between NCLB and the upward trend in
the state scores. In an article written in 2007, Hursch described a report authored for the Harvard
Civil Rights project (Lee, 2006) as evidence of NCLB’s failure. The report presents evidence
that, although NCLB increased the state test scores of minority students, it did not improve the
math or reading scores of such students in the National Assessment of Educational Progress
(NAEP), an index considered by many to be America’s report card. This lack of improvement
has also been witnessed in international examinations. Both the TIMSS (Trends in International
Mathematics and Science Study) and the PISA (Program for International Student Assessment)
revealed that American students have certain educational gaps, such as the absence of
mathematical reasoning skills, which are not being identified by state examinations (Fuller,
Wright, Gesicki, and Kang, 2007).
Identified as a critical component of any 21
st
century tuition, mathematical reasoning is
now tested as its own cognitive domain by a number of well-known international and national
examinations. For example, about 20 percent of the math questions in the 4
th
grade TIMSS
DEVELOPING MATHEMATICAL REASONING 35
examination were designated to assess mathematical reasoning (Mullis, Martin, Foy, Arora,
2012). Likewise, the 2011 NAEP examination identified mathematical reasoning as its newest
subtopic. According to the National Assessment Governing Board, this new subtopic arose
from the need to prepare 12
th
grade American students to effectively participate in “post
secondary education and training” (National Assessment Governing Board, 2010, p. 2). In short,
about 25 percent of questions in the NAEP were characterized as being of “high mathematical
complexity” (National Assessment Governing Board, 2010). Consequently, both international
and national tests include sub-sections specifically designed to evaluate how students reason,
plan, analyze, and/or construct creative paths to solving non-routine mathematical problems.
State examinations such as the California Standardized Test (CST), on the other hand, only
embed such skills rather than allocate specific sections, or items, to assess them (Appendix C).
The choice to embed rather than to clearly identify mathematical reasoning within state
examinations gives American teachers the option to not instruct such skills, which resulted in
low international and national mathematical reasoning scores. For instance, international tests
show that “in terms of comparisons with other countries, U.S. fourth graders [tend to] perform
relatively better on average in the knowing domain than [in] the applying and reasoning domains”
(Gonzales, et al., 2008, p. 10). Moreover, the 2007 TIMSS examination revealed that, when
American students were asked to solve items requiring the use of lower order thinking skills such
as recollection of facts, recognition of objects, and computation of algorithms, students from
only 5 international educational systems, Hong Kong SAR, Singapore, Chinese Taipei, Japan,
and Kazakhstan, scored significantly higher (Gonzalez, et al., 2008). Conversely, when
examining the application of mathematical knowledge, students from 11 educational systems
scored statistically higher than did students in the United States. And, when isolating questions
DEVELOPING MATHEMATICAL REASONING 36
designed to assess mathematical reasoning skills like determining and describing relationships
between variables, using proportional reasoning, and decomposing geometric figures, students
from 7 educational systems scored higher than did students in the United States. Therefore,
regardless of the increased focus that mathematical reasoning experienced amongst national and
international examinations, the data shows that American students continue to struggle with
these skills (Gonzales, Williams, Jocelyn, Roey, Kastberg, & Brenwald, 2008). Such results
present a serious pedagogical gap that needs to be addressed. That is, while the need to reason
mathematically became an increasingly important aspect within the life of a 21
st
century student,
the American educational systems continues to struggle in instructing such cognitive skills.
These discrepancies in scores could be due, in part, to the type of methodologies that
American teachers employ ever since the enactment of NCLB. In fact, studies show that the
increase in accountability had a direct impact in the pedagogical and methodological choices
teachers made when instructing their students (Valli & Buese, 2007; Weiss, Pasley, Smith,
Banilower, & Heck, 2003). For example, while studying the way elementary teachers are
influenced by national, state, and local educational policies, Valli and Buese (2007) reported that,
during NCLB, teachers demonstrated “a decline in every category that signified cognitively
complex instruction (p< .01) and either a rise or constancy in instruction that placed little
cognitive demand on students” (p.546). They also noticed a drop in higher order questions and
thought-provoking assignments, the two factors that Darling-Hammond (2007) proposed were
behind major pedagogical differences existing between American schools and their international
counterparts. And, when interviewing teachers and observing classrooms, Weiss et al. (2003)
found that “entire lesson[s] were focused on having students perform well on high stakes tests
without also focusing on student understanding” (p. 47). In short, these studies point to the fact
DEVELOPING MATHEMATICAL REASONING 37
that teachers in America have been more inclined to teach the skills required to perform well in
multiple choice tests than on developing the higher order thinking skills their students will
require long beyond that test.
Besides the decrease of higher-order thinking skills, the accountability movement also
brought about a resurgence of teacher-centered pedagogies. Having lost ground to competing
constructivist ideologies during the 1990’s, behaviorist (didactic) pedagogies experienced
resurgence during the first decade of the new millennium. Evidence of this can be seen in the
growth of companies like DataWORKS, an educational consulting company specializing in
curriculum calibration and Explicit Direct Instruction (EDI). Founded in 1997, a few years prior
to the enactment of NLCB, DataWORKS increased their clientele dramatically during its first
thirteen years of business (DataWORKS, n.d.).
DataWORKS asserted that the company’s success has been, in part, due to the empirical work of
past educational researchers. One of the researchers the company points to is Jeanne Chall.
According to DataWORKS, Chall’s (2000) meta-analysis offers a century of statistical evidence
identifying what kind of pedagogical approaches (teacher or student-centered) offer the greatest
capacity to increase student learning. According to Chall, the answer was clear. In most cases,
and especially for students considered “at-risk”, the best pedagogical approaches employed a
teacher-centered model of instruction grounded on a behaviorist model of learning. The
company also presents the research of Rosenshine and Stevens (1986) as evidence that
successful teachers always “start [their] lessons by reviewing prerequisite learning”, “providing a
short statement of goals”, “giving clear and detailed instructions and explanations”, “checking
for understanding”, and “providing explicit instruction and practice for seatwork exercises”
(DataWORKS, n.d.) – which, again, are all characteristics of a teacher-centered model of
DEVELOPING MATHEMATICAL REASONING 38
instruction that arises from a behaviorist model of learning. Furthermore, the company also
offers the work of Baker, Gerstern, and Lee (2002) as evidence of how effective direct
instruction can be and the work of Rittle-Johnson (2006) as proof of the capacity offered by
teacher-centered pedagogies to raise student achievement. Hattie’s (2008) extensive examination
of 800 meta-analyses is also put forth as proof that “Direct Instruction [can be] very significant
[in] influencing student achievement” (DataWORKS, n.d.). In short, according to the review
presented by DataWORKS, the literature is very clear; in order to increase their students’
academic performance as well as their test scores, teachers must employ teacher-centered
pedagogical approaches that employ a behavioral model of learning.
But not everyone agrees with this position. Contrasting this behaviorist approach of
learning is the constructivist view. This view of learning begins with the idea that children bring
with them a breath of knowledge that can, and should, be employed to build further
understanding. Furthermore, the constructivist view assumes that children desire to understand
the world and, as a result, are inquisitive about it. Consequently, rather than perceiving children
as tabulae razae upon which information can be imprinted, constructivist see children as
interpreters of past, present, and future knowledge. This is why constructivists’ models often use
thought provoking questions to prime a learner’s meta-cognitive skills, reflecting upon one’s
thinking, resulting in the acquisition of deeper understanding. Hence, a constructivist educator
must always be a facilitator of knowledge rather than an imparter of it. As a result, a
constructivist classroom must always be characterized by students who understand what they are
saying and why they are saying it, rather than illogically echoing what their teacher uttered.
Wakefield (1997) explained this approach in more detail. According to Wakefield, the types of
higher order thinking that a constructivist approach can offer a learner are the following: active
DEVELOPING MATHEMATICAL REASONING 39
thoughts, social interaction, autonomy, and an understanding of the value of errors – the last one
being similar to Kapur’s (2010) work with productive failure. Wakefield explains that teachers,
as well as parents, can become so concerned with wrong answers that they fail to recognize what
the child might actually be doing right based on his/her development. Wakefield describes this as
“developmental errors [on] the way to being right” (p. 236). But Wakefield does not stand alone
in this perspective. Anthony and Walshaw’s (2009) booklet, written for the International
Academy of Education, describe how effective mathematics teachers perceive of incorrect
answers as “understanding in progress” (p.11). This allows such teachers to see “their students’
thinking as resources for further learning” (p.11), instead of as empty buckets that lie idle
waiting for a teacher to fill them with the knowledge required by the state examination.
As can be seen from the information presented, the debate between which pedagogies are
best is yet to be resolved. Within the domain of mathematics education, this difference of
opinion is known as “the math wars” (ED.gov, n.d.). Former director of the Institute of
Education Sciences within the U.S. Department of Education, Doctor Grover Whitehurst, once
explained the situation in this fashion. “[American education has been in the middle of] a
philosophical war about what math children should be taught and how they should be taught it…
It would be very satisfying if I could [declare] that the math wars have been resolved based on
high quality research… unfortunately [the truth is] far from that” (ED.gov, n.d.). This lack of
resolution generates national economic problems.
The Economic Connection
Although the lack of resolution between mathematical pedagogies would appear to most
individuals as an inconsequential pedagogical debate, the reality is in fact much different.
America’s inability to resolve such indifferences generates the potential for future economic
DEVELOPING MATHEMATICAL REASONING 40
problems. In fact, some economic theorists believe that nations who fail to teach higher order
thinking skills to their children will not be able to economically compete in the global markets of
the 21
st
century. Peters (2010) and Putnam, Lambert, and Peterson (1990), for example, assert
that 21
st
century students will require skills that extend beyond that of the behaviorist model to
which American teachers have grown so accustomed (Weiss, Pasley, Smith, Banilower, & Heck,
2003). The argument is a simple one, contemporary educational systems need to provide
students with more than the ability to recall factual and/or procedural information, two of the
knowledge types that have recently engulfed American education (Porter, McMaken, Hwang, &
Yang, 2011). Instead, educational systems will have to make sure their students have the skills,
and the knowhow, to employ higher order thinking skills in order to innovatively solve the
unique and unforeseen problems that their futures will present. Those problems will be so
unprecedented and complex, that no current individual is endowed with the factual, procedural,
or conceptual understanding that their solutions will require (Anderson, Greeno, Reder, &
Simon; Bereiter, 2002; Botstein, 1997; Kuhn, 2005, 2007). As a result, Wakefield suggests that
schools should “provid[e] environments that encourage children to think and figure things out for
themselves” (p. 236), two skills that are deeply embedded within a constructivist (student-
centered) approach of learning. The following words by Kuhn (2007) elucidate this need more
clearly: “a resolution has been [made] in the direction of undertaking to teach not simply
knowledge itself but the skills of knowledge acquisition – skills what will equip a new
generation to learn what they need to know to adapt flexibly to continually changing and
unpredictable circumstances” (p.110).
Two researchers who saw the need for such a pedagogical shift were Sawyer (2006) and
Resnick (2007). Their work illustrates the necessity that schools face to teach students to think
DEVELOPING MATHEMATICAL REASONING 41
creatively rather than retain and recall information. Their work offers playing, imagining, and
idea sharing as possible vehicles to assist students in developing the innovative thinking. In
essence, they concur with Peters’ (2010) perspective and affirm that only by increasing students’
higher order thinking skills will nations provide their future citizenry with the necessary skills to
compete in the new “knowledge economies”.
However, when one examines the structure of a Direct Instruction (DI) lesson like
identifying a learning objective, activating prior knowledge, developing the skill and conceptual
understanding of the student via step by step instructions, stating the relevance of the day’s
objective, checking for understanding, closing the lesson, and having the students practice
independently, it is evident that it leaves no room for any kind of creativity, problem solving, or
critical thinking process. DI simply asks students to retain and recall information in order to
regurgitate it at a later time. This, of course, is not problematic if a teacher is trying to impart
factual or procedural information. However, the students of the 21
st
century will require more
than the ability to recall facts (Peters, 2010). Their participation in the new knowledge
economies will demand skills that are not part of the DI model, such as innovation, creativity,
adaptation, communication, and the ability to analyze information.
Peters (2010), for example, who investigated education via an economics lens, claimed
that, because much of the world shifted away from the Industrial Economical model into a
Creative Economical model, schools should focus on a new set of cognitive skills. Learning
institutions should be involved in teaching their students how to be more inventive with their
knowledge, rather than simply assimilating it. He states that “there is now widespread agreement
among economists, sociologists and policy analysts that creativity, design and innovation are at
DEVELOPING MATHEMATICAL REASONING 42
the heart of the [new] global knowledge economy” (p.69). In agreement with Peters, Sahlberg
mentions that a change is required in the educational systems of industrialized nations. He says:
…a knowledge society is grounded upon the power to think, learn, and innovate both
individually and collectively. Learning to think, to learn and to innovate requires more
than orderly implementation of externally mandated regulations. The high-demand
features of modern schooling – learning together, creating new ideas, and learning to live
with other people peacefully, best occur in an environment decidedly different from what
our schools offer young people and their teachers today (p.47 & 48).
Peters and Sahlberg are not alone in this perspective. Almost twenty years earlier, the
Carnegie Commission on Science, Technology and Government (1991) expressed that “the
national interest [was] strongly bound up in the ability of Americans to compete
technologically. . . . All young people, including the non-college bound, the disadvantaged, and
young women, [would have to] be [presented with] the opportunity to become competent in
science and mathematics” (p. 15). The above perspectives, consequently, point to following four
factors regarding 21
st
century educational systems: (1) they need to generate creative individuals,
(2) they need to teach higher order thinking skills in the domains of technology, math, and
science, and (3) they need to enact pedagogical models offering the ability to develop students’
communication, technological, reasoning, creative, social, and psycho-emotional skills (e.g.,
diligence).
In addition to the development of higher order thinking skills, advocates of student-
centered pedagogies affirm that constructivist’s methodologies can also assist in the acquisition
other skills that will be in great demand during the 21
st
century, social and emotional skills.
Michael (2006) presents evidence of this. According to Michael, research in the learning
DEVELOPING MATHEMATICAL REASONING 43
sciences, as well as in cognitive science and educational psychology, supports the use of active
learning educational techniques to increase a student’s ability to not only gain greater and deeper
knowledge between content domains, but also to learn how to communicate what they know to
others. Consequently, pedagogies through which students are continually being asked to express
their thinking ( student-centered approaches of learning) should work to increase the students’
ability to (1) work as a team by offering and listening to ideas, (2) communicate most effectively
with others, and (3) develop the ability to expand and improve on the ideas of others. It is
important to note that these are some of the skills that Peters (2010) and Sahlberg (2010)
discussed as critical to the success of students and, in turn, nations during the 21
st
century.
In short, 21
st
century schools will have to alter the way that they carry out their work.
That is, on top of teaching students all the factual and procedural knowledge they will require to
pass their nation’s summative high stake tests, they will also have to instruct students on how to
work with one another as well as individually, to evaluate, analyze, and to synthesize critical
information (Ertmer & Newby, 1993). Moreover, schools will also have to construct situations
in which students gain the creative skills to generate innovative solutions to unusual problems, as
these are skills that should serve them well in a future characterized by complex and unusual
world dilemmas like overpopulation, sources and consumption of energy, technological advances,
and unforeseen moral and ethical predicaments (Kuhn, 2005, 2007).
Recent Shifts in Educational Policy
Policy makers have begun to understand the need for schools to instruct metacognitive
knowledge alongside factual and procedural knowledge. They have also begun to understand
that teaching metacognition will require a systematic change in the way American educational
institutions have recently carried out their work. Consequently, in order to renegotiate the way
DEVELOPING MATHEMATICAL REASONING 44
teachers instruct students, a new set of standards (The Common Core) was introduced. These
new standards demand that students not only remember the facts within each content domain, but
also understand higher order thinking skills such as (1) how to evaluate their own performance
and the performance of others, (2) how to identify and utilize patterns to accomplish tasks, (2)
how to reason and critique the reasoning of others, (3) how to persevere when confronted with
difficult endeavors, (4) how to use tools appropriately, and (5) how to attend to precision in their
work (Common Core Standards Initiative, 2014). However, these skills cannot be taught by the
pedagogical techniques that tend to characterize teacher-centered instructional models. As a
result, new pedagogical approaches have to be identified and implemented.
The shift to develop such schools will require great institutional change, which the United
States seems poised to make. Ed. Magazine, a publication prepared by the Harvard Graduate
School of Education, published an article discussing a few of the policies and methodologies that
influenced mathematics education in the United States, the latest one being the Common Core
Standards (Wilson, 2013). Amongst some other changes Wilson states, the new standards bring a
departure from the broad yet shallow educational approach that was established during the
accountability movement (Common Core State Standards Initiative, 2010b, p. 3). As a result, the
Common Core will decrease the number of standards that teachers have to cover, which will
allow educators to spend more time developing their students’ conceptual understanding and
higher order thinking skills (Porter, Mc Maken, Hwang, & Yang, 2011; K–8 Publishers’ Criteria
for the Common Core State Standards for Mathematics, n.d.). Additionally, this will also force
teachers to replace their previous drill and kill behaviorist methodologies (Silver, Mesa, Morris,
Star, & Benken, 2009) for more conceptually-based pedagogical models.

DEVELOPING MATHEMATICAL REASONING 45
How has Mathematics been Taught during Past and Present Times?
This first theme focuses on the historical lineage of mathematical education.
Consequently, it begins by discussing the work of ancient mathematicians in order to see how
such individuals viewed, understood, and carried out their mathematical instruction. Following
that historical account, the review explores the philosophical and pedagogical differences among
behaviorism, cognitivism, and constructivism - the three theoretical frameworks that were
critical to the study. To do so, a comparison of some of the major contributors to each theory,
Thorndike, Skinner, Watson, Piaget and Dewey, is discussed ( ). Such comparisons naturally
lead to a discussion of the different pedagogical approaches attached to each theory. This section
concludes by focusing on a third set of scholars who offered a middle ground within this
historical dicothomic perspective.
The Ancient Past
Thus it appears that many of the pedagogical techniques and principles that are favored
today have experienced a long and diverse history. That history not only spans time but
also cultures and testifies to the fact that while mathematics is universal, to a large extent,
so is its pedagogy. The need to communicate mathematical understanding spawned its
own devices. They were conceived to help popularize mathematics: to more easily reveal
its concepts, to facilitate and strengthen a memory of its processes and operations and
insure an efficient application of the knowledge learned. [Hence] mathematics and good
teaching were bound together thousands of years ago… (Swetz, 1995, p.85-86)
Although a very lengthy quote, the above statement was included in this section because
it eloquently explains what the focus of it is, which is that, for thousands of years, people have
been performing mathematical computations based on their personal necessities and desires. And,
DEVELOPING MATHEMATICAL REASONING 46
for as long as they have done such computations, they have also developed ways by which to
pass that knowledge on to one another.
Evidence of the connection among the practical needs of individuals, the math they
perform, and pedagogy they create to share what they know, was discovered all over the world.
Gullberg (1997), for example, asserts that “the earliest traces [of mathematics}... date from about
a hundred centuries before our time… [exemplified by] a form of counting where markers,
counters, or tokens corresponded directly with the things or goods they represented” (p.5).
Moreover, the Rhind Papyrus (1550 BCE), an ancient mathematical manuscript, was found to
contain 84 practical problems dealing with numerical operations, problem solving, and
geometrical shapes (The British Museum, n.d.) whereas “notches on a 25,000-year-old fossilized
bone found in Zaire may have represented phases of the moon...” (Smith, 1996, p.1). In
accordance, ancient Pythagoreans viewed numbers as the fundamental components of nature,
and Plato believed that mathematics would allow one to understand the unadulterated world of
the universal forms (Campbell, 2001). Thus, it is clear that the history of mathematics is
inextricably related to the practical needs humans have had to count and/or represent objects.
As such, historically, math appears to have been perceived and defined much like the
constructivist perceive and define it today: an expression of someone’s experiences rather than a
distant and alien esoteric activity. An example of this can be seen in the book Rewriting the
History of School Mathematics in North America 1607-1861. In it, Ellerton and Clements (2012)
discuss the history of the ciphering tradition - an ancient arithmetic pedagogy that endured for
about 600 years. Their work found that the method of instruction employed by this tradition was
not lecture, but, rather, the transcription of knowledge from one ciphering book to another, which
begs the question. Why would any child desire to carry out the tedious task of transcribing math
DEVELOPING MATHEMATICAL REASONING 47
problems over and over from one book to another? The answer is found in the extrinsic reward
that was waiting for anyone who would complete the tuition. According to Ellerton and
Clements, “the curriculum within the tradition had a strong mercantile emphasis, and was most
suited to boys who wished to gain employment as clerks or reckoners, or as navigators or
surveyors” (p. 7). Consequently, it would seem that the practical necessity of gaining an
occupation provided the necessary motivation to carry out the process. In essence, math, to such
boys and, sometimes, girls, had a very real and palpable meaning.
Another historical illustration of the connection between mathematical curricula and
practical necessities is found in the work of Howson (1982). According to Howson, Robert
Recorde (ca. 1512-1558), a Welsh mathematician and contemporary to the ciphering approach,
wrote a series of works that successfully catered to the learning necessities of 16
th
century
craftsmen and tradesmen. With his work, Recorde departed from the pedagogy of his time and
created a new way of learning mathematics (Johnson & Larhey, 1935) that did not require the
traditional lecture-based approach to which most pupils and teachers of his time were
accustomed. Recorde’s genius, therefore, was to envision a curriculum that would offer those not
able to attend formal training a way to understand math on their own. “The result was a set of
texts of astonishing freshness, vivacity, and great pedagogic insight, which put many textbooks
of the subsequent 400 years to shame” (Fauvel, 1989, p. 2). As a result of Recorde’s work,
curricula became malleable pedagogical tools that could be shaped and adjusted in order to meet
the needs of students – a perspective which is now identified as individualized instruction. It is
important to note, that in doing so, Recorde placed the responsibility of learning upon the
individuals themselves, which forced them to construct their own knowledge. No teacher was
DEVELOPING MATHEMATICAL REASONING 48
there to either guide them or provide the necessary motivation to accomplish the task. Their
motivation came from their necessity to understand mathematics to perform their daily work.
In addition to allowing for greater accessibility and individualized instruction, ancient
texts also include examples of how important social interaction was considered for mathematical
instruction. Mathematical conversations appear to have been used as a valuable pedagogical tool
during ancient times. Examples of such techniques can be found in Recorde’s dialogues between
a master and his student, as well as in the Zhoubi Suanjing (ca 100 BCE), which presents a
mathematical dialogue between the “Duke of Zhou and his Grand Prefect Shang Gao (Swetz,
1995, p. 75). The ciphering tradition can also be considered highly communal. According to
Ellerton and Clements (2012) “students wrote initial solutions to problems on scraps of paper…
and if and when the tutor indicated that they were correct, they were inscribed into ciphering
books” (p. 38). Consequently, there seems to be an abundance of historical evidence pointing to
the importance of social interaction (teacher to student) and/or mathematical conversation
(teacher to student, text to student) as a sound pedagogical method.
A perfect example of such a dialectical process is found within Plato’s Meno (Plato, 380
B.C./1961). In it, Socrates is found dialoguing with a man named Meno about how people can
become virtuous. During their discourse, Socrates shows Meno how individuals may be assisted
in understanding concepts hidden deep within them. To demonstrate this point, Socrates
questions a slave boy about the characteristics of a square. By the end of their conversation, the
unlearned boy understood a number of complicated truths about the mathematical attributes of
the shape and its area, which, as Freydberg (1993) pointed out, paints a picture of what is to be
considered good quality pedagogy.
DEVELOPING MATHEMATICAL REASONING 49
On top of the revelation to employ dialogue as a strong pedagogical technique, the Meno
also presents evidence as to the value of other student-centered methodologies, discovery
learning, problem-based learning, and guided learning. One could argue that the Meno, as well
as most of the other interactions that Socrates had, are examples of problem- and/or inquiry-
based learning. In other words, both problem and/or inquiry-based learning, much like the
Socratic Method, are grounded on unanswered questions designed to create cognitive dissonance
in order to guide the learner to greater understanding. For example, in trying to answer Socrates’
questions about the square, the slave boy is forced to evaluate all his prior knowledge, examine it
for weaknesses and either confirm or dismiss his current schema, which, ultimately, helps him
arrive at the newer more complex understanding of the shape. In fact, most of Plato’s writings
include Socrates asking his audience rhetorical questions designed to generate uncertainty about
their current cognitive state. Thus, Socrates’ techniques appeared to parallel those employed by
contemporary constructivists. That is, rather than simply showing or telling what he knows
about a subject, Socrates tried to create an environment in which the pupil was forced to build or
discover her own knowledge.
Along with cognitive dissonance, ancient texts also include examples in which learners
are asked to use a high degree of personal mathematical insight. For example, Swetz (1995) talks
about Babylonian clay tablets (originating around 2000 BCE) that include a number of exercises
“require[ing] a high degree of geometric intuition” (p.77). Such exercises point to the fact that
ancient scholars were concerned with what contemporary researchers identify as higher order
thinking skills (Lewis & Smith, 1993). That is, Ancient students were not only asked to
memorize algorithms, but were also asked to manipulate their knowledge in novel and complex
ways, which represents an approach that resembles the techniques espoused by the supporters of
DEVELOPING MATHEMATICAL REASONING 50
problem and inquiry-based learning (e.g., Carpenter, Fennema, & Franke, 1996; Wakefield,
1997; Erickson, 1999, Zemelman, Daniels, & Hyde, 2005).
Not all ancient techniques appear to align themselves with student-centered pedagogical
approaches. A number of historical methodologies seem to fit better within a teacher-centered
model: information processing theory, cognitive load theory, direct instruction. For example,
ancient texts present evidence that lead us to believe that early mathematics instructors thought
of “learning… mathematics [as] an active process involving the doing of mathematics [and] the
solving of problems…” (Swetz, 1995, p.77). To accomplish this, a number of ancient authors
produced works primarily composed of various mathematical exercises meant to be studied and
analyzed. An example of this was found in a cuneiform Babylonian tablet containing a sequence
of geometrical problems (Swetz, 1995). The tablet was originally composed of 40 problems that
were specifically arranged hierarchically based on their difficulty. As the learner proceeded
through the appointed series of geometric problems, he/she would encounter “an increased
degree of conceptual difficulty [where] solutions to latter problems [would] require the transfer
and adaptation of principles learned or used in obtaining solutions [of] … previous problems”
(p.78). This technique appears to follow the contemporary methodological approaches of
Information Processing Theory (IPT) and Direct Instruction (DI). That is, instead of assisting the
student conceptually, the main concern of the tuition is the process: how to do mathematics.
Another example of such controlled learning experience in which easier problems incrementally
become more difficult is found in the Jiu Zhang Suan Shu, a Chinese mathematical text from
around 100 BCE. Swetz (1995) mentions that the problems within it offered “complex
mathematical situations that [had to] be unraveled by the reader” (p.78). Siu (1993) concurred in
mentioning that the problems in the Jiu Zhang Suan Shu fail to offer any conceptual justification
DEVELOPING MATHEMATICAL REASONING 51
or explanation regarding their solution. Given this lack of conceptual guidance, therefore, the
process appears to follow a cognitivist approach to learning, which is one in which learning the
steps to solve math is more important than developing the conceptual and metacognitive
knowledge of the student. Consequently, such works simply offer a step-by step-guide of the
process (procedural and factual knowledge) and ignore the other two knowledge types.
Still, historical evidence shows that ancient scholars understood that simply presenting
students with an excessive number of problems did not guarantee their learning. Consequently,
ancient texts also show evidence that their writers employed the technique of scaffolding. That
is, some ancient teachers appear to have deliberately chosen and organized problems in ways to
maximize the acquisition, retention, and recall, of mathematical information, which, again,
parallels the contemporary techniques used in the Information Processing and Direct Instruction
models. For example Siu (1993) comments that the 246 problems divided amongst 9 chapters
found in the Jiu Zhang Suan Shu were “typical of most ancient oriental mathematical texts: [that
is] a few problems of one particular type along with the answers, followed by a description of the
method used to solve them” (p. 346). Moreover, the last chapter of the book is composed of 24
problems that are ordered based on the specific mathematical principle that the author was trying
expose and the level of conceptual difficulty that was required to solve them (Swetz, 1993, 1995).
Such evidence, therefore, points to the type of controlled methodological approaches that were
used by some ancient educators.
The final historical example of a more controlled way of instruction comes from the
abaci manuscript. The contents of this text include a prescriptive series of mathematical
problems that also appear designed to help the student gain greater procedural knowledge, an
approach that approximates the techniques described by Engelmann’s (1980) direct instructional
DEVELOPING MATHEMATICAL REASONING 52
model. That is, along with the student-centered pedagogical approaches previously discussed,
ancient manuscripts also present evidence of very controlled and guided pedagogical techniques.
Pedagogies that appear to follow many of the same instructional techniques currently employed
by those who support cognitivists (Sweller, 1988, 2006; Tuovinen, & Sweller, 1999; van Gog,
Paas, & Sweller, 2010) and behaviorists’ models of learning, Engelmann’s direct instruction).
Along with the dichotomic methods described above, ancient texts also include a number
of techniques that support both teacher and student-centered pedagogical approaches. For
instance, in order for pupils to “do” mathematics, ancient texts urged their readers to employ
manipulatives and visual aids – a scaffolding technique espoused by both pedagogical camps.
Swetz (1995) mentions that the first European books to include images were mathematical
manuscripts. He also asserts that ancient Chinese mathematical texts were full of colorful
illustrations meant to assist readers in their learning. Another example of such visual scaffolding
can be seen in the work of Zhao Suang (ca 300 CE). The author of 21 geometrical theorems,
Suang “used color to elucidate the areas in his geometrical-based proofs” (Swetz, 1995, p. 81).
The use of manipulatives was also common among ancient works. According to Ang
(1977) color rods were used by ancient instructors as assistive tools to help students compute
their calculations (Ang, 1977). Whereas Liu Hui (ca 263 CE), an ancient Chinese teacher,
advised his learners to draw diagrams on paper, cut them, and rearrange them in order to
augment their understanding (Switz, 1995), a hands-on technique that appears to have been quite
common in ancient China. In fact, according to Sui (1993), using paper as an assistive tool was
so normal that, when Euclid’s Elements was finally translated, the version of the work came with
three dimensional paper-manipulatives (Swetz, 1995).
DEVELOPING MATHEMATICAL REASONING 53
Ancient mathematical teachers were also concerned with their students’ motivational
levels, which alludes to the fact that, much like today, they might have had an understanding of
the connection between motivation and learning. An example of this is found in the work of
Ahmes (1650 BCE) who, in order to increase the possibility of catching and holding his reader’s
attention – a topic discussed by contemporary researchers such as Dewey (1913), Hidi & Baird
(1986), and Mitchell (1993) – wrote a motivational sentence in the preface of his Rhind papyrus.
The work states that its contents include “a thorough study of all things, insight into all that
exists [and] knowledge of all obscure secrets” (Burton, 1985, p. 36). Although a bit fantastical at
first glance, the words do hold a persuasive weight. Clearly, this was Ahmes’ way to increase his
readers’ extrinsic and intrinsic motivational levels in order to generate enough interest in his
work to create the required action potentials (Kandel, Schawartz, & Jessel, 2000) that would
have been required to learn mathematics. Other ancient scholars also included words meant to
motivate their readers. Sun Zi, for example, writes within the Sun Zi suanjing that “if one
neglects mathematics study, one will not be able to achieve excellence and thoroughness…when
one becomes interested in mathematics, one will be fully enriched…” (Lam & Ang, 2004, p.
190). Much like Ahmes, Sun Zi seems to implore to his students’ intrinsic motivational factors.
Similar techniques have also been found in the writings of Robert Recorde (1512-1558). His
work, The Declaration of the Profit of Arithmeticke Recorde (1956) declares the importance and
value of acquiring mathematical knowledge. Consequently, along with the use of manipulatives
and visual-aides, ancient teachers also thought about their students’ ability to motivate
themselves to begin, carry out, and complete their mathematical training. As a result, they
included both extrinsic and intrinsic motivational factors to help their students achieve greater
mathematical knowledge.
DEVELOPING MATHEMATICAL REASONING 54
In summary, this short historical overview offered a number of mathematical techniques
used by ancient teachers as good pedagogical tools. These techniques, moreover, present
evidence that historically mathematics was characterized by both student and teacher-centered
pedagogies. Examples of this amalgamation are found in the way ancient mathematics teachers
(1) connected math to practical necessities, (2) considered and adjusted to the cognitive, cultural,
social, and emotional needs of their pupils, (3) used the interactive approach ( teacher to student,
text to student, student to student, or self to student), (4) took into account human motivational
factors, (5) included the use of work samples, (6) employed manipulatives and visuals-aides, and
(7) involved ways of developing procedural, conceptual, and metacognitive knowledge ( lower
order and higher order thinking skills).
A number of these characteristics appear to have been forgotten during present times. In fact,
according to Mesa, Morris, Star, Benken (2009):
research studies have characterized mathematics classrooms…as places where students
often work alone and in silence, with little or no access to classmates or to suitable
computational or visualization tools, on low-level tasks that make little or no connection
to the world outside of school, in order to produce answers quickly and efficiently
without much attention to explanation, justification, or the development of meaning. (p.
504-505)
So how did American mathematical pedagogy move so far away from what history
demonstrated to be best practices? This argues that the answer lies in the dichotomous schools
of thought generated during the 20
th
century. Simply put, by adopting, and arising from, various
philosophical and psychological perspectives, 20
th
century pedagogies became dichotomous and
incongruous rather than compatible with one another. The resulting situation was primed for
DEVELOPING MATHEMATICAL REASONING 55
competition rather than collaboration, which, in turn, led to the enduring swinging pendulum that
many contemporary educators describe.
The Recent Past
The educational landscape of the 20
th
century has been inextricably connected and
molded by a number of philosophical perspectives. Among these perspectives are the schools of
idealism, realism, pragmatism, and existentialism (Johnson, Dupuis, Musial, & Gollnick, 2003).
Although it would be extremely fascinating to analyze the history and influence these
philosophical points of views had on education, the present study follows the path laid out by
Johnson et al. (2003) and combines the four philosophical perspectives into two categories,
authoritarian or non-authoritarian. However, for the purpose of clarity and continuity of terms,
and, because they are easily interchangeable, those two terms are replaced by teacher-centered
(authoritarian) and student-centered (non-authoritarian).
One of the psychological perspectives attached to a teacher-centered pedagogical
approach is that of behaviorism. Behaviorists argue that the external environment, rather than
the internal cognition, should be understood as the crucial factor affecting the actions of the
subject (Ertmer & Newby, 1993). Consequently, an external stimulus has the ability to control
how a subject may act in the future. In other words, learning can be conditioned by the subject’s
surroundings.
Two types of conditioning processes associated with behaviorism are classical and
operant, and, although the two share many similarities, there are a few differences that are
paramount to their pedagogical application. Classical conditioning, for example, deals with
involuntary behaviors; hence, it tends not to be employed – as much – by classroom teachers.
Operant conditioning, on the other hand, is concerned with voluntary behavior, making it a much
DEVELOPING MATHEMATICAL REASONING 56
more useful technique within educational milieus. That is, because operant conditioning uses an
incentive system of rewards and punishments in order to alter a subject’s chosen behavior, offers
a very functional way by which educators can alter their students’ conduct and/or learning.
Schools that adopt such a framework, therefore, strive to become places where lessons
are highly organized and controlled, where complex information is broken down into sequential
steps, and where teachers assess whether students gained knowledge by observing changes in
their behavior (Ertmer & Newby, 1993). They uphold that “periodic practice or review serves to
maintain a learner’s readiness to respond” (Ertmer & Newby, 1993, p. 55), and, in turn, to their
ability to transfer the knowledge to other areas of their learning.
Names that are synonymous to the behaviorist movement are Pavlov (1849-1936),
Thorndike (1874-1949), Watson (1878-1959), and Skinner (1904-1990). Their work focused on
the “study of overt behaviors that [could] be observed and measured” (Mergel, 1998, p.2). For
example, Pavlov is most famous for his experiments with dogs that were conditioned to salivate
upon the sound of a bell whereas Thorndike introduced the theory of connectionism, which
maintained that learning is nothing more than the connection between stimulus and response.
Watson, on the other hand, is best known for his infamous experiments with a boy named Albert
in which the child was conditioned to fear and avoid a rat - a feeling that became so strong that it
transferred to other small animals as well. The last advocate of behaviorism was Skinner. His
focus, unlike those of his behaviorist colleagues, did not look into stimulus-response reflect
reactions. Instead, Skinner was more intrigued by operant voluntary behavior. That is, his
approach worked to condition a person’ action by rewarding the small steps the person took
towards a pre-identified end.
DEVELOPING MATHEMATICAL REASONING 57
Because behavioral techniques have been empirically proven to succeed with struggling
students (Adams & Engelmann, 1996), Skinner’s theory of operant conditioning is widely used
as the theoretical framework on a number of academic and behavior curricula. Nevertheless,
research showed Skinner’s approach to have a number of pedagogical limitations. Studies
demonstrate that, although behavioral techniques are most useful when instructing the
discrimination and generalization of facts like new vocabulary, association of ideas, and/or
procedural knowledge (Schunk, 1991, as cited by Ertmer & Newby, 1993), the educational
techniques that arise from the theory are not as powerful at developing higher order thinking
skills.
Another school of psychology that can be placed within a teacher-centered authoritarian
view of education is that of cognitivism. Unlike the outward focus espoused by behavioral
theories, cognitive theorists concentrate on the inner workings of the mind (Ertmer & Newby,
1993). “Cognitive theorists view learning as involving the acquisition or reorganization of the
cognitive structures through which humans process and store information” (Good and Brophy,
1990, p.187). Consequently, the three major tenets of cognitivism are the efficiency of
processing, organizing, and recalling information within the mind. According to cognitivist
pedagogical theories, such as Information Processing Theory (IPT) or the Cognitive Load
Theory (CLT), teachers need to consider the amount of information their students are asked to
process at any given time. This is due to the fact that the sensory and working memories tend to
be limited in two particular ways: (1) by time, information tends to degrade rapidly in our
memories, and (2) by quantity, only a few bits of information can be processed at any given time.
In short, according to cognitivists, humans suffer from limited attentional resources (Anderson,
2000; Neath, 1998). Consequently, the shift away from behavioral pedagogies and into more
DEVELOPING MATHEMATICAL REASONING 58
cognitive ones focuses more on “procedures for manipulating the materials to be presented by
the instructional system to procedures for directing student processing and interaction with the
instructional design system” (Merrill, Kowalis, & Wilson, 1981, as cited by Ertmer & Newby,
1993, p.58).
Nevertheless, regardless of the previously mentioned limitations of the memory system,
people still manage to remember complex ideas long after the limited time allotted by the
sensory and working memory. So how does this occur? Supporters of IPM assert that there is a
third memory type, the long term memory. This memory stores and retrieves as much
information as possible regardless of the amount of time elapsed between encoding and retrieval.
However, information will not be stored into long term memory unless the subject creates a
proper schema for it. This is accomplished by acting upon the sensory information in some
meaningful way. In other words, the subject must somehow transform the value of the
information from inconsequential to critical by organizing it, making inferences with it, or
elaborating upon it. In addition, the information will not undergo either one of these three
encoding processes without the aid of selective processing, which is the cognitive “act of
intentionally focusing one’s limited resources on stimuli that [is] most relevant to the task at
hand” (Information Processing Theory, 2009).
Teachers who employ information processing models, therefore, tend to design lessons
that are highly controlled. They, much like the behaviorist tradition, work at managing all the
aspects of their students’ learning environment by (1) continually maintaining their students’
attention on what they think is pertinent, (2) working to minimize extraneous load of the lesson,
and (3) assessing the lesson’s intrinsic load in order to control the amount of information their
students are asked processed. Essentially, IPM designers, as well as teachers who employ such
DEVELOPING MATHEMATICAL REASONING 59
models, must control all the external factors that could potentially overtax the working memory
of their learners. They can do this by embedding, within their lessons, the appropriate
scaffolding tools, such as graphic organizers, to allow the students to schematize the new
information into long term memory. Consequently, “instructional explanations, demonstrations,
illustrative examples and matched non-examples are all considered instrumental in guiding
student learning” (Ertmer & Newby, 1993, p. 58) Moreover, assessing the students
understanding becomes a process of checking whether they attained the information the teacher
imparted. Consequently, teachers must continually check for understanding throughout the
lesson in order to affirm that the students are tracking along.
The last school of thought examined in this section is that of constructivism. Extending
from both the pragmatic and existentialist philosophical views, constructivist pedagogies are
very much student-centered. Although a general understanding of what constructivism is has
hovered throughout philosophy, psychology, and education, a clear and concise definition has
not yet been worked out (Gijbels, van de Watering, Dochy, & van den Bossche, 2006; Loyens,
2007). For the purpose of the present work, constructivism will be defined as the “view of
learning that considers the learner as a responsible, active agent in his/her knowledge acquisition
process” (Loyens & Gijbels, 2008).
Some of most recognizable names attached to this school of thought are Thomas Kuhn,
Ludwig Wittgenstein, Richard Rorty, Jean Piaget, Jerome Bruner, Lev Vygotsky, and John
Dewey. However, the tenets of constructivism appear much earlier in history. Cunningham and
Duffy (1996) talk about 17
th
century thinkers such as Vico and Rousseau who wrote passages
describing a very constructivist view of learning. For example within his book Emile, Rousseau
(1955) argues that “the senses where the basis of intellectual development and that the child’s
DEVELOPING MATHEMATICAL REASONING 60
interaction with the environment was the basis for constructing understanding” (Cunnigham &
Duffy, 1996, p. 4). Consequently, much like today’s constructivist, Rousseau perceived of
teachers as guides whose primary job is to present problems that can catch the students’ interest
and, in turn, generate learning. In essence, Rousseau’s ideas place one’s experience at the center
of knowledge acquisition.
As a whole, constructivist thinkers such as those presented above, argue for a never-
ending spirit of understanding that should be grounded on the processes of innovation,
construction, and the re-interpretation of ideas, knowledge, and truth. A spirit well depicted by
Rorty (1991) when he explains that it is hard to…
…imagine a moment at which the human race could settle back and say, “Well, now that
we’ve arrived at the Truth we can relax”. We should relish the thought that the sciences
as well as the arts will always provide a spectacle of fierce competition between
alternative theories, movements, and schools. The end of human activity is not rest, but
rather richer and better human activity (p.39).
Although the above paragraph is a perfect example of the dichotomic perspective that this
work is arguing against; it also serves as a demonstration of how constructivist thinkers perceive
of learning. That is, they see human experience, understanding, and knowledge acquisition as a
social and communicative process, in which the learner interacts with his environment in order to
make sense of it, alter it, and/or be altered by it. This perspective can also be seen within
Piaget’s (1973) book, To Understand is to Invent: The Future of Education. In it, Piaget
mentions that the programmed instruction that follows Skinner’s psychological model is, indeed,
“conducive to learning, but by no means to inventing” (p. 7) – a point that is essential to the
present work. Constructivist also perceive of learning not as a compartmentalized act that one
DEVELOPING MATHEMATICAL REASONING 61
attends to while in school, but rather as at outflow of being human. This is why Dewey writes
within My Pedagogic Creed that education is not preparation for life, but life itself (1897).
Consequently, this tradition emphasizes hands-on activity-based learning techniques (Johnson,
Dupuis, Musial, & Gollnick, 2003) that spring out of interesting problems or questions used as
departure points to initiate student learning. Unlike behaviorist and cognitivist classrooms,
constructivist educational environments are places where children have the opportunity to
“discover their own answers to important questions” (Johnson et al. 2003, p. 292), rather than to
follow an adult’s step-by-step instructions towards a solution (Stipek, Givvin, Salmon, &
MacGyvers, 2001). In essence, the ultimate goal of a constructivist classroom is to develop
critical thinking skills. As a result, the pupils’ focus is not so much on getting the right answer,
but on developing as thinkers (Stipek et al. 2001), which will ultimately allow them to become
individuals “who can create meaning as opposed to acquiring it” (Ertmer & Newby, 1993, p.62).
Such metacognitive skills can only be gained by personalizing the instruction. In other
words, a constructivist classroom should be a place where students have the opportunity, the time,
and the support to deeply delve into and reflect upon their authentic learning experience. Brown,
Collins, and Duguid (1989), for example, argue that the learner’s environment and the
experiences that environment induces are co-producers of a student’s knowledge along with his
cognition. “The emphasis [therefore] is not on retrieving intact knowledge structures, but on
providing learners with the means to create novel and situation-specific understandings by
‘assembling’ prior knowledge from diverse sources appropriate to the problem at hand” (Ertmer
& Newby, 1993, p. 63). Moreover, constructivist classrooms should be characterized by social
interaction, social negotiation, understanding of multiple perspectives, self-reflection and self-
evaluation, and metacognition. In short:
DEVELOPING MATHEMATICAL REASONING 62
“the role of instruction in the constructivist view is to show students how to construct
knowledge, to promote collaboration with others, to show the multiple perspectives that
can be brought to bear on a particular problem, and to arrive at self-chosen positions to
which they can commit themselves, while realizing the basis of other views with which
they may disagree.” (Cunningham, 1991, p.14)
As a result of all these attributes, assessing how much a child learned within a
constructivist classroom is approached differently than the type of summative examinations to
which most people have grown accustomed during NCLB. Understanding knowledge as a
developmental instead of as something that must be obtained requires assessments become more
formative. To do so, teachers have to perceive of their students’ actions as moments by which to
capture their understanding. Furthermore, because the knowledge gained by students in a
constructivist classroom is expected to be diverse, the methods by which to assess them should
also be of a more malleable nature. Accordingly, knowledge cannot, and should not, be assessed
by the limited information that can be gathered through a multiple choice test. Rather, it should
be collected through the evaluation of portfolios, journal entries, and hands on activities, all of
which offer ways by which students can demonstrate a deeper more comprehensive
understanding of what they have learned.
As has been previewed, the philosophical, psychological, and pedagogical perspectives
introduced up to this point have jostled for greater space within both academia and the classroom.
This interaction is often characterized by a set of oppositional features. That is, while some of the
theories focus on the outward behavior of students, others look towards their inner cognitive
development. While some demand control over the educational environment, others allow
students the freedom to investigate and make mistakes. In addition, while some are considered
DEVELOPING MATHEMATICAL REASONING 63
better at developing higher order thinking skills, others seem to be better adapted to increasing
the procedural and factual knowledge of the child.
These differences have been used by some scholars to generate a third pedagogical option.
Such scholars have envisioned a middle ground in which both perspectives are given their due
space by affirming that each approach can offer its own valuable contribution to the development
of the child (Klahr, 2009, Kuhn, 2007). For example, Stipek, Givvin, Salmon, and MacGyvers
(2001) mention that “proponents of inquiry approaches are not proposing that teachers dispense
altogether with developing students’ fluency in mathematics computations, but rather that
learning rules and applying them efficiently should not be the exclusive focus of mathematics
education, as it is in many American classrooms” (p. 215). This sentiment is concurred by Harel
(2008), who states that “while knowledge of and focus on subject matter is indispensable for
quality teaching…it is not sufficient. Teachers should also concentrate on conceptual tools such
as problem-solving approaches…” (p. 267).
This need to focus on the entire set of cognitive skills, rather than on a mere few can be
furthered substantiated by Slavin and Lake’s (2008) meta-analysis of mathematical programs. In
it, the authors point out that, of 87 studies reviewed, the strongest positive effects arose from
programs that employed “cooperative learning, classroom management and motivation programs,
and supplemental tutoring programs” (p. 427). Hence, their results show positive features arising
from programs that infuse the tenets of both student-centered (cooperative learning) and teacher-
centered approaches (classroom management). Slavin and Lake conclude that the research
suggest that the most successful programs they investigated tended to focus on “how teachers use
instructional process strategies, such as using time effectively, keeping children productively
engaged, giving children opportunities and incentives to help each other learn…” (p.475). In
DEVELOPING MATHEMATICAL REASONING 64
short, these words by Slavin and Lake appear to support the idea that, although teachers need to
provide freedom to their students, they also need to be concerned with time on task – one of the
major tenets found within teacher-centered pedagogical approaches.
Although Slavin and Lake’s work concentrated on math, there are those who have argued
for a more balanced approach in other content areas as well. While discussing a successful
writing strategy (Self-Regulated Strategy Development [SRDR]), Harris, Santangelo, and
Graham (2008) also presented the idea that best practices need to exemplify principles of both
teacher and student-centered techniques. They maintain that the premises of SRDR “are not
advocating a return to a primarily skills-oriented, back-to-basics curriculum. Rather, [they] are
arguing that explicit, focused, and at times isolated instruction needs to be provided in response
to individual student’s needs” (p. 401).
The last supporting evidence of the need to infuse teacher and student-centered
pedagogies comes from Ertmer and Newby’s (1993) work. While comparing the critical
instructional design features that arise out of behaviorism, cognitivism, and constructivism, they
concluded that learners maintain different levels of knowledge; as such, different approaches
should be used to teach them. They maintain that one would not design the same lesson for a
novice learner as for an expert one. This perspective was corroborated by Kalyuga, Chandler,
Tuovinen, and Sweller’s (2001) work. They found that the effectiveness of work samples or
problem-based learning depended on the level of prior knowledge the subjects’ possessed, which
is based on the findings that “as people acquire more experience with a given content; they
progress along a low-to-high knowledge continuum” (p. 51). Consequently, a teacher should not
employ the same pedagogical approach when teaching facts to a new learner as when trying to
instruct creative problem-solving strategies to someone who knows most of the facts. In other
DEVELOPING MATHEMATICAL REASONING 65
words, as people attain more knowledge of a subject, they tend to move away from being able to
identify and apply the necessary facts and rules toward thinking about how to transfer such
factual and procedural knowledge into new situations, and, finally, into the creation of more
sufficient ways of performing the task. In the end, Kalyuga et al. (2001) contend:
“the behavioral approach can effectively facilitate mastery of the content …(knowing
what)…cognitive strategies are useful in teaching problem-solving tactics where defined
facts and rules are applied in unfamiliar situations (knowing how); and constructivist
strategies are especially suited to dealing with ill-defined problems through reflection-in-
action” (p. 68).
In addition, Ertmer and Newby offered a second reason as to why we should follow a
dynamic rather than dichotomic pedagogical model. They maintain that different tasks can be
instructed most efficiently using different pedagogical techniques. Discrimination and/or
memorization, for example, tend to align themselves better to a behavioral approach whereas
heuristic problem solving and/or self regulations appear to work better with constructivist
approaches.
Ertmer and Newby’s perspective finds a great deal of parallelism with the work of
Anderson et al. (2001). They argue that, rather than having pre-established pedagogical
perspectives, teachers should identify which particular pedagogical approach would work best
for their lesson. This can be accomplished by using a two dimensional taxonomy that focuses on
a verb and nouns of a lesson’s objective. Anderson et al. explain, “the verb generally describes
the intended cognitive process; [whereas,] the noun generally describes the knowledge students
are expected to acquire or construct” (p.4-5). Consequently, pedagogy becomes more fluid,
demanding that one lesson be taught via memorization, while another one is taught using
DEVELOPING MATHEMATICAL REASONING 66
reflective questions. In short, “different types of objectives require different instructional
approaches; that is, different learning activities, different curricular materials, and different
teacher and student roles” (Anderson, et al. p.8). The present work, therefore, presents a
research investigation that follows this type of thinking. It maintains, as one of its underlying
assumptions, that no individual philosophical, psychological, or pedagogical approach should be
considered adequate in every learning situation. As a result, each pedagogical approach should
hold its own burden of proof as the reasons why it needs to be employed during a specific lesson.
What Pedagogical Differences Exist between EDI and CGI?
This second theme introduces and juxtaposes the two educational methodologies were
employed during the study (explicit direct instruction [EDI] and cognitively guided instruction
[CGI]). To accomplish this, the first section discusses the connections each methodology
maintains with its respective philosophical and psychological schools. In addition, the
pedagogical approach of each methodology will be discussed. To conclude, this section delves
into the professional development of each approach and what that entails regarding students’
experiences in the classroom.
Cognitively Guided Instruction (CGI) and Explicit Direct Instruction (EDI) are two very
distinct ways of teaching mathematics. While the former comes out of a constructivist tradition
of learning (Carpenter, Fennema, Franke, Levi, and Empson, 1999; Michael, 2006), the latter
follows the tenets laid out by supporters of behaviorism and cognitivism (Adams & Engelmann,
1996; Becker & Gersten, 1982; Meyer, 1984, Rosenshine, 2012, Tarver & Jung, 1995).
Therefore, CGI tends to be a much more student-centered approach than EDI is. Regardless of
that difference, both of these methodologies are considered professional development approaches
rather than curriculum programs. As a result, they both serve to increase the knowledge of the
DEVELOPING MATHEMATICAL REASONING 67
teacher via a series of trainings and observations instead of offering a commercial package of
lessons designed to be implemented by a teacher.
To better understand the differences between these two approaches, one must look at
their methodological family trees. The history of EDI arises from one of the largest educational
research projects ever conducted by the United States Government, Project Follow Through. As
part of President Lyndon Johnson’s war on poverty, Project Follow Through’s main purpose was
to identify pedagogical techniques that could close the academic achievement gap exhibited by
low-income minority students. The study’s results ultimately showed that the most effective
instructional approach was Engelmann’s Direct Instruction (DI) model (Adams & Engelman,
1996; Hattie, 2008). Consequently, after the enactment of NCLB, when many educational
leaders sought pedagogies and methodologies that would assist them in meeting their Adequate
Yearly Progress (AYP) goals, this approach to instruction experienced a vibrant institutionalized
rebirth – although some would claim the teacher-centered strategies espoused by DI have been
ubiquitously used in the American classrooms for quite a long time (Hiebert & Stigler, 2000;
McKinney & Frazier, 2008; National Council of Teachers of Mathematics, 1989Putman,
Lampert, & Peterson, 1990).
Supporters of Direct Instruction maintain that it is the theory’s framework that allows
teachers to attain such instructional successes. “Direct Instruction (DI) programs are… based on
the explicit or direct approach to teaching that consists of effective instructional design, effective
presentation techniques, and a logical organization of instruction” (Przychodzin, Marchand-
Martella, Martella & Azim, 2004, p. 64). The framework follows the following eight criteria: (1)
the lesson should be teacher directed, (2) the lesson should actively present information in that
students' must be attentive, teachers must provide motivational cues, lessons should incorporate
DEVELOPING MATHEMATICAL REASONING 68
advance organizers, teachers should expose students to essential content, and teachers should
assist students in retrieving prior relevant knowledge, (3) the information being presented should
be clearly and coherently organized, (4) the teacher must employ a step-by-step progression from
subtopic to subtopic, (5) the teacher must employ numerous examples, visual prompts, and
demonstrations in order to mediate between concrete and abstract concepts, (6) the teacher must
perform constant assessments ( check for understanding) in order to find out if the students have
acquired the information that was presented or required more teaching (7) the teacher must
change the speed of instruction in order to match student understanding, and (8) the teacher must
maintain control of the students’ behavior and attention in order to maximize time on task (Huitt,
1996). In essence, “All details of instruction must be controlled to minimize the chance of
students' misinterpreting the information being taught [as well as] to maximize the reinforcing
effect of instruction” (National Institute for Direct Instruction, 2012).
As we have already established, this type of instruction tends to be rooted in the
behaviorist model of learning (Magliaro, Lockee, Burton, 2005) espoused by psychologists such
as Skinner, Thorndike, and Watson. Therefore, it associates learning with changes in one’s
performance or behaviour (Ertmer & Newby, 1993). Hence, the model forces the learner to react
to his/her surroundings rather than actively participate with them to gain knowledge.
Taking a different theoretical approach, CGI lies more in the constructivist side of learning.
Consequently, this methodology is mostly grounded in the works of Piaget, Bruner, Goodman,
and Dewey. It views “the notion of organisms [i.e. students] as ‘active’ – not just responding to
stimuli, as in the behaviorist rubric, but engaging, grappling, and seeking to make sense of things”
(Perkins, 1991, p.49). Accordingly, learners do not just input and output information, but rather
DEVELOPING MATHEMATICAL REASONING 69
generate, elaborate, and test the interpretations of the experiences that their environments provide
(Duffy & Cunningham, 1996).
It is critical to comprehend that CGI should not be categorized as ‘pure’ discovery
learning. That is during a CGI lesson, the teacher should be guiding the students towards better
understanding. As a result, the approach tends to avoid all the draw backs that have been
associated with pure discovery learning (see Mayer, 2004). To explicate this point in more detail,
the following description of a CGI lesson is put forth.
During a CGI lesson, students are presented with a word problem, which they are asked
to answer (Carpenter, Fennema, & Franke, 1996; Carpenter, Fennema, Franke, Levi, & Empson,
1999). No direct instruction is provided by the teacher for the students to follow. Instead, the
students are presented with manipulatives that act as assistive tools by which to grapple with the
proposed dilemma. While the students are involved in the task, the teacher is supposed to
interact with them and guide by asking clarifying questions about the way each one chose to
approach the problem’s solution. By doing so, the teacher can encourage the students to reflect
upon what they have chosen to do and, in turn, increase their metacognition and/or their
mathematical reasoning. It must be reiterated that the teacher is not to instruct the students at
any point during the CGI lesson. Instead, she is simply there to cognitively guide the students to
their own understanding. If the students are not able to answer the problem correctly, the lesson
is not considered a failure, but a necessary step in their journey towards greater and deeper
mathematical knowledge.
The last critical step in a CGI lesson is the act of student sharing. That is, while the
teacher walks around the room conferencing with individual students, s/he is supposed to look
out for different solution patterns that, when shared, would increase the understanding of the rest
DEVELOPING MATHEMATICAL REASONING 70
of the class. However, the sharing session is not carried out by the teacher him/herself, but rather
by each student whose work is shared. During this time, other students can ask questions such as,
how did you do that? Why did you do that? Or could you have done it this way? Through these
questions, students are, once again, presented with the opportunity to reflect upon their own
understanding. As a result of this dialoguing that occurs while the teacher conferences with each
student and the dialoguing that occurs while the students present their strategies to one another,
supporters of CGI claim that teachers gain a greater understanding of their students’
mathematical development. In other words, teachers will begin to see how each student
perceives of numbers, understands mathematical functions, and approaches various types of
word problems, which will enhance the teacher’s ability to meet all of his/her students’
individual needs (Carpenter, Fennema, Franke, Levi, Empson, 1999).
As one can imagine, the two previous approaches go about professional development
(PD) in completely different ways. The PD involved in augmenting a teacher’s DI skills is one
in which teachers are instructed on the different parts of the lesson and how to carry them out
efficiently. The students are seen as superfluous to the process. A well-structured and
performed lesson, therefore, is one in which the parts are broken down into manageable bits of
information that the students can process in an appropriate and controlled manner (Rosenshine,
2012).
The structure of the DI approach, as well as the way teachers are professionally
developed to carry it out, led many people to criticize the model as rigid, spurious, anti-
developmental, scripted, adequate for low performers only, promoting passive learning, and
lacking in individualization. Although all these criticism have been answered by Adams and
Engelmann (1996), this work will only present their response to the accusation that DI is too
DEVELOPING MATHEMATICAL REASONING 71
scripted. Regarding this allegation, Adams and Engelmann maintain that “once the teacher has
mastered the basics and is able to achieve attainable results, the teacher has a basis for being
creative” (p.28). In short, according to Adams and Engelmann, most people seem confused
about the way the DI asks teachers to carry out their lessons; that is, the model was only meant to
be scripted for those teachers who were just beginning to use it; for those who feel comfortable
with it, more latitude is granted. Still, the claim can avoid the fact that the model is designed to
be scripted and authoritarian in nature, forcing teachers to create classroom cultures that exude
such characteristics.
Conversely, professional development for CGI is characterized by teachers gaining more
knowledge about the development of their students’ mathematical thinking processes. In CGI,
the child, rather than the process, becomes the centrepiece of the lesson. Consequently, training
in CGI should always involve the students themselves. By doing so, teachers are presented with
the following two opportunities: (1) they are allowed to experience the way students perceive of
mathematics and (2) they are allowed the chance to listen and observe rather than instruct the
child. These two actions are quite difficult for educators, especially when the child in front of
them is in the process of making a mistake. As a result of this type of training, educators begin
to perceive that learning occurs within a child in spite of them or the errors being made.
Research shows that teachers’ beliefs can play a significant role in the way they carry out
their work (Thompson, 1984). Based on the divergent beliefs that these two types of trainings
will generate, teachers attending such professional development will return to their classrooms
with different perspectives about how to carry out their lessons and how to interact with their
students. The DI teachers will be very much focused on attending to the lesson (the DI lesson
format) in order to see if the students are following the step by step directions that are being
DEVELOPING MATHEMATICAL REASONING 72
provided. The constructivist, an, in our case, CGI teachers, on the other hand, will be focused on
how students are developing and constructing their knowledge, as well as on how to best guide
this development (Wood, 1993). But how does this affect the students themselves? The next
section of this review investigates this very question.
How Do American Students Perform in Mathematical Reasoning?
This third theme focuses on mathematical reasoning. It begins by presenting a working
definition of the construct. Next, it presents a short explanation, grounded in the context of
globalization, as to why the development of higher order thinking skills, like mathematical
reasoning, became imperative to the United States. Following that, presents the reality of how
American students performed within this particular content and cognitive domain. To accomplish
this, scores from international (TIMSS and PISA) and national (NAEP) tests are presented. In
addition, this theme provides details as to the differences between the knowledge types and
cognitive sub-skills tested during the standards movement of No Child Left Behind’s (NCLB)
and the type of outcomes these assessments have regarding instruction. It closes by identifying
what research says are the best practices to increase mathematical reasoning and how each of the
two professional development programs compared in this study (Cognitive Guided Instruction
and Explicit Direct Instruction) align to such research.
What is mathematical reasoning? The answer to this question spurred a number of
definitions. For example, the description offered by the developers of the Trends in International
Mathematical and Science Study (TIMSS) is one of performance. That is, a student asked to
reason mathematically should be able to go “beyond the solution of routine problems to
encompass unfamiliar situations, complex context, and multi-step problems” (TIMSS, 2011, p.
24). This study defines mathematical reasoning as the “attainment of abilities to construct
DEVELOPING MATHEMATICAL REASONING 73
mathematical conjectures, develop and evaluate mathematical arguments, and select and use
various types of representations” (Kramarski and Mevarech, 2003, p.2). Yet, regardless of the
plethora of definitions that have been provided in the literature, two ideas appear ubiquitously
shared by most researchers, policy makers, and practitioners regarding this construct: (1) that the
development of mathematical reasoning is imperative to 21
st
century students and (2) that
American students lag behind in this domain. Each one of these ideas will be discussed below.
In centuries past, imperialistic nations battled to control the trade routes of the world;
during the 20
th
century, countries used political ideologies to identify economic and political
allies (the cold war). The 21
st
century, on the other hand, has been defined, so far, by
technological advances that nations, individuals, and multinational conglomerates have
employed to connect with one another in ways never seen before. This technological
interconnectedness was described by Friedman (1999) at the turn of the century when he stated
that the world was becoming characterized by “the inexorable integration of markets, nation-
states, and technologies … that enable individuals, corporations and nation-states to reach around
the world farther, faster, deeper and cheaper than ever before . . .” (p. 7-8). Fourteen years after
Friedman’s description of globalization, the world appears to be very much as he depicted it.
Entities are now ubiquitously connected to each other via technological advances such as the
internet, skype, facebook, and mobile phones, that have led to greater economic opportunities
like the ASEAN economic community and the North American Free Trade Agreement, and
stronger geographical unions such as the European Union).
These technological, economic, and geographical connections have generated both
possibilities as well as difficulties for all entities around the world. One positive note that came
from such a globalization process is the increase of Gross Domestic Product (GDP) for
DEVELOPING MATHEMATICAL REASONING 74
developing nations. On a more troublesome note, developed nations, such as the United States
have to reinvent their economies in order to move away from the manufacturing industries that
are now part of the overall Gross Domestic Product (GDP) of less developed nations.
This new economic reality forced developed nations to increase the number of skilled
laborers they produce. Peters (2010) explained this modification by pointing to the effect that
new technologies have on the world. He stated that, “…modernity [has moved nations] to a form
of post-industrial economy that focuses on the production and consumption of knowledge and
symbolic goods as a higher-order economic activity…” (p.65). Peters is not the only one who
recognized this transition. The Organization for Economic Co-operation and Development
(OECD) (1996) and Jimenez, Nguyen, and Patrinos (2012) also explicated the need that
industrialized and middle-income nations have to move away from manufacturing goods and into
the production, management, and distribution of knowledge and ideas. In short, the technological
advances of the 21
st
century made it possible for many entities to instantaneously interact with
one another, which, in turn, forced some nations to seek new economic routes in order to not
only increase, but also maintain their GDP levels.
So how does this play into the development of mathematical reasoning skills at the elementary
schools? The answer is simple. In order to collaborate and compete with one another in the new
globalized market, international entities must first develop and maintain an adequate labor force
(human capital). However, a strong labor force can only be achieved when one improves the
factual, procedural, and conceptual knowledge of the individuals who make up that labor force,
and that can only be accomplished by establishing a stronger and more successful educational
system (OECD, 1996, Bhatiasevi, 2010). In other words, 21
st
century businesses and nation-
states will not be able to adequately participate in the modern global markets until the individuals
DEVELOPING MATHEMATICAL REASONING 75
who compose them begin to possess the necessary knowledge and skills that such markets
demand. Consequently, entities desiring to compete in the new “knowledge-based economies”
of the 21
st
centuries (KBE) (Peters, 2010, p. 1; Bhatiasevi, 2010, p.1) must begin to see education
as a critical national commodity rather than a privilege afforded to a few (Bhatiasevi, 2010) – an
idea well expressed within the endogenous growth models of Romer (1990), Lucas (1988), and
Aghion and Howitt (1992).
Dynamic interplays exists between a nation’s schools, its future labor force, and its
ability to compete in the new knowledge-based economies (Jimenez, Nguyen, & Patrinos, 2012).
Consequently, international test-scores are now being used to identify knowledge gaps that can
potentially affect a nation’s productivity levels (Bhatiasevi, 2010). Peters (2010) argued that
contemporary educational systems need to provide their students with more than the ability to
recall information; they must also present them with opportunities to learn how to innovatively
use technology in order to creatively and collaboratively solve problems. Consequently,
countries wishing to participate in the global markets of the 21
st
century will have to develop
such skills in their citizens. The question then becomes, how has the United States performed
with regards to such skills? The answer is not a positive one.
Proof of this lack of higher order thinking skills has be seen in international exams, like the
Trends for International Mathematics and Science Study (TIMSS) and the Program for
International Student Assessment (PISA), as well as national tests like the National Assessment
of Education Progress (NAEP). For instance, although the overall average scores for American
4
th
graders in the TIMSS appear to have grown considerably between the years 1995 and
2011(Kastberg, Ferraro, Lemanski, Roey, & Jenkins, 2012), when scores are segregated using
the different cognitive sub-skills that the test evaluates, mathematical reasoning becomes a major
DEVELOPING MATHEMATICAL REASONING 76
point of concern (Appendix D). That is, although American 4
th
graders remained 6 points above
the international average on the TIMSS 2011 examination, 15 other countries, many of which are
direct economic competitors of the U.S., performed better or equal to the United States.
However, there are those who say that economic competition should not dictate educational
policy (Hursh, 2007). The reply to this point comes in the form of a question. That is, even after
setting aside a country’s need to compete globally, are not these skills which a country should be
providing its current and future citizenry anyway?
Along with the TIMSS results, other evaluations demonstrated a similar lack of higher
order thinking skills amongst American students. The PISA, a test that examines the ability of
15 year-olds to apply their mathematical knowledge to real-world situations (Koretz, 2009),
continues to demonstrate American students’ lack higher order thinking. For instance, results
from the PISA 2009 showed that, after experiencing a drop in performance, the latest scores have
shown improvement, but are still not above the results attained back in 2003. Moreover, the
average score for a United States student is lower than the OCDE average, which places
American students below the results of 24 other countries; again, many of these are direct
economic competitors of the U.S. (Fleischman, Hopstock, Pelczar, & Shelley, 2010). Regarding
the ability to carry out mathematical reasoning, PISA 2009 results show that “27 percent of U.S.
students scored at or above proficiency level 4… [which] is the level at which students can
complete higher order tasks such as ‘solving problems that involve visual and spatial reasoning
in unfamiliar contexts’” (p.20). This percentage was lower than the 32 percent averaged by other
OECD countries.
In spite of this low performance on international tests, many mathematical state scores
climbed during the accountability era of No Child Left Behind (NCLB) (Azzam, 2007; Dee &
DEVELOPING MATHEMATICAL REASONING 77
Jacob, 2010). Both national tests, such as the NAEP, and state tests like the California
Standardized Test (CST) have shown significant improvement amongst all assessed students.
How is this possible? One possibility is the different foci of the test. For instance, while state
and national examinations focus on academic topics, the PISA asks students more contextualized
questions. A second difference is the way that the tests identify and assess the different
cognitive sub-skills. That is, while the TIMSS and the PISA examinations include items that are
specifically designed to evaluate higher order thinking skills (mathematical reasoning), the
NAEP only recently began to assess such skills and the CSTs simply embed them rather than
specifically assess them. However, one is hard pressed to find too many items in the CSTs that
ask students to “identify and understand the role that mathematics plays in the world, to make
well-founded judgments and to use and engage with mathematics in ways that meet the needs of
that individual’s life as a constructive, concerned and reflective citizen” (OECD, 2009, p.84, as
cited by Fleischman et al., 2010). As such, by failing to clearly identify items demanding
mathematical reasoning, the CSTs might have provided absolution for teachers wishing not to
involve themselves in such complicated matters – a notion that found empirical support (Silver et
al., 2009). Add to this the fact that many district and school administrators have looked to,
adopted, and supported the use of Direct Instructional strategies as a viable way to increase their
scores, and one should not be confounded about the lack of American students’ higher order
thinking skills.
So what should educators do about this problem? Research is indefinite about the
solution. Some studies support the idea that the best way to increase the higher order thinking
skills of children is by using more constructivist approaches (Carpenter, Fennema, & Franke,
1996; Erickson, 1999; Wood, Williams, & McNeal, 2006). An example of this can be seen in the
DEVELOPING MATHEMATICAL REASONING 78
study conducted by Barak, Beh-Chaim, and Zoller (2007) in which they saw a statistical
significance in the reasoning skills of students who were instructed using less traditional methods
(teacher-centered). Other studies, meanwhile, maintain that a more cognitive and/or behavioral
approach can also increase students’ abilities to carry out higher order thinking. One illustration
of a more cognitivist approach to generating “cognitive flexibility” for problem solving is that of
“inter-examples” (Atkinson, Derry, Renkl, & Wortham, 2000, p.208). Atkinson et al. maintain
that, by presenting students with a range of examples, the teacher instills in his/her students the
ability of “expert thinking” (p. 208). That is to say, the students will go beyond the procedural
knowledge commonly associated with information processing models and creatively solve
problems by employing metacognitive skills. Consequently, the literature offers only ambiguous
answers regarding which theoretical approach should be considered the best for developing the
mathematical reasoning of children.
Changes in order to fix the lack of higher order thinking within American classrooms
have already begun. With the transition into the new Common Core Standards, teachers are now
asked to include a higher number of metacognitive skills within their lessons. Nevertheless, only
time will tell if the new policies can successfully increase the scores of those tests that measure
higher order thinking skills. In the meantime, given the ambiguity of the literature regarding best
practices and given the imperative need that countries have to understand how to develop the
higher order thinking skills of their citizenry, further research should be conducted on the matter.
Consequently, the present study was designed to juxtapose two professional development
programs in order to see which approach (EDI or CGI+EDI) offered the best way of developing
the mathematical reasoning of elementary students.

DEVELOPING MATHEMATICAL REASONING 79
What Does Research Say about How Students Like to be Taught?
This fourth theme focuses on the students themselves. To be specific, investigates
whether students’ perceptions of math can be influenced by the different types of pedagogical
models used by teachers.
According to research, when students perceive math to be intrinsically important to their
lives, they will be more inclined to actively participate in the learning subject, resulting in
superior and deeper understanding of the concepts being taught (Benware & Deci, 1984). This
statement, of course, should come as no surprise. Common sense dictates that, if someone enjoys
an activity, s/he will probably spend more time involved in it, which should result in greater
knowledge of it. In fact, the literature is emphatically clear in this regard. International
investigations have shown that students with a more positive attitude towards mathematics tend
to perform better than those who perceive of mathematics as boring, insignificant, or tedious
(Mullis, Martin, Foy, & Arora, 2012). This connection between mathematic performance and
motivation is clearly documented by Hattie’s (2009) meta-analysis. In it, he describes 288
studies in which students’ attitudes regarding a specific academic subject were associated to
higher academic performance.
In addition, there is a second factor to the equation that is not discussed by previous
studies: the milieu in which the learning is experienced. The following story is presented as an
illustration of this point. A child is trying to learn about different types of candies in two
different situations. In the first, the child is asked to go into a room day in and day out in order
to listen to an expert talk about different types of candies: how they look, how they taste, how
they feel, and how to combine them to make them even sweeter. Yet, the child is never presented
with the opportunity to see, touch, or taste a candy. In the second situation, the same child is
DEVELOPING MATHEMATICAL REASONING 80
present. However, this time, the child is found in a room full of sweets where he can touch, taste,
and combine as many varieties of candies as s/he desires. During this exploration, an adult who
is an expert in candies is available to answer any of the child’s questions regarding the plethora
of colors, smells, and flavors with which he has been presented. The question then becomes,
which environment would be most motivating to a child?
Although very limited as an illustration of the complexities that exist in the nexus
between students, motivation, and learning, the above scenarios do succeed in illuminating one
important aspect of this study. They highlight how significant pedagogical methodology
becomes to motivation and, ultimately, learning. That is, unlike the one dimensional approach
described by the studies presented above, where the content is the only deciding factor by which
to evaluate if someone is motivated, our illustration includes a second dimension, the way in
which someone is asked to relate to the content at hand: the pedagogy involved in teaching the
content. This second dimension is an important aspect of investigating how children view and
experience a specific academic subject. Because, as one could imagine, even learning about a
topic as interesting as candy would become very laborious if one was not presented with the
opportunity to experience the wonders of the subject. Likewise, to ask a child to gain a sense of
wonder for the subject of mathematics, ( how it is put together, how it can be manipulated and
controlled, and how it offers uncountable benefits to one’s life, without ever permitting him to
meaningfully interact with the subject is a bit limiting.
The way teachers choose to instruct, therefore, appears to be a critical factor of the
motivation equation, and, the way they choose to instruct can be considered a physical
manifestation of the deep philosophical, psychological, and pedagogical beliefs they hold. In fact,
studies show that teachers who believe in a more student-centered approach to mathematics
DEVELOPING MATHEMATICAL REASONING 81
increased their students’ mathematical academic performance (Staub & Stern, 2002).
Furthermore, theoretical papers have proposed calling for the investigation of how different
“mathematics teaching strategies [can] foster optimal kinds of situation-specific engagement”
(Goldin, Epstein, Schorr, Warner, 2011). Consequently, the rest of this theme focused on how
different types of pedagogical approaches appear to have better alignment with certain types of
motivational theories, which, in turn, can spur a more conducive environment for the
development of mathematical reasoning.
Research showed that schools are failing to motivate students, especially those in urban
settings (Galloway & Lasley, 2010). However, studies have also pointed out that there are a
number of ways in which teachers can successfully motivate their pupils. Amongst such
methods are the use of student-centered pedagogical approaches (City, Elmore, Fiarman, &
Tietel, 2009), the use of mastery and/or performance goals (goal oriented theory) as well as the
four-phase model of interest development (Hidi & Renninger, 2006). The next few paragraphs
will touch upon each one of these approaches within the context of our study.
Student-centered pedagogical views have been shown to increase motivation and learning.
One study that has exhibited this was conducted by Tsuei (2012). He demonstrated that by
employing a combination of technological advances and cooperative learning, students were
assisted in improving their mathematical understanding and reasoning skills. Similarly to
Tsuei’s results, Cornelius-White’s (2007) meta-analysis found that learner-centered
environments helped improve myriad educational factors like critical thinking, math
achievement, positive motivation, IQ, and grades. Unfortunately, since most educators seem to
have a behavioral understanding of teaching, student-centered pedagogies are not commonly
employed (Galloway & Lasley, 2010). This is exemplified in the teacher-centered lessons that
DEVELOPING MATHEMATICAL REASONING 82
have historically characterized American education. Nevertheless, Galloway and Lasly assert
that, unlike the educators of the first half of the 20
th
century who were entrenched in the
behavioral techniques of Watson, Thorndike, and Skinner, today’s educators have the ability and,
in turn, the responsibility to be more dynamic in their teaching. They must also work to include
what the literature presents regarding the theories of Piaget, Bruner, and Vygotsky. Accordingly,
in order to increase the motivation of their students, contemporary teachers must “personalize”
the learning milieu and invite students to “think and read with a critical eye, [as well as] to solve
mathematics problems through understanding” (p.274) rather than through repetition. Even so,
classroom realities do not always align to such intrinsic values and teachers are often left
employing more extrinsic approaches.
Most teachers know that not every student is captivated by intrinsically motivating
factors. Consequently, at times, students will require a greater amount of extrinsic motivation. In
such situations, as direct instruction and cognitive load theory supporters contend, the techniques
offered by the behaviorist and cognitivist traditions can become useful. Hidi and Renninger
(2006), for instance, offered a model of motivation that includes aspects of both the behavioral
and constructivist traditions. Their four-phase model of interest development, gives teachers a
process by which to understand the broad spectrum of motivational catalysts that can exist within
a classrooms. It also offers teachers a way to systematically move students away from their need
to be extrinsically motivated and into a much more personal intrinsic state, which was defined by
the authors as a state of “well-developed individual interests” (p. 115). Consequently, teachers
do not have to choose between the teacher-controlled efforts found in the first phase of the model
or the student-centered inertia that is found in the last phase. Rather, they must dynamically
evaluate where each student lies, and how to best motivate him/her at that present moment either
DEVELOPING MATHEMATICAL REASONING 83
by extrinsic or intrinsic ways. However, the question remains, are there specific pedagogical
approaches that can assist teachers in this activity? The last model of motivation examined helps
answer this question.
Goal orientation theory is a well-established and successful approach to motivating
students (Ames, 1992; Anderman & Anderman, 1999; Dweck, 1999; Pintrich & de Groot, 1990;
Turner et al., 2002). The theory is mainly characterized by two ends of the same spectrum,
mastery and performance goal orientations. Students who maintain a mastery orientation tend to
focus on the intrinsic rewards of learning, such as understanding the subject more deeply, as well
as in becoming more adaptive to their environment (Anderman & Wolters, 2006). Conversely,
students who maintain a performance goal orientation focus more on extrinsic rewards like
stickers, candy, and teacher recognition. However, Ames (1992) believes, and many other
scholars support her position, that maintaining a mastery orientation is more beneficial.
Furthermore, she claims that teachers can increase the chances that their students will adopt such
a mastery view of learning by altering the way they conduct their daily affairs. Her focus on a
growth/developmental model of understanding aligns well with student-centered pedagogical
methodologies. She states that “within a mastery goal orientation, the focus is on effort, not
ability, and belief in the efficacy of one’s effort mediates approach and engagement patterns”
(p.268). Consequently, it would seem that mastery orientation is enhanced by a teacher who
perceives of errors not as something to be avoided, but as a function of learning, and as a way in
which both the teacher and the student can gain a greater understanding of each other and the
subject in which they are involved, as these two factors that are very much embedded in the CGI
process.
DEVELOPING MATHEMATICAL REASONING 84
One must not forget that there is another end to this goal-oriented spectrum, that of
performance orientation. Performance orientation focuses less on growth and improvement and
more on outcomes. Thus, students upholding such a view will tend to avoid any situation which
will generate what they perceive to be a negative result. Furthermore, performance oriented
students tend to be more concerned with comparing their work with the work others, rather than
seeing how they can improve as they grow in their knowledge. Hence, at first sight, it seems as
if this approach could add no value to the work of a teacher. Nonetheless, there are some
researchers who found performance orientation to offer plenty of value. Their research
demonstrated that students with performance goals have higher achievement, effort, and are
more connected to accomplishing the task at hand (Elliot, Shell, Bouas Henry, & Maier, 2005;
Harackiewicz, Barron, Pintrich, Elliot, & Thrash, 2002).
In conclusion, both the student-centered and the teacher-centered approaches appear to
offer motivational and academic benefits. For instance, classrooms that focus on constructivist
(student-centered) seem to present their students with a greater opportunity to develop mastery
goal orientations or well-developed individual interests, and such interests have been associated
with the ability to be more adaptive, which, in turn, should help with the development of
mathematical reasoning. Conversely, educational environments that are grounded on
performance, such as those espoused by behavioral or cognitivist pedagogical approaches
(teacher-centered) have also presented their own positive motivational and academic outcomes.
Consequently, the author asserts that this study would be complete without examining which
pedagogical approach used herein ( EDI or CGI+EDI) would offer the greatest capacity to
develop students’ positive perceptions of math and, in turn, mathematical reasoning.

DEVELOPING MATHEMATICAL REASONING 85
Summary
American education is characterized by various, and often conflicting, pedagogical
perspectives (Wilson, 2013). Supporters of each of these diverse perspectives maintain that their
particular approach is the most efficient for the entire gamut of content domains, knowledge
types, and/or cognitive sub-skills. This state of affairs created an excess of educational
methodologies and curricula, some of which stand in contradiction to one another. As a result, a
pendulum has been generated that finds itself in perpetual motion. Unfortunately, practitioners
are often asked to take the brunt of the pendulum’s fluctuations, forcing them to swing back and
forth with all the theories, pedagogies, and institutional demands that characterize it.
Among such divergent perspectives stand teacher-centered and student-centered
approaches. One of the most publicized and used teacher-centered pedagogies is Direct
Instruction (DI). Direct Instruction gained recognition during “Project Follow Through”, a
federally funded research study that evaluated the ability of nine different pedagogies to instruct
students from economically disadvantaged backgrounds (Meyer, 1984). With the success of DI
came a number of other teacher-centered pedagogical approaches such as DataWORKS’ Explicit
Direct Instruction (EDI).
EDI, much like its predecessor, finds its theoretical roots in the behavioral theories of
Thorndike, Watson, and Skinner as well as in the Information Processing Model (IPM) of the
cognitivist theorists. As such, it is based on a more teacher-centered approach to education, an
approach that designed to be rigid and prescriptive rather than exploratory in its nature.
Consequently, EDI does not ask teachers to understand their student’s Zone of Proximal
Development (Chaiklin, 2003), but, rather, to assume students have little prior knowledge of the
subject, facts, or processes being taught. Furthermore, EDI’s prescriptive approach also
DEVELOPING MATHEMATICAL REASONING 86
disaffirms the need for students to have opportunities to investigate or develop concepts on their
own. As a result of these pedagogical premises, many of the model’s critics point out that EDI
students never gain the required capacities to know how to cognitively labor their way through
questions or problems in which they had no prior instruction.
CGI, on the other hand, is grounded on a constructivist model of learning that focuses on
the students, rather than on the teachers, as the principal source of knowledge. Accordingly,
CGI’s main purpose is to educate teachers on the way students develop their mathematical
schemas, giving them the developmental knowledge necessary to assist each child with his/her
own construction of mathematics. Consequently, during a CGI lesson, a teacher must constantly
assess each student’s strengths and weaknesses in order to identify the appropriate level of
engagement the child requires in a process Vygotsky (1978) referred to as the Zone of Proximal
Development (ZPD). Accordingly, if the CGI model is faithfully and successfully implemented,
by its very nature, it will offer students the opportunity to develop and practice their
metacognitive skills: reasoning, analysis, creativity, and problem solving.
Regardless of the dichotomic approach that has generally characterized both the theory
and practice of American schools, there are some scholars that argue for a much more
comprehensive and inclusive view of learning. Three studies that followed such a
comprehensive approach are: (1) Alfieri, Brooks, Aldrich, and Tenenbaum’s (2011) investigation
of discovery-based learning, (2) Matthews and Rittle-Johnson’s (2008) comparison of self-
explanations, concepts, and procedures as pedagogical tools, and (3) Khun’s (2006) investigation
of the long-term effects of direct instruction versus discovery learning. Each one of the studies
points to the need for more complex models of investigation and instruction, alluding to the fact
DEVELOPING MATHEMATICAL REASONING 87
that the exclusive dichotomic models that are often employed in schools and research cannot
cover the entire gamut of learning.
Following these studies, this review argues for a pedagogical investigative model that can
be more inclusive rather than exclusive in its approach. That is, it argued that, teacher-centered
pedagogies are not the most efficient and successful educational approaches for the improvement
of some knowledge types or cognitive sub-skills. Conversely, student-centered approaches are
not the most efficient and successful educational approaches towards the acquisition of other
knowledge types or cognitive sub-skills. Accordingly, the present work identified one content
domain and one cognitive sub-skill, mathematical reasoning, and investigated which
pedagogical approach, EDI or CGI+EDI, could offer students the best opportunity to develop
such knowledge. Moreover, because of the connection in the literature between motivation and
performance, the study also investigated whether the students’ experiences with each
pedagogical approach increased their positive perception of mathematics and, in turn, their
mathematical reasoning as well.

DEVELOPING MATHEMATICAL REASONING 88
CHAPTER THREE: METHODOLOGY
Introduction
International tests, such as the TIMSS, demonstrated that American students struggle in
mathematical reasoning, especially when compared to their counterparts in other developed
nations (Mullis, Martin, Foy, & Arora, 2012). In addition, national tests, such as National
Assessment of Educational Progress (NAEP), have shown that state assessments do a poor job of
identifying the true range of academic abilities of American students. This is based on the fact
that many students perform well on state examinations, which mostly focus on lower order
thinking skills, such as facts and procedures, while they continue to struggle on national and
international tests, which focus on both lower and higher order thinking skills like reasoning,
analysis, and evaluation. Accordingly, this study evaluated the capacity of two distinct
pedagogical models, Cognitively Guided Instruction and Explicit Direct Instruction, to determine
which one can best assist elementary students in their development of mathematical reasoning.
Other entities also attempt to ameliorate the cognitive limitations of American students.
For example, the federal government’s development and implementation of the Common Core
Standards enacted a major pedagogical shift that focuses, amongst other things, on developing
the mathematical conceptual understanding of American students. Consequently, this study
comes at a point when knowing how best to instruct and develop such skills is a priority. Hence,
the outcomes of this investigation will offer educators vital information regarding what type of
curriculum should be employed to reach these new pedagogical aspirations.

DEVELOPING MATHEMATICAL REASONING 89
Research questions
1. What are the mathematical performance differences that are exhibited by students who
were instructed using an amalgamation of student and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
2. What are the mathematical reasoning differences that are exhibited by students who were
instructed using an amalgamation of student and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
3. What are the mathematical perception differences that are held by students who were
instructed using an amalgamation of student and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
4. What are the mathematical reasoning differences that are exhibited by students when
curriculum type and mathematical perception are combined in order to generate a linear
composite?
Research Hypotheses
 Research question 1 maintains a non-directional hypothesis: A difference exists between
the mathematical performance (higher order thinking skills) of students taught using
student-centered pedagogies (Cognitive Guided Instruction) and that of students taught
using teacher-centered pedagogies (DataWORKS’ Explicit Direct Instruction model).
 Research question 2 maintains a non-directional hypothesis: A difference exists between
the mathematical reasoning abilities (higher order thinking skills) of students taught using
DEVELOPING MATHEMATICAL REASONING 90
student-centered pedagogies (Cognitive Guided Instruction) and that of students taught
using teacher-centered pedagogies (DataWORKS’ Explicit Direct Instruction model).
 Research question 3 maintains a non-directional hypothesis: A difference exists in the
mathematical perceptions of students taught using constructivist student-centered
pedagogies (Cognitive Guided Instruction) and those of students taught using behaviorist
teacher-centered pedagogies (DataWORKS’ Explicit Direct Instruction model).
 Research question 4 maintains a non-directional hypothesis: The interaction that is
created between the type of curriculum to which students have been exposed and the
perception of mathematics that they maintain makes a difference in the students’ ability
to reason mathematically.
Nature of the Study
The study used a retrospective casual-comparative research model, an ex post facto
approach (Cohen, Manion, & Morrison, 2007; Gay, Mills, Airasian, 2006; Lodico, Spaulding,
Voegtle, 2006), in order to analyze the mathematical reasoning and mathematical perceptions of
upper elementary students (Figure 1). Because this was an ex post facto approach, participants
in the study were neither randomly selected nor randomly assigned to an intervention group.
Rather, the participants in each group had previous exposure to the independent variables
investigated (EDI or CGI+EDI). Moreover, because this is a non-experimental model, no
interventions were carried out during the study.

DEVELOPING MATHEMATICAL REASONING 91

Independent variables

Two distinct curricula

EDI students

CGI + EDI students

ex post facto
Accounted for Confounding variables Not accounted for

Participants’ academic performance

Classroom differences
Participants’ ethnicity Leadership differences
Participants’ primary
language Parental differences
Teachers’ experience and
credentials

School culture differences
Socio-economic status

Dependent variables

Mathematical reasoning Mathematical perception

Instruments

 twelve question test  nine-question questionnaire
 six CST and six TIMMS items  eight 4-point Likert scale items
 TIMSS = mathematical reasoning  one open ended response

Figure 1. Model of the causal-comparative approached used in the study.

Interaction
DEVELOPING MATHEMATICAL REASONING 92
As a result, no direct correlation or causation effect was identified (Gay et al., 2006; Lodico et al.,
2006), which lowers the external validity of the study. Nevertheless, if done correctly, the model
does present strong preliminary causal evidence, which, in turn, can eventually be used to create
an experimental study appropriately designed to investigate the causality or correlation between
the identified variables.
The study followed Lodico, Spaulding, and Voegtle’s (2006) recommendations about
how to conduct a causal comparative investigation. Consequently, it instituted the following
steps. First, a construct was chosen that dealt with a practical, rather than an esoteric, problem in
education: mathematical reasoning), what Lodico and his colleagues identified as real world
problems. Second, the study identified all the dependent and independent variables. Only one
independent variable was explored: the type of curriculum to which the participants had been
previously exposed, CGI or EDI. The three dependent variables examined were (1) the students’
overall mathematical ability, (2) the students’ ability to reason mathematically, and (3) the
students’ perception of mathematics. Third, it identified measurement tools that could assess the
relation between the variables. The first two variables, for example, were measured using a 12-
question test, whereas the last variable was measured using a nine-question survey. Clear
definitions of each variable were also provided. Mathematical reasoning, for example, was
defined as “the attainment of abilities to construct mathematical conjectures, develop and
evaluate mathematical arguments, and select and use various types of representations”
(Kramarski & Mevarech, 2003, p.2) to solve mathematical word problems. Also, as stated in
chapter one, mathematical perception was defined as the combination of the following factors:
(1) how much a student is interested in mathematics, (2) how much a student enjoys mathematics,
DEVELOPING MATHEMATICAL REASONING 93
(3) how the student interprets the mathematical enjoyment and interest of others, and (4) how
much the student and others enjoy solving mathematical word problems
Subjects
Connecting With the School-Sites
The study was conducted using 5
th
grade students at two different school sites in
Southern California (N= 81). In accordance with IRB specifications, the children, teachers, and
school sites that participated in the study have been kept anonymous. Consequently, each site
involved in the process was given a pseudonym: School A or School B.
School A was chosen because it was recognized as the CGI flagship of its district. As a
result, the school’s entire teaching staff received extensive training in how to carry out a CGI
lesson. Because of this, the school was used as the go to site whenever teachers needed to learn
how to successfully carry out a CGI lesson. Nevertheless, the majority of the Title I teachers
working within this particular school district had been trained in the EDI lesson format, and,
since School A is considered a Title I school, many of its teaching staff members had also been
previewed to EDI teaching strategies. Consequently, a large number of the teachers from School
A were employing both CGI and EDI strategies in their lessons. This study, therefore, should
not be understood as a comparison between two isolated and distinct models of instruction, EDI
versus CGI), but, rather, as a comparison between a more dynamic model of instruction,
CGI+EDI, that employs both student- and teacher-centered pedagogies, and a second approach
characterized by a singular pedagogical view, EDI. Although classroom observations would have
been beneficial to the validity of the study, no observations were carried out. However, most
teachers claimed, when they were asked, that they used the CGI strategies once or twice a week,
depending on time constraints.
DEVELOPING MATHEMATICAL REASONING 94
Teachers in School B, on the other hand, had never been trained on CGI. Instead, they
had received extensive training on the EDI model. In fact, every teacher in school B was
provided with, and asked to display somewhere in their room, a poster delineating the different
parts of an Explicit Direct Instruction lesson. Moreover, every formal and informal observation
carried out by the school’s administration had to follow the EDI lesson framework, which
ensured that the teachers and the students at the site were very well versed in the EDI process.
The process of gathering subjects followed all IRB regulations. As a result, the first
contact with each site occurred via a face-to-face meeting between the schools’ principals and
the principal investigator (PI). Information regarding the reasons for, the subjects of, and the
methodology that would be employed during the study were provided during this first encounter.
Each site’s principal was asked to affirm approval of the study via a letter of confirmation
written on the school’s letterhead. Once the principals approved the study, all 5
th
grade teachers
at each site were contacted and briefed regarding the methodology and the reasons for the
investigation. All teachers wishing to participate were asked to sign a letter of consent stating
their willingness to assist in the study (Appendix E). Every teacher who was contacted (n = 8)
decided to partake in the study, which substantiated the idea that the topic of mathematical
reasoning was of interest and relevance to their work. Still, it is important to note that each
teacher volunteer would receive two movies tickets as a show of gratitude for the time and effort
they would contribute to the investigation, a factor that could have influenced their decision to
participate.
Once the teachers accepted to participate in the study, the principal investigator attended
each classroom to present a mini-lecture regarding the importance of conducting research. This
was a critical step in trying to maintain the validity and reliability of the study given the age of
DEVELOPING MATHEMATICAL REASONING 95
the participants. In other words, because students of this age, upper elementary, often have a
tendency to look at each other’s papers and/or give up too quickly when presented with questions
that may appear to be too difficult, a presentation explaining how important it would be to do
their own work became essential. Accordingly, the mini-lecture included a series of PowerPoint
slides that explained how researchers use the scientific method to investigate interesting
questions and how important it is to gather clean and accurate data. The presentation also
included a slide describing how students who participate in the study would enter a raffle to win
a $25 gift certificate. Only one card was raffled per school.
Given schedule conflicts between two of the participating teachers and the principal
investigator, only six out of the eight participating classrooms were available for the presentation.
This created a lack of understanding regarding the purpose of the study among those classrooms,
which ultimately affected their participation rates. For instance, in one of these classrooms, only
five out of thirty-four students chose to participate in the study, which is a very small
participation rate given that the other classrooms attained a rate of 40% or higher.
Because the participants of the study were minors, permission to participate in the study
included both parental consent and personal assent. Two different letters were drafted for each
of these purposes. The consent form followed the template provided by the Institutional Review
Board at the University of Southern California (Appendix F). It included 4 pages that detailed
the rights of each participant, including the freedom to drop from the study at any point without
personal repercussion. Both the consenting adults and assenting minors were asked to sign this
form. Due to the Hispanic/Latino background of the subjects, the information included in both
documents was written in both English and Spanish. The assent form, conversely, was a much
shorter document written using language simple enough for an upper elementary student to
DEVELOPING MATHEMATICAL REASONING 96
understand (Appendix G). Although not strictly necessary due to the fact that prior written
participation had been given in the consent form, participants were asked to sign this form as
well. Not all participants used in the final analysis of the study turned in a signed assent form.
Still, all participants who were included in the final analysis did turn in a consent form signed by
themselves and a consenting adult.
Controlling for Confounding Variables
Confounding variables such as the students’ socioeconomic status, primary language,
ethnic background, expertise and experience of the classroom teacher, and education level of
their parents, can drastically increase the error term of a study. To try to control for these
confounding variables, researchers wanting to employ a causal-comparative model have to make
sure to match as many of those variables, between the group-samples, as possible (Lodico,
Spaulding, & Voegtle, 2006).
Consequently, the two school-sites used in the study where purposefully chosen based on
their ability to provide students and teachers who possessed equivalent characteristics. For
example, all the students who were part of the final analysis were Hispanic/Latino 5
th
graders
(70% for school A and 99% for school B) from Title I schools located about 2 miles away from
each other (Ed-Data, n.d.). Furthermore, all of the teachers employed in both schools were fully
credentialed at the time of the study. Additionally, the average experience of the teachers was
very comparable: 14 years for school A and 9 years for school B. The most significant factor,
however, was the resemblance in the Academic Performance Index (API) attained by each of the
schools during the previous academic year. That is, both schools scored an 824 in their API,
demonstrating that the overall performance of the teachers and students at both sites was
extremely comparable.
DEVELOPING MATHEMATICAL REASONING 97
Regardless of the researcher’s best intentions to account for as many confounding
variables as possible, some distinctions still remained. For example, at the time of the study,
school A had a total student population of about 780 students, which made it larger than school
B’s population of 612 students. Furthermore, although both schools scored above the state
average in their mathematics state tests ( 15% above the average for school A and 7% above the
average for school B), the 5
th
graders in school A were able to outperform their counterparts in
school B by a margin of 8 percentile points (Great Schools, 2013). Fortunately, the disparity was
not as wide when the Hispanic/Latino students from each site were isolated from the rest of the
population. When this was done, the gap between the schools narrowed to only 5 percentile
points. That is to say, 83% (72% advanced and 15% proficient) of Hispanic/Latino students in
school A were able to attain a score adequate enough to be considered advanced or proficient in
their state examinations and 78% (59% advanced and 20% proficient) of such students in school
B were able to attain a similar status (star.cde.ca.gov., 2013). It is evident that, even after
isolating the Hispanic/Latino population, a considerable difference still remained between the
students who scored advanced in each school, which is an issue that could have generated
problems in the ecological validity of the study.
To control for between-groups disparities, such as the imbalances exhibited in the CST
scores, one can either perform an ANCOVA analysis or design a measuring tool to investigate
whether differences exist between the groups. Because an ANCOVA would have required
access to the students’ CST math scores in order to create the covariate variable required in such
an analysis, a step that could have potentially reduced the participation rate of the study, running
the ANCOVA never became an option. As a result, the second alternative was employed. That is,
along with the six TIMSS question included to measure mathematical reasoning, the test also
DEVELOPING MATHEMATICAL REASONING 98
included six CST questions. The inclusion of these items provided a way to compare whether
the mathematical abilities of the two groups were indeed different from one another as the
previous year’s test score had shown. In the end, a one-way ANOVA analysis demonstrated no
statistical significance in the CST items between the two groups, giving weight to the idea that
the samples employed within the study were indeed similar to one another (Table 1).
In the end, any 5
th
grader wishing to participate in the study was given the opportunity to
do so, which turned out to be 122 students (N= 57 for EDI, N= 65 for EDI+CGI). Nevertheless,
based on the need to control for as many confounding variables as possible, all gifted and/or
talented ( GATE) students who were not Latino/Hispanic were excluded from the final analysis
( N= 30 for EDI+CGI).
Table 1
Test Items Between-Groups Comparison

M

EDI

EDI + CGI

Sig. (2-tailed)

CST items

4.53

4.96

.11

TIMMS items 2.74 4.05 .0002

A second factor that decreased the number of students who had originally wanted to
participate in the investigation was the pilot study. A number of the students in School B had, the
previous spring, partaken in the pilot, which made them ineligible to be included in the final
results given their prior exposure to the exam questions. This decision to exclude these students
from the final analysis lowered the EDI group from 57 students to 51.

DEVELOPING MATHEMATICAL REASONING 99
The Data Collection Process
The entire data collection process took about three days. Therefore, no considerable
attrition rate was experienced during the study. Missing data resulted from students’ absences
rather than by their choosing to drop out of the study. Consequently, anyone who was absent
during the collection of the data was appropriately identified as having missing data. An
accentuating factor was the choice a few of participating teachers made to collect their data
during two different days. That is, they gave the test on one day and the survey on another. This
differentiation of dates resulted in some participants having one set of data, but not the other,
generating inconsistencies in the number of subjects included in each one of the performed
analyses. Beyond that fact, the discrepancy in testing dates that was experienced by some of the
subjects could have decreased the study’s internal validity. In the end, only one student from
school A and three students from school B were identified as having missing data for the test,
and two students from school A and one student from school B were identified as having missing
data for the survey. This resulted in a participation rate of 94% for the test and 96% for the
survey (N = 81). Due to time and pedagogical constraints, no make-up sessions were carried out
with students who missed the testing period.
Instruments
The study used two measurement tools to collect data (Table 2). The first tool was a 12-
item test composed of six California Standardized Test (CST) questions and six questions from
the Trends in International Mathematics and Science Study (TIMSS). The second tool was a
nine-question survey that followed the parameters of a survey used to assess mathematical
perception during the 2011 TIMSS investigation. The process of constructing and assessing each
tool’s validity and reliability is explained in the paragraphs below.
DEVELOPING MATHEMATICAL REASONING 100
Table 2
Research Questions with Corresponding Measurement Instruments and Supporting Literature

Research questions

Measurement tool
used

Supporting Literature

1. What are the differences in
mathematical performance
exhibited by students
instructed via Cognitively

A 12-question test
composed of 6 CST
and 6 TIMSS
questions.

McEwan & McEwan,
2003;Creswell, 2009; Kurpius &
Stafford, 2006; Miller, Linn,
Gronlund, 2009)

2. What are the differences in
mathematical reasoning
exhibited by students
instructed via Cognitively
Guided Instruction versus
Explicit Direct Instruction?

A 12-question test
composed of 6 CST
and 6 TIMSS
questions.

McEwan & McEwan,
2003;Creswell, 2009; Kurpius &
Stafford, 2006; Miller, Linn,
Gronlund, 2009)
3. What are the differences in
mathematical perception held
by students instructed via
Cognitively Guided
Instruction versus Explicit
Direct Instruction?
A 9-question survey
model after items
used in the 2011
TIMSS
investigation.
(McEwan & McEwan, 2003;
Creswell, 2009; Fink, 2013; Kurpius
& Stafford, 2006; Miller, Linn,
Gronlund, 2009, Deci’s,1980) self
determination theory; Deci & Ryan,
1985; Ryan & Deci, 2000)
Cognitive Evaluation Theory

4. Does pedagogical
approaches make difference in
students’ mathematical
perceptions as well as their
ability to reason
mathematically when analyzed
as a linear composite?

A 9-question survey
model after items
used in the 2011
TIMSS
investigation.

(McEwan & McEwan, 2003;
Creswell, 2009; Fink, 2013; Kurpius
& Stafford, 2006; Miller, Linn,
Gronlund, 2009)

The Test
All the questions that comprised the test designed for the study had been previously
employed in large scale assessments (CST and TIMSS). As a result, the study assumed the
validity and reliability of all of the questions that were included in the study’s test. Nevertheless,
DEVELOPING MATHEMATICAL REASONING 101
given the fact that the study’s test was created using an amalgamation of two distinct
examinations, analyzing its reliability became critical.
During the pilot study, the results of this analysis were less than moderate (α =.57).
However, given the low number of participants who were involved in the pilot (n = 23) and the
absence of one item (a question was not included in the reliability analysis because it lacked an
appropriate response), the low reliability score was considered adequate enough. More
importantly, an item-total analysis of the pilot data demonstrated that the deletion of only one
item (question 8) would have improved the test’s reliability, albeit just slightly (α = .602). Based
on these factors, no questions were omitted from the test during the actual study. As a result, the
final measurement tool included all 12 questions that had been part of the pilot study, but with a
few necessary modifications. In the end, the combination of having a greater number of
participants and the improvements made to the test, this tool’s reliability score was elevated to an
adequate level (α = .70) during the study.
Each of the questions chosen for the assessment were selected to accomplish the
following goals: (1) to assess whether the two groups partaking in the study differed in their
mathematical abilities with regards to mathematical reasoning, (2) to assess whether differences
exist between the way American students perform in state and international examinations, and
(3) to test which instructional approach (CGI+EDI or EDI) can be more suitable for developing
the mathematical reasoning of upper elementary students. As previously discussed, the TIMSS
questions that were included were selected because of their ability to assess the construct of
mathematical reasoning (Appendix H). An equal identification and selection process was not
possible for the CST questions because, according to the California Department of Education,
mathematical reasoning is embedded within the CST test rather than being evaluated by specific
DEVELOPING MATHEMATICAL REASONING 102
items. Several mathematical topics and key standards made up both the pilot and final test.
Amongst them were whole numbers, reading and interpreting data, patterns and relationships,
organizing and representing data, using standard algorithms with multi-digit numbers,
understanding and using formulas to solve perimeter problems, visualizing models of geometric
solids, and mathematical reasoning. This plethora of topics validated the idea that the test was
broad enough to be used as a tool to assess someone’s overall mathematical ability.
The Pilot Study. The pilot study was conducted in the spring of 2013. A group of 23
students, from one of the sites used in the study, were used as the pilot’s subjects. From that
group of students, five were randomly selected to be interviewed about the difficulty, format,
readability, and process of the test. The following five questions were used to carry out the
interviews: (1) what were the 3 hardest questions in the test and why? (2) What were the 3
easiest questions on the test and why? (3) What would you change about the test to make it
better? (4) Was there any part of the test that you did not understand? How would you change it
to help other students understand it? (5) Did you use any things (knowledge, techniques, and
processes) that your teacher taught you this year to solve this test? Tell me the question number
and how your teacher taught you to solve it.
The interviews were able to identify a couple of design errors in the test. The first
problem had to do with the format of the test. That is, students pointed out that some of the
questions started in one page and finished in another, forcing them to flip back and forth between
the pages. This turning of pages increased the difficulty of the test based on the increased
extrinsic load the process created (Mayer, 2011) This, in turn, affected the ability of the test to
approximate the participants’ true scores ( true knowledge of the construct being measured).
Moreover, an item analysis of the pilot test corroborated the perspective of the students,
DEVELOPING MATHEMATICAL REASONING 103
confirming the need to reformat the tool. The new version of the test did not include any
questions that were split between two pages. Unfortunately, this forced all the TIMSS questions
to be placed at beginning of the test (items 1, 2, 3, 4, 6, 7) and all the CST questions to be at the
end (items 5, 8, 9, 10, 11, 12). Even so, statistical analyses comparing the scores in the
reliability and normal distribution between the pilot and the actual study showed that the changes
to the test appeared to be positive (Table 3).
Table 3
12-Question Test Comparison Using Results from the Pilot Study and the Actual Study

Reliability Normal distribution

Tool n α Shapiro-Wilk Skew Kurtosis

Pilot 23 0.43 0.03 0.34 -1.18

Actual 76 0.70 0.17 -0.48 0.05

The second problem that the pilot study revealed was the absence of a correct response in
one of the multiple choice questions (item #7). This predicament was quickly resolved by editing
the new version of the test to include a correct response. In the end, the problem appeared to
have been resolved given that 95% of the participants were able to answer this question correctly
during the actual study.
The pilot study was also able to confirm a difference in the level of difficulty between the
TIMSS and CST questions. That is, the pilot students were able to achieve a composite average
score of .76 in the CST items, while only achieving a composite average score of .41 in the
TIMSS questions. Supporting the idea that the TIMSS questions were, indeed, more difficult to
answer, a fact that was also corroborated by the composite results of the actual study (TIMSS
DEVELOPING MATHEMATICAL REASONING 104
= .53 and CST = .79). This discrepancy of scores also validated the claim that the TIMSS
questions assessed a different type of cognitive domain (mathematical reasoning). In short, by
combining the reliability scores, the item-analysis scores, and the results attained when items
were segregated and evaluated by each examination type, the pilot study lends support to the idea
no item had to be omitted from the final version of the test.
The Survey
The main reason for the survey was to assess the mathematical perceptions of the
students. Mathematical perceptions were defined in this study as the combination of the
following factors: (1) a student’s evaluation of how much s/he is interested in mathematics, (2)
how much s/he enjoys mathematics, (3) how he/she interprets the enjoyment and interest of
others regarding mathematics, and (4) how much he/she and others enjoy solving mathematical
word problems. This construct was of central importance to the study because research shows a
correlation between students’ enthusiasm for a subject matter and how well they perform in it
(Hattie, 2009). Moreover, when such research is coupled with the idea that children who are
instructed using student-centered pedagogies tend to hold a more positive perspective of learning
than those who are not (Deci, 1980; Deci, & Ryan, 1985; Goldin, Yakov, Schorr, Warner, 2011),
investigating the interplay between perception and performance became a critical part of the
study. Consequently, a survey was created to examine whether the students’ perceptions of
mathematics was affected by the pedagogies they experienced in their classrooms (via CGI or
EDI) and, in turn, positively affected their mathematical reasoning skills.
All items in the survey were model after questionnaires used during the 2011 TIMSS
investigation. Namely, they followed the word choice and formatting style of two specific
surveys that had been used to assess a student’s enjoyment and confidence of mathematics (e.g.,
DEVELOPING MATHEMATICAL REASONING 105
I enjoy learning mathematics, I wish I did not have to study mathematics, I learn many
interesting things in mathematics, and I am good at working out difficult mathematics problems).
In the end, the survey was composed by nine-questions, eight Likert style questions (that used a
4 point scale) and one open ended response that was included to provide students with the
opportunity to offer more detailed and comprehensive responses.
Due to constraints in resources and time, the survey was only piloted with a small number
of participants (n = 3). The participants who partook in the pilot did not attend the schools that
would ultimately participate in the final study nor did they share the same ethnicity or social
economic status, but they were of the same age as the study’s participants (5th graders) and
attended the same school district.
Regardless of the limitations in sample size and lack of subject parallelism, the pilot
study revealed vital information regarding the survey’s format. For instance, the pilot study
identified a problem in the wording of the answer choices. That is, during the 2011 TIMSS
investigation all survey items were constructed using the following four-point Likert scale: agree
a lot, agree a little, disagree a little, and disagree a lot. Still, by conducting the pilot study the
investigative team noticed that these values were too difficult for the students to use accurately.
Consequently, the final version of the survey altered the answer choices to always, mostly,
sometimes, and never. Furthermore, each one of these answer choices was attributed a specific
point value: 4 points for Always, 3 points for Mostly, 2 points for Sometimes, and 1 point for
Never - the scale was inverted for the reverse coded items ( 4 points for Never, 3 points for
Sometimes, 2 points for Mostly, 1 point for Always). By providing such values the words
embedded in the survey were quantified from ordinal into interval variables providing the
DEVELOPING MATHEMATICAL REASONING 106
opportunity to carry out different statistical analyses requiring such type of data (e.g., T-tests,
ANOVAS).
These changes made to the survey appeared to be the correct ones, as demonstrated by
the high level of internal consistency (α = 0.74). Furthermore, an item-total statistical analysis
demonstrated that the deletion of only one item would have slightly improve the survey’s
reliability score (α = 0.75 for item 2). It is important to note that this item was a reverse coded
question asking participants to evaluate how other students, rather than themselves, perceived
mathematics at their school, which points to the notion that students of this age (upper
elementary) may have difficulty processing all the cognitive requirements needed to answer such
complex items.
Research procedure
Teachers were given the choice to collect data during the Wednesday, Thursday or Friday
before Thanksgiving break, so the majority of the data was collected in the span of three days.
However, one of the eight participating classes (n = 3) was allowed to administer and return data
at a later time, the Monday after Thanksgiving break. The extra time was given due to a
misunderstanding between the principal investigator and the participating teacher regarding the
pre-established window for testing. The class that was given the extra time belonged to the DI +
CGI group; however, only three students from this class were included in the results of the study.
Teachers were responsible for administrating and collecting their own surveys and tests. To
maintain validity across teachers and schools, very simple written instructions were provided for
the supervision of both the test and the survey. The instructions were sent to their personal work
emails a few days before the testing window opened. No training was provided for the teachers
to carry out either the test or the survey. Essentially, the e-mail directions told the participating
DEVELOPING MATHEMATICAL REASONING 107
teachers to offer no guidance or clarification during the test and to provide as much guidance and
clarification as needed during the survey. That is, no student was supposed to be provided with
any explanation regarding the instructions, process, or responses of any kind during the test.
Conversely, students were to be provided with as much assistance and guidance as possible for
them to accurately answer how they view and felt about the items in the survey. The decision to
offer as much clarification as possible was due to the fact that students in 5
th
grade are usually
not versed in Likert scale surveys that include reverse coded questions. Consequently, in order
to increase the validity and reliability of the survey teachers were asked to give as much
clarification as needed during the process.
Data reduction
The study followed a causal-comparative model of research. In spite of its name,
controversy still remains regarding the model’s ability to identify causal relationships between
variables (Brewer & Kuhn, 2010). This is due to a number of inherent limitations within the
model. For example, causal-comparative research does not provide for random selection of
subjects. Instead, it employs purposeful sampling to identify groups that already exhibit the
independent variable under investigation. Furthermore, investigations using this type of model
are ex post facto (after the fact), which makes it impossible for a researcher to manipulate the
variables on a real-time basis. This fact, in turn, diminishes the model’s ability to make causal
inferences. As a result of all these limitations, causal-comparative studies are considered quasi-
experimental rather than experimental.
Even so, one of the model’s strengths is its ability to compare the means of two groups
that cannot, or should not, be randomly assigned. Given this strength, all of the statistical tests
used for this study were chosen based on their ability to compare the means of two or more
DEVELOPING MATHEMATICAL REASONING 108
distinct groups (independent-samples T-tests and two-way analyses of variances). The level of
significance for each of the analytical test followed the conventional wisdom (p < .05). Besides
its common usage, this level of significance was chosen based on the fact that this study was
carried out using a fairly small sample size (N < 100).
Because causal-comparative models cannot offer strong evidence of cause-and-effect
and/or correlations between variables, all 4 hypotheses within the study were non-directional.
Research question number 1, for example, hypothesized that differences did indeed exist
between the mathematical performance of students using CGI + EDI and that of those taught
using EDI alone. Consequently, the statistical analysis that was chosen to investigate this
question was an independent samples T-test in which the raw scores of the test were employed as
the dependent variable (what SPSS identifies as the testing variable) and the curriculum
employed by each school became the independent variable (what SPSS identifies as the grouping
variable). In the end, the T-test provided a comparison of the two group means with regards to
their overall score in the test.
Research question number 2 also assessed mathematical performance. However, this
question focused specifically on the students’ performance regarding mathematical reasoning.
The question hypothesized that differences existed in the way the two populations would
reasoned mathematically. Six TIMSS questions that had been specifically designed to assess
students’ ability to reason mathematical were used to investigate this construct. To accomplish
this investigation, all the individual scores from each TIMSS question were compounded into
one aggregate score (TIMSS total score). This variable became the dependent variable in the
second independent-sample T-test, while the curriculum employed by each school remained the
independent variable.
DEVELOPING MATHEMATICAL REASONING 109
Similarly to research question 2, research question 3 also maintained a non-directional
hypothesis. That is, this question was aimed to investigate whether difference existed between
the mathematical perceptions of the group samples based on the curriculum they experienced at
their schools. The hypothesis, therefore, claimed that a difference did exist in the way students
perceived of math based on the type of pedagogy they experienced at their school (student-
centered or teacher-centered approaches). To accomplish this, a total score was computed using
all eight Likert scale items included in the survey. The open ended question (item 9) was not
included in the total score because of the difficulty of quantifying the variety of responses that
the students provided. The values attributed to each response ( 4 points for Always, 3 points for
Mostly, 2 points for Sometimes, and 1 point for Never) were added together to attain a total raw
score in the survey. The values of the reverse coded questions were inverted in order to maintain
parallelism in the scores of the reverse coded and regular questions (4 points for Never, 3 points
for Sometimes, 2 points for Mostly, 1 point for Always). Consequently, a total score of 25 or
higher on the survey meant that that particular student maintained a high perception of math
whereas, a score of 7 or below meant that the student held a pretty low perception of math. This
hypothesis was also tested via an independent-samples T-test where the curriculum employed by
each site was used as the independent variable and the total score in the survey became the
dependent variable.
Research question 4 was no different than its three predecessors. That is to say, it
maintained a non-directional hypothesis purporting that the linear-composite generated through
the combination of pedagogy and mathematical perception makes a difference in a student’s
ability to reason mathematically. However, unlike its predecessors, this question required the
inclusion of two independent variables, rather than the single variable used within T-tests.
DEVELOPING MATHEMATICAL REASONING 110
Consequently, this hypothesis was investigated using a two-way ANOVA in which the subjects’
scores on the TIMSS items of the 12-question test were used as the dependent variable and the
total scores in the survey along with the type of curriculum used to instruct them became the two
independent variables.
To accurately carry out analyses, each one of these two approaches (independent samples
T-test and two-way ANOVAs) requires affirmation of certain assumptions about the data used.
An independent-samples T-test, for example, assumes (1) that the dependent variable is being
measured as continues value, (2) that the independent value involves two distinct groups, (3) that
no subjects from one group also belong the other group, (4) that there are no outliers within
either of the groups being compared, (5) each independent variable should render a normal
distribution with regards to the dependent variable being investigated, and (6) the within group
variance for the independent variable is fairly equal. Similarly, a two-way ANOVA also requires
certain assumptions about the data: (1) that there are no outliers in any of the groups being
investigated, (2) that each group’s data is normally distributed, and (3) that each group’s data
maintains homogeneity of variance. Each one of these assumptions will be tested and discussed
during the results section of this work.
Summary
In short, this study employed an ex post facto causal-comparative approach of
investigation to determine whether differences existed in the mathematical reasoning and
mathematical perception of students who had been instructed using a combination of student and
teacher-centered pedagogies ( CGI + EDI) and those of students who were taught using simply
teacher-centered ones ( EDI). Given that the study fell within a quasi-experimental
DEVELOPING MATHEMATICAL REASONING 111
classification, all the hypotheses were non-directional. As a result, the study used analytical tests
designed to show differences between means, rather than causal or correlational inferences.
In order to control for as many confounding variables as possible, the participants of the
study were chosen from two very similar schools. That is, both schools were geographically
close to one another, characterized as Title I while maintaining large populations of
Latino/Hispanic students, and had attained the exact same scores on both their CSTs and school
rankings during the data collection year. Furthermore, the experience and credentials attained by
the teaching staff at both sites were analogous.
Regardless of the attempt to control for as many confounding variables as possible, some
critical distinctions still remained between the two group-samples. For example, there was an
eight percentile point difference in the number of 5
th
grade students (the grade chosen for the
study) who scored proficient or advance in their CST math examinations the previous year. This
disparity was lowered to five percentile points by focusing on the Latino/Hispanic populations of
each school site rather than the entire 5
th
grade student body. Furthermore, there was a
considerable difference in the number of students who attended the two sites chosen for the study.
That is, one site was larger than the other by about 120 students. More importantly, while
maintaining its identification as Title I school, one of the sites was also a GATE clustered school,
meaning that this site had a small population of students who qualified for the gifted and talented
program. However, it must be reiterated that, once the choice was made to focus upon the
Hispanic/Latino population of each school, only three GATE students remained part of the study.
DEVELOPING MATHEMATICAL REASONING 112
CHAPTER FOUR: RESULTS
Statement of the Problem
Historically, mathematics instruction has been characterized by a plethora of
permutations. Amongst these stand Cognitively Guided Instruction (CGI) and Explicit Direct
Instruction (EDI). These two approaches arise from two different theoretical camps,
behaviorism and constructivism. The premises that encompass each one of these camps direct
their supporters to enact different pedagogical models. CGI, for example, tends to be
characterized by the principles of a student-centered pedagogical approach. Consequently, it
makes the student’s prior knowledge and experiences its main pedagogical focus. Conversely,
EDI, which follows a more teacher-centered view of learning, makes the individual learner a
secondary feature of a well-crafted and delivered lesson.
Another critical distinction between these two views of learning is the amount of teacher
guidance demanded. That is, while student-centered pedagogies like inquiry-based learning,
problem-based learning, activity-based learning, and Cognitively Guided Instruction are less
prescriptive in their approach, teacher-centered pedagogies like Direct Instruction, Explicit
Direct Instruction, Cognitive Load Theory, and Information Processing Theory are much more
rigid and didactic. This disparity in their rigidity has been identified by some scholars as one of
the catalyst for the failure to develop higher order thinking skills of students in teacher-centered
classrooms. In other words, the factors that make a teacher-centered lesson successful, its very
directed and controlled approach of learning, are the same factors that detract from the
development of the creative, reasoning, and metacognitive skills students need to enhance their
higher order thinking skills. This is a notion corroborated by the scores of national and
international tests. These examinations demonstrate that American students tend to lack higher
DEVELOPING MATHEMATICAL REASONING 113
order thinking skills such as mathematical reasoning, yet these skills will be in great demand
during the 21
st
century. Hence, it is now vital to identify which specific pedagogies can
successfully increase the cognitive sub-skills as well as the higher order thinking skills that
American children lack. This goal is the focus of this study.
The results of each statistical analysis used to compare the two different pedagogical
approaches found within this study are presented below. Consequently, each one of the four
research questions used to frame the study is discussed in order. However, prior to delving into
each question, the chapter presents the demographic statistics of the two groups were involved in
the process. This is followed by a discussion of the assumptions connected to each one of the
analyses as well as the results each one attained. For those analyses in which the data did not
meet one of the necessary assumptions, explanation is given as to the type and reasons a
transformation of the data was employed.
Research Questions
1. What are the mathematical performance differences that are exhibited by students who
were instructed using an amalgamation of student and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
2. What are the mathematical reasoning differences that are exhibited by students who were
instructed using an amalgamation of student and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
3. What are the mathematical perception differences that are held by students who were
instructed using an amalgamation of student and teacher-centered pedagogies
DEVELOPING MATHEMATICAL REASONING 114
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
4. What are the mathematical reasoning differences that are exhibited by students when
curriculum type and mathematical perception are combined in order to generate a linear
composite?
Demographics
Table 4
Demographics of Each School Site
Curriculum Category n Minimum Maximum M (SD)
EDI
(School B)
Participants 51
Score on test 49 1.33 11.66 7.23 (2.18)
Score on survey 50 11.00 31.00 23.94 (4.70)
Valid N (listwise) 48

EDI + CGI
(School A)
Participants 30
Score on test 27 4.00 12.00 9.02 (2.07)
Score on survey 28 12.00 30.00 22.68 (4.56)
Valid N (listwise) 27

As can be observed in Table 4, the study included 81 subjects. This total number of
participants does not coincide with the total number of scores generated by each measurement
tool. This is due to the fact that a few students were either absent during the administration of
the test and/or the survey. The table is divided using the two independent variables, the type of
math curriculum employed to instruct the participants, used throughout the study. Each
curriculum included within the table, therefore, represents one of the sites involved in the study
(EDI+CGI for school A and EDI for school B).
The results demonstrate a number of parallelisms between the sites. For example, a wide
range of scores, both in the survey as well as in the test, were displayed for each school.
DEVELOPING MATHEMATICAL REASONING 115
Moreover, the standard deviations displayed for each of the schools were comparable. That is,
School A exhibited a standard deviation of 2.10 in its test scores while school B exhibited a
standard deviation of about 2.20. The standard deviations of the questionnaire were not much
different, given that there was only a 0.09 difference between the sites. Such similarities among
two different samples of students point to the effectiveness of the study’s ability to control for
confounding variables as well as to the validity and reliability of the measurement instruments
that were employed.
Two descriptive factors failed to exhibit the same interrelatedness demonstrated by the
others. The first factor was the mean score displayed by School A. This site’s test results were
larger than those of School B by a margin of about 2 points. Conversely, the mean score attained
by School B in the survey was larger than that of School A by a margin of about a point and
quarter (1.24 points). Consequently, the rest of this chapter focuses on these discrepancies in
order to investigate whether these differences in group-means are significant enough for
educators, policy makers, and researchers to take account of them. This is accomplished by
focusing on each research question individually.
Research Question #1
Research question #1 investigated whether differences existed in the mathematical
performance of students who were instructed via Cognitively Guided Instruction as opposed to
those who were taught via Explicit Direct Instruction. Given the characteristics of the question,
the comparison of two independent groups, results pertaining to this question were analyzed
using an independent-samples T-test.
In order to effectively carry out such an analysis, certain assumptions have to be met.
One of these assumptions is the need to have only one dependent variable measured using a
DEVELOPING MATHEMATICAL REASONING 116
continuous scale. This assumption was met by using the 12-question test – a single variable that
provided a range of scores from 0 to 12. Another assumption that needs to be met before
adequately interpreting the results of an independent-samples T-test is the need to exhibit
dichotomy in the independent variable. That is, the independent variable must be characterized
by two distinct groups. The present study met this assumption by comparing the scores of
students who had been instructed using distinct curricula (CGI+EDI and EDI).
The third assumption inherent to an independent-samples T-test is that of independence
of observation. This refers to the idea that participants representing one of the group-samples
cannot share any relationship with the participants who represent the other group-sample. To be
specific, participants cannot belong to both group-samples, personally know someone in the
other group-sample, or have the ability to influence a participant in the other group-sample. The
present study accounted for this assumption by selecting two school sites that were
geographically separated, and, given that the schools involved in the study were community
schools and that the subjects involved were in 5
th
grade, the possibility of participants interacting
with one another were low. Nonetheless, given the numerous variables that were not accounted
for, such as students’ having extended family relationships, parental relationships, or belonging
to the same sports teams, a small possibility did remain that this assumption had been violated.
Regardless of this possibility, no measure was adopted to examine whether participants had any
form of interaction with one another. Consequently, although the likelihood of its violation was
small, the affirmation of it could not be guaranteed either.
The fourth assumption dealt with the existence of outliers. According to this assumption,
subjects whose scores are greatly beyond that of the majority of the other participants generate
problems when trying to use group means as a comparison tool. The validation of this
DEVELOPING MATHEMATICAL REASONING 117
assumption is critical in order to guard against the possibility of a type I error where the means of
the two groups appear to be significantly different, but are significantly different only because a
few outliers make it so. Boxplots were used to check for this affirmation of this assumption
(Figure 2).

Figure 2. Box plots of the 12-question test categorized using the curriculum type that was
employed to instruct subjects.
The results revealed one outlier amongst the two group-samples, subject #3), violating
the assumption. Consequently, prior to running the independent-samples T-test, a Wilcoxon-
Mann-Whitney test was carried out. The decision to include such a test was based on its ability
to be less affected by outliers than an independent-samples T-test is.
Similarly to an independent-samples T-test, the Wilcoxon-Mann-Whitney test also holds
its own assumptions about the data that it is analyzing. The first three assumptions of the test are
analogous to the first three assumptions of the independent-samples T-test. Each of these
DEVELOPING MATHEMATICAL REASONING 118
assumptions have already been discussed and affirmed in the previous paragraphs, so no further
elucidation will be offered regarding their affirmation at this point. The fourth assumption deals
with the shape of the distribution for each variable used in the analysis. To be more precise, it
demands that each independent variable maintain a similar shape with regards to the dependent
variable. This assumption was affirmed. That is, the distributions of the scores in the test for the
DI and the DI+CGI groups were similarly shaped as affirmed by visual inspection The results of
the Mann-Whitney U test demonstrated that the scores for the DI group (Mdn = 7.33) were
statistically significantly lower than for the CGI+DI group (Mdn = 9.33), U = 963.50, z = 3.28, p
= .001. These results were in concurrence with the results of the independent-samples T-test,
discussed below, which alluded to the fact that the single outlier present within the DI group was
not having a strong effect on the outcome of either analytical test (T-test or Mann-Whitney U).
This allowed the investigative team to continue using the independent-samples T-test to
investigate this research question.
The fifth assumption that needs to be checked in order to carry out an independent-
samples T-test has to do with the shape of the data’s distribution. According to requirements of a
T-test, each independent variable must maintain a normal distribution with regards to the
dependent variable. For this particular question, it meant that the test scores for both the DI and
CGI+DI groups had to be normally distributed with regards to the total scores of the test. This
assumption was checked visually, via histograms and Q-Q plots (see Figure 3), and numerically,
using the Shapiro-Wilk test (DI group’s p = .20 and CGI+DI group’s p = .35). Both the test and
the graphs confirmed that the data was normally distributed for both groups.
DEVELOPING MATHEMATICAL REASONING 119

Figure 3. Histograms and Q-Q plots for the 12-question test categorized using the curriculum
employed to instruct the subjects.

The last assumption required of any T-test analysis is that of homogeneity of variance.
This assumption requires that the population variance for each group of independent variable be
similar. Violating this assumption is not too problematic if the two group-samples used in a
study are of equal sizes. However, if they are uneven, as in the case of the present study
(N = 27 for DI+CGI and 49 for DI), the violation of this particular assumption can inhibit the
interpretation of the T-test. Levene's test of equality of variances was employed to check for this
DEVELOPING MATHEMATICAL REASONING 120
assumption. The results of the test confirmed the null hypothesis that both groups were indeed
similar to each other, F (1,74) = .01, p= .91.
Having affirmed all the necessary assumptions, an independent-samples T-test was
carried out to see whether differences existed in the mathematical performance of the two groups
of students. The results of the test indicated that, indeed, there was a significant difference in the
mathematical performance of the two groups the null hypothesis could be rejected, t (74) = -3.49,
p = .001. These results imply that individuals who were instructed using a combination of
CGI+EDI (n= 27; M= 9.02; SD= 2.07) tended to display a higher overall mathematical
performance than those individuals who had been instructed using EDI alone (n=49; M= 7.23;
SD= 2.18). To be more precise, on average, the CGI+EDI group attained a mean test score that
was 1.79 points above that of the EDI group (Table 5). In addition, a Cohen’s effect size value
(d= .84) confirms the practical significance of the findings.
Table 5
Independent Samples T-test Using All the Items Included in the 12-Question Test (Assessment of
Subjects’ Overall Mathematical Performance)

Curriculum

M (SD)

M difference

p
95% Confidence
Interval
Lower Upper

Test
scores

CGI+EDI

9.02 (2.07)

1.79

.001

.77

2.81

EDI

7.23 (2.18)

-1.79

.001

-2.81

-.77

Research Question #2
Research question #2 investigated whether there was any difference in the ability of the
two group-samples to reason mathematically. Consequently, it maintained a non-directional
hypothesis. An independent-samples T-test was used to investigate this question as well. The
DEVELOPING MATHEMATICAL REASONING 121
subsequent paragraphs will discuss whether the assumptions of the T-test were met, as well as,
the results of the test itself.
Given that this question used the same dependent and independent variables as research
question #1, the first three assumptions required of an independent-samples T-test are be
discussed any further than to say that, as previously seen, they were not violated. The fourth
assumption, which looks for the presence of outliers resulting in the interaction between the
dependent and independent variables, required further investigation. That is, given that the items
used to measure this research question focused on the mathematical reasoning of the students,
rather than their overall math ability, and given that only half the test was strictly designed to
measure that particular construct, it was vital to investigate whether outliers were present when
the study focused solely upon mathematical reasoning. The presence of any outliers was
investigated using a box plot (Figure 4). The plot demonstrated an absence of outliers in either
group, affirming the assumption.

Figure 4. Box plots using only the TIMSS items in the 12-question test designed to assess
mathematical reasoning.

DEVELOPING MATHEMATICAL REASONING 122
Next was the assumption of the data’s distribution. As previously stated, each independent
variable within an independent-samples T-test must maintain a normal distribution with regards
to the dependent variable employed during the analysis. With regards to this particular question,
it meant that the TIMSS items’ scores for both the EDI and CGI+EDI group had to be normally
distributed. This assumption was checked visually via histograms and Q-Q plots (Figure 5) and
numerically using the Shapiro-Wilk test. The visual inspection displayed contrasting
distributions for each sample. Namely, the EDI group was normally distributed while the
CGI+EDI group was moderately, negatively skewed. These results were corroborated by the
Shapiro-Wilk test (p > .05), which confirmed a normal distribution for the EDI group (p= .56),
but not for the CGI+EDI group (p= .04). Consequently, the assumption was violated.
To account for this violation, the data was transformed using a reflect and square-root
transformation. This transformation was chosen because of its utility in adequately working with
moderately, negatively skewed data such as that displayed by the CGI+EDI group. The
transformation worked accordingly, furnishing a Shapiro-Wilk test (p > .05) of p =.23 for the
EDI group and p =.18 for the CGI+EDI group. However, the process of transforming the data
generated other dilemmas. First, it produced an outlier within the EDI group, creating another
violation of the assumptions required of a T-test. Second, it inverted the means of the groups.
This inversion was due to the mathematical process required in a reflect square root
transformation. That is, a reflect square root transformation will augment scores that are below a
point-value of one and diminish scores that are above a point-value of one. And, given that one
of the groups was characterized by a number of scores that were below one, the transformation,
although successful generating a normal distribution, was still problematic when trying to
compare the mathematical reasoning of the two groups. Because of these two complications, the
DEVELOPING MATHEMATICAL REASONING 123

Figure 5. Histograms and Q-Q plots for TIMSS items total scores

final analysis was run using the original TIMSS scores, even if the data was moderately skewed,
rather than with the scores generated by the reflect square root transformation. This decision was
made based on the T-test’s utility in working with non-normal distributions, rather than outliers,
and to the fact that the new transformation scores were hindering rather than assisting with the
interpretation of the data.
DEVELOPING MATHEMATICAL REASONING 124
In the end, this analysis employed 64 DI and 27 CGI+EDI participants. An independent-
samples t-test was run to determine whether there were differences in the mathematical
reasoning of the two groups. The data showed no outliers, as assessed by inspection of the two
groups’ box plots. Mathematical reasoning scores were normally distributed for the EDI
participants (N = 49), but moderately, negatively skewed for the CGI+EDI participants (N = 27),
as assessed by Shapiro-Wilk's test (p > .05). The scores also displayed homogeneity of variances,
as assessed by Levene's test for equality of variances (p = .69). Results showed mathematical
reasoning to be stronger in the CGI+EDI group (M = 4.05, SD = 1.49) than in the EDI group (M
= 2.74, SD = 1.32), allowing for the rejection of the null hypothesis. Accordingly, the difference
between the two groups turned out to be statistically significant (see table 6), M = 1.32, 95% CI
[0.66, 1.98], t (74) = 3.98, p = .0002. In addition, a Cohen’s effect size value (d= .93) confirmed
the high practical significance of the findings.
Table 6
Independent Samples T-test Using Only the TIMSS Items (Assessment of Subjects’ Mathematical
Reasoning)

Curriculum

M (SD)

M difference

p
95% Confidence
Interval
Lower Upper

TIMMS
scores

CGI+EDI

4.05 (1.49)

1.32

.0002

.66

1.98

EDI

2.74 (1.32)

-1.32

.0002

-1.98

-.66

Research Question #3
The third research question investigated the differences in mathematical perceptions
between students instructed by a combination of Cognitively Guided Instruction and Explicit
DEVELOPING MATHEMATICAL REASONING 125
Direct Instruction (CGI+EDI) (N= 28) and students who were instructed using only Explicit
Direct Instruction (EDI) (N= 50). Consequently, research question #3 maintained a non-
directional hypothesis as well. It stipulated that a difference did exist in the mathematical
perceptions of students who were instructed using student-centered pedagogies (CGI) versus
students who have been instructed using teacher-centered ones (EDI). Therefore, similarly to
the two previous questions, this question was also examined using an independent-samples T-test.
Each assumption required in this type of analysis is reviewed below.
The dependent variable used in this analysis was the score achieved when all 8 Likert-type
questions in the survey were added together. Each question had a maximum possible score of 4;
therefore, the maximum possible score for the entire instrument was 32. This factor affirmed
one of the assumptions required by a T-test – having one dependent variable that is measured at
the continuous level. The second assumption was also met by the design of the investigation.
That is, the assumption that a T-test will compare two independent groups was affirmed by the
two groups that were part of the analysis, EDI and CGI+EDI. The third assumption that all T-
tests require is the need to have independence of observations, which, as previously seen, is the
need for each sample-group to be unrelated to one another. As previously seen, although the
affirmation of this assumption could not be entirely confirmed, it is highly probable that the
assumption was not violated. Regarding the presence of outliers, the data had no evidence of any
scores that were far apart from that of the mean.
The fifth assumptions required of a T-test analysis did not fare as well as the first four. In
other words, as appraised visual by a histogram and numerically by the Shapiro-Wilks test (p
> .05), the EDI group was found to be not normally distributed (it was moderately, negatively
skewed). To attain normality, a reflect square root transformation was performed on the original
DEVELOPING MATHEMATICAL REASONING 126
data. The transformation worked accordingly (Figure 6) and the new data that was outputted
displayed the necessary normality required to perform a T-test. Furthermore, the new
transformed data also met the last assumption required of the T-test: homogeneity of variance.
This assumption was assessed and confirmed using Levene's test for equality of variances (p
= .64).
Neither difference, that of the original scores nor that of the transformed scores, turned
out to be statistically significant. In other words, the difference in the mean scores attained by
the two groups was not large enough to state that the students’ mathematical perceptions had
been altered by the type of curriculum to which they had been exposed. For instance, the
original data showed that the EDI group (M = 23.94, SD = 4.70) had a slightly higher
mathematical perception score than CGI+EDI group (M =22.68, SD =4.56). However, the
difference between the scores was not large enough to be significant, M = 1.26, 95% CI [-0.93,
3.45], t(76) = 1.15, p = .25 (see Table 7). Furthermore, a Cohen’s effect size value (d= .27)
suggested a low practical significance. In addition, although the transformed data displayed a
reversal in the order of the means in that the CGI+EDI group now displayed the highest mean
score, a condition that known to occur during square-root transformations, the results of the
transformation still paralleled that of the original data. That is to say, even though the
transformed data now portrayed the CGI+EDI group as having a slightly higher mean (M = 2.94,
SD = .79) than that of the EDI group (M = 2.72, SD = .84), the difference between the groups
was still not statistically significant, M = .22, 95% CI [-.17, .61]. t(76) = 1.12, p =.27 (see Table
8).

DEVELOPING MATHEMATICAL REASONING 127

Figure 6. Histograms, Q-Q plots, and Box plots displaying the reflect square-root transformation
scores of the survey designed to assess students’ mathematical perceptions.

DEVELOPING MATHEMATICAL REASONING 128

Table 7
Independent Samples T-test Using Data from the Survey’s Total Score (Assessment of Subjects’
Mathematical Perception)

Curriculum

M (SD)

M difference

Sig. (2-tailed)
95% Confidence
Interval
Lower Upper

Survey
Scores

EDI

23.94 (4.70)

1.26

.25

-.93

3.45

CGI+EDI

22.68 (4.56)

-1.26

.25

-3.45

.93

Table 8
Independent Samples T-test using the Survey’s Reflect Square-Root Transformation Data
(Assessment of Subjects’ Mathematical Perception)

Curriculum

Mean (SD)

Mean difference

Sig. (2-tailed)
95% Confidence
Interval
Lower Upper

Survey
Scores

CGI+DI

2.94 (.79)

.22

.27

-.17

.61

DI

2.72 (.84)

-.22

.27

-.61

.17

Research Question #4

Research question #4 investigated whether a difference existed in the mathematical
reasoning of the subjects when mathematical perception and type of curriculum were combined
to create a linear composite. Consequently, this question maintained a non-directional
hypothesis. It stipulated that the curriculum experienced by the students in conjunction with the
mathematical perceptions generated by their academic experiences would generate a difference
DEVELOPING MATHEMATICAL REASONING 129
in their ability to reason mathematically. This question was investigated using a two-way
ANOVA in which the total score from the eight Likert-type questions and the type of curriculum
employed by each school were used as the independent variables, and the total score in the
TIMSS items was used as the dependent variable. In order to use the results of the survey, each
student’s total score in the survey was turned from a numerical continuous variable into an
ordinal one. That is, each student was given a mathematical perception designation of low,
medium, high, or very high based on the aggregate score s/he attained in the nine-question
survey. The results of this process are presented in Table 9.
Like other statistical tests, a two-way ANOVA must meet a number of assumptions
before it can provide valid results. Subsequently, each assumption, as well as how those
questions fared with regards to this particular analysis, is discussed below.
Table 9
Descriptive Statistics for the Linear Composite Curriculum/Perception Using TIMSS Total Score
as the Dependent Variable

Curriculum

Designation

n

M

SD

EDI
Low 2 2.00 1.41
medium 21 2.92 1.28
high 19 2.41 1.46
very high 6 3.44 .84
total 48 2.74 1.33

EDI+CGI
Low 1 4.00 -
medium 18 4.04 1.42
high 5 3.86 1.87
very high 3 4.44 2.14
total 27 4.05 1.49

Total
Low 3 2.67 1.53
medium 39 3.44 1.44
high 24 2.71 1.62
very high 9 3.77 1.36
total 75 3.21 1.52
DEVELOPING MATHEMATICAL REASONING 130

One of the assumptions required by a two-way ANOVA is the absence of outliers. The
majority of the groups displayed no outliers (Figure 7), as assessed by inspection of a box plot
for values greater than 1.5 box-lengths from the edge of the box. Still, there was one outlier
(subject 53) within the EDI group who was characterized by a very high perception of
mathematics.
Although violating the assumption, the outlier was not extracted from the sample group.
In addition, no transformation was applied to the data in order to ameliorate the discrepancy
between the outlier and the rest of the scores. This choice was made based on the uniqueness of
the outlier, which would have made it difficult for it to affect the outcome of the ANOVA.
The second assumption required by a two-way ANOVA is the normality of each group’s
distribution. That is, the distribution of the TIMSS scores had to maintain normality with
regards to every group that can be generated between the linear composite and the independent
variable. Examples of these groups are DI and low mathematical perception, DI and medium
mathematical perception, and DI+CGI and low mathematical perception. Results showed that
TIMSS scores were normally distributed for all group combinations of curriculum and
mathematical perception, as assessed by Shapiro-Wilk's tests (p > .05) (see Table 10).
Nonetheless, it is important to note that two groups were composed of only a limited number of
subjects: EDI and low mathematical perception, and CGI+EDI and low mathematical perception.
This fact eliminates the chances of displaying a Shapiro-Wilk score. Furthermore, two subjects
were identified as having missing scores, which eliminated their chances of displaying a Shapiro-
Wilk score as well. Lastly, the data displayed homogeneity of variances, as assessed by Levene's
DEVELOPING MATHEMATICAL REASONING 131
Test of Homogeneity of Variance (p = .29), thus meeting the last assumption required to carry
out a two-way ANOVA.

Figure 7. Box plots of the linear composite perception and curriculum used as the independent
variable of the dependent variable TIMSS total scores.

DEVELOPING MATHEMATICAL REASONING 132
Table 10
Normality Test Results Using the Linear Composite Curriculum/Perception to Analyze the
Subjects’ Ability to Reason Mathematically.

Curriculum type

Perception
Designation
Shapiro-Wilk

Statistic

df

Sig.

CGI+EDI
Low - - -
Medium .92 18 .15
High .97 5 .85
Very High .87 3 .30

EDI
Low - - -
Medium .99 21 .99
High .94 19 .28
Very High .81 6 .07

The results of the analysis showed no statistically significant difference in the
mathematical reasoning of the groups. That is, the linear composite created by combining each
school’s chosen curriculum and the mathematical perception of each students did not attain an
appropriate level of significance to affirm that a difference existed between the two groups, F(3,
82) = .14, p = .94, partial η
2
= .01 (see Table 11). Furthermore, a partial eta-square effect size
value (η
2
= .01) confirmed a low level of practical significance for the findings.
Table 11
Two-way ANOVA results for Linear Composite Curriculum/Perception
Source MS F p Partial η
2

Curriculum 13.23 6.23 .01 .09

Survey Designation

1.35

.67

.57

.03

Curriculum*Survey
Designation

.27

.13

.94

.01

DEVELOPING MATHEMATICAL REASONING 133
However, as previously seen in the individual-samples T-test, the between-subjects
effects did display a statistically significant difference between the type of curriculum used by
each school and the students’ ability to reason mathematically, albeit a small effect size, F(1, 82)
= 5.54, p = .02, partial η
2
= .09. In contrast, no statistically significant difference was observed
between the students’ mathematical perception and their mathematical reasoning, F(3, 82) = .67,
p = .57, partial η
2
= .03. Consequently, since it appeared as if curriculum had more of an
influence on reasoning than perception did, the principal investigator decided to take out the
interaction term. When the analysis was conducted in this way, the effect sizes did experience a
slight increase. Yet, the increase was still not large enough to make the variable of mathematical
perception a statistically significant factor with regards to mathematical reasoning.
The inconsistency of scores exhibited by the results of the two-way ANOVA – based on
the linear composite of curriculum employed by each school and the perception scores from the
survey – can be observed in the succeeding bar graph (Figure 8). It presents the randomness in

Figure 8. Clustered bars of the two-way ANOVA results categorized by both the curriculum
experienced and perception designation given to the students.
DEVELOPING MATHEMATICAL REASONING 134

the perception of scores that both the EDI and CGI+EDI groups displayed. This visual
representation can help one understand the lack of relation that was displayed by the linear
composite and the students’ ability to reason mathematically. To be specific, the results
demonstrate that the mathematical reasoning ability of the CGI+EDI group was not influenced at
all by their mathematical perceptions. The EDI group did demonstrate a slight interaction
between mathematical perception and mathematical reasoning. That is, these students’ reasoning
scores tended to improve when their perception of mathematics increased. Moreover, it is also
interesting to note that the students who held a very high perception of mathematics on average
had a higher mathematical reasoning score than their counterparts, albeit at times only slightly
higher for the CGI+EDI group (Figure 9).

Figure 9. Graphs depicting the mean score attained in mathematical reasoning based on students’
mathematical perceptions and curriculum experienced.

DEVELOPING MATHEMATICAL REASONING 135
Lastly, it is also interesting to note that the patterned attained by the linear composite
generated by combining curriculum experienced and mathematical perception was exactly the
same for both groups (Figure 9). In other words, the lowest scores in mathematical reasoning
were attained by students who maintained a low perception of math, followed by those
maintaining a high perception of math, proceeded by those with a medium perception, which
were outscored by those who held a very high perception. In addition, it is also interesting to
notice that the medium perception group outscored the high perception group in both groups – a
fact that future studies should examine in more detail.

Summary
To summarize, this chapter presented the results of four statistical tests, three
independent-samples T-tests and one two-way ANOVA. The first T-test investigated whether
differences existed in the mathematical performance of students who were instructed via a
combination of Cognitively Guided Instruction and Explicit Direct Instruction (CGI+EDI) and
those who were taught solely through Explicit Direct Instruction (EDI). The results showed a
statistically significant difference between the two groups, with students who had been taught
using the combination CGI+EDI scoring higher than their counterparts by a mean difference of
1.79 points. The second T-test investigated whether there were any differences in mathematical
reasoning between the two groups. Paralleling the results of the first T-test, this analysis showed
a statistically significant difference between groups, with the CGI+EDI group scoring higher
than the EDI group by a mean difference of 1.32 points. The third research question investigated
whether there were any differences in the mathematical perception of the two groups. Results of
the independent-samples T-test showed no statistically significant difference between the groups.
The last research question investigated how the interplay between curriculum and mathematical
DEVELOPING MATHEMATICAL REASONING 136
perception could generate a difference in the mathematical reasoning of the students. Similar to
the results of the previous question, no statistical difference was observed between the groups.
Nonetheless, interesting patterns did arise between the groups’ ability to reason and their
perceptions of mathematics. These patterns suggest the need for further investigation.

DEVELOPING MATHEMATICAL REASONING 137
CHAPTER 5: DISCUSSION
Introduction
The focus of this study was to compare two different types of pedagogies: Explicit Direct
Instruction (EDI) and the amalgamation of Cognitively Guided Instruction and Explicit Direct
Instruction (CGI+EDI). This comparison was designed to examine which pedagogy offered the
greatest opportunity to increase the mathematical reasoning abilities of elementary school
students, as these abilities comprise a combination of content domain and cognitive sub-skills in
which American pupils have shown signs of weakness. Towards this end, the study compared
the examination scores of students who attended two different school sites, School A and School
B. The schools were purposefully chosen based on the methods through which faculty instructed
their students. For instance, students in School A had been exposed to an amalgamation of CGI
and EDI (n = 27), whereas, students in School B had only been instructed using EDI (n = 47).
Furthermore, the study also examined whether a connection existed between the mathematical
perceptions held by the students and the type of instruction they had previously received.
Moreover, the study also investigated whether particular combinations between mathematical
perception and type of curriculum would generate differences in the students’ mathematical
reasoning.
Two different measurement tools were used to accomplish these investigations, a 12-
question test and a 9 question survey. The test was constructed using previously released items
from the 2011 TIMSS and CST examinations. The survey was modeled after two surveys that
had been employed to assess the mathematical perception of students during the 2011 TIMSS
investigation. Three individual-samples T-test and a one-way ANOVA were used to compare
the means of the two groups. The main findings, a number of limitations that were discovered in
DEVELOPING MATHEMATICAL REASONING 138
the process of the study, as well as future implications to research and practice are described in
the rest of this section.
Research Questions
1. What are the mathematical performance differences that are exhibited by students who
were instructed using an amalgamation of student and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
2. What are the mathematical reasoning differences that are exhibited by students who
were instructed using an amalgamation of student and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
3. What are the mathematical perception differences that are held by students who were
instructed using an amalgamation of student and teacher-centered pedagogies
(Cognitively Guided Instruction and Explicit Direct Instruction) rather than only teacher-
centered pedagogies (Explicit Direct Instruction)?
4. What are the mathematical reasoning differences that are exhibited by students when
curriculum type and mathematical perception are combined in order to generate a linear
composite?
Summary of the Findings
The results of the study substantiated two out of the four hypotheses that were proposed
prior to the investigation. For example, it was hypothesized that a difference would exist between
the mathematical performance of students who had been taught using a combination of student
and teacher-centered pedagogies (CGI + EDI) and that of students who had simply been taught
DEVELOPING MATHEMATICAL REASONING 139
using teacher-centered pedagogies (EDI). Indeed, results showed that a difference did exist in
the mathematical performance of the two groups. That is, on average, CGI+EDI students scored
higher than EDI students in their 12-question test.
The second hypothesis was also substantiated by the results of the study. It stated that
difference exist in the mathematical reasoning abilities (higher order thinking skills) of the two
aforementioned groups. Results showed that, on average, students who had been instructed using
a combination of CGI+EDI attained higher mathematical reasoning scores than their EDI
counterparts.
Conversely, the two remaining hypotheses, which were designed to assess the subjects’
perceptions of math, were not affirmed by the data. That is, results of the independent samples T-
test show no difference in the scores attained by the two groups with regards to mathematical
perception. Similarly, no difference was observed between the groups with regards to the linear
composite that was generated in order to examine the amalgamation of mathematical perceptions
and curriculum type with regards to mathematical reasoning.
Key Findings
Four different questions were proposed at the outset of this investigation: (1) What are
the differences in mathematical performance exhibited by students instructed via CGI+EDI as
opposed to EDI? (2) What are the differences in performance exhibited by students instructed via
CGI+EDI as opposed to EDI when assessing their mathematical reasoning skills? (3) What are
the differences in mathematical perception held by students instructed via CGI+EDI as opposed
to EDI? (4) What are the mathematical reasoning differences that are exhibited by students when
curriculum type and mathematical perception are combined in order to generate a linear
composite? These questions frame the remainder of this section. A discussion of each question
DEVELOPING MATHEMATICAL REASONING 140
is carried out by juxtaposing existing literature with the results of the study. To conclude, this
section discusses the study’s limitations as well as the research and practical implications that
arise from its results.
Research Question # 1
As previously seen in the literature review, historically, mathematicians have used a
variety of techniques to instruct their students on the wonders of their field (Switz, 1995). More
important to the purpose of this work is the fact that ancient records exhibit aspects of both
student and teacher-centered pedagogies. Consequently, the history of mathematics points to the
supremacy of a dynamic rather than a particular approach to mathematic instruction.
Nevertheless, with the rise of various psychological theories (behaviorism, cognitivism, and
constructivism) during the 19
th
and 20
th
century, schools began to experience the dichotomy of
competing pedagogical approaches. Amongst such competing approaches one finds Direct
Instruction (DI), Cognitive Load Theory (CLT), Explicit Direct Instruction (EDI), and
Cognitively Guided Instruction (CGI), which are the four pedagogies which have been touched
upon the most by this work.
Each one of these pedagogies is supported by empirical studies. For instance, supporters
of Direct Instruction point to the outcomes of Project Follow Through as clear evidence that
their approach can be one of the most powerful ways of educating students, especially those who
have traditionally been known to underperform (Adams & Engelmann, 1996). Yet, this
approach is criticized by constructivist theorists because of its inability to develop students’
higher order thinking skills (Ertmer & Newby, 1993). To amend this, constructivists offered the
student-centered models of problem and/or inquiry based learning. However, critics of those
approaches maintain that such unguided learning situations create too much cognitive load,
DEVELOPING MATHEMATICAL REASONING 141
which stifles knowledge acquisition (Kirschner, Sweller, & Clark, 2006). Consequently, they
argue for a more controlled environment in which a person’s cognitive architecture becomes the
central focus of the lesson – even if/when a teacher instructs complex skills meant to be
employed within “applied domains” (Vogel-Walcutt, et al., 2011, p. 133).
Not all cognitivists agree on their model’s superiority. Almost ten years before, a study
conducted by Kalyuga et al. (2001) showed that the students’ prior knowledge of a subject was
an important factor to consider when choosing which pedagogical approach would be optimal
(Tuovinen & Sweller, 1999). Since then, other articles have also argued against Kirschner,
Sweller, and Clark’s position (Hmelo-Silver, Duncan, Chinn, 2007). Given all these
perspectives, one of the factors this study set out to investigate was which of the approaches,
used herein, would spawn the greatest overall mathematical performance of elementary students.
The study’s outcomes were consistent with the views of those who support student-
centered pedagogical models. Such individuals claim that allowing students to have authentic
experiences can amount to deeper and greater understanding of the subject. For instance, CGI
supporters assert that generating a milieu in which students are allowed to be inquisitive and
creative about a problem will not only increase their computational skills, but also increase their
conceptual and metacognitive skills as well (Carpenter, Fennema, Franke, Levi, & Empson,
1999; Cunningham, 1991; Staub & Stern, 2002). The results of this study show that such claims
can be substantiated. To be exact, the comparison between the groups displayed a mean
difference score of 1.54, which meant that, on average, CGI students’ scored 12.8% higher than
their EDI counterparts with regards to their overall mathematical ability.
Yet, these results should be taken temperately. This is due to the fact that statistical
significance was not obtained amongst all 12 items of the test. That is, when the questions were
DEVELOPING MATHEMATICAL REASONING 142
categorized using each of the examinations from which they had been taken (CST and TIMSS
questions), the only statistical significance observed between the two groups was amongst the
TIMSS items rather than the CST items. This alludes to the notion that teacher-centered
instructional approaches, such as EDI, can be as successful as student-centered pedagogies
(CGI+EDI) at developing factual and procedural knowledge. This should come as no surprise
given the fact that the two knowledge types most used in the CSTs tend to be factual and
procedural knowledge (Porter, Mcmaken, Hwang, & Yang, 2011), which tend to be
characterized by less demanding cognitive sub-skills ( remember, understand, and apply rather
than analyze, evaluate, and create). In addition, the absence of statistical significance between
the two groups should not be discarded as inconsequential. Rather, it should be seen as evidence
that different types of pedagogies can, and do, as many have previously argued (Ertmer &
Newby, 1993), develop different types of knowledge.
The results of the study, therefore, point to the need for schools to become more balanced
in their pedagogical approaches. That is to say, rather than employing purely constructivist or
behavioral educational techniques, schools should infuse these pedagogical perspectives in order
to offer their students all the necessary cognitive skills they will require to participate in the
knowledge and creative economies of the 21
st
century. In short - and as so many have argued in
the past - the study’s results seem to support the idea that the amalgamation, rather than the
isolation, of constructivist, behavioral, and cognitivist theories is the most efficient way to
increase the overall mathematical ability of students.
To accomplish this, district, school administrators, and teachers will have to begin
tracking a different kind of academic data. Educators, for instance, will have to gather
information about the types of pedagogies they use. This type of data can be gathered using
DEVELOPING MATHEMATICAL REASONING 143
Anderson et al.’s (2001) taxonomy. That is, by asking teachers to fill out the taxonomy during a
month long series of lessons, teachers and administrators could gather data regarding the types of
cognitive sub-skills and knowledge types teachers tend to employ. This, in turn, could be used to
create professional development that is more individualized based on the needs of each teacher,
as well as to identify gaps in the students’ cognitive development.
In addition to that pedagogical information, teachers and administrators will also have to
try to track and identify other more intangible data. Educators will need to know about their
personal educational philosophies and how such philosophies can, consciously and/or
subconsciously, affect their students’ academic performance (see Staub and Stern, 2002). Hence,
districts should carryout surveys to gather data about the way their teachers perceive of learning
and instruction and how those perceptions translate into particular pedagogies. As a result of
such information, districts will be able to adequately balance each teacher’s distinct pedagogical
approach (with its appropriate counterpoise) by offering more pin-pointed professional
development.
Research Question # 2
The second research question examined the mathematical reasoning ability of the
subjects. Specifically, the question asked if there was a difference in the mathematical reasoning
of students who had been instructed using a combination of CGI and EDI as opposed to those
students who had experienced EDI.
Paralleling the outcome observed by the first research question, the analysis of the second
question also showed a statistically significant difference between the two groups. That is, the
mathematical reasoning abilities of the groups were, indeed, caused factors other than random
chance. The data showed that students who belong to the CGI+EDI group scored, on average,
DEVELOPING MATHEMATICAL REASONING 144
significantly higher than the EDI group by a difference of 1.32 points. That is, on average, the
students who were taught using the amalgamation of CGI and EDI scored 22% higher on their
reasoning questions than their EDI counterparts. This substantial difference between the groups
helps corroborate the constructivist arguments that student-centered instruction leads to a deeper,
more innovative, and more creative understanding of the subject at hand (Staub and Stern, 2002).
This paper established that the CGI approach ought to not be identified as either a “pure”
inquiry or discovery way of learning. Rather, CGI lessons are highly guided, as the name
implies, although not in the manner in which supporters of direct instruction and cognitive load
theory would want. Consequently, although CGI tends to be more constructivist in nature, the
approach should not be placed within the same pure discovery learning category that Kirschner,
Sweller, and Clark (2006) argued against. Instead, it should be identified within the same
category as the methods discussed by Hmelo-Silver et al. (2007).
As a result of the need to place CGI within such a special categorization, a hybrid
between discovery learning and guided instruction, the outcomes attained by this study should be
viewed with caution. That is, the results of this study cannot be generalized to all constructivist
pedagogies, especially since, as Gijbels et al. (2006) and Loyens (2007) pointed out, the
literature seems to be missing a clear and concise definition of the term.
A second way to reach beyond the results of this study is by highlighting only the
constructivist side of CGI. To avoid this, it is important to note that students instructed via CGI
tend to experience both student and teacher-centered pedagogical techniques. In other words,
although CGI students have the liberty to solve problems in whatever manner they see fit, the
last part of any CGI lesson is characterized by a time of sharing in which the answers of a few of
the children are presented to the rest of the class. It is during this time when the teacher can
DEVELOPING MATHEMATICAL REASONING 145
direct the conversation and/or learning of the presenters in a direction to maximize learning for
the rest of the class. In other words, although the ones sharing are the students, and what is being
shared is all the different ways in which they arrived at the solution, the teacher still focuses on
certain aspects of the information shared in order to highlight whatever factual, procedural, or
conceptual knowledge the other students might lack. Consequently, the results attained by the
CGI group during the study do not support the use of a distinct and solitary pedagogy, but, rather,
the use of a more dynamic approach of learning characterized by both student- and teacher-
centered pedagogies instead of the dichotomic approach that schools have employed for many
years. Hence, rather than understanding the results of the study as saying that constructivist
approaches are, in general, better for developing mathematical reasoning, it would be more
prudent to view the results as supportive of a more dynamic view of learning. This is a view
proposed by Anderson et al. (2001) and one in which the instruction of both higher and lower
order thinking skills becomes an integral part of the curriculum.
In short, although the data does present evidence as to the value-added that guided-
constructivist pedagogies can offer, the study maintains that merely using such pedagogies could
generate major knowledge deficiencies. Accordingly, rather than arguing for the superiority of
constructivism, this work proved the necessity to institute a hybrid model of instruction
characterized by a cornucopia of pedagogies where teachers choose specific methodologies
based on specific learning objectives. In other words, in order to afford students the best chance
to develop the entire gamut of cognitive sub-skills, remembering, understanding, applying,
analyzing, evaluating, and creating and knowledge types, factual, procedural, conceptual, and
metacognitive, educators will have to use a more dynamic model of instruction.

DEVELOPING MATHEMATICAL REASONING 146
Research Question #3
The third research question investigated whether there were any differences in the
mathematical perception of the two groups. The Results of the independent-samples T-test
showed no statistically significant difference between the groups, hence the null hypothesis
could not be rejected. This was an unanticipated outcome given the vast amount of research
supporting the motivational effectiveness of student-centered pedagogical approaches, such as
the work done with mastery orientation of goal theory and the notion of having a well-developed
individual interest within the four phase model of motivation (Schunk, Pintrich, & Meece, 2008).
Adding to this perplexity is the fact that results of the EDI group turned out to be negatively
skewed, meaning that these students had a tendency to view math in a more positive way, which
seems to counteract the claims of a number of motivational theories.
To make sense of such results, it is important to note that teacher-centered pedagogies
have also been known to exert positive effects upon a student’s motivation and academic
performance. For example, both the performance orientation within the goal theory of motivation
and the extrinsic values embedded within the first two stages of the four-phase model of
motivation have been known to assist students in their academic tasks. Consequently, although a
bit surprising, it is not entirely startling that the investigation found no statistical difference in the
mathematical perception of the two groups. Still, given the vast pedagogical differences that
exist between the two approaches, the fact that the data showed no difference between the groups’
mathematical perceptions seems unusual.
A factor that could explain the lack of a statistically significant difference between the
two groups could be the pedagogical amalgamation that characterized the CGI+EDI group. That
is, these students were exposed to both student and teacher-centered instructional approaches;
DEVELOPING MATHEMATICAL REASONING 147
consequently, their mathematical perceptions might have been heavily influenced by the EDI
part of their experience. As a result, their perceptions of math could have become more
analogous to the perception maintained by the students who only experienced the EDI approach.
Perhaps if the group representing the constructivist view of learning would have been
characterized by individuals who had only been exposed to student-centered pedagogies like
discovery learning, problem-based learning, then the results might have been more drastic
between the groups – an idea that should be investigated by future research.
Research Question #4
The last research question investigated how the interplay between curriculum and
mathematical perception can generate a difference in the mathematical reasoning of the subjects.
This question was investigated using a two-way ANOVA in which the total score from the
survey was used in conjunction with the type of curriculum experienced by the students as the
two independent variables and the total score of the six TIMSS items of the 12-question test was
used as the dependent variable. No statistical significance was observed in this analysis, thus
failing to reject the null hypothesis.
Regardless of this lack of significance, it was interesting to notice that the highest
mathematical reasoning scores, both in the EDI and the CGI+EDI group, were exhibited by those
students who maintain high perceptions of math. When these results are coupled with the fact
that the distribution of mathematical perception for both groups were identical, patterns do begin
to emerge that might require further investigation. For instance, the group who was designated
as having a low perception of math, on average, attained the worst mathematical reasoning
scores. This group was followed those students who held a high perception of math, which were,
in turn, outscored by students who maintained a medium perception of the domain. The pattern
DEVELOPING MATHEMATICAL REASONING 148
displayed by the two groups alludes to two interesting questions on which to base future
research: (1) is this a pattern that is common to the construct of mathematical reasoning? And if
it is, (2) why do students with a medium perception of mathematics attain higher mathematical
reasoning scores than their peers who maintain a high perception of the domain, Although
statistical significance was not obtained, such questions call for further investigation using the
variables employed in this question.
Limitations of the Study
A number of limitations were identified prior and during the study. The first limitation
has to do with generalization of the study. As discussed in the first chapter, based on the
specificity of this investigation, the results attained should not be generalized to the entire
population of elementary school students without further investigation. That is, based on the
purposeful sampling and specific pedagogical approaches used to carry out the study, one cannot
say that all student-centered approaches have the ability to enhance the mathematical reasoning
of elementary students. In fact, due to the number of confounding variables involved in a causal
comparative study as well as the number of confounding variables this study failed to control, it
is difficult to affirm that CGI should be considered the sole differential factor that increased the
mathematical reasoning of the students. However, the results were strong enough to affirm that
further work is necessary using this study’s parameters.
Another limitation of this study was the difference in the academic capacity of the two
groups involved. To control for this confounding variable, the study only employed students
who were Hispanic/Latino at schools that were geographically close to one another and had
attained the same math score in their CST scores (824). Still, even after making the two groups
as homogenous as possible, the percentage of Hispanic/Latino students who scored proficient or
DEVELOPING MATHEMATICAL REASONING 149
advanced in School A (the CGI+EDI group) were a few percentile points higher than those in
School B. Consequently, future studies should control for this discrepancies by making the
students’ CST scores, or whatever other standardized score may be available, into a covariate
variable. Hence, the study would not be analyzed using T-tests or ANOVAs, but ANCOVAS in
order to control for such between-group differences. By doing so, future studies could better
account for initial discrepancies between the two groups being compared and, as a result, attain
more accurate data about the differences that exist between the groups.
A third limitation of the study was due to the quantity and quality of the CGI instruction.
This limitation could have been avoided by carrying out classroom observations. However, due
to the extensive resources required to accomplish such observations, the study was conducted
under the assumption that the teachers were faithful to the professional development training
they received to become the district’s CGI model school. Consequently, assumed second
assumption was that the amount of time, as well as the quality of instruction conducted by the
teachers, was in alignment to the requisites shown by CGI trainers. Most teachers, as well as the
school’s math coordinator, asserted that CGI was faithfully carried out two times a week. Again,
no observation protocol or survey was implemented to affirm whether the quality or quantity of
the instruction was in accordance to such statements. Accordingly, any future investigation
should include, if resources permit, observations of the instructional approaches carried by the
teachers of each school site. Furthermore, a survey could be designed to gather preliminary data
about each teacher’s ability and faithfulness to the pedagogical approach they represent in the
study. By doing so, future investigations would be able to discuss the differences between the
two groups in a more specific and comprehensive manner.
DEVELOPING MATHEMATICAL REASONING 150
The fourth limitation of this study deals with the measuring tools involved to collect the
data. It is important to note that, although pilot studies were employed to validate the two
instruments used in this investigation, the pilot studies only included a small number of
participants and, in the case of the survey, the participants did not display the same exact
characteristics as the subjects of the study. Consequently, even though the reliability of both
measuring tools employed in the study turned out to be adequate enough, both tools could have
benefited it from having more participants during their pilot studies.
The last limitation springs from the number of groups compared in the study. That is,
this particular study only compared two groups: EDI and CGI+EDI. As a result, it was only able
to gather data pertaining to either a pure teacher-centered pedagogical approach (EDI) or for a
hybrid approach (CGI+EDI) that infused aspects of both teacher and student-centered
methodologies. However, the inclusion of a third possible category such as the pure
constructivist approach of discovery learning would have enriched the outcomes of the study.
To be specific, the study would have benefited from including an approach that could accurately
oppose the direct instruction perspective of EDI. It may be that the best pedagogical approach
for building the mathematical reasoning skills of elementary school students is neither EDI nor
an amalgamation of EDI and CGI, but an even more constructivist approach of learning; that is a
notion this particular study was not properly designed to ascertain.
Recommendations for Research and Practice
Historically, mathematics curricula have been characterized by both student- and teacher-
centered instructional methodologies. The 20
th
century brought the burgeoning of number of
diverse instructional theories that sprung from different perspectives of human learning. As a
result, a research and practice pendulum characterized by discourses grounded on dissimilarities
DEVELOPING MATHEMATICAL REASONING 151
rather than conformities began to swing. Consequently, specific learning theories are sometimes
generalized far beyond what their pedagogical specifications imply, forcing practitioners to
implement lessons using instructional techniques not suited to the development of certain
cognitive sub-skills and/or knowledge types. In turn, this generated lower capabilities in certain
academic areas, such as mathematical reasoning).
Regardless of the dichotomic approach that has generally characterized both the theory
and practice of American schools, some scholars argue for a much more comprehensive and
inclusive view of learning. One of these scholars is Jerome Bruner. Although a strong supporter
of discovery learning, a student-centered pedagogical approach, Bruner (1979) has also been a
strong proponent of acquiring factual knowledge, which is a knowledge type that can be attained
most efficiently via teacher-centered methodologies. In fact, Bruner stated that “it is rarely, on
the frontier of knowledge or elsewhere, that new facts are ‘discovered’ in the sense of being
encountered, as Newton suggested, in the form of islands of truth in an uncharted sea of
ignorance…Discovery, like surprise, favors the well-prepared mind” (p. 82). Bruner’s words
indicate that discovery learning, or the higher order thinking skills demanded by it, does not
come easily in the absence of factual and procedural knowledge. Conversely, knowing facts and
processes does not guarantee that a student will move forward into discovering or creating new
ways of manipulating, using, or interpreting the information. A middle approach, therefore, must
be seen as the key to a comprehensive well-rounded education.
However, while affirming the need for a more balance approach, one should not forget
the reason study was required in the first place: the need for American schools to increase their
instruction of higher order thinking skills such as mathematical reasoning. Consequently any
attempt to generate a more balanced pedagogical approach between higher and lower order
DEVELOPING MATHEMATICAL REASONING 152
thinking skills should begin with a strong focus upon Galloway and Lasley II’s (2010) words,
“educators [need] to know more about how children learn, how the mind works when learning …,
and how adults can facilitate [that] learning” (p.273). One way of doing so is to design and carry
out research studies that evaluate the capacity that different pedagogical approaches offer with
regards to the development of specific knowledge types, cognitive sub-skills, and content
domains. This, of course, is the dynamic model of instruction espoused in Anderson et al.’s
(2001) work.
The time has now come to leave no child’s cognitive sub-skill behind. Evidence of this
shift can be observed in the emphasis that the Common Core placed on developing students’
higher order thinking skills, such as the analysis of information, the demonstration understanding,
and the solution of non-routine word problems, rather on the memorization of facts and
procedures (Porter et al., 2011). Still, it needs to be reiterated that teachers should not abandon
their current direct instructional methodologies for another swing of the pendulum in the
direction of more student-centered approaches. Instead, they should adopt a more balanced
approach to instruction that parallels the way literacy skills have been taught once scholars
realized that neither an isolated phonics nor a whole language approach was best for students.
That is, rather than teaching students reading using only phonics or whole language, many
schools opt for a more balanced literacy approach in which the amalgamation of the two is seen
as best.
Based on the necessity to increase higher order thinking skills while still maintaining the
strong foundation gained from factual and procedural understanding, the present work argued for
and implemented a more complex model of investigation framed in the work of Bloom and
Krathwohl (1956) and Anderson and Krathwohl (2001), who assert that a lesson’s objective,
DEVELOPING MATHEMATICAL REASONING 153
rather than policy or personal beliefs, should be the ultimate determinant for each day’s
pedagogy. Nonetheless, while affirming the need for a more balanced approach, it is important
to note the reasons behind this study: American students’ lack of higher order thinking skills.
Consequently, any attempt to generate a more balanced pedagogical approach between higher
and lower order thinking skills should focus upon the words of Galloway and Lasley II (2010):
“educators [need] to know more about how children learn, how the mind works when learning …,
and how adults can facilitate [that] learning” (p.273).
One way of doing so, of course, is by designing and carrying out research that evaluates
the capacity that different pedagogical approaches offer with regards to the development of
specific knowledge types, cognitive sub-skills, and content domains. This study aimed to present
a theoretical framework for such a dynamic and focused approach. Consequently, it argued for a
third pedagogical option to offset the dichotomous perspective maintained by student- and
teacher-centered approaches. That is, it assumes that an amalgamation of both student- and
teacher-centered pedagogical techniques can increase the academic performance of students, for
mathematics in general and mathematical reasoning in particular, beyond that of any exclusive
approach. Consequently, this work maintained that, rather than allowing educational policies or
personal epistemologies to dictate what the daily pedagogy should be, teachers must use the
lesson’s objective to guide such decisions (Anderson, et al. 2001). This, in turn, generates a
broad acceptance and affirmation of all pedagogical approaches, which allows the model to
maintain a dynamic nature that can adjust to the requirements of a lesson’s objective.
Nevertheless because it assumes an identical capacity for all pedagogies, it also must place an
equal burden of proof upon them. That is, each pedagogical approach should be tested to
determine how successful its use is at developing not only content knowledge, but also specific
DEVELOPING MATHEMATICAL REASONING 154
knowledge types and cognitive sub-skills. Accordingly, this approach identified by the author as
“Dynamically Focused Pedagogy”.
The idea of Dynamically Focused Pedagogy should be understood as research and
practice that investigates and uses the ways in which specific pedagogies can enhance the
learning of students with regards to the combination of specific content domains, knowledge
types, and/or cognitive sub-skills. This study, therefore, presented, and carried out, such an
investigation. Following Anderson et al.’s (2001) lead, it used an ex post facto approach to
investigate which pedagogical method, EDI or CGI+EDI, generated the greatest capacity for
mathematical reasoning in elementary students. The results showed that the hybrid approach of
using the CGI+EDI model was more apt at developing the specific combination of content
domains, knowledge type, and cognitive sub-skills exhibited by mathematical reasoning. Of
course, more research still needs to be carried out in order to maximize the value of this
particular model of investigation. Nevertheless, at first sight, the model does present a viable
approach by which to increase what Hiebert, Gallimore, and Stigler (2002) identified as the
knowledge base of the teaching profession.
In short, the study’s results point to two different conclusions. The first is the need for
teachers to maintain greater pedagogical balance in their mathematics instruction and, perhaps, in
other subjects as well. Simply teaching students how to follow someone else’s, even the
teacher’s, thoughts will not work to increase the mathematical reasoning or higher order thinking
skills of the learner. Consequently, all mathematics classrooms should include times in which
students are asked to work towards the solution of a problem without the assistance of an expert.
By doing so, students will have the opportunity to develop solution pathways on their own,
DEVELOPING MATHEMATICAL REASONING 155
which, in turn, can assist them in gaining a number of cognitive skills they will utilize in their
academic and personal lives, what Kapur (2010) referred to as the process of productive failure.
Second, the study showed that pedagogical research, especially those studies meant to
enhance practice rather than advance theory, should not only prove their worth with regards to
content domains, but also their ability to increase specific knowledge types and cognitive sub-
skills. That is, instead of asserting that a particular pedagogical approach is generally better for
the entire gamut of content domains, knowledge types, and/or cognitive sub-skills, applied
research models must investigate the added value that specific pedagogies can offer each of these
educational categories. Of course, this will only be accomplished if more researchers and
professional educators use the model of investigation and practice put forth herein.
In closing, it needs to be reemphasized that the approach this study found most fruitful
towards the development of mathematical reasoning and mathematical perception should not be
generalized as the best approach for all knowledge types, cognitive sub-skills, or motivational
approaches. After all, as the literature review discussed, “good teachers are not exclusively
direct instruction or inquiry oriented; [but rather] they understand the needs of students in ways
that cause them to structure instruction to the unique learning needs of the students they teach”
(Galloway & Lesley II, 2010, p. 274). Much like this quote, this study, was designed with the
assumption that the general use of a particular pedagogy will limit a student’s ability to learn.
Instead, different pedagogical theories should be employed for different educational occasions.
Consequently, researchers and practitioners should work towards determining which pedagogical
approaches are most suitable when confronted with a particular objective and/or a particular
learning situation, a process which this study undertook.
DEVELOPING MATHEMATICAL REASONING 156
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DEVELOPING MATHEMATICAL REASONING 178
Appendix A
12-QUESTION TEST
Question 1

In a soccer tournament, teams get:

3 points for a win
1 point for a tie
0 points for a loss

Zedland has 11 points.

What is the smallest number of games Zedland could have played?

Answer_______________________
------------------------------------------------------------------------------------------------------------
Question 2

The graph shows the number of students at each grade in the Pine School.

In the Pine School there is room in each grade for 30 students.
How many more students could be in the school?

A. 20

B. 25

C. 30

D. 35
-----------------------------------------------------------------------------------------------------------
Question 3

Pine School
0
5
10
15
20
25
30
35
1 2 3 4 5 6
Grade
Number of Students
DEVELOPING MATHEMATICAL REASONING 179
The scale on a map indicates that 1 centimeter on the map represents 4 kilometers on the land.
The distance between two towns on the map is 8 centimeters. How many kilometers apart are
the two towns?

A. 2

B. 8

C. 16

D. 32

------------------------------------------------------------------------------------------------------------

Question 4

Bill is arranging squares in the following way:

Figure 1 Figure 2 Figure 3

A. Draw Figure 5.

B. How many squares would Bill need to make Figure 16?

Answer:_______________________________

------------------------------------------------------------------------------------------------------------

Question 5

DEVELOPING MATHEMATICAL REASONING 180
Jonathan read 541 pages during his summer reading program. In order to reach his goal of 650
pages, how many pages does he need to read?

A. 99

B. 109

C. 119

D. 199

------------------------------------------------------------------------------------------------------------

Question 6

Three thousand tickets for a basketball game are numbered 1 to 3000. People with ticket
numbers ending with 112 receive a prize. Write down all the prize-winning numbers.

Prize winning numbers: ____________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

------------------------------------------------------------------------------------------------------------

Question 7

DEVELOPING MATHEMATICAL REASONING 181
John was given the following table by his teacher and was asked to identify the graph that
correctly displays the data. Which graph below should he choose?

Name Savings
Sara 22 zeds
Peter 15 zeds
Pamela 17 zeds
Chris 10 zeds

A. B.

C. D.

------------------------------------------------------------------------------------------------------------
Question 8

There are 9 rows of seats in a theater. Each row has the same number of seats. If there is a total
of 162 seats, how many seats are in each row?

A. 17

B. 18

C. 19

D. 20
------------------------------------------------------------------------------------------------------------

Question 9

DEVELOPING MATHEMATICAL REASONING 182
The figure below is made of three squares joined together.

3 in.

3 in.

3 in.

3 in.

What is the area of the figure in square inches?

A. 9 square inches

B. 18 square inches

C. 27 square inches

D. 81 square inches

------------------------------------------------------------------------------------------------------------

Question 10

Roy has a bag with 8 red marbles, 4 blue marbles, 5 green marbles, and 9 yellow marbles all the
same size. If he pulls out 1 marble without looking, which color is he most likely to choose?

A. red

B. blue

C. green

D. yellow

------------------------------------------------------------------------------------------------------------

DEVELOPING MATHEMATICAL REASONING 183
Question 11

Which figure can form a pyramid when folded on the dotted lines without overlapping?

A. C.

B. D.

------------------------------------------------------------------------------------------------------------

Question 12

At a local school, the fourth, fifth, and sixth graders sold flowers as a fundraiser. The bar graph
below shows how many flowers were sold by each grade.

How many flowers did the students sell in all?
A. 20
B. 35
C. 40
D. 70

DEVELOPING MATHEMATICAL REASONING 184
Appendix B
9 QUESTION SURVEY

Please circle just one option

1. I _________ enjoy learning math.

Always ----- Mostly ----- Sometimes ----- Never

2. I think the other kids in school don’t like doing math.

Always ----- Mostly ----- Sometimes ----- Never

3. My favorite part of math time is the word problems.

Always ----- Mostly ----- Sometimes ----- Never

4. Math time in my school is ____________ exciting and fun.

Always ----- Mostly ----- Sometimes ----- Never

5. I think other students in my school don’t like solving math word problems.

Always ----- Mostly ----- Sometimes ----- Never

6. During math I am _______________ interested in what the teacher is saying.

Always ----- Mostly ----- Sometimes ----- Never

7. I enjoy learning how to solve different types of math word problems.

Always ----- Mostly ----- Sometimes ----- Never

8. I wish I did not have to study math

Always ----- Mostly ----- Sometimes ----- Never

9. When you think back on previous years, what do you remember most about how your prior
teachers taught you math?

DEVELOPING MATHEMATICAL REASONING 185
Appendix C
EXAMPLES OF HOW REASONING IS EMBEDDED IN THE CSTs (GRADES 2 & 3)
CALIFORNIA CONTENT STANDARDS: GRADE 2
# of
Items
%

Mathematical Reasoning
Embedded
Standard Set 1.0 Students make decisions about how to set up
a problem:

1.1 Determine the approach, materials, and strategies to be
used.
Embedded
1.2 Use tools, such as manipulatives or sketches, to model
problems.
Embedded
Standard Set 2.0 Students solve problems and justify their
reasoning:

2.1 Defend the reasoning used and justify the procedures
selected.
Embedded
2.2 Make precise calculations and check the validity of the
results in the context of the problem.
Embedded
Standard Set 3.0 Students note connections between one
problem and another.

California Standardized Test grade 2 mathematics (Blueprint adopted by the State Board of
Education 10/02)

CALIFORNIA CONTENT STANDARDS: GRADE 3
# of
Items
%
Mathematical Reasoning
Embedded
Standard Set 1.0 Students make decisions about how to
approach problems:

1.1 Analyze problems by identifying relationships,
distinguishing relevant from irrelevant information,
sequencing and prioritizing information, and observing
patterns.
Embedded
1.2 Determine when and how to break a problem into
simpler parts.
Embedded
Standard Set 2.0 Students use strategies, skills, and concepts in
finding solutions:

2.1 Use estimation to verify the reasonableness of calculated
results.
Embedded
DEVELOPING MATHEMATICAL REASONING 186
2.2 Apply strategies and results from simpler problems to
more complex problems.
Embedded
2.3 Use a variety of methods, such as words, numbers,
symbols, charts, graphs, tables, diagrams, and models, to
explain mathematical reasoning.
Embedded
2.4 Express the solution clearly and logically by using the
appropriate mathematical notation and terms and clear
language; support solutions with evidence in both verbal
and symbolic work.
Embedded
2.5 Indicate the relative advantages of exact and approximate
solutions to problems and give answers to a specified
degree of accuracy.
Embedded

2.6 Make precise calculations and check the validity of the
results from the context of the problem.
Embedded
Standard Set 3.0 Students move beyond a particular problem by
generalizing to other situations:

3.1 Evaluate the reasonableness of the solution in the context
of the original situation.
Embedded
3.2 Note the method of deriving the solution and
demonstrate a conceptual understanding of the derivation by solving
similar problems.
Embedded
3.3 Develop generalizations of the results obtained and apply
them in other circumstances.
Embedded
California Standardized Test grade 3 mathematics (Blueprint adopted by the State Board of
Education 10/02)

DEVELOPING MATHEMATICAL REASONING 187
Appendix D
RESULTS FOR THE THREE MAJOR COGNITIVE DOMAINS TESTED BY THE TIMSS
INVESTIGATION (U.S. IS AT THE BOTTOM0

Country
Mathematics Cognitive Domains
Knowing Applying Reasoning
Korea, Rep. of 79 74 65
Japan 74 70 63
Singapore 81 75 61
Hong Kong SAR 80 75 61
Chinese Tapei 75 72 59
Russian Federation 65 60 51
Netherlands 61 60 51
Northern Ireland 71 66 49
England 66 61 49
Lithuania 61 60 49
Portugal 63 60 48
Germany 60 58 48
United States 67 60 46
DEVELOPING MATHEMATICAL REASONING 188
Appendix E
TEACHER’S CONSENT FORM

University of Southern California
Rossier School of Education
Waite Phillips Hall
3470 Trousdale Parkway, Los Angeles, CA 90089

Assessing Mathematical Reasoning: A casual comparative study of upper
elementary students instructed via Cognitive Guided Instruction and Explicit Direct
Instruction

TEACHERS
Because you are a 5
th
grade teacher at your school site, you are being invited to
participate in a research study conducted by Luciano Cid (MAT, M.Ed) and Sandra Kaplan
(Ed.D) for the University of Southern California. Your participation is voluntary. You should
read the information below, and ask questions about anything you do not understand, before
deciding whether to participate. Please take as much time as you need to read the consent form.
If you decide to participate, you will be asked to sign this form. A copy of this form will be
provided for your records.

PURPOSE OF THE STUDY
International assessments continue to demonstrate that mathematical reasoning is one of the
skills upon which American students need improvement. Consequently, this study will analyze
and compare two different ways of teaching math to see which way is better for developing the
mathematical reasoning of elementary students.

STUDY PROCEDURES
As a teacher volunteering to participate in this study, you will be asked to perform two
duties: (1) supervise a test and (2) administer a short nine-question survey regarding your
students’ perception of math.

The test
As the test’s supervisor you will be asked to monitor that all your students are working
alone, quietly, and without disruptions during the test. Moreover, in order to maintain the validity
and reliability of the test, you will be asked to provide no clarification for any of your students
during the test. If a student does ask for help, you are to remind him/her that he/she is taking this
test for research purposes rather than for a grade and, as a result, no help will be provided. The
test should take about 30 minutes give or take 15 minutes to complete.

The Survey
In opposition to the supervision of the test, during the administration of the survey you
will be able to provide clarification to any student that may require it. However, similarly to the
test you must monitor that all students are working alone, quietly, and without disruptions while
they answer their surveys. The survey should take about 5 to 10 minutes to complete.
DEVELOPING MATHEMATICAL REASONING 189

In short, your participation in the study will be minimal. As a classroom teacher you will
only interact with your own students during the study. The entire data collection process should
take about 45 minutes to an hour. No interventions beyond what you normally do will be
required for this study. The whole process should happen in your classroom, students will not be
allowed to take either the test or the survey home.

POTENTIAL RISKS AND DISCOMFORTS

No foreseeable risks or discomforts will be part of the study.

POTENTIAL BENEFITS TO PARTICIPANTS AND/OR TO SOCIETY

There are two possible benefits that you might experience from seeing the analysis of the
data. First, you could get a sense of how your students perceive mathematics in your own
classroom as well as in general. Second, you will be able to see the capacity that your students
have to reason mathematically when exposed to problems without prior

Regarding the benefits to the larger society, the results of this study may inform teachers
across the world on the type of instruction that can increase the mathematical reasoning of
elementary students. This possible shift in understanding has the potential to increase the
capacity of students to succeed in high school and college math courses, as well as, to increase
their ability to be creative about problems in general. Of course, all the benefits previously
mentioned are all conditional upon the results obtained in the study.

PAYMENT/COMPENSATION FOR PARTICIPATION

A small gift, in the form of two movie tickets, will be provided for your participation in
the study. The tickets will be given to you at the end of the data collection process. You will not
be paid cash for participating in the study. Since the data will be collected at your work site, no
reimbursement will be provided for your travel or parking expenses.

CONFIDENTIALITY

We will keep your records for this study confidential as far as permitted by law. However,
if we are required to do so by law, we will disclose confidential information about you. 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.

The data will be stored inside a locked briefcase during transit and inside a locked filing
cabinet at all other times. After results are analyze all students, parents, and school staff will
have the ability to view them. However, the raw data will not be viewed by, or released to,
anyone besides the principal investigator ( Luciano Cid) and the faculty advisors overseeing the
study.

DEVELOPING MATHEMATICAL REASONING 190
In order to maintain anonymity each participating student will be given a number only
known to the principal investigator. Along with this number, each participating school will be
identified by an upper case letter and each participating teacher will be given a lower case letter.
Again only the principal investigator will have access to how each participant has been coded.
Consequently, rather than using participants’ ( students and teachers) names, identification will
follow the following pattern: Aa1, Aa2, Aa3, Ba1, Ba2, Ba3, Ab1, Ab2, Ab3, depending on
which school, which teacher, and which child is being analyzed. The raw data will not be viewed
by, or released to, anyone besides the principal investigator ( Luciano Cid) and the faculty
advisors overseeing the study.

The raw data will be kept for the required minimum of three years after the completion of
the study. After the three years are over the principal investigator will destroy the data beyond
any possible recognition.

CERTIFICATE OF CONFIDENTIALITY

Any identifiable information obtained in connection with this study will remain
confidential, except if necessary to protect your rights or welfare (for example, if you are injured
and need emergency care). A Certificate of Confidentiality has been obtained from the Federal
Government for this study to help protect your privacy. This certificate means that the
researchers can resist the release of information about your participation to people who are not
connected with the study, including courts. The Certificate of Confidentiality will not be used to
prevent disclosure to local authorities of child abuse and neglect, or harm to self or others.

When the results of the research are published or discussed in conferences, no identifiable
information will be used.

PARTICIPATION AND WITHDRAWAL

Your participation is voluntary. Your refusal to participate will involve no penalty or loss
of benefits to which you are otherwise entitled. You may withdraw your consent at any time and
discontinue participation without penalty. You are not waiving any legal claims, rights or
remedies because of your participation in this research study.

If the principal investigator observes any misconduct, dishonesty, or unethical behavior
on the part of any participant, he reserves the right to terminate participation immediately.

INVESTIGATOR’S CONTACT INFORMATION

If you have any questions or concerns about the research, please feel free to contact
Luciano Cid by calling (714) 292-1631 or via email at lcid@pylusd.org.

RIGHTS OF RESEARCH PARTICIPANT – IRB CONTACT INFORMATION
DEVELOPING MATHEMATICAL REASONING 191
If you have questions, concerns, or complaints about your rights as a research participant
or the research in general and are unable to contact the research team, or if you want to talk to
someone independent of the research team, please contact the University Park Institutional
Review Board (UPIRB), 3720 South Flower Street #301, Los Angeles, CA 90089-0702, (213)
821-5272 or upirb@usc.edu

SIGNATURE OF RESEARCH PARTICIPANT

I have read the information provided above. I have been given a chance to ask questions.
My questions have been answered to my satisfaction, and I agree to participate in this study. I
have been given a copy of this form.

Name of Participant

Signature of Participant Date

SIGNATURE OF INVESTIGATOR

I have explained the research to the participant and answered all of his/her questions. I
believe that he/she understands the information described in this document and freely consents to
participate.

Name of Person Obtaining Consent

Signature of Person Obtaining Consent Date
DEVELOPING MATHEMATICAL REASONING 192
Appendix F
PARENTAL CONSENT FORM

University of Southern California
Rossier School of Education
Waite Phillips Hall
3470 Trousdale Parkway, Los Angeles, CA 90089

Assessing Mathematical Reasoning: A Causal Comparative Study of Cognitive
Guided Instruction and Explicit Direct Instruction

PARENTS
Because your child is in 5
th
grade, he/she is being invited to participate in a research
study conducted by Luciano Cid (MAT, M.Ed) and Sandra Kaplan (Ed.D) for the University of
Southern California. Your child’s participation is voluntary. You should read the information
below and ask questions about anything you do not understand before deciding whether your
child should participate in the study. Please take as much time as you need to read the consent
form. If you decide to participate, you will be asked to sign this form. A copy of this form will be
provided for your records.

PURPOSE OF THE STUDY
International assessments continue to demonstrate that mathematical reasoning is one of
the skills in which American students need assistance. Consequently, this study will analyze and
compare two different ways of teaching mathematics in order to see which one is better equiped
for developing the mathematical reasoning of elementary students.

STUDY PROCEDURES
Your child’s participation in this study will involve two basic requirements: (1) he/she
will be asked to take a twelve question math test and (2) he/she will be asked to take a short nine-
question survey regarding his/her perception of mathematics.

The test
Your child will be asked to take a test composed of 12 questions that have been taken
from previous state and international assessments. The questions are designed to test the
mathematical reasoning of your child. The test should not take longer than 30 minutes to
complete, however, each child will be given as much time as he/she may need to finish it.

The Survey
Your child will also be asked to complete a 9 question survey designed to assess his/her
perceptions of mathematics. The survey should not take more than 5 to 10 minutes to complete.
Examples of questions included in the survey are: (1) I enjoy learning math, (2) I think the other
kids in my class don’t like math time, and (3) my favorite part of math time is the word problems
I get to solve.

DEVELOPING MATHEMATICAL REASONING 193
In short, your child’s involvement in the study will not ask him/her to do anything out of
the ordinary.

During the study your child will only interact with his/her classroom teacher and the
principal investigator (i.e. Luciano Cid). At no point will anyone involved in the study ask your
son/daughter to be left alone with an adult. The entire data collection process should take about
45 minutes to an hour. No interventions beyond what your teacher is normally doing to teach
his/her students will be required for this study. The whole process should happen in the
classroom of your son/daughter’s teacher.

POTENTIAL RISKS AND DISCOMFORTS
No foreseeable risks will be part of the study; however, anxiety can be a discomfort that
some children might experience during the math test.

POTENTIAL BENEFITS TO PARTICIPANTS AND/OR TO SOCIETY
There are two possible benefits that you might experience from seeing the analysis of the
data. First, you could get a sense of how your child perceives mathematics. Second, you will be
able to see the capacity that your students have to reason mathematically when exposed to
problems with which they have had no prior experience.

Regarding the benefits to the larger society, the results of this study may inform teachers
on what is the best instruction to increase the mathematical reasoning of elementary students.
This possible shift in understanding has the potential to boost the capacity of students to succeed
in high school and college math, as well as, to augment their ability to be more creative about
solving mathematical problems in general. Of course, all the benefits previously mentioned are
conditional upon the results obtained in the study.

PAYMENT/COMPENSATION FOR PARTICIPATION
For participating in this study your child will be entered into a drawing in which a $25
gift certificate card will be raffled. There will be one winner from each participating school. The
lucky participant will receive his/her prize at the end of the data collection process. Your child
will not be paid cash for participating in the study. Since the data will be collected at your child’s
school site, no reimbursement will be provided for your travel or parking expenses.

CONFIDENTIALITY
We will keep the records of your child’s participation for this study confidential as far as
permitted by law. However, if we are required to do so by law, we will disclose confidential
information about your child. 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.

All data collected will be stored inside a locked briefcase during transit and inside a
locked filing cabinet at all other times. After results are analyze all students, parents, and school
staff will have the choice to view the results. However, the raw data will not be viewed by, or
released to, anyone besides the principal investigator ( Luciano Cid) and the faculty advisors
overseeing the study.
DEVELOPING MATHEMATICAL REASONING 194

In order to maintain the anonymity of your child, each participating student will be given
a number that is only known by the principal investigator. Along with this number, each
participating school will be identified by an upper case letter and each participating teacher will
be given a lower case letter. Again only the principal investigator will have access to how each
participant has been coded. Consequently, rather than using participants’ ( students and
teachers) names, identification will follow the following pattern: Aa1, Aa2, Aa3, Ba1, Ba2, Ba3,
Ab1, Ab2, Ab3, depending on which school, which teacher, and which child is being analyzed.
The raw data will not be viewed by, or released to, anyone besides the principal investigator
( Luciano Cid), the faculty advisors overseeing the study, and each parent’s child if so desire.

The raw data will be kept for the required minimum of three years after the completion of
the study. After the three years are over the principal investigator will destroy the data beyond
any possible recognition.

CERTIFICATE OF CONFIDENTIALITY
Any identifiable information obtained in connection with this study will remain
confidential, except if necessary to protect your rights or welfare (for example, if you are injured
and need emergency care). A Certificate of Confidentiality has been obtained from the Federal
Government for this study to help protect your privacy. This certificate means that the
researchers can resist the release of information about your participation to people who are not
connected with the study, including courts. The Certificate of Confidentiality will not be used to
prevent disclosure to local authorities of child abuse and neglect, or harm to self or others.

When the results of the research are published or discussed in conferences, no identifiable
information will be used.

PARTICIPATION AND WITHDRAWAL
Your child’s participation is voluntary. Your refusal to allow your child to participate will
involve no penalty or loss of benefits to which your child is otherwise entitled. You may
withdraw your consent at any time and discontinue participation without penalty. You are not
waiving any legal claims, rights or remedies because of your participation in this research study.

If the principal investigator observes any misconduct, dishonesty, or unethical behavior
on the part of any participant, he reserves the right to terminate participation immediately.

INVESTIGATOR’S CONTACT INFORMATION

If you have any questions or concerns about the research, please feel free to contact
Luciano Cid by calling (714) 292-1631 or via email at lcid@pylusd.org.

RIGHTS OF RESEARCH PARTICIPANT – IRB CONTACT INFORMATION
If you have questions, concerns, or complaints about your rights as a research participant
or the research in general and are unable to contact the research team, or if you want to talk to
DEVELOPING MATHEMATICAL REASONING 195
someone independent of the research team, please contact the University Park Institutional
Review Board (UPIRB), 3720 South Flower Street #301, Los Angeles, CA 90089-0702, (213)
821-5272 or upirb@usc.edu

SIGNATURE OF RESEARCH PARTICIPANT

I have read the information provided above. I have been given a chance to ask questions.
My questions have been answered to my satisfaction, and I agree to allow my child to participate
in this study. I have been given a copy of this form.

Name of Parent

Signature of Parent Date

Name of student

Signature of Student Date

SIGNATURE OF INVESTIGATOR
I have explained the research to the participant and answered all of his/her questions. I
believe that he/she understands the information described in this document and freely consents to
participate.

Name of Person Obtaining Consen

Signature of Person Obtaining Consent Date

DEVELOPING MATHEMATICAL REASONING 196
Appendix G
STUDENT ASSENT FORM

University of Southern California
Waite Phillips Hall
3470 Trousdale Parkway, Los Angeles, CA 90089

Assessing Mathematical Reasoning: A Causal Comparative Study Using Cognitive Guided
Instruction and Explicit Direct Instruction

Luciano Cid wants to learn about how elementary students solve math problems that are
challenging; as well as, how they feel about learning math in general. One way to learn about it is to do a
research study; the people doing the study are called researchers.

Your mom/dad/Legally Authorized Representative (LAR) have told us we can talk to you about
the study. You also can talk this over with your mom or dad. It’s up to you if you want to take part, you
can say “yes” or “no”. No one will be upset with you if you don’t want to take part.

If you want to be part of the study, you will be asked to do a 12 question test and a 9 question
survey. The test will ask you to solve math word problems that are challenging. The survey will ask you
questions about your feelings and your friends feelings about math, like: how much do you enjoy math, or
do you think other kids in your class enjoy math.

Researchers don’t always know what will happen to people in a research study. We don’t expect
anything to happen to you, but you might not like answering challenging math questions. Keep in mind,
however, that the test you will take will not be part of your grade. And if you participate you will be
entered in a drawing to win a $25 gift certificate.

Your answers will not be part of your grade. You will be asked to put your name on the test,
however, the research team will be the only people who will know who answer what. This is because
every student will be given a code to hide their identity. Your teacher and your principal will have access
to the overall scores of your school, but your parents will be given access to your own personal score. If
you have any questions about anything, you can ask the researchers.

If you want to take part in the study, please write and then sign your name at the bottom. You
can change your mind if you want to, just tell the researchers.

_________________________________
Name of Participant
____________________________________ ____________________
Participant’s Signature Date
___________________________________
Name of person consenting
__________________________________ ____________________
Signature of person consenting Date

DEVELOPING MATHEMATICAL REASONING 197
Appendix H
DESCRIPTION OF EACH ITEM EMPLOYED TO CONSTRUCT THE 12-QUESTION TEST
Question # Test Domain Topic
1 TIMSS Number Whole numbers
2 TIMSS Data Display Reading and Interpreting
3 TIMSS Number Whole numbers
4 TIMSS Number Patterns and Relationships
5 CST Number Number sense
6 TIMSS Number Whole Numbers
7 TIMSS Data Display Organizing and Representing
8 CST Number Number sense
9 CST Measurement and
Geometry
Perimeter and
Area
10 CST Statistics, Data Analysis,
and Probability
Predict simple probability
situations

11

CST

Measurement and
Geometry
Understanding of plane and
solid geometric objects and
use this knowledge to show
relationships and solve
problems
12 CST Statistics, Data Analysis,
and Probability
Organize, represent, and
interpret numerical and
categorical data and clearly
communicate their findings

Item by item explanation of the 12-Question test

Question 1- TIMSS Content Domain Number Topic – Area: Whole Numbers

Question 2 - TIMSS: Content Domain Data Display Topic Area: Reading and Interpreting

Question 3 - TIMSS Content Domain Number Topic – Area: Whole Numbers

Question 4- TIMSS Content Domain Number Topic Area: Patterns and Relationships

Question 5 - CST Number sense 4NS3.1 (Key standard) Demonstrates an understanding of, and
the ability to use, standards algorithms for the addition and subtraction of multi-digit numbers.

Question 6 - TIMSS Content Domain: Number- Topic Area: Whole Number

Question 7 - TIMSS Content Domain: Data Display - Topic Area: Organizing and Representing

DEVELOPING MATHEMATICAL REASONING 198
Question 8 CST Number sense 4NS3.4 (Key standard) Solve problems involving division of
multi-digit numbers by one-digit numbers.

Question 9 - CST Measurement and Geometry (Students understand perimeter and area):
4MG1.4 Understand and use formulas to solve problems involving perimeters and areas
ofrectangles and squares. Use those formulas to find the areas of more complex figures by
dividing the figures into basic shapes.

Question 10 - CST Statistics, Data Analysis, and Probability Students make predictions for
simple probability situations 4PS2.2 Express outcomes of experimental probability situations
verbally and numerically (e.g., 3 out of 4; 3/4).

Question 11 - CST Measurement and Geometry (Students demonstrate an understanding of plane
and solid geometric objects and use this knowledge to show relationships and solve problems):
4MG3.6 Visualize, describe, and make models of geometric solids (e.g., prisms, pyramids) in
terms of the number and shape of faces, edges, and vertices; interpret two-dimensional
representations of three-dimensional objects; and draw patterns (of faces) for a solid that, when
cut and folded, will make a model of the solid.

Question 12 - CST Statistics, Data Analysis, and Probability (Students organize, represent, and
interpret numerical and categorical data and clearly communicate their findings): 4PS1.3
Interpret one- and two-variable data graphs to answer questions about a situation.

Abstract (if available)

Abstract Focusing on the discrepancies that exist between student and teacher-centered pedagogies, this study analyzed the ability of two professional development (PD) models, Explicit Direct Instruction (EDI) and Cognitively Guided Instruction (CGI]), to increase the mathematical performance, the mathematical reasoning, and the mathematical perception of elementary students. A causal-comparative ex post facto model of investigation was employed to carry out the study

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Asset Metadata

Creator Cid, Luciano (author)

Core Title Developing mathematical reasoning: a comparative study using student and teacher-centered pedagogies

School Rossier School of Education

Degree Doctor of Education

Degree Program Education (Leadership)

Publication Date 10/27/2014

Defense Date 05/17/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Cognitively Guided Instruction,Explicit Direct Instruction,inquiry based learning,mathematical reasoning,OAI-PMH Harvest,student-centered instruction,teacher-centered instruction

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Kaplan, Sandra (committee chair), Gallagher, Pat (committee member), Hasan, Angela Laila (committee member)

Creator Email lcid@pylusd.org,lcid@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-510035

Unique identifier UC11288143

Identifier etd-CidLuciano-3033.pdf (filename),usctheses-c3-510035 (legacy record id)

Legacy Identifier etd-CidLuciano-3033.pdf

Dmrecord 510035

Document Type Dissertation

Rights Cid, Luciano

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA