IMPLEMENTING KEYMATH APPROACH FOR ASSESSMENT, LEARNING, AND
TEACHING FOR AN INCLUSIVE MIDDLE SCHOOL CLASSROOM IN BRITISH
COLUMBIA
by
Trevor Stovel
BSc, Biology, University of Northern British Columbia, 2008
BEd, University of Northern British Columbia, 2010

PROJECT SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF EDUCATION
IN
SPECIAL EDUCATION

UNIVERSITY OF NORTHERN BRITISH COLUMBIA
April 2018
© Trevor Stovel, 2018

ii
Abstract
Mathematical pedagogy has a large body of research as it pertains to both typically
achieving students and learners with special needs, particularly math disabilities. The
label of mathematical disability is dependent on several types of assessments that
measure various aspects of cognition related to mathematical skill. Rural school districts,
such as the one where this project started, have limited resources to assess and instruct
learners with mathematical disabilities. This project made use of the KeyMath-3
diagnostic assessment in the construction of classroom math units using the KeyMath-3
diagnostic assessment that guides diagnosis of math disability. This diagnostic
assessment was used as a focus for the language and types of questions used in various
mathematical units. The British Columbia Ministry of Education math curriculum,
KeyMath-3 diagnostic assessment, and IXL.com math program were all analyzed to find
common language and goals as the focal points for the lessons. Probability, Pythagorean
Theorem, Algebraic Expression, and Surface Area units were constructed at the Grade 8
middle school level using this approach to make learning accessible to learners with math
disabilities while simultaneously allowing stronger math learners to fully express their
levels of mathematical understanding. The combined use of diagnostic assessment,
curricular goals, and support programs analyzed in this project allows for the construction
of math units that could improve the understanding of all math learners, especially those
students with math disabilities.
Keywords: KeyMath, assessment, mathematics, diagnosis, math disability

iii
Table of Contents
Abstract

ii

Table of Contents

iii

Acknowledgements

vi

Chapter 1

Introduction
Significance of the Project
Background of the Project
Personal Location
Purpose of the Project
Researcher Context
Project Overview
Chapter Summary

1
2
3
4
5
5
6
8

Chapter 2

Current Understanding of Math Pedagogy
Use of Computer Assistive Instruction
Math Pedagogy in Special Education
Reinforcing active memory
Reinforcing attention
Training executive functions
Developing and reinforcing visuospatial perception
Reinforcing skills related to speech and language in
math
Criteria for Diagnosing Math Disabilities
Validity of IQ tests in the measure of math disability
Underlying cognitive factors or an overall cognitive
discrepancy
Valid detection of mathematically relevant cognitive
domains
Affected domains of cognition and other
characteristics of math disability
Test for Determining Math Disabilities
KeyMath-3 diagnostic assessment
The IXL math program
Chapter Summary

9
15
18
22
23
23
23

Chapter 3

Research Methods
Content Selection of KeyMath-3 Assessment
Construction of Units Around KeyMath-3 Assessment
Use of Computer Assisted Technology in KeyMath
Chapter Summary

Chapter 4

Teaching Units

24
32
32
33
34
35
41
45
46
48
50
52
54
54
57

iv
The Math Concepts
The Probability Unit
Table 1
Lesson 1: How many choices do you have?
Lesson 2: How many of each option do you have?
Lesson 3: How many ways can I do this? How many
ways can I get there?
Lesson 4: If I do this # of times, how many times
should I get x-results?
Lesson 5: Experiment creating your own data using
dice
Lesson 6: Playing cards and the likelihood of winning
or losing
Lesson 7: Which option should I choose?
Lesson 8: Games of chance and the math behind
them
Project: Game of Chance
The Pythagorean Theorem Unit
Table 2
Lesson 1: Square numbers
Lesson 2: Create the square number rainbow sheet
Lesson 3: Use of the rainbow sheet
Lesson 4: Estimating square roots
Lesson 5: Introduction of types of triangles
Lesson 6: Pythagorean theorem introduction
(concrete and representational)
Lessons 7-9: Variations of the Pythagorean Theorem
Project: Pythagorean art assignment
Algebraic Expressions Unit
Table 3
Lesson 1: Basic expressions and simplification
Lesson 2: Order of operations
Lesson 3: Substitution into expressions using
variables
Lesson 4: Simplifying algebraic expressions with
variables
Lesson 5: The distributive property
Lesson 6: The steps of simplifying expressions
Project: The math map
Surface Area Unit
Table 4
Lesson 1: Folding 2D nets into 3D shapes
Lesson 2: Geometry vocabulary
Lesson 3: Anatomy of 3D objects
Lesson 4: Creating picture using formulas
Lesson 5: Surface area of rectangular prisms

59
59
60
62
64
64
65
66
67
67
68
68
69
69
72
74
74
74
75
76
77
77
79
79
81
82
84
85
87
88
89
89
90
91
93
93
94

v

Chapter 5

References
Appendix A
Appendix B
Appendix C

Lesson 6: Surface area of triangular prisms and
pyramids
Lesson 7: Surface area of cylinders
Project: Applications of surface area
IXL Reinforcement of Units
Chapter Summary
Reflection and Conclusion
Reflection
Conclusion
Sample Rainbow Squares Table
Samples of Algebraic Container Models
Samples of Pythagorean Art Projects

95
96
97
98
98
99
100
101
101
105
106
116
117
119

vi
Acknowledgements
The idea for this project came from my belief that research should inform
teaching practice and that this leads to the best possible experience for all learners. I have
created units with other teachers based around current trends in research such as universal
design and backwards design and this has resulted in exceptional units that have students
engaged across multiple difficulty levels, especially those with individualized education
plans. This project would not have occurred without those educators who supported me in
the construction of these units and those students who found that they could be good at
math with the right mindset.
I would like to thank my wife, Janilee, for keeping an eye on our daughter while I
was off listening to lectures or writing papers. Her input about topics for papers was also
invaluable and she always pushed me to add that extra reference or statement that made
projects and papers that much better.
This project would not have been possible without my supervisor, Dr. Lantana
Usman, whose patience with my changing topics and focus for this project is greatly
appreciated. My committee members, Dr. Andrew Kitchenham and Ms. Tracy Connell
were also patient enough to tolerate several drafts of this project and still give assistance.
Their feedback and advice helped to shape the paper into its final form.

1
Chapter 1: Introduction
The instruction of mathematics constitutes a cornerstone in the educational
systems of an industrial and global society, and the importance of adult numeracy is a
concern for many educators and governmental bodies (Coben, 2003; Gal, 2000; Johnston,
2002; National Numeracy Review Report Panel, 2008; Parsons & Bynner, 2005). The
ability of citizens to participate in everyday finances also requires a reasonable level of
mathematical competence (Northcote & McIntosh, 1999). The use of mathematics
permeates modern society and the correct application of mathematical algorithms occurs
in the lives of citizens on a daily basis (Northcote & McIntosh, 1999). Correct, efficient,
and fluent use of mathematical concepts is a stated target of mathematical curricula (BC
Ministry of Education, 2016). This importance has led to debates on the proper and
effective instruction of mathematics at the junior and high school levels (Hutchinson,
2017; Winzer & Mazurek, 2011).
Scholars of mathematics education have reiterated the competence and ability of
the instructor, which has significant effects on the success of students (Brownell,
Sindellar, Kiely & Danielson 2010; Bouck & Flanagan, 2009). In addition, diagnoses of
students’ relevant academic abilities can also lead to different methods of teaching
mathematical concepts and other contents of the curriculum (Ketterlin-Gellar, Chard &
Fien, 2008; Seo & Bryant, 2012; Zheng, Flynn, & Swanson 2013). Modern technological
applications in the classroom have greatly impacted the ability to teach mathematical
concepts and have the potential to make even abstract concepts more understandable for
all learners (Loong, 2014; Tezer & Kanbul, 2009; Watson 2016). These factors are in turn
influenced by the philosophical beliefs and values of the teachers’ involved (Winzer,

2
2007) while the “correct” philosophical teaching approach has undergone shifts over the
history of mathematics education (Haggarty and Pepin, 2002; Katz, Jankvist, Fried &
Rowlands, 2014).
Significance of the Project
Current trends in mathematics education have been readily adopted by teachers in
many schools, especially at the elementary and high school levels so as to ensure success
of learners (BC Ministry of Education, 2016). The idea of having all students access the
material and express their understanding in unique ways has been seen as the way to
bring all students into a single learning environment in an inclusive and successful way
(Dieker & Rodriguez, 2013; Sayeski & Paulsen, 2010). Further significance of the
developed lesson units could provide both support for the continued use of the KeyMath3 assessment and place the assessment in the role of informing teaching practice in a
more direct and innovative way, thereby providing variety for the teachers and learners.
In addition, through the developed units, students’ learning needs can be predicted
by the teachers and can be compensated with appropriate strategies, resources and
technologies (BC Ministry of Education, 2016). The developed lesson units using the
KeyMath approach could alleviate time constraints on teachers and allow for more
universal participation for learners challenged with mathematic concepts across the
middle school classroom, the target of the project. These learners that would typically be
isolated from what the rest of the class is working on will be able to participate in a
relevant and constructive manner (Daugherty, Campana, Kontos, Flores, & Shaw, 1996;
Pliner & Johnson, 2004).

3
The middle high school math units constructed and examined in this project have
been used for several years but will now be presented in the context of current assessment
practices for learners with special needs.
In sum, due to the emphasis on Math learning success, acquisition of math and the
need for applying math concepts in practical real-life situations for students as referenced
in the new BC Curriculum (BC Ministry of Education, 2016), this project can hopefully
be of significance to students’ successful learning.
Background of the Project
This project is based on the researcher’s teaching experiences at a middle school
in northern British Columbia. It is the only middle school in this rural BC school district.
The school has approximately 450 students with 150 students in Grades 6, 7, and 8
respectively. The Grade 8 class has five divisions that are taught by three math teachers.
The demographics of the student population are predominantly First Nations, with the
remaining students being from a variety of other backgrounds (Kelley & Legge, 2006).
The school has a population of students who are predominantly from a lower
socioeconomic status and there is a high proportion of students with Individualized
Education Plans (IEPs) (Kelley & Legge, 2006). There are many students on waiting lists
to be assessed for possible IEP designations.
This project has grown out of teachers reporting increased student success and
participation when using the methods of Universal Design for Learning (UDL) and a
computer practice program, IXL, in the teaching of math. The math teachers of the school
found success in increasing student engagement across skill levels and have seen an
increase in student initiative in math projects. These projects have allowed students to

4
display their individual abilities to a much greater extent than traditional math projects
that limited student options for expression. With the focus of this project on the inclusion
of the KeyMath-3 assessment diagnostic tool, it is hoped that more student learning
problems and challenges will be addressed during unit construction which was the
purpose of this project.
Personal Location
For eight years I have been a math and science teacher in a rural British Columbia
school district. I have taught adults and middle school students. In both environments, I
have encountered individuals with barriers to their education. Some students had
diagnosed disabilities while others likely had undiagnosed disabilities. Either scenario
would have led to difficulties in the classroom and in the math classroom in particular.
Through multiple professional development workshops and district initiatives I
was exposed to the idea of Universal Design for Learning (UDL) and the idea of
removing educational barriers for students at the level of unit construction. The school
district and teachers started initiatives to develop units that used the UDL approach to
create several units. Success was found when these units were implemented and the units
have been improved over the years.
Both classroom and special education teachers in the school have experience
using the unit construction in Math subjects as a means of addressing the challenges of
learners with special needs. There are still some questions as to whether the approach is
truly effective. Some teachers have been hesitant to begin the process of recreating units
from a student perspective. With this project, I hoped to provide innovative examples of
approaches to unit construction using the KeyMath diagnostic assessments, which will

5
have a significant impact on Special Education teachers in the production of Math
curriculum units in middle school classrooms.
Purpose of the Project
The primary purpose of this project was to develop units using the KeyMath
diagnostic as an approach for teachers to improve the learning environment of learners in
Math classrooms, especially for special education learners at the middle school level.
The secondary purpose of the project is to circumvent this problem not through
repeated practice of multiple methods but through altering the entirety of the middle
school BC math curriculum unit to focus on fewer, but more universally applicable,
mathematical algorithms.
Additionally, the project aims at improving teaching practices for classroom math
teachers of middle school through the developed units from the current British Columbia
Mathematics Curriculum that corresponds with the KeyMath-3 diagnostic assessment.
Other aims of the project are to create lessons that include exceptional learners
and their potential math disabilities (MD) as recommended by many Canadian scholars of
exceptional learners (Hallahan, Kauffman & Pullen, 2015; Hutchinson, 2017; Winzer,
2007).
Researcher Context
For this project, the research context was based on documentary and content
analysis of relevant literature related to math assessment and pedagogies for inclusive
classrooms. I used the BC middle school math curriculum (BC Ministry of Education,
2016) at the Grade 8 level to develop my units using the KeyMath-3 diagnostic
assessment approach.

6
As the creator of the units, I synthesized a set of lessons that would target
potential deficits in math ability and understanding. By using a diagnostic tool and
current research to guide unit construction and directly influence my approach to lessons,
I created student-centered lessons that promote accessibility to the material by
determining what criteria students will be assessed on in a diagnostic assessment such as
the KeyMath-3.
As earlier stated, in my current middle school, I created math teaching units using
the UDL approach and continue to be an active proponent of the IXL computer program
being used. The other teachers and I have combined our expertise to create lessons, notes,
and assignments that make use of modern math pedagogy such as the Universal Design
approach. By using my experience in a special education program, I hope to further refine
and enhance my ability to create units that make math accessible to all learners. A major
focus of this project is bringing together innovative modern math teaching and
assessment tools in both general math and special education pedagogy to implement the
best teaching practice possible for the math concepts being taught.
Project Overview
The following project will provide examples of the thought process behind
lessons and units created using diagnostic assessments with special needs learners in
mind. The units that are developed will greatly improve both student and teacher
understanding of mathematical concepts by using the KeyMath approach. From the
student perspective, lessons have been made more accessible and assignments have been
made open-ended and have allowed for more personal expression. Learners that typically
face a large number of barriers can now feel some learning satisfaction, motivation and

7
success in the math classroom with the KeyMath approach. Student projects, such as the
Pythagorean Art assignment or the student created game of chance give students a basic
set of criteria that can be expanded on according to student interest. The build up to these
projects includes assignments focused around concrete understanding of the fundamental
concepts while allowing for all learners to express their understanding to the best of their
ability. Students are taught how to manipulate formulas but are also given heuristics to
complete questions quickly. An example of this is the multiple uses of the Pythagorean
Theorem to find both a leg and a hypotenuse. Students are all taught how to manipulate
the formula but are also given key words and clues to watch for that allow them to know
which variation of the formula n to use. Approaching the unit from the KeyMath
perspective has highlighted common questions that students are assessed on and this has
focused the lessons towards ensuring that all students are given the tools to complete a
given question.
From the perspective of teachers, the units could replace rote practice lessons with
more involved lessons focusing on understanding and discussion of the concept being
taught. Use of computer assistive technology is what frees the teacher to engage with
students more about understanding rather than simply as a source for correct answers.
Using the KeyMATH diagnostic to highlight the basic skills that are assessed in the
diagnostic will focus the teachers’ efforts and allow the teacher to work on areas of need
and be more efficient with the limited class time available.
The next chapter of this project provides a review of literature as it relates to
general math pedagogy, special education math instruction, diagnosis of math disabilities,
and the use of technology in assisting mathematics instruction.

8
Chapter Summary
This project has come out of a perceived need to improve the math learning
success of students at a British Columbia middle school that services a large number of
low socioeconomic status and at-risk students. I have found that many students struggle
with not simply the mathematical operations that are taught in math class, but with the
language of the questions as well. Knowing that language can be a stumbling block for
students, I make efforts in class to ensure that the language of the tests and quizzes
parallels the language used in class to limit the confusion for students. I examined the
KeyMath-3 diagnostic test to check for the language that is used in the assessment in the
hopes of further focusing the language that I use in my math classes so that students who
are assessed using this diagnostic will be able to more clearly express their understanding
of mathematics. Math fluency has the potential to improve the lives of students as
mathematical operations are a common part of daily life (Northcote & McIntosh, 1999).
It is with this idea in mind that I am creating units that can be accessed by all learners so
that they not only learn the material, but are able to express their learning clearly. Using
diagnostic test as the starting point for unit construction gives me the language that will
focus the unit towards areas commonly assessed by the KeyMath-3 assessment. By using
this KeyMath approach, both teachers and learners could benefit. Students may benefit
from consistent use of language that parallels the diagnostic that tests their math ability.
Teachers may benefit from the KeyMath method by having their efforts focused towards
areas of need and by improving the quality of student instruction in the classroom.

9
Chapter 2: Literature Review
This literature review goes through the current understanding of math pedagogy in
both the general classroom and in special education services. The use of modern
technology to assist in these learning environments is a topic of interest in both the
general and special education classrooms. Common math disabilities and their solutions
are also examined. The diagnosis of these math disabilities and the assessments that are
used to determine if a learner has a math disability are then discussed with emphasis
placed on the KeyMath assessment that is the focus of this project.
Current Understanding of Math Pedagogy
There is a broad consensus on mathematical knowledge for teaching (MKT) and
what comprises this body of knowledge: subject matter knowledge and pedagogical
content knowledge (Brownell, Sindellar, Kiely, & Danielson, 2010). Additionally,
Brownell, Sindellar, Kiely & Danielson (2010) proposed that these two categories can be
broadly translated into “what it is” and “how to teach it” respectively with regards to
mathematical concepts. The authors further stated that with the current existing body of
research into mathematics education, it is important to ensure that effective practices are
improved and less effective practices are removed from teacher education. Brownell,
Sindellar, Kiely, & Danielson (2010) stated that research should be conducted into
effective mathematics instruction for the identification of practices that may not only be
ineffective but also have a net negative effect on children’s understanding. Sullivan and
Field (2013) suggested that current methodologies used in pre-school special education,
which have a statistically significant negative effect on learners’ math progress. The
existence of possible threshold effects for certain approaches to mathematics instruction

10
could change the amount of time allocated to certain tasks or approaches that may
traditionally have made up a great deal of the instructional time in a math classroom
(Doabler, Clark, Stoolmiller, Kosty, Fien, Smolkowsky, & Baker, 2017). Furthermore,
Doabler et al., (2017) reported that rote practice of skills and time spent on certain tasks
reaches a limit of effectiveness and that time could have been spent on other tasks that
would have led to greater gains in understanding (p. 107).
Another division of mathematical knowledge that also exists includes conceptual
and procedural knowledge, often seen as two distinct areas of math knowledge (Agrawal
& Morin, 2014). Current focus of conceptual knowledge is the general idea of a
mathematical concept in the course of teaching and learning. Agrawal and Morin (2014)
stated this knowledge as the linking of new phenomenon with known phenomena in an
attempt to see patterns. Procedural knowledge is the following of a series of steps to solve
a problem and it is mainly the ability to correctly apply and use an algorithm to produce
an answer (Agrawal & Morin, 2014). A student may have an understanding of how a
concept relates to others but cannot properly follow the steps to produce a correct answer.
Conversely, as is the case for many students, procedural knowledge may be given in a
vacuum. The students may be able to correctly follow a procedure to produce an answer
but may have little knowledge or ability to see how this procedure relates to other
mathematical applications. It is when students are taught, and become proficient in, both
procedural and conceptual knowledge that real growth and understanding occurs.
Powell (2015) identified some current factors of effective mathematics instruction
such as general awareness of mathematical concepts and the relevant pedagogy. The
common misconceptions related to math concepts are also important factors so that these

11
misconceptions can be kept from taking root in the habits and understanding of math
learners. Other current pedagogical practices in the teaching of math are impacted by
awareness of legislation and standards as they pertain to the grade being taught and for all
grades (BC Ministry of Education, 2016). This becomes expressly important when
emphasizing skills and algorithms for those students with math disabilities and focusing
on areas of the greatest relevance to future interests.
Agrawal and Morin (2016) reported that one of the current practices prevalent in
the teaching of mathematical concepts is that of CRA or concrete-representationalabstract. In this teaching method, hands-on activities are introduced to give students
direct experience in the concept being taught (Agrawal & Morin, 2016). Representations
of the manipulatives used are then given alongside abstract notation on a board at the
front of the class. In this way, students are given scaffolding of the abstract concept in the
form of direct interaction (concrete) and diagram and picture examples (representations)
(Agrawal & Morin, 2016). Particular success has been found in teaching algebra to
learners with special needs using the CRA methods (Witzel, 2005).
There is a lack of math knowledge background given by teacher educators to new
teachers which affects the varied nature of mathematics instructional competence which
correlates with poor or weak student academic performance (Stotsky 2009). Current
emphasis on ways of how to teach math in teacher education programs is given attention
in the numeracy and technology courses of teacher education programs to solve this
problems (Kitchenham, 2011). Teacher confidence and competence is seen as a key
factor in student academic achievement (Ekstam, Korhonen, Linnanmaki & Aunio, 2017;
Kitchenham, 2011).

12
Current research supports and has influenced the shift of math education away
from rote practice skills in isolated situations to a more student centered and problem
solving set of skills (Boyd & Bagerhuff, 2009; Sayeski & Paulsen, 2010; Shin & Bryant,
2015). Furthermore, Boyd and Bagerhuff (2009) explained the use of relevant problems
as having the ability to maintain student interest while allowing for the practice of
applying math skills to real world situations that students are likely to encounter. This
method has also been shown to improve student performance and application of learned
processes and allows them to “see the point” of math and gives the students impetus to
put in the necessary effort to internalize the procedures learned (Boyd & Bagerhuff,
2009). Teachers adopting this approach must relate the mathematical curriculum to real
life situations and situations that hold personal relevance and importance for the student
as part of the classroom practice (Hallahan, Kauffman, & Pullen, 2015; Sayeski &
Paulsen, 2010). Current studies reported that by giving students mathematical problem
solving situations related to the real world, the students will not question the importance
or use of the material they are being taught and will connect the math to their daily lives
(Sayeski & Paulsen, 2010).
The process of providing personal motivation is doubly important for special
needs students as they are already at risk of low motivation due to the increased difficulty
with mathematics and other academic areas (Willcutt, Petrill, Wu, Boada, DeFries, Olson
& Pennington, 2013).
A classical approach to mathematics education could have very real application in
modern special needs education (Katz, Jankvist, Fried & Rowlands, 2013). According to
the author’s report, classical mathematical proofs, especially for those concepts taught in

13
early to middle grades, was involved, hands-on, concrete and often showed mathematical
patterns in everyday life. These practical uses lend themselves to the kind of
mathematical literacy that is being sought for students. An added advantage to this
approach is the relevance mathematics would have on other disciplines, i.e. social studies
curricula and the math of a culture, which will allow for the creation of cross-curricular
units. An example of this would be the mathematics involved in the creation of
monuments constructed by ancient peoples (pyramids and temples). While not
necessarily a comprehensive solution to the problem of student interest, this broadens the
application and relevance of mathematical processes to culturally significant or at least
culturally recognized aspects of the world (Katz, Jankvist, Fried, & Rowlands, 2013).
Such interdisciplinary techniques in mathematics at the secondary school level is not
highly encouraged, especially by math teachers with less teaching experience and
background knowledge (Hallahan, Kauffman, & Pullen, 2015; Winzer, 2007).
Current math pedagogical approaches by teachers in high schools center on the
use of textbooks in line with most Canadian provincial and territorial math curriculum
policies (BC Ministry of Education, 2016; Hutchinson, 2017). Textbooks are often
viewed as the sentinels that can determine what is elite or appropriate information to pass
on to pupils (Haggarty & Pepin, 2002). Such important documents as math textbooks
should be held to a high standard but textbooks can vary greatly in their approaches to
mathematical content. How students are taught to do math will determine the type of
math textbooks used, from simple math books with illustrations and pictorials for junior
elementary grades to a more social story textbook for middle school students. The

14
selection of appropriate texts will be especially important for those learners challenged by
and diagnosed with dyscalculia (Hutchinson, 2017).
Recent developments in math pedagogy involve moving away from sole reliance
on text-based resources, due to the fact that evidence of multiple sources of
representation is advantageous to the development of mathematical understanding
(Loong, 2014). The increased availability of online resources and the prevalent access
students have to information from multiple sources requires that students be better
prepared to filter information on their own. A single text that must be taken as fact is
counterproductive to the development of this skill (Tezer & Kanbul, 2009).
The use of visual representations (VR) has been shown to be effective in teaching
mathematical concepts in today’s math classroom at all levels. A visual representation is
generally regarded as a physical representation of a student’s mental processing during
problem solving (van Garderen, Sindellar, Kiely, & Danielson, 2016). These are used by
both special education and mainstream classroom teachers.
Emphasis towards lower-level questions as opposed to higher-level questions is a
common feature of math classrooms, particularly for new teachers. This teacher approach
may seem logical given the deficits of special needs learners but is counterproductive to
the argument that real-world multi-step problems improve student overall understanding
(Griffin, Jitendra, & League, 2009). The focus of mathematics classrooms is shifting
towards fewer in number but more complicated real-world problems. The emphasis of
special needs math teachers towards basic questions is widening the ability gap between
special needs student and their peers without math disabilities (Bjorklund, 2012; Schulte
& Stevens, 2015).

15
Fraction operations are challenging for teacher delivery methods as well as
student learning engagement (Shin & Bryant, 2015). This area of mathematics, and the
proper way to teach it, has been the focus of current math research (Agrawal & Morin,
2016; Shin & Bryant, 2015). Effective strategies often included a combination of
concrete and visual representations; explicit and systematic instruction; and the use of
real world problems (Agrawal & Morin, 2016; Shin & Bryant, 2015).
Use of Computer Assistive Instruction
Other strategies currently used by math teacher include the use of computer
assisted learning and the general increase in creating problems that are directly relevant to
students (Bouck & Flanagan, 2009). Currently, the study of probability is being used as a
means to relate fractions to real world mathematics and forms the basis for one of the
units developed in this project.
Research has highlighted some key factors in successful implementation of
technology in mathematics education and among these is the ability to maintain student
attention. This has been shown to be a key factor in the early years of mathematics
instruction (Tezer & Kanbul, 2009). The internalization of the key concepts of math
learned in early grades has the benefit of improving the ability of math learners to
participate in the more complicated problems of advanced mathematics. Without these
basics at a high enough level of competency, if not mastery, math learners cannot fully
participate or conceptualize the real-world problems that are increasingly the focus of
mathematics classrooms (Tezer & Kanbul, 2009).
Powell (2015) also noted that a key factor of mathematics instruction is assessing
and responding to assessment and that current technology can greatly assist in this

16
particular endeavor. Increasing the amount of self-correcting practice and immediate
feedback that occurs with computer instruction can therefore be a valuable asset to
teaching mathematics. In addition to the advantages of using computers in math
instruction, all learners (including learners with dyscalculia) benefit from immediate and
constructive feedback. Unfortunately, constraints on achieving this level of feedback
exist in the classroom (Hutchinson, 2017). Using computer programs has the potential to
alleviate misconceptions that can develop. The ability of students to independently
practice outside the classroom is also opened up with computer based math instruction
(Hutchinson, 2017), as well as allowing the development of additional practice time.
Additional practice time is seen as a key factor in the approaches used to improve the
skills of learners with special needs (Hutchinson, 2017; Winzer, 2007).
In terms of manipulating a computer representation or manipulating a physical
object, it was found by Loong (2014) that a combination of both methods yielded better
results than either option on its own. A variety in the presentation of mathematical
concepts, so long as it involves manipulation and participation, improves student
understanding of mathematics. The selection of these virtual manipulatives needs to have
three (3) key factors considered, which include mathematical, cognitive, and pedagogical
fidelity must be considered for the manipulatives (Loong, 2014). Mathematical fidelity is
the degree to which the virtual environment represents the mathematical properties of the
objects being manipulated and this can be seen as the mathematical validity of the
representation being shown (Loong, 2014).
Cognitive fidelity is the degree to which the manipulatives represent student
understanding and thinking (cognition) of the math concept; this is a measure of how

17
much the manipulative parallels the student’s own understanding of the concept (Loong,
2014).
The third type of fidelity, pedagogical, is the degree to which the manipulative
represents the teachers own methods and therefore the understanding that has been
conveyed in the classroom (Loong, 2014). While these factors are being applied to virtual
manipulatives, the same principles to apply to physical manipulatives as well and are
important factors to consider when using any manipulative or instructional example.
Any computer program or game that is used will need to have great care given
during its programming to create something that uses proper math etiology and pedagogy
(Bakker, van den Heuvel-Panhuizen, & Robitzsch, 2016). An important fact to consider
is the use of games as a math pedagogical approach and that games can prove to be
statistically significant in the improvement of student understanding, but caution must be
taken to properly create such a program from its inception to its clinical practice (Bakker,
van den Heuvel-Panhuizen, & Robitzsch, 2016). Indeed, teachers should take the time to
familiarize themselves with any program that is used and be able to base their decision on
pedagogy that they use in the classroom.
Just as important as the choice and use of technology in the teaching of math is
the teacher training in such technology (Bouck & Flanagan, 2009). The use of technology
as part of education should be infused throughout teacher education services as a means
of broadening its application in teaching practice (Bouck & Flanagan 2009; Kitchenham,
2011).
Shin and Bryant (2017) reported that computer programs with corrective feedback
and sequential difficulty have been shown to improve student performance. The authors

18
reported that this factor would be a deciding factor in the selection of math programs for
classroom use. To this end, the IXL math program was chosen to support students in this
project.
Math Pedagogy in Special Education
The mathematical instruction of students in special education is a growing
concern due to different philosophical and pedagogical approaches to mathematics
instruction, the support given to mathematics instruction, and the reinforcement of the
mathematics instruction (Bjorklund, 2012; Dieker & Rodriguez, 2013; Ekstam,
Korhonen, Linnanmaki, & Aunio, 2017; Griffin, Jitendra & League, 2009; Sayeski &
Paulsen, 2010).
Students in special education are already at a deficit when compared to other nonspecial-education students in elementary, middle, and high school grades. More
traditional math pedagogies are not inclusive of these special needs, especially for those
students who have IEPs (Hutchinson, 2017; Winzer, 2012). For most teachers, the focus
of math pedagogies has been the constructivist approach which continues to current times
(Boyd & Bagerhuff, 2009). The traditional process for instructing higher-need
exceptional students has traditionally been approached from a behaviourist perspective
(Boyd & Bagerhuff, 2009). These two approaches can be seen as contradictory in several
respects.
Chong and Siegel (2008) conducted a longitudinal study of a group of 214
children across the second and fifth grades. The study investigated two types of
computational deficits: procedural deficits and fact fluency deficits. Both of these deficits
are strongly associated with math learning disability. The constructivist approach of

19
many math teachers involves establishing an algorithm or formula that requires certain
pieces of information (Chong & Siegel, 2008). This approach focuses on the procedural
skills of mathematics and it was found that the gaps in procedural knowledge of the
students lessened between the second and fifth grades. The author stated that if these
pieces of information are correctly discerned and entered into the algorithm, the correct
answer is given. Initially, only the relevant pieces of information are given to allow for
practice. As students become more proficient, more information is given that requires
additional algorithms to create the relevant pieces of information (Chong & Siegel,
2008). These added steps often build on the math knowledge gained in previous lessons
or school years. Additionally, questions can begin to contain extraneous pieces of
information that need to be filtered out of the question to obtain the correct solution.
These multistep approaches and extraneous pieces of information can cause problems for
special needs learners who struggle with short-term and working memory deficits (Chong
& Siegel, 2008; Garrett, Mazzocco & Baker, 2006). The fact fluency deficits were found
to be more stable when compared to the progress made in the procedural deficits for the
same children (Chong & Siegel, 2008).
A discrepancy exists between the more constructivist approach in the main-stream
classroom and the support that is given in the special needs classroom. Math support
provided by teachers to learners with special needs in classrooms is focused on practiced
behaviour that can be internalized through enough practice (Swanson, 2015; Zheng,
Flynn, & Swanson, 2013). For special education students in lower grades, most
pedagogical practices of teachers often involve the use of manipulatives in teaching basic
concepts and algorithms with information given in a very consistent manner, with little or

20
no deviation throughout the unit (Griffin, Jitendra, & League, 2009). While this method is
in line with older methods of teaching mathematics it has had very little success with
providing special needs students with the ability to close the gaps in their education and
allow those students to join with their peers in advanced level mathematics courses or
pursue more mathematics oriented careers (Swanson & Jerman, 2006).
Approaching mathematics instruction from both perspectives and amalgamating
the methods of both constructivist and behaviourist approaches could have the potential
to benefit special needs learners in the classroom (Boyd & Bagerhuff, 2009). This may
also have the benefit of leveling the playing field and circumventing the need to retrofit
lessons based on IEP goals for special education learners in all grades. Although pullout
instruction may be beneficial for some special education learners in math lessons,
especially as added practice, it can also limit further interaction with peers, which may be
a limitation to learning socialization and collaboration skills (Zheng, Flynn, & Swanson
2013).
Teacher use of more complex problems, and the learning of skills necessary to
sort out relevant information which can be used to solve a desired solution, is a typical
approach to teaching many math topics. This is seen as the most “real world” approach to
mathematics instruction (Shin & Bryant, 2015). The mathematical demands of the
workforce require that students have these skills and this method is becoming much more
common place in the math classroom, especially at the high school level, where
community living transitioning strategies are taught to special needs students (Hallahan,
Kauffman, & Pullen, 2015). This differs from the traditional teacher approach of
providing a set of basic learning problems for students to practice in isolation of the

21
situations where algorithms and formulas might apply. The literature supports the idea of
authentic practice and real world application in math class (Shin & Bryant, 2015).
Math that is authentic has meaning for the student, regardless of practical use
(Shin & Bryant, 2015). The drawing of proportional shapes on graph paper may have a
purely aesthetic appeal for students, but it provides the representation of math with an
artistic meaning, thereby linking the discipline to other disciplines such as the arts, one of
the goals of the New BC Curriculum (BC Ministry of Education, 2016). This has
relevance to this project because this approach was used in the creation of several
assignments and activities in the units that were created. Artistic expression of
mathematical concepts was used to improve student interest and investment in the
mathematical concepts of the lessons.
Real world math has a clear and obvious practical purpose for the student,
especially Low Incident Exceptionality (LIE) special needs students at the middle and
high school ages, who are prepared for community transition living. Using Math in real
life situations, such as the purchasing of items at the best price, is a practical learning
outcome for the students (Hallahan, Kauffman, & Pullen, 2015; Hutchinson, 2017). Math
content such as multiplying the sides of shapes to determine area has practical purposes
for construction and other trades, where some special needs adolescents at the middle and
high school level may find themselves (Hallahan, Kauffman, & Pullen, 2015). The above
are both authentic and real world practice for students, which helps to maintain student
interest in math topics (Bjorklund, 2012; Boyd & Bagerhuff, 2009).
Intervention at the neuropsychological level that target specific cognitive deficits
rather than tackling mathematical deficits directly has been shown to improve

22
mathematical performance (Faramarzi & Sadri, 2014). The researchers examined a group
of 30 second grade girls with dyscalculia. Using a pretest, posttest and control group, the
researchers used the Wechsler Intelligence Scale for Children (WISC-III-R) and the
KeyMath assessment. The study found a statistically significant different between the
experimental and control groups at the p ≤ 0.001 level. This idea that neuropsychological
interventions can assist students with dyscalculia opens the door to several types of
interventions that can be undertaken in the special education classroom. Math exercises
could be done at home with the support of family members and would not be
academically demanding for parents or guardians of students. Faramarzi & Sadri (2014)
found the following four domains to influence math interventions for learners, especially
the special education students.
Reinforcing active memory. Poor active to non-active memory are some of the
exceptionalities experienced by many students experiencing Math difficulties as well as
those diagnosed with dyscalculia in middle to high schools across Canada (Hutchinson,
2017; Winzer, 2013). Various studies recommend learning activities that support this
cognitive aspects which include recognition memory of hidden objects, while other
teaching-learning strategies include the improvement of visual and auditory memory.
Active use of memory in the forms of matching games using flashcards can strengthen
this area of cognition and improve the ability of students to remember math facts quickly
and automatically (Faramarzi & Sadri, 2014). By improving automaticity for memory
recall, the working memory limitations of MD students can be eased and allow for the
advancement of mathematical understanding (Faramarzi & Sadri, 2014).

23
Reinforcing attention. These activities include looking at a picture where an
object is obscured by other objects; books and games that involve looking for objects in a
cluttered image help to improve a student’s ability to attend to certain features amongst
seeming chaos (Faramarzi & Sadri, 2014). For auditory deficits, a similar activity can be
undertaken using recordings where a particular message needs to be interpreted amidst
background noise (Faramarzi & Sadri, 2014). The applications of these teaching and
learning strategies are beneficial to special needs learners, especially those challenged by
attention deficit as ADHD, Indigo and Spinal Bifida students (Hutchinson, 2017; Lipson,
2004; Rose & Holmbeck, 2007; Stegeman & Aucion, 2018).
Training executive functions. Most special needs learners are challenged by
limited executive functions such as organizing and planning (Faramarzi & Sadri, 2014).
These functions can be reinforced in math pedagogy by having students at the lower
grades group blocks according to size, colour, thickness, or any other relevant criteria,
and by constructing structures according to a reference model. This will facilitate selftraining of some of these executive functions (Faramarzi & Sadri, 2014). Such math
teaching-learning strategies will not only benefit special needs learners executive
function deficits, but engage them in hands on learning, which enhances the psychomotor
level domains of Blooms taxonomy of learning (Anderson & Krathwol, 2001).
Developing and reinforcing visuospatial perception. Faramarzi and Sadri
(2014) reported several ways that visuospatial perception can be developed, especially in
students with special needs. These include balancing and hand-eye coordination exercises
that will improve this perception. Identifying objects without the use of the eyes (with the
exception of visually impaired special needs learners) can serve to improve visuospatial

24
perception (Faramarzi & Sadri, 2014). Graph paper is commonly used in the math
classroom and the practice of copying models using graph paper can increase perception
while making students familiar with the use of graph paper to maintain organization in
drawing and writing (Faramarzi & Sadri, 2014). While these math teaching strategies are
good and effective, it excludes some special needs learners as mentioned above, hence
alternative inclusive strategies must be sought by teachers.
Reinforcing skills related to speech and language in math. Games that improve
phonological awareness are abundant and are often played by parents and children. In this
theory of intervention, these phonological games not only serve their directly intended
purposes of improving language, but may also improve the student’s math ability
indirectly (Faramarzi & Sadri, 2014). Simply improving knowledge of the meaning of
mathematical words and concepts by reading about them can serve as a means of
improving the familiarity with the language of mathematics (Faramarzi & Sadri, 2014).
As students improve their fluency in the language of mathematics, they will improve their
ability to converse in and manipulate this language to achieve a solution (Faramarzi &
Sadri, 2014). Furthermore, Breaux et al. (2017) reported that naming automaticity of
objects was found by to be a strong predictor of math computation ability. The author’s
report supports the link between language and math ability.
Building all of these previous exercises and practices into the curriculum of preelementary, elementary, and to some extent middle and high school programs creates and
environment that promotes student success in both reading and mathematics without
drastically increasing academic demands (Faramarzi & Sadri, 2014). I found the use of
these neurological interventions that were seemingly unrelated to math useful as a means

25
of approach mathematical concepts with students who considered themselves to be
innately poor at math. Practicing seemingly unrelated skills allows for students to feel
success and progress in my classroom and the research by Faramarzi and Sadri (2008)
strongly influenced the development of several lessons and activities in the units created
in this project.
When cooperative teaching is used, emphasis should be placed on the “why” of
particular pedagogy. This is important so that a compromise can be met between the
methods of the teachers that makes full utilization of both teacher’s areas of expertise
(Speijer, Gray, Peirce & Doherty, 2016). Both the classroom and special needs teacher
need to consider the reasoning behind the type of instruction being given so that
achievable and relevant goals can be made. When the “why” is addressed, then
consideration can be given to the various ways to achieve the final goal (Dieker &
Rodriguez, 2013). How students demonstrate their learning will vary based on the
essential goals outlined by the teachers. When both parties explain to each other why
certain math methods are used, there can be an appreciation of the perspectives that will
be important in reconciling any apparent contradiction in instructional methodology. If a
student is receiving contradictory math teaching methods of instruction this will lead to
increased confusion for the student, especially for learners with special needs (Ekstam,
Korhonen, Linnanmaki & Aunio, 2017). This is detrimental to any student but is
exacerbated by a special needs student’s inherent difficulty in discerning key and salient
information (Sayeski & Paulsen, 2010).
Special education teachers and classroom teachers need to recognize that special
education need not be a permanent designation. In many cases students are in special

26
education for only part of their educational career, either as a result of successful
intervention, time limits, or late diagnosis (Schulte & Steven, 2015). The authors
conducted a statewide longitudinal study using the database available to examine students
diagnosed with a disability and students with a diagnosis across the third and seventh
grades. Gaps between the groups widened over time as students were diagnosed over
their time in school and were then placed in the diagnosed group. The use and
implementation of successful strategies for students can have a very real impact on their
educational trajectories. Any educational plans made for students should be made not
only for the specific class or school year but put in the context of the student’s entire
mathematical education (Schulte & Steven, 2015). I found that this study highlighted a
need to be aware of the diagnostic tests that students are subject to so that I as an
instructor can be better prepared to spot discrepancies in student achievement and tailor
my lesson to better prepare students for diagnostic assessment.
There is a need to alter the approach of special educators towards special
education. The ability gaps between learners with and without special needs widen as the
students proceed through the school system. Some research suggests that traditional
methods of mathematics instruction or special education instruction in general have netzero or even detrimental effects on special needs students’ ability in mathematics
(Speijer, Gray, Peirce, & Doherty, 2016). This study was professional learning project for
Grade 9 and 10 Applied Mathematics classroom with the intent of improving
communication between the Regular Classroom Teacher and the Special Education
Resource Teacher. Improving communication between these main-stream classroom and
special education teachers was seen as a possible way improve the understanding of

27
students and make the most use of teachers relevant fields of expertise. For the purposes
of this project, this study highlighted the need for myself as a Regular Classroom Teacher
to bring the knowledge I have gained in my studies in a Special Education Program and
attempt to bridge the gaps between the two positions.
The current trend towards co-teaching is not without both merit and concern. The
one lead/one teach method is the most common in the secondary years (Dieker &
Rodriguez, 2013). In this situation one teacher has the specific content knowledge and the
other assists either during instruction or immediately after during student practice (Dieker
& Rodriguez, 2013). A drawback of this method is that one teacher is often relegated to a
subordinate role that can be due to ignorance or a lack of comfort with the material being
taught. This can be a problem when considering that teacher comfort and confidence with
material is correlated with student success (Ekstam, Korhonen, Linnanmaki, & Aunio,
2017). In these situations special needs teachers often fall into the role of soft-skill work
such as homework practice and organizational skills (Rosa and Holmbeck, 2007). While
these skills are necessary, they are not immediately relevant to the content at hand and
can distract from the immediacy of what is being taught. A work-around for this problem
can be the improved implementation of co-operation during the planning phase rather
than the instructional phase. This avoids having the special needs educator being
relegated to soft-skill work (Dieker & Rodriguez, 2013).
It is important for teachers to predict and recognize specific problems that student
with disabilities may have and which assessments and interventions are available for
those disabilities (Boyd & Bagerhuff, 2009). Technology can play a significant role in
these interventions (Brownell, Sindelar, Kiely, & Danielson, 2010).. The ability and

28
willingness to scale back on concepts to the level of the student has shown success in
improving student understanding of math concepts in special education settings
(Björklund, 2012).
The method and level of communication for students is an important factor in
special education (Hallahan, Kauffman, & Pullen, 2015; Winzer, 2010). There are many
speech and communication deficits that students may have and this may have a greater
impact than just the language barrier to communicating mathematical concepts. Use of
egocentric speech and talking through problems to themselves is one way that students
can learn mathematical concepts, especially at the lower elementary and middle school
grades (Stegemann & Aucion, 2018). In this way, a student slowly internalizes external
ideas as the speech becomes internal and automatic (Vygostky, 1978). Some students
may communicate through gestures and these gestures can serve as a type of egocentric
speech (Hutchinson, 2017; Stegemann & Aucion, 2018). These gestures may be smaller
and more subtle than other types of gesture for communication and it is important to
encourage the use of such gestures as a means of working through math concepts towards
improved understanding. Many of the more visuospatial concepts of mathematics, such as
fractions, may need such external movements to allow for understanding to occur (Zurina
& Williams, 2011).
Based on a study using meta-analyses by Ketterlin-Geller, Chard & Fien (2008),
six (6) instructional strategies emerge as potentially beneficial for students with
disabilities: visual and graphic depictions; systematic and explicit instruction; student
think-alouds; peer-assisted learning; formative assessment data provided to teachers;
formative assessment data provided directly to students (p. 35). These instructional

29
strategies were used in the creation of two intervention groups for a study involving 52
fifth-graders in the Pacific Northwest. Low-achieving students were divided into three
groups: small group instruction aimed at highlighting student’s misconceptions, extended
time and support for concepts taught, and a control group.
The strongest support for the above six areas of intervention can be found for
visual and graphic depictions, systematics and explicit instruction, and formative
assessment data that is quickly relayed to both teachers and students (Ketterlin-Gellar,
Chard & Fien, 2008). Peer assisted instruction as a math pedagogy failed to have large
effect sizes during a literature analysis by Zheng, Flynn, and Johnson (2013). However,
whether peer-assisted instruction has an impact on learners with special needs was not
clarified by the authors.
There has been evidence provided in a paper by Elliot, Kurz, Tindel and Yel
(2016) that indicates special education students are receiving equal opportunity-to-learn
time in mathematics when compared to their peers. This study used a group of 78
teachers and had them keep instructional logs daily with regards to a group of 162
students with learning disabilities and 165 students without disabilities. The instructional
logs were examined to determine opportunities that students had to learn. There is
evidence that not all of the math instruction given to special needs learners is positive or
beneficial (Sullivan & Field, 2013).
The beneficial assistance that students with math disabilities receive can be
limited. When compared to out-of-class tutoring that is received, students with a reading
disability report far more tutoring assistance than their peers with a math disability
(Willcutt, Petrill, Wu, Boada, DeFries, Olson & Pennington, 2013). This highlights the

30
need to address the problems of students with math disability in-class and improve the
quality of in-class math instruction.
The net negative effect seen in mathematics achievement may be the result of the
more social aspects of education focused on in special education. Because there is an
emphasis on social skills development, math skills are not at the forefront and so suffer
from less time and practice. This could result in the net negative effects seen in special
education at the preschool level, which will have a growing effect at the elementary,
middle, and high school levels (Sullivan & Field, 2013). This study made examined
groups of preschool students who received special education services and compared the
achievement of these students to children who did not receive special education services
at the pre-school age. The authors reported that part of the net negative effect on math
achievement can be seen as a result of the behaviourist approach of special needs
teachers. The students in question did see a decrease in social behavior problems that are
the focus of the behavior intervention. Sullivan and Field (2013) stated that the emphasis
on the behavior problems left less time for intensive and direct mathematics instruction,
which could have impacted the gains in mathematical understanding. A notable finding
was that there was a statistically significant increase in learning related behaviours that
would not be directly reflected in academic achievement (Sullivan & Field, 2013). Again,
this discrepancy in achievement could be reflective of an underlying behaviourist
approach to special education (Morgan, Frisko, Farkas & Hibel, 2010). While this
approach is not invalid and does have some positive effects on student learning
behaviour, the approach needs to be adjusted if math performance is to be improved. I
used this concept of potentially doing harm with my instructional strategies to carefully

31
examine my approaches in the construction of my math units. I took into account the
difficulties and misconceptions that learners with special needs may have in their
understanding of mathematics. It was important for me to create lessons that did not
hinder a student’s future progress or ability to achieve in mathematics.
Emphasizing the most universally applicable algorithms and teaching as few of
these algorithms as possible has the potential to improve student performance (Sayeski &
Paulson, 2010). Such algorithms include cross-multiply and divide for the solving of
many types of proportions. The breaking down of questions into clearly distinguished
steps can ease the demands on working memory for students with a math disability
(Swanson, 2015; Swanson & Jerman, 2006). The generalizability of students with a math
disability may be limited and applying learned processes in novel ways may also be
limited. Explicit instruction of the steps required may assist these students in overcoming
mental deficits in problem solving ability (Seo & Bryant, 2012). Teachers should
emphasize the instruction of the most applicable algorithms to improve student
understanding and allow them to apply what they know to the widest range of situations.
While mastery of algorithms is not the intention of the math classroom, there has been
success when lowering the amount of information necessary for the special needs
students to remember. Allowing for mastery of fewer skills may lead to more math
competency and literacy (Sayeski & Paulsen, 2010; Shin & Bryant, 2015). If done at an
early enough level, there is the potential to lessen the ability gap and improve confidence
to the point where more students with math disabilities can participate in higher level
math courses without the need for severe time and resource investment on the part of
schools, families and students.

32
Criteria for Diagnosing Math Disabilities
It is estimated that between 5 and 10 percent of all students have a math disability
(Xin and Tzur, 2016, p. 196). Other research estimates the number of students with MD
at between 3 and 8 percent or possibly between 6 and 7 percent (Swanson & Jerman,
2006, p 250). The diagnosis of mathematics disability has a long history in both its
diagnosis and treatment. It is important to recognize the growing schism between the
treatment or pedagogy associated with mathematics disability education and the means by
which we determine if an individual has such a need for treatment (Xin & Tzur, 2016).
Traditionally, a math disability, or MD, diagnosis was the result of a discrepancy between
math score assessment and overall intelligence scores, specifically the results of IQ tests
(Garret, Mazzocco, & Baker 2006, p. 79). This has been found to not have strong validity
in two literature analyses (Hoskyn and Swanson, 2000; Stuebing, Fletcher, LeDoux,
Lyon, Shaywitz & Shaywitz, 2002). These faults are the result of the validity of IQ tests
as they relate to clinical teaching practice; the inability to detect discrepancies between an
underlying cognitive factor related specifically to math as opposed to an overall cognitive
discrepancy; and the assessment itself not being built to validly test specific domains of
cognition as they relate to various aspects of mathematics ability.
Validity of IQ tests in the measure of math disability. The overall use of testing
student IQ has had questionable use in teaching practice (Dekker, Ziermans, & Swaab,
2016). A general intelligence test does not directly inform teaching practice and the use
of the score to the classroom teacher as a guiding tool is limited as a result of this
(Dekker, Ziermans, & Swaab, 2016). Dividing mathematical intelligence into several
factors allows for examination of the interaction of these aspects and this is useful from a

33
research perspective. The most effective predictor of math ability is the combination of
these factors into a general intelligence score (Markon, Chmielewski, & Miller, 2011;
Parkin & Beaujean, 2012). In addition, Parker and Beaujean (2012) suggested that this
general intelligence score can then be used to guide decisions for instruction or
assessment, especially when developing IEPs for students with math disabilities
(Stegemann & Aucion, 2018).
Underlying cognitive factors or an overall cognitive discrepancy. Two studies
have examined the differences in ability between students with an MD (math disability)
diagnosis, RD (reading disability) diagnosis, MD + RD combined, and students without
any diagnosis (Foster, Sevcik, Romski & Morris, 2015; Willcutt, Petrill, Wu, Boada,
DeFries, Olson & Pennington, 2013; Zheng, Flynn, & Swanson, 2013). The concerns are
that there may be a relationship between cognitive domains affected for math and reading
diagnoses (at least in certain cases) and that these domains are not being tested for in
current methods. Attempts to improve the diagnosis and address the underlying cognitive
domain could improve teaching practice and allow for more gains in mathematical
understanding. Comorbidity with other disabilities also makes accurate diagnosis difficult
(Branum-Martin, Fletcher & Stuebing, 2013). The authors have suggested that there is a
continuum of severity and this calls into question the need to separate certain learning
disorders into categories. This was supported by Chong and Siegel (2008) who also noted
that math disability be seen as more of a continuum that an orthographic category. This
opinion resulted from findings that memory-retrieval deficits were relatively stable over
time while procedural deficits improved for learners with a math disability over the same
period of time. Research methodologies that allow for multiple means of solving

34
problems may allow for differential procedures that allow learners with a math disability
to compensate for cognitive deficits through the application of particular math algorithms
(Chong & Siegel, 2008). Though there is a high level of relationship between reading
disability and math disability, not all students show both disabilities. Students diagnosed
with only a reading disability report lower academic success that those with only a math
disability (Willcutt, Petrill, Wu, Boada, DeFries, Olson & Pennington, 2013).
An interesting connection between reading and math disabilities was reported by
Foster, Sevcik, Romski and Morris (2015) where a significant relationship between
phonological awareness and naming speed with mathematical achievement was found.
While naming speed has been seen as subsidiary to phonological awareness in
determining math ability, both had varying degrees of significance in different areas of
mathematics. The naming speed of colours was consistently related to success on addition
and subtraction tasks. Math fact retrieval was seen to trigger the same area of the brain as
language processing and this could strongly influence the ability of students in certain
areas of mathematics such as Time, Money, and Geometry units that have a strong verbal
language component. Even the ability to detect rhyme and alliteration at age 3-5 years
was related to math achievement at age 6 years (Foster, Sevcik, Romski, & Morris,
2015).
Valid detection of mathematical relevant cognitive domains. There needs to be
consideration of the clinical applications of a diagnostic test and the focus on improving
student achievement. Basing the diagnosis on a variety of factors has been suggested by
Branum–Martin, Fletcher, and Stuebing (2013). The authors compared this diagnosis of
math disability to the diagnosis of those factors that determine obesity, which is

35
diagnosed around multiple factors that determine if a given individual is obese (height,
weight, and other biometrics). Scores may be similar between non-diagnosed and
diagnosed individuals but it is the set of scores which warrant diagnosis and therefore
treatment. For mathematics, these domains may include factors such as fact retrieval,
procedural knowledge, verbal understanding, etc. Students with a math disability may
score similarly on a given task to students without a math disability but the suite of scores
taken together would indicate a need for intervention (Branum-Martin, Fletcher, &
Stuebing, 2013).
In addition to assessments not necessarily detecting the proper cognitive domains,
there are a variety of assessment tools that are used to determine if a learner qualifies as
having a math disability. These tests are often used in research projects and achievement
criteria are set to determine groupings of students for analysis. The specific criteria for
determining a math learning disability can vary between and even within studies (Chong
& Siegel, 2008). The authors reported that test cut-off criteria for determining whether
students do or do not have a disability can vary from 10th percentile to 11th-25th percentile
and that this shift can significantly affect results (Chong & Siegel, 2008, p. 309).
While not denying the existence of math disabilities, there is concern over both
the range of characteristics tested for in diagnosing math disability (Branum-Martin,
Fletcher, & Stuebing, 2013) and the relatively fluid achievement range and cut-off points
that determine whether learner is considered to have a math disability (Chong & Siegel,
2008).
Affected domains of cognition and other characteristics of math disability.
Math disabilities are seen as being persistent deficits in mathematical achievement

36
separate from other academic areas. Any math deficit would be separate from overall low
intelligence (Garrett, Mazzocco, & Baker, 2006). The most common deficits of
individuals with an MD diagnosis are visual-spatial deficits and the inability to switch
between different mathematical operations. These are only two of the most common
deficits noted in for students with MD but have severe implications for mathematical
ability.
Visual-spatial awareness has severe implications for the more concrete aspects of
fraction understanding and geometry. A common method of mathematics education
pedagogy is that of concrete/representational/abstract with the implication that concrete is
the area most easily understood and forms the basis and bedrock of further understanding.
The hampering of concrete understanding by lack of visual/spatial understanding causes a
disconnect with the commonly accepted and empirically supported method of teaching
most mathematical concepts. This issue leads to the need to fundamentally change the
way that a concept has been taught successfully in the past. Basic numerical processing
shares resources in the brain with areas that are commonly associated with spatial tasks
(Stanescu-Cosson, Pinel, van De Moortele, Le Bihan, Cohen & Dehaene, 2000). Lack of
spatial awareness would highlight a cognitive deficit that is directly linked to cognitive
resources needed for efficient mathematical ability.
The inability or delayed ability to switch between operations is a significant
detriment to the math ability of students with a math disability (Garret, Mazzocco, &
Baker 2006). This diminishes the student’s ability to solve the complex real-world
mathematical problems that are increasingly becoming the focus of math classrooms at
all levels of instruction. Current math pedagogy increasingly focuses on the idea that real-

37
world situations and the relevant math will have the greatest impact and connection to
students and increase student interest and therefore student mathematical ability. A
difficulty arises from this approach based on the reality that “real-world” problems have
multiple steps with numbers that are not always clear and may require rounding with the
need to understand the need for multiple operations (Garret, Mazzocco, & Baker 2006).
An inherent difficulty or inability to switch between multiple operations will greatly
hinder the ability of individuals with a math disability to independently complete such
work, even if the individual were given minimal assistance (Garret, Mazzocco, & Baker
2006). A subset of this problem is an inability to filter out relevant or salient information
from a situation. Again this hampers an individual’s ability to complete a complex realworld problem as there is often an abundance of information in the real world and the
filtering of such information is an important skill in daily mathematical functioning.
Mathematical awareness can be split into the two distinct groups of symbolic and
non-symbolic numerical information (Furman & Rubinsten, 2012). Symbolic structures
include such things as Arabic numerals where the visual information is not directly linked
to the magnitude of the concept or object. Non-symbolic numerical representation is
recognizing the amount of objects in a group or differences in groups of objects and
seeing that one group is larger than another (Furman & Rubinsten, 2012). Non-symbolic
representation can further be broken into two (2) different ranges; the subitizing range
includes small numbers (1-4) where the recognition is automatic and quick and the
counting range which includes larger numbers and recognition is accomplished serially
and slowly (Furman & Rubinsten, 2012). Students with MD seem to have difficulty in
automatically associating symbolic and non-symbolic understanding (Furman &

38
Rubinsten, 2012). This is to say that a symbol representing even a small subitizing
amount is not automatically processed as quickly as seeing the corresponding number of
objects.
Neurological dyscalculia is the term used when a defect or deficiency may have
developed at the neurological level (Faramarzi & Sadri, 2014). Support for this type of
dyscalculia can be seen in the presence of abnormal brainwaves in students with learning
disabilities. There are several subtypes of this dyscalculia. There may be a deficit in
verbal semantic memory and connecting meaning to words. This makes it difficult for the
student to quickly or accurately recall math facts or conceptual meanings. A second
subtype of neurological dyscalculia is using developmentally immature methods of
problem solving and making frequent errors in basic calculation. Examples of
developmentally immature methods would be finger counting or simple repeated addition
to solve larger multiplication problems. The third subtype of dyscalculia is a visuospatial
deficit that leads to improper column placement and place value errors. This type of error
can lead to problems in mathematical calculations especially for longer problems where
organization is key to correctly solving a problem.
A specific area of difficulty for students with MD is the telling of time (Burny,
Valcke, & Desoete, 2012). These problems can be the result of multiple number scales
and also deviation from the base-10 numbering system (Burny, Valcke, & Desoete,
2012). Telling time requires a great deal of verbal knowledge and so is strongly linked to
a student’s phonological processing. Burny, Valcke, & Desoete (2012) studied 725
students from Grades 1-6 and found that students diagnosed with math disabilities
consistently had difficulty with the tasks of telling time. These tasks included stating

39
analog reading times, stating digital reading times, transforming between digital and
analog times, and writing times represented by written or spoken phrases (quarter past
12) (Burny, Valcke, & Desoete, 2012, p. 356). The authors reported that being alert to
time telling ability can serve as a basic warning sign of math disability at elementary and
middle school ages (Burny, Valcke, & Desoete, 2012).
An individual’s available working memory can determine what problems or level
of complexity they are best able to work at. Focusing on working memory from a lesson
planning perspective can give students an ability to approach subjects that may otherwise
be overwhelming and frustrating for them. Use of tools that supplement working memory
may also be an important intervention for those students that have deficits in this area
without the need for major interventions that differentiate them from their peers. A
notable finding of some of the research was that students without a math disability
showed no sizeable difference in effect size between methods of instruction (Swanson
2015). This may prove important as it shows the possibility that some teaching
methodologies may be harmful or overly difficult for MD students while other methods
greatly assist them but neither method disadvantages those students without a math
disability.
According to the studies of Stanescu-Cosson, Pinel, van De Moortele, Le Bihan,
Cohen and Dehaene (2000) specific areas of the brain that are active during rote
mathematical calculation are also active during verbal communication. Approximation
and exact calculation of large numbers activated different areas of the brain, which would
typically encode numbers in a non-verbal format. This strongly implies that there are two
systems at work in the mind. One system exists for fast approximation processes and

40
another for slower and more exact calculation processes. Certain subtypes of dyscalculia
can be explained by lesions affecting one of these networks disproportionately (StanescuCosson, Pinel, van De Moortele, Le Bihan, Cohen & Dehaene, 2000).
The mathematical research of Hale, Fiorello, Dumont, Willis, Rackley and Elliott
(2008) highlights the need to know a learner in all aspects. One of these aspects is the
domain of time. Teachers need to monitor a student’s progress across not only the current
school year but across several years and to plan accordingly. The inherent persistence of
MD, which is also a key factor of its diagnosis (Chong and Siegel, 2008), means that
gains will be small but significant and any gains should be examined and expanded upon
to make the best use of the child’s inherent strengths. Patterns of performance are key to
ideographic interpretation of interventions that will work for that child. These are ignored
if examining general scores or assessments in isolation (Hale, Fiorello, Dumont, Willis,
Rackley & Elliott, 2008).
A number of studies as reported in Zheng, Flynn and Swanson (2013) have
reported an overall pattern in effective math instruction for MD students. The authors
conducted a literature review of eight group and seven single-subject studies. The
participants of the study were organized into learners with a math disability, learners with
both a reading disability and a math disability, and a control group. While there is
consensus on limited research in the area of math disability diagnosis and treatment, it is
important to collate and make sense of what information currently exists. In general, it
was reported that high effect sizes included the following components: stating
instructional objectives and/or directing students to focus on particular information
(advanced organizers); fading prompts and/or providing necessary assistance (control

41
difficulty); explaining underlying concepts and/or providing repetition within text
(elaboration); distributing review and practice (explicit practice); engaging students in
dialogue and asking questions (questioning); sequencing short activities (sequencing);
skill modeling; reminding students to use instructed strategies or procedures (strategy
cues); breaking down the targeted skill into small units (task reduction) (Zheng, Flynn, &
Swanson, 2013, p. 108). I found that this study highlighted some of my own practices in
my classroom. Seeing how each of the key factors was reflected in my own teaching also
highlighted those factors that I tend to not focus on. I have increased my focus on all of
these areas as a result of this paper and have taken them into account in the construction
of my teaching units.
The listed factors above have been explained and reported by Zheng, Flynn, and
Swanson (2013) as making their way into math classrooms that make use of evidence
based pedagogy. These factors can be found to support several of the methods used in
this project to improve mathematics learning and understanding for middle school aged
learners with special education diagnoses.
Tests for Determining Math Disabilities
The assessments used to determine a math disability can be seen as belonging to
one of two groups: diagnostic tests or screening tests. Diagnostic tests are thorough
construct that determine a student’s mathematical ability across a wide range of math
topics. These tests determine that a student has a math disability. Screening tests are
simpler and briefer tests that determine the need for further assessment of an individual.
These tests do not determine that an individual has a math disability.

42
Diagnostics tests, because of their educational implications, must be properly
constructed and administered. A properly constructed diagnostic assessment will directly
test the differing underlying cognitive domains that can be connected to math
achievement. A test should examine a student’s level in these domains and create a
profile of what deficits there may be in the student’s cognition so as to guide intervention.
As discussed earlier, a diagnosis of an individual should also take into account the
treatment for that individual.
Screening tests are used at the beginning of the assessment process to guide the
decision making process (Erford & Biddison, 2006). The authors reported that decisions
for further assessment using diagnostic tests or more careful classroom monitoring by
teachers, educational assistants, or special education teachers can be made using
screening tests. Early interventions can have significant impacts on future learning and
these screening tests can allow for interventions before serious problems can manifest
(Erford & Klein, 2007). It was reported by Erford and Klein (2007) that intensive and
effective elementary education can lead to improved mathematical performance later in
school.
A distinguishing factor between screening and diagnostic tests is the level of
achievement that the test detects. Screening tests are generally briefer and are designed to
discriminate between low-achieving and typically-achieving students (Erford & Biddison,
2006). High achieving students will score similarly to average achieving students on
these tests and will not usually be distinguished. To separate out the high-achieving
students, it was reported by Erford and Biddison (2006) that more difficult items are
necessary than those typically used in screening tests. Diagnostic tests, because they are

43
much more thorough and specifically examine cognitive domains (Erford & Klein, 2007;
Parker & Beaujean, 2012), can distinguish between all 3 groupings. This ability to
determine more levels of skill comes at the price of being longer tests that require a
greater investment of time to administer and score.
A second difference between diagnostic and screening tests as reported by Erford
and Biddison (2006) is the type of error that is of concern. False positives, identifying
students as needing intervention that do not actually have a math disability, are less of a
concern for screening tests. These students would simply be identified as possibly
needing further assistance and additional observations would likely conclude that only
moderate classroom intervention would be needed instead of a full diagnostic or special
education assistance (Erford & Biddison, 2006). False positives for diagnostic tests result
in a great deal of professional effort to assist a learner that may have only needed
moderate intervention.
Both screening and diagnostic tests are concerned with false negative errors where
students needing assistance are misidentified as not needing assistance (Erford &
Biddison, 2006; Erford & Klein, 2007). It was reported by the authors that false negatives
are important at the screening level and diagnostic level as it eliminates a student from
being a concern for special education services. This can be especially detrimental in
educational systems that already have limited resources and cannot continually reassess
or screen all students regularly (Erford & Biddison, 2006).
Within assessment tests there is the possibility of improving student performance
through the interspersing of easier test items that reinforce understanding (Robinson &
Skinner, 2002). The authors studied thirty Grade 7 Students in a rural southern school

44
district. This study made use of subtests from the KeyMath-R diagnostic assessment. The
subtests were altered to allow for interspersed easier test items. Students were shown to
answer items correctly on tests that had interspersed easier items when they had answered
those same items incorrectly on tests without interspersed easier items. It is important to
note that this method can have differing effects for level of task demand. Verbal
responses, lack of time constraints, and mental computation can be significantly affected
by the interspersing of easier items in the assessment. Test items that fall in the
challenging range (those questions that are not too easy or too hard) were most affected
by the interspersing method. An important educational implication of this research is that
typically low performing students may have a means of showing their true understanding
by having easier items mixed into the test that are related to those challenging questions.
Summative, screening, and diagnostic assessments may thus have their validity increased
by the use of this method of interspersing easier test items. I have applied the information
gained from this study in the construction of my own tests and also in how I approach
assisting students. When students ask for assistance on a given math task, I emphasize the
basics of the question by breaking the problem down into steps. This is a common
method of assisting math learners but is difficult to do during test administration to keep
the testing fair to other students. By placing easier test items before the larger and more
complex questions, I hope to improve student performance on the more complex
problems that immediately follow these simpler and related test items.
KeyMath-3 diagnostic assessment. The KeyMath-3 assessment is a series of 13
subtests split into the 3 general categories of Basic Concepts, Operations, and
Applications. Basic concepts is made up of the Numeration, Rational Numbers, and

45
Geometry subtests. Operations consists of the Addition, Subtraction, Multiplication,
Division, and Mental Computation subtests. Applications consists of the Measurement,
Time and Money, Estimation, Interpreting Data, and Problem Solving subtests.
Combined, these subtests form a robust measurement of a student’s overall math ability
(Williams Fall, Eaves, Darch, & Woods-Groves, 2007). As an overall measurement of
math ability, the KeyMath-3 assessment also aligns with the new BC curriculum goals
with its ability to examine the mathematical literacy that allows students to become
competent citizens (BC Ministry of Education, 2016). The new BC curriculum has
increased emphasis on the development of financial and problem solving ability that the
KeyMath-3 explicitly examines. The KeyMath-3 subtest sections are very closely aligned
with the British Columbian educational outcomes for Grade 8 (BC Ministry of Education,
2016). This makes this grade ideal for using the diagnostic as a planning tool for units for
the purposes of this project.
The KeyMath-3 assessment has two developed forms that have been normed for
Canadian students, which makes it possible to give the assessment twice over a relatively
short amount of time and to examine differences between the two assessments. The
KeyMath approach is useful in any proposed project that will see students receiving a
pre- and post-assessment of math ability after two math units. The different forms of the
assessment will limit the practice effects that could occur if the same version of the test
were used in such a short period of time.
The KeyMath-3 assessment shares many characteristics with the Wechsler
Intelligence Scale for Children (WISC), such as being averaged around a score of 100
with standard deviations of 15 (Parker & Beaujean, 2012; Williams, Fall, Eaves, Darch,

46
& Woods-Groves, 2007). The KeyMath test is also administered in a similar manner with
the establishment of basals and ceilings similar to those found in the WISC (Parker &
Beaujean, 2012; Williams, Fall, Eaves, Darch, & Woods-Groves, 2007).
The IXL math program. This is an online program that offers math curriculum
from grades K-12 that is separated by grade, topic and unit (IXL.com). The program is
currently in use in several schools in the school district where the units of this project
were developed. A student is given an account and their own login and password which is
controlled directly by the classroom teacher. The student accounts are attached to a
teacher administrator account. This allows the teacher to monitor progress and offer
assistance if students are not making progress in a given area. Teachers may also monitor
the progress of students over the course of their time with the program and examine areas
of prior success or difficulty. Schulte and Stevens (2015) reported that the ability to
examine a student’s progress through education is an important factor in making
appropriate educational and instructional decision with regards to mathematics education.
The student selects a topic and grade level and is then given a series of questions. The
questions can be answered as multiple choice or written answer depending on the grade
level selected. Points are awarded as the student gives a correct response. An incorrect
answer results in a loss of points. The goal is to get to 100 points and achieve “mastery”
of the topic. Questions are randomized so there can be no memorization of the correct
answer in a particular order. The ability to memorize answers after trial and error is a
downside of many online computer assisted instruction programs. The validity of the
achievement score in such programs is questionable if the score is used as a measure of
math ability (Bakker, van den Heuvel-Panhuizen, & Robitzsch, 2016). Not only are the

47
questions randomized but they become increasingly difficult. Questions also reward
fewer points as the student increases their score. This allows a teacher to gauge a
student’s relative ability with confidence in a given math topic area based on where the
student finishes at the end of a practice period.
The hundreds of assessments that make of the IXL program are focused around
single concepts that target specific mathematical algorithms or procedures. Zheng, Flynn,
and Swanson (2013) reported that repetitive, guided, and relatively short activities that
make use of task reduction are among the practices that produce the greatest effect sizes
in mathematics classrooms.
A key part of the potential benefits of IXL.com is the immediate feedback upon
answering a question. Immediate and instructive feedback is important in effective
formative assessments for mathematics instruction (Ketterlin-Gellar, Chard & Fien,
2008). The student knows immediately if their response to a question is correct and does
not develop misconceptions that may result in many questions being answered
incorrectly. An incorrectly answered question results in an explanation appearing
onscreen to highlight the error and then explain the full process again. While useful for
all students, this feature has the benefit of being available to parents so they can work
through the solution with their students. The ability to assist parents at home opens the
possibility for more assistance and practice out of the classroom, which can be a problem
for mathematics instruction and practice (Brownell, Sindelar, Kiely, & Danielson, 2010;
LeFevre, Polyzoi, Skwarchuk, Fast & Sowinsky, 2010). Increased practice can help some
learners and a major barrier to this can be that the only adult available with the fluent
knowledge to complete the math may be the math teacher. By having the explanation

48
highlighted and available on screen, more parents will be able to assist their children and
this removes this barrier of lack of access to an expert or proficient individual (LeFevre,
Polyzoi, Skwarchuk, Fast & Sowinsky, 2010).
The IXL program can provide immediate feedback and support for students,
which is an important factor in the selection of a classroom math program (Shin &
Bryant, 2017). The broad scope of the IXL program ensures that it covers those topics
that are assessed in the KeyMath-3 diagnostic and can assist in the reinforcement of
concepts and improve the quality of instruction in the classroom (Ketterlin-Gellar, Chard,
and Fien, 2008). I have found that the language used in the KeyMath-3 and IXL.com can
differ for similar concepts and that this can occasionally cause confusion for learners with
math difficulties. While providing more context and vocabulary for high-achieving
learner, the difficulties caused for other learners should not be overlooked.
Chapter Summary
The pedagogy of mathematics is a varied topic. This literature review narrated a
general picture of the effective methods of mathematics instruction for average achieving
students and special needs students.
The importance of making mathematical lessons relevant to students’ lives is a
main concern in many teaching methodologies. This importance is reflected in the
curricula created by provinces and is evident in the learning outcomes of the curriculum.
Teaching methodologies commonly make use of manipulatives to illustrate concepts and
improve student understanding. Modern computers are increasingly used to make math
concepts more accessible for learners. The IXL math program is an example of a

49
computer program that addresses pedagogical concerns of immediate and constructive
feedback and is currently in use by school districts to support mathematics instruction.
Mathematics in special education has been the result of constructivist and
behaviorist approaches and this discrepancy has led to differing pedagogical approaches
to mathematics instruction. Instruction that is applicable to student’s lives and that
addresses the specific cognitive deficits of the learner has proven to be effective in
improving mathematical understanding.
The diagnosis of math disabilities determines interventions for students and these
interventions guide teaching practice. Valid assessment of math disabilities ensures
efficient use of resources and that teaching methods will address the correct cognitive
domain that is affected by the disability. The KeyMath-3 diagnostic assessment is used in
schools to determine mathematical disability and the assessment is broken into several
categories that correspond to different domain of mathematical understanding.
Current trends in math pedagogies were explained with the suggestion that
planning units around the special needs students first and taking their barriers into
account at the time of planning may allow for the creation of units that make math
accessible for all students regardless of designation. These units could receive substantial
support from modern technological advances and make improved math achievement a
realistic goal for all learners. With improved efficiency in unit construction, teachers will
have increased time for direct student interaction and this will improve the quality of
mathematics education.

50
Chapter 3: Research Methods
As a teacher in the school where this project is based, I have worked with learning
services teachers as part of a referral process for students who are having difficulty in
math classes. These teachers often use the KeyMath-3 assessment to gauge a student’s
ability in mathematics. I understand the need to test students based on language that is
familiar to them and after examining the test that was administered to students I realized
that the language of the test differed from my own in-class usage or those of my
colleagues who also teach math. It was my concern that this confused students and
therefore did not validly test their understanding of the concept.
This project comes from this realization that students may be scoring lower than
expected on assessment tests that can be used to determine if a math disability is a likely
diagnosis for the student. The success that has been found using universal design for
learning approaches will be combined with the KeyMath-3 assessment to produce units
where students are able to access the material of the lesson and express their learning in
an authentic manner that truly reflects their understanding of the underlying math
concept.
The project is based on qualitative research orientation that lends itself to an
exploration of teacher practices (Creswell, 2015) especially in the teaching and
assessment of mathematics inclusive classrooms reaching learners with special needs and
those challenged by math disabilities (Hallahan, Kauffman, & Pullen, 2015). This
research project is a non-subject type, hence the method used is primarily the
documentary method of analysis (DMA), described as “the technique used to categorize,
investigate, interpret, and identify the limitation of physical sources, most commonly

51
written documents whether at public or private domains” (Payne & Payne, 2004, p. 23).
Historically, educational researchers have and still use DMA in understanding and
reporting government policies and documents on school practices (Creswell, 2015). The
choice of DMA for this project includes the application of the method in educational
research based on its flexibility and availability (Creswell, 2015; Scott, 2005). The
method has and is still used by researchers to investigate and report studies that will
improve the learning success of special needs students with math disabilities
(Hutchinson, 2017).
Specific government documents such as the BC Ministry of Education middle
high school curriculum policy document, and the BC Ministry of Education Special
Education Unit, were analyzed as part of the literature and the development of the select
math teaching units using the KeyMath approach in chapter four.
There is an explanation of types of primary documents as “eye-witness accounts
produced by people who experienced the particular event or the behaviour we want to
study… they are documents that are produced by individuals and groups in the course of
their everyday practices and are geared exclusively for their own immediate practical
needs…” (Scott, 2005, classified p. 10). This explanation applies to me as a practicing
teacher seeking to understand my teaching practice and student success in Math subject
teaching practice for over eight years. As part of my professional practice, I access,
reflect, and evaluate student records such as report cards. My position allows access to
school records such as report cards to analyze the math performance of students across
my middle school classroom. Access to student documents and records is what allowed
me to see patterns in the problems faced by students and shaped my development of using

52
the KeyMath approach. This was done in an attempt to improve my teaching practice, the
learning flexibility for students and improving the overall learning success of the
students, which are goals of the new BC curriculum (BC Ministry of Education, 2016).
Further consideration of using DMA approach for this project was based on the
methods’ measures of validity as authenticity (evidence as genuine and from an
impeccable source), credibility (free from errors and distortions) and meaning (clear and
comprehensible evidence) (Scott, 1990). As noted by educational researchers, Creswell
(2015) confirms the validity of DMA process through a triangulation process whereby
researchers corroborate with other sources of data to ensure validity of a study.
Additionally, this project also used the qualitative content analysis method (CAM)
that examines documents, especially texts that are related to the phenomenon under
investigation (Mayring, 2000). Content analysis is defined by Krippendorff (1969) as “the
use of replicable and valid methods for making specific inferences from text to other
states or properties of its source” (p. 103). To this effect, I engaged in the content analysis
of various texts (books and journals) related to the scope of the project on not only
KeyMath policies and implementation, but on Math pedagogies in elementary and middle
schools, diagnosis of math disabilities (MDs), technology and the teaching and
assessment of math disabilities amongst others.
Content Selection of KeyMath-3 Assessment
Content analysis of the KeyMath-3 (2008) along with the BC Math Curriculum
for Grades 8-10 (BC Ministry of Education, 2016) serves as the content being analyzed
and interpreted in the development of teaching units in chapter 4 of this project.

53
In directed content analysis, there is an existing body of knowledge that can be
extended to validate a theory or theoretical framework (Hsieh & Shannon, 2005). Patterns
in the types of questions asked by the KeyMath-3 assessment were examined to
determine the types of questions and language that should be used in math units that teach
math concepts. By knowing the types of questions and language used by the assessment,
students will be better situated to display their learning if they are being assessed using
this tool. This forms the basis of the research questions of this project. The units and
lessons are constructed using language that parallels the KeyMath-3 assessment in the
hopes of improving student understanding of the questions that determine or otherwise
guide their diagnosis of having a math disability. The IXL program was also examined to
an extent so that the language of the lessons also appropriately lined up with the language
of the practice program.
The KeyMath-3 assessment is currently used to assess mathematical ability at the
middle school where this project was developed. The KeyMath-3 assessment looks at a
range of math skills in its subtests and many of these correspond with the current BC
curriculum for Science 8 (BC Ministry of Education, 2016; Connolly, 2008). Because of
the close alignment with currently taught units and its use as a diagnostic assessment of
math disability at the middle school, the KeyMath-3 assessment was chosen to guide the
construction of units directed towards increased mathematical learning and understanding
for learners diagnosed with math disability.
The KeyMath-3 assessment may guide the diagnosis of the students as having a
math disability and familiarity with its language and approaches to mathematical
understanding would be important in ensuring the validity of the assessment.

54
Examination of the KeyMath-3 assessment showed the types of questions asked of
learners during administrations of the test and this would allow for in class math practice
to parallel the types of questions that would ultimately determine if a learner receives a
diagnosis of possessing a math disability. Using the wording and types of examples used
in the assessment could potentially remove any confusion or misunderstandings that may
arise during the administration of the test. Students will have encountered similar
language in the classroom and therefore be better prepared to display their understanding
of the math concept, rather than their understanding of the test language.
Construction of Units Around KeyMath-3 Assessment
The teaching units created for this project are built from the very beginning with
the special needs learner in mind. Not only is the learner considered from the viewpoint
of difficulties that they may have with math, but the assessment used to diagnose them,
the KeyMath-3, will be used as a guide for the domains of math that are commonly
assessed to determine math disability. This approach specifically targets deficits found in
the assessment and directly teaches to the problems that students may encounter. By
directly addressing typical problems encountered on the test, it is hoped that students will
directly improve those skills most relevant to their diagnosis and work towards removing
their MD designation.
The stated goal of removing the MD designation has several implications for
learners. The first is that the designation is not inherent to the person that has the
diagnosis and that through practice and study an individual’s understanding can improve
and be compensated for. The second implication is that diagnoses are not permanent and
that this will give students a reason to work hard at improving skills that deemed low so

55
as to improve as learners. A student need not accept the diagnosis of low math ability and
give up on mathematics based learning or career goals. By working to remove this
designation, student can prove their ability to overcome obstacles in spite of cognitive
deficits and move on to more academic pursuits that would otherwise have been shut off
to them if they had merely accepted their diagnosis of low math ability.
The KeyMath-3 assessment is divided into multiple sections that test a range of
skills. These skills are taught to student across multiple grades and are also tested at
increasing levels of difficulty. By using this diagnostic tool as a guide for unit
construction, teachers can create math problems with multiple levels of difficulty that
students can attempt to answer according to their own ability. This means allowing
students to choose how much of a question is answered. This will allow lower achieving
students to answer to the best of their ability (basic concepts) and will give higher
achieving students the choice to attempt to answer in greater detail and with more
precision. This has the effect of not changing the question but having students choose the
approach to the questions that reflect their own understanding. By opening up the criteria
for answering a question, teachers do not have to provide separate work for low and high
achieving learners and can just set a universal set of criteria and place the impetus on
students to choose how they will answer, giving students greater control of their learning
and the expression of their understanding.
Use of Computer Assisted Technology in KeyMath
The units are constructed based on the idea that rote practice and basic skills
related to each lesson would be reinforced using a computer program that provides
automatic and immediate feedback to improve student understanding (Ketterlin-Gellar,

56
Chard, & Fien 2008; Shin & Bryant, 2017). The IXL program will be providing much of
the feedback necessary to improve basic skills and the in-class lesson will focus on the
concepts that improve understanding rather that basic skill mastery.
When providing students with technologies that automatically calculate outcomes,
such as fraction calculators or other programs, it is emphasized that these tools do not
replace, or make up for, a lack of understanding of the concept being taught. The
technology makes some aspect of the problem easier that would otherwise cause a great
deal of frustration, even though the skill that the technology compensates for may be
unrelated to the math concept that is of primary concern in the lesson. This frustration
might lead to student giving up on a math concept that they actually understand because
of some cognitive deficit in an unrelated area of mathematics. By using tools that
supplement weakness during the application of math knowledge, students can see that
they do possess some knowledge of mathematics and that their deficits can be overcome
with perseverance and assistance.
All of the units will have supplemental support from an online math program,
IXL.com, that supports the lessons with basic drill practice as well as providing virtual
manipulatives. This will be especially helpful for the more visuospatial tasks as a means
of alleviating the need to have fine motor skills necessary in graphing or the drawing of
diagrams. The immediate feedback and corrective nature of the IXL program will free up
class time for the teacher to focus on the thought processes of the math concepts, rather
than basic practice and marking. This will have the effect of allowing for more valuable
and constructive class time while making the practice more meaningful, similar to those
effects reported by Seo and Bryant (2012). The students will know immediately if they

57
are incorrectly completing problems and these misconceptions can be dealt with before
they become embedded in the student’s understanding.
Chapter Summary
The use of documentary method of analysis (DMA) is a technique used to
examine government documents and programs, which was a central focus of this project.
The qualitative content analysis method (CAM) was also used examine texts that were
related to the phenomenon under investigation, namely the diagnosis and related math
pedagogy related to special needs learners with a math disability diagnosis. The
KeyMath-3 diagnostic assessment and the BC Ministry of Education middle high school
curriculum document were the main documents examined using this method. It was the
examination of these documents that guided unit and lesson construction with special
needs learners with a math disability diagnosis being the main target of the lessons
produced.
The KeyMath approach to unit construction makes use of the diagnostic test that
will determine if a student has a math disability and uses this assessment to guide
construction of units. The questions and topics covered in the assessment are examined in
order to make sure that lessons accurately reflect the test items. In this way, students will
become familiar with the concepts and language of the questions that will determine their
diagnosis. Students will be given the opportunity to improve their understanding of the
material that determines their diagnosis. This improves the validity of the diagnostic test
by ensuring that student understanding of the math concept is what is being tested, not the
language or approach of the test administrator. Student understanding will be reinforced
with a program that provides immediate corrective feedback to ensure the teachers can

58
address problems or misunderstandings in a timely fashion. Students with IEPs will have
equal access to the materials and will be part of the class as a whole rather than receiving
different work or standing out from amongst their peers. By taking all of these factors
into consideration during unit construction, mathematical understanding may be
increased for all learners in the classroom.

59
Chapter 4: Teaching Units
The Math Concepts
The units for the project are based around the Pythagorean Theorem, Probability,
Algebraic Expressions, and Surface Area portions of the KeyMath-3 and Grade 8 British
Columbia Curriculum Goals (BC Ministry of Education, 2016; Connolly, 2008). These
four units have a great deal of real world application and include multistep problems that
are traditionally difficult for learners with special needs. All four of these units also have
classical roots which means the concepts will have multiple means of expression for
students to learn and approach the concept (Katz, Jankvist, Fried, & Rowlands, 2014).
This is also a benefit for teachers who may adjust their approach based on the form they
find most useful or with which they are most comfortable. Use of a variety of research
based methods will be used to improve students understanding of not only the specific
concept of these units but will be focused around general math literacy and ability that is
assessed in the KeyMath-3 diagnostic.
The construction of the units takes into account the varying levels of student
ability and the assessments and activities can be accessed in a variety of ways. The
concern of students standing out because they are given separate work will be dealt with
from the outset by having all learners producing the same assignment. Opening up the
criteria for assignments will ensure that learners of all levels can express understanding of
the concepts. Those students with IEPs will also have immediate access to all BC
curricular topics available on the IXL program and can be guided towards math that is at
their current level of understanding. Students will be encouraged to try the given
assignments given to the class as a whole because the initial questions of an IXL section

60
are meant to test basic understanding of a concept. If a student is unable to complete the
basic initial questions, the instructor will already be receiving valuable information about
the level of the student and will quickly be able to assign work at the student’s level to
ensure quality use of class time.
The Probability Unit
The concept of probability can be simplified to a series of choices and which
choice is likely to occur. The options and their likelihood are often expressed as a ratio,
most commonly written as a fraction. There is a subtle distinction between ratio and
fraction but the two are very similar. Ratios based off of spinners or dice can be used to
more clearly illustrate what is meant by a fraction. There are a limited number of
outcomes available that can be directly observed by the learner. This level of concrete
reference makes understanding ratios a more achievable goal for many learners,
especially those diagnosed with a math disability.
Determining the numbers in a probability fraction is a case of simply counting the
total number of likely results and counting the number of desired outcomes. The two
numbers are then placed one on top of the other with the total number of results always
going on the bottom. This simple and consistent algorithm is what makes the probability
unit an appealing start for teaching fractions as a whole and limits the number of
algorithms that a learner needs to remember. Students will gain an understanding of what
numbers in a fraction mean in the context of mathematics and that there is consistency
and logic to the number being placed in that position.
Probability makes for a very real-world introduction to the idea of ratios and
fractions. Fractions is a traditionally difficulty unit for learners but lends itself well

61
towards the use of manipulatives to illustrate concepts. The school where this project was
based had multiple sets of manipulatives for the instruction of fractions. The abstract
ideas of fraction math (the bottom number representing a whole) can be more concretely
represented in probability, which allows for increased understanding for the learners.
Probability also has the advantage of being an experimental topic where students can
undertake trials and create their own data. This increases scientific literacy and allows for
the active creation of a student’s own data that can then be analyzed. These are valuable
skills in a learner and their development will be key to increasing scientific as well as
mathematical literacy.
Probability and statistical analyses is explicitly examined across a range of
difficulty levels in the KeyMath-3 diagnostic. The situations and problem proposed in the
diagnostic follow the common understanding of probability taught to students of middle
school grades. As skill level can vary greatly at an individual level, establishing a
baseline of understanding will help to make sure that all students are accessing the
material. The emphasis of this unit as an introduction to the structure of fractions means
that the unit draws direct parallels between fraction manipulation as it relates to the
probability of given events. The fraction problems encountered in the diagnostic
assessment are often encountered in the probability unit this familiarity should improve
performance and understanding of the particular types of questions that are typically
asked when discussing operations with fractions. The explicit connections made between
this unit and fractions allows for the use of consistent language and increases the time
that students can spend with the language. This increased time may help learners with
special needs solidify their understanding of the concepts.

62
The follow unit was created using the curriculum goals of Grade 8 Math in the BC
Ministry of Education New Curriculum. The following table contains the key ideas of
each lesson along with math difficulties that can be encountered in topic area.
Table 1
Probability Unit Summary

Lesson
Lesson 1: How
many choices do
you have?

Lesson 2: How
many of each
option do you
have?

Key Idea(s)
The denominator of
a fraction
corresponds to the
total number of
probabilities
The total number of
possibilities does
not vary for the
same given situation
(spinner or data
table)
Assistive
technology that does
not remove the
necessary
understanding to
complete a problem
should be
encouraged

Math Difficulties Approached by the
Lesson
Students can commonly have difficulty in
seeing the meaning behind parts of a ratio.
This lesson gives context from a particular
perspective for the denominator of a
fractions as a means of solidifying this
concepts in the minds of learners

Students can commonly have difficulty in
seeing the meaning behind parts of a ratio.
This lesson gives further context for the
parts of a fraction by showing the
numerator as the outcome of the probability
and the denominator as the total number of
options.

Basic multiplication is a skill that is often
taught by rote memorization without
Lesson 3: How
context. This lesson looks to provide
many ways can I
context for a basic math skill that can be
do this? How many
applied to real world context. The use of
ways can I get
tree diagrams helps to illustrate this
there?
concept visually
A whole number
This lesson shows the idea of expected
multiplying a
values from a probability. This is the
Lesson 4: If I do
probability is the
application of the expected probability over
this # of times, how
many times should number of times you repeated trials to determine an expected
repeat the event
value. Again, this gives students an
I get x-results?
Cross multiply and application for basic math skills in a
By multiplying the
number of options
given, you can get a
number for how
many combinations
are possible

63
divide can be
applied to this
situation
The experimental
and theoretical
Lesson 5:
values should get
Experiment
closer together as
creating your own
you increase the
data using dice
number of trials
Probabilities related
Lesson 6: The odds to colour, suit, and
value are examined
found in a deck of
with a diagram of a
playing cards
deck of cards
There are scenarios
that exist with a
logical pro/con list
that includes
mathematics and
Lesson 7: Which
this can help
option should I
students choose the
choose?
best option.
Logic can be used to
reduce the number
of negative or
incorrect options
Lesson 8: Games of
chance and the
math behind them

Many games have
different levels of
rewards based on
the rarity of certain
events occurring

Project: Game of
Chance

Students can create
scenarios that have
certain probabilities
make games of
chance. These
games are pending
the approval of the
teacher to ensure
probabilities are not
overly complicated

particular, real-world context.
MD students (and typically achieving
students) can have difficulty with the
abstract idea of approaching a theoretical
value with enough trials. This lesson has
students directly creating their own data to
prove this concept.
The visual of the deck of cards illustrates
the different ways that a deck can be
organized and provides a visual for the
fractions/probabilities to help MD learners
clearly see the values of the probability

Applying logical algorithms to decision
making processes can help students to
directly see the importance of math in
everyday life. This lesson helps to solidify
this concept for MD students who may see
less of a need for probability in their
everyday lives.

This lesson is meant to give more realworld applications of probability. With
enough real-world references for
probability, students will have a clearer
picture of the application of this area of
mathematics.
In this project, students take some
probability scenario that has been
previously discussed and make a game out
of that scenario. This is meant to give
students a unique means of expressing
understanding of probability and make
their learning more personal.

64
Lesson 1: How many choices do you have? This is an exercise in knowing the
number of different outcomes available. The specific math involved is simple counting
but the concept can be seen as complex. This lesson has the additional goal of teaching
students that basic math can be used to describe everyday situations. A single number
answer is produced by counting the options available. This idea of knowing the total
number of options forms the basis of the rest of the unit. In the concept of fractions, the
number being produced in this lesson becomes the denominator of the probability
fraction.
Lesson 2: How many of each option do you have? This lesson requires students
to know the total number of options, as discussed in the previous lesson, but also to know
how many of each intended option there is. Students are putting option into subgroups
from the whole. This is an important concept in fractions and marks the beginning of
having a numerator over a denominator.
The activities of this lesson have student looking at spinners or data tables to
determine a total number of outcomes. It is stated explicitly to student that this number is
consistent for all probabilities that apply to that situation, disregarding fractions put into
lowest terms. Placing fractions into lowest terms should not be considered a priority for
this lesson as it is not the main goal and can lead to confusion, particularly for MD
student who have trouble with multiple additional steps. Students should be instructed on
the meaning of each of the terms in the probability. Writing the meaning of each term
relative to the question being asked will help students to see the meaning behind the
numbers. As an example, a four-section spinner has two even number options. If the
student is asked to give the probability of getting an even number, the correct answer is

65
2/4 with a simplified version of ½. If students are encouraged to write the meaning of
each number, they would then write that the 2 represents the number of even values while
the 4 represents the total options. The reduced form of ½, while still correct, has no direct
meaning to the spinner but stronger math students could be encouraged to take this extra
step. The amount of automatic math knowledge necessary to easily put fractions into
lowest terms is a significant limiting factor for many math students, especially students
with a math disability.
Assistive technology in this situation can include several modern apps that
automatically place fractions into lowest terms. As the main goal of this lesson is to
determine the number of options that meet certain criteria over the total number of
available options, assistive technology that only places fraction in lowest terms does not
oversimplify the task being asked of the students. A student must have understood the
concept to be able to create the initial, unreduced fraction.
Lesson 3: How many ways can I do this? How many ways can I get there?
This is a straightforward multiplication when the math is examined but explains the idea
of combinations of options. This has very applicable real world application as many
choices are afforded consumers in the modern economy and students will have likely
encountered scenarios where this math has applied in the past.
The key idea of this lesson is that there are a set number of combinations possible
from a limited number of selections. Students should be explicitly told that while the
math is straightforward multiplication, the concept that the math is describing has real
world implications. The use of branching tree diagrams will give a visual for students to

66
see the choices available at splitting of the branch. The total number of options at the
final tier of the diagram illustrate the total number of options for a given set of options.
The assessment for this lesson includes questions with varying number of options,
such as how many sandwiches can be made from 3 cheeses, 4 kinds of bread, and 2 types
of meat. The multiplication gives the answer but this concept can also be shown more
concretely through the use of tree diagram that show the branching nature of the options.
Tree diagrams also help to illustrate that the order will not affect the answer of the
question as the final result of 24 will be achieved no matter the order that the diagram or
multiplication is completed.
Lesson 4: If I do this # of times, how many times should I get x-results? This
lesson is based around a probability of an event occurring over many trials. It is a fraction
multiplied by a whole number but can be shown using concrete examples. If a 6-sided die
is rolled a total of 24 times, how many times can we expect to get the number 4? This
works out to 24 (the number of trials) multiplying 1/6 (the probability of getting a 4)
which gives the value of 4. We can expect to get the number 4 a total of four times.
Students should be explicitly told the whole numbers multiplying the probability values
represent number of trials. Use of this vocabulary is important in tying the concepts of
this lesson with the upcoming lab and the idea of repeating events to gain more data.
When discussing the examples, emphasis on the ability of cross-multiply and
divide to give an answer will give all students a powerful and nearly universally
applicable tool to determine an accurate result. The use of cross-multiply and divide and
the ability to set up ratios to form proportions where this method applies can be an
extremely powerful tool for mathematics. This tool comes up in many mathematical

67
situations and meets the needs of many MD students to have fewer but more applicable
algorithms. For this lesson specifically, the students should be taught that the bottom
numbers of each ratio represent total amounts (for both probability and number of trials)
and that the only top number known is for the probability of the given event. As there is
only 1 missing value from the fractions, this gives students the pattern required for crossmultiply and divide to be used.
Lesson 5: Experiment creating your own data using dice. Giving expected
(theoretical values) versus experimental (actual values) after a series of trials using
multisided die. Students complete several sets of trials and patterns in the experimental
value are compared with theoretical values and the two should become closer as the
number of trials increases. Students will actively observe the difference between the two
values and how this difference of what “should” happen and what “does” happen can be
reconciled by increasing the number of trials for an event.
Students are creating their own data and examining the discrepancies between
what they theoretically know should happen and what is actually happening as they
increase the number of trials.
Lesson 6: The odds found in a deck of playing cards. This lesson makes use of
the probabilities found in a deck of playing cards. Students are taught about the
probability of drawing the various cards based around colour, suit, number, and
combinations of these results. A handout is given to students that has the cards of the
deck organized in rows according to suit in ascending order of card value. This gives a
visual reference for the various probability questions that can be asked (What are the
odds of drawing a 7?) that will aid students in answering the question but also aid

68
teachers and educational assistant in explaining questions to students. Having the cards in
a diagram rather than using an actual deck of cards helps to keep the information clear
without becoming overwhelming or requiring the organization of an entire deck of cards
each time a card problem needs to be solved.
Lesson 7: Which option should I choose? Scenarios of choice are given to the
students the outcome that has the most likely beneficial outcome is explained. This lesson
focuses around the ideas of logic and using pros and cons from a mathematical standpoint
to make choices based around numbers rather than preferences. Game theory makes use
of this type of math to determine the best possible choice in a given situation. While the
scenarios described may seem isolated to students, making analogies to choices they
make in their own lives will help to connect this math and allow students to make
mathematically informed decisions in the future.
Activities for students mainly involve the use of logic to reduce the number of
choices and improve the likelihood of a correct choice. Many of the example involve
multiple choice problems where there are some obviously incorrect choices. The change
in the probabilities due to the removal of incorrect choices is mathematical proof of the
advantage that using logic on multiple-choice problems can have. Many tests in the
future, including KeyMath assessments, will have multiple-choice questions to which this
process can be applied.
Lesson 8: Games of chance and the math behind them. The main focus of this
lesson is the explanation of games of chance and how their probabilities are determined.
Games such as roulette and poker are examined with emphasis being placed on why
certain events are less common and therefore have higher payouts. A key take-away from

69
this lesson is the idea that a less likely event should have more value associated with it.
Roulette is an example of one such game. Many students encounter these games and this
will give some logic and reasoning behind these games. False beliefs about gambling
should also be discussed in this lesson to dispel any misconceptions of gambling, such as
the gambler’s fallacy. This concept is related to the idea of independent events found in
the curriculum.
Project: Game of chance. Students will create and describe a game of chance.
Several examples are given and the project leads to a carnival day where all of the games
are played in class and students move about taking turns playing the games. The game
needs to have its probability stated. Many modern games, both board games and
computer games, make use of probability. Students are encouraged to research these
games and make connections to their own hobbies and interests. The knowledge that their
hobbies make use of the basic mathematical principles they have been studying will give
the students a greater appreciation of both mathematics and the role it plays in the
activities they enjoy.
The Pythagorean Theorem Unit
This unit is an excellent example of the concrete/representational/abstract
approach due to its classical discovery and use. The math unit makes extensive use of a
visual tool of the students’ own creation, involving them in the process of learning the
Pythagorean Theorem. The theorem can be shown using manipulative and videos that
clearly illustrate the concept. This unit has traditional been seen as quite difficult but
recent success has been found using the tools built in this unit with these assignments as
assessment. The theorem’s classical origins have allowed centuries of examples to be

70
developed along with application to many real world situations including the trades and
more academic applications; this allows for appeal to many typically achieving
individuals from a range of backgrounds.
Assignments do not deviate from a set idea or require students to alter the
algorithm of the day in any significant fashion. All students will have equal access to the
lessons and will be taught as a single group.
Higher achieving students are encouraged to find more accurate answers or to
reinforce concepts on IXL. An example of this is the number of decimal places in a
square root. An acceptable answer for the square root of 150 would be “between 12 and
13” because the square roots of 144 and 169 are 12 and 13 respectively. This can be
found using the visual tool that the students create. A more accurate answer would
indicate that the answer is closer to 12 than it is to 13. A decimal answer would also be
acceptable but is not considered the “best” answer due to the fact that a calculator can be
used and may not reflect understanding. The explanation of the answer is what makes it
better, not the accuracy of the number.
The importance of providing proof is reinforced by the marking scheme used for
the questions. This is also an opportunity to teach the importance of showing organized
thinking through proper mathematical reasoning in a way that can be easily followed by a
reader. Marks are given for the inclusion of the formula, the work of substitution,
squaring, and taking the square roots, as well as a correct final answer. In the case of
longer word problems, both a diagram depicting the question and a sentence answer are
also required. Students become aware of the many aspects of the questions but also see
the basics that are repetitive throughout the application of the theorem. The basics of the

71
theorem’s formula and substitution as well as the labeling of a right angle triangle are
present in all questions and give students the potential to never receive “0” on any
question by always providing at least these basic pieces of knowledge. Receiving partial
marks has the dual effect of increasing student confidence and allowing the teacher to be
aware of at least a basic understanding of the concept.
The unit makes extensive use of the visual tool that is created in the first several
days. This constantly ties the questions using the theorem to the understanding of what it
means to be a square number and how the roots of these numbers operate. This will create
a conceptual reference point for students and prevent the more complex problems from
becoming too abstract for students to understand.
While not explicitly examined in the KeyMath-3 diagnostic, all of the requisite
mathematical operations and understanding necessary for the Pythagorean theorem are
separately examined across various subtests. Student are asked to solve the sum of two
square numbers (Ex. 32 + 52) and there are multiple questions regarding students
understanding of angles, particularly of 90° angles. Awareness of type of triangles is also
tested for. Ability to take a square root is also tested. Use of formulae in a variety of
applications is also examined. Students will be given clues to watch for in the question
that allow them to understand if they are looking for a leg or a hypotenuse. These clues
are explicitly connected to variations of the formula given to them in the notes. Students
are shown how to manipulate the initial formula but the use of clues and pre-manipulated
formulas takes some of the burden off of struggling learners and allows them to more
actively participate in the topics of the day. All of these separate pieces of mathematical

72
knowledge are utilized during any given theorem question and fluency with these terms is
major goal of this unit.
The follow unit was created using the curriculum goals of Grade 8 Math in the BC
Ministry of Education New Curriculum. The table gives the keys ideas for each lesson as
well as some math difficulties that can be encountered in the subject area.
Table 2
Pythagorean Theorem Unit Summary
Lesson

Lesson 1: Square
Numbers
Lesson 2: Create
the Square
Number Rainbow
Sheet
Lesson 3: Use of
the Rainbow Sheet

Key Idea(s)
The reason we refer
to numbers as
“square” is because
they literally create
perfect square
shapes
Square numbers
have a pattern that
can be shown
visually for easy
reference
The creation of a
tool can help to
reinforce concepts
and facts

Lesson 4:
Estimating Square
Roots

Square roots do not
have to have exact
decimals and have a
reasonable range
based on their
values in relation to
perfect squares

Lesson 5:
Introduction of
Types of Triangles

There are various
types of triangles
with consistent
characteristics

Math Difficulties Approached by the
Lesson
Students with MD can have difficulty in
understanding the reasoning for certain
terms. This lesson gives explicit instruction
for the meaning of square numbers, which
forms the basis for the unit.
The tool created in this lesson is a strong
visual tool that simplifies and illustrates the
relationship between squares and square
roots while helping students to memorize
the pairings of root and square.
This lesson further uses the tool to allow
students to become proficient and
internalize the number pairs of the tool.
This lesson tries to make it easier for MD
student to represent an estimate by giving a
range for the values of a root, rather than
giving exact number or just simply using a
calculator to give a number they don’t truly
understand. This gives an accurate gauge of
the students understanding of roots and the
ability to use the tool they constructed in
previous lessons.
Visuospatial awareness can be a problem
for MD students and this lesson seeks to
highlight the differences between
seemingly similar objects (triangles) and
how they can be grouped together into

73

Lesson 6:
Pythagorean
Theorem
Introduction
(Concrete and
Representational)

The theorem has
many concrete
proofs that show the
relationship between
the legs and
hypotenuse of a
right-angle triangle
When given the legs
of a right-angle
triangle, you add the
squares and then
take a root to
determine the length
of the hypotenuse

Lessons 7-9: Use of
scaffolding and
single ideas
(manipulations of
the algorithm to
solve various kinds
of Pythagorean
problems)

Project:
Pythagorean Art
assignment

When given the
hypotenuse and a
leg, subtract the
square of the leg
from the hypotenuse
square to determine
missing leg
During word
problems, determine
if you have a
hypotenuse or not to
determine if the
question is an
addition or
subtraction problem
Right-angle
triangles can be
placed together in a
multitude of ways to
form images with
mathematical
meaning

subgroups based on certain characteristics.
General math instruction can be improved
by using concrete examples. This makes
these ideas more accessible for MD
learners and helps to improve their
understanding and involvement in the class.
The Pythagorean has many such concrete
examples available in online videos or
through the use of math manipulatives.
This lesson does not deviate from a set
pattern of being given the legs of a triangle
to determine the hypotenuse. It can be
difficult for MD students to constantly
change algorithms and this lesson was
meant to focus and improve understanding
of a single application of the theorem.
Again, this lesson focused around a single
concept to leave less room for confusion of
MD student who have difficult with
multiple types of questions. Giving
students a list of key terms or hints to
watch out for will help them learn to
understand when and where they can apply
the various forms of the theorem
The main focus of this lesson was to give
student the ability to determine the
presence of an hypotenuse either through
drawing a diagram or recognizing language
in a question. This ties in to the previous
lessons and allows students to identify the
algorithm necessary to solve the problem.
This assignment allows for a great deal of
student independence while creating
achievable goals for learners who are
having difficulty. MD students can show
their understanding at whatever level they
feel most comfortable. This assignment
also allows students to create their own
simple problems and then solve them.

Lesson 1: Square numbers. Students are given a table and create a list of
numbers that are multiplied by themselves (1x1, 2x2, 3x3, etc.). Each expression is then

74
drawn as a set of rows where 1 is 1 row of 1, 2 is 2 rows of 2s, etc. This has the effect of
creating a square and shows the reason for calling these numbers “square”. The amount
that can be used to create the square is referred to as a square number and the number of
rows is the square root. Students are given a visual representation of the definition with
this method and this will help to internalize the concept.
Lesson 2: Creating the square number rainbow sheet. This is a visual
representation developed as a tool to use in solving square number and square root
problems. Students use graph paper to create a graph that shows the relationship between
increasing numbers (1, 2, 3…) on the horizontal axis and their respective squares (1, 4,
9…) on the vertical axis. The graph ends up create a square that consists of the exact
number that corresponds to the root. There is a square of 9 blocks at 3 and a square of 25
blocks at 5. Students are able to see the increasing value and proportions of square as the
numbers increase. This tool also allows for easily finding the root of a number or the
square of a number. The lines are coloured and students need only start at the
corresponding root or square and then follow the line back to the corresponding axis. This
method also works for non-perfect squares as an estimation tool. Students may not
determine the exact root but will be confident that it is between two square roots and will
likely be able to say which of the roots the actual solution is closer to. A sample of this
tool can be found in Appendix-B.
Lesson 3: Use of the rainbow sheet. In this lesson the sheet created during the
previous class is utilized to determine squares of numbers and their respective roots. Use
of the tool may highlight any errors used in the construction of the tool. Lines may not be
lined up properly and so may give incorrect answers. This lesson serves as a calibration

75
of the tool that the students have developed. Students are made aware of this fact and are
therefore taught the skill of testing to determine accuracy of a tool or concept.
Lesson 4: Estimating square roots. Use of the developed tool is further refined
in this lesson. Students are given notation to show non-perfect roots and squares.
Showing that a root is between two numbers allows students to not simply recite a
memorized answer or do basic calculator work. This type of answer requires use of the
tool and a thought process that shows an understanding of what the true answer might be.
This lesson also shows students that the square root estimates have a narrower range than
square estimates and the idea of amplifying error through multiplication.
The non-decimal method of displaying square roots can be given by the following
example. If a student is attempting to determine the square root of 45, they would locate
the relative position of 45 on the rainbow squares sheet. As 45 falls in between 36 and 49
on the table, the student takes the square root of 36 and 49 and determines that the root of
45 must fall between 6 and 7, the roots of 36 and 49. As a means of displaying this result
the student could write 6-----7, indicating a range that the root must fall in. As a further
display of understanding, the student could then indicate that the root likely falls closer to
7 because 45 is closer to 49. That can be indicated with 6----x-7. The student has shown
several levels of understanding and has not resorted to using a calculator or relying on
decimals. For students with a lack of symbolic understanding of the relative value of
decimals, this method provides them the means to show their level of understanding. This
method also gives other learner another means of expressing their own understanding of
an estimated square root.

76
Lesson 5: Introduction of types of triangles. The major groupings of triangles
are discussed with images and key characteristics displayed. This is done to show that not
all triangles will work with the theorem and how to identify a right angle triangle
compared to other types of triangles.
The specific types of triangles mentioned in the lesson include, scalene, obtuse,
right-angle, isosceles, oblique and equilateral. The triangles are visually displayed and
students are shown the relationship of angles and the importance of the right-angle
triangle in the context on this unit. Identifying a right-angle triangle and the ability to
infer that a right-angle triangle is being discussed are key components of this lesson. An
example of such a situation could be a boat that leaves a dock and goes south for 8 km
and then east for 6 km. It is implied by this question that the turn would have been a
right-angle even if it is not explicitly stated. Inferring this sort of information will be
important in setting up diagrams and knowing that the theorem can be applied. An
advantage of IXL reinforcement at this stage is the access to assignments that can quickly
reinforce these terms with immediate feedback to students. The program will allow for
more practice that a worksheet would allow and will give rapid feedback to ensure that all
learners have a strong foundation for the terms that will be used in the unit.
The visuospatial deficits of some MD learners will keep them from being able to
recognize or recreate some of these triangles. Focus should be given to identifying rightangle triangles over the ability to recognize the other types of triangles. The right-angle
triangle forms the basis of this math unit and this should be made explicit to all learners.
Lesson 6: Concrete and representational forms of the Pythagorean Theorem.
The theorem itself can be illustrated using a manipulatives to clearly and concretely

77
illustrate the concept of the a2 + b2 = c2. There are many videos that illustrate the concept
in a fluid manner. Students are given several example questions with numbers small
enough to be solved using both manipulatives and the square numbers tool. Those
triangles that do not have a perfect square hypotenuse are estimated using the tool.
During this lesson, various videos are shown displaying scenarios that display
how the theorem functions. A key aspect of this lesson is the multiple means of providing
evidence that the theorem functions. It is not enough for students to simply accept that a
theorem works in the modern curriculum. Emphasis is being placed on the ability of
students to prove that an answer is what they say it is by providing accurate evidence.
The visual aids that have been developed for the theorem provide a body of evidence that
gives increasing support for the theorem. In this way students are exposed to the idea of
math as not a set construct but an ever expanding body of knowledge that can be
examined in new ways, even after thousands of years.
Lessons 7-9: Variations of the Pythagorean Theorem. These lessons show the
manipulation of the Pythagorean Theorem and then make consistent use of that single
change to the theorem.
The first scenario is the basic a2 + b2 = c2 where both a and b are given in the
equation. The squares are added and the root determined to obtain c. This method is then
consistently used for the remainder of the lesson to avoid confusion. Questions are kept
within the range of the rainbow sheet initially to keep material simple for students. After
the initial assignment, moving to larger numbers that require calculator work with require
training so that student use the square root functions properly. It is possible to maintain
the line method with calculator work to help reinforce the idea of the decimal being

78
between two square roots. Students can give their answers in the form of the line method
rather than simply transcribing the number from a calculator. This requires that students
take the time to actively think about the decimal and determine which number it is closest
too. This exercise can help reinforce the ideas of estimation for all learners but will be
particularly useful for MD learners who may not have fully developed the concept.
The second lesson’s scenario adjusts the formula to c2 – a2 = b2 and students are
given the hypotenuse of the triangle and then asked to determine the missing value. How
this formula is arrived is shown explicitly through the manipulation of the original a2 + b2
= c2 as student are shown how equation can be manipulated to determine the values for
other variables. The formula c2 – a2 = b2 is then consistently used for the remainder of the
lesson to avoid confusion. This prevents students who have difficulty in manipulating
formulas from having a barrier to their progress that is not reflective of their ability to use
the formula provided and complete the assigned work.
The third lesson and any further practice sessions that are deemed necessary
examine the use of the theorem in word problems that reflect real world scenarios and
applications. It is during this lesson that patterns and sets of numbers that meet the
criteria of the theorem are highlighted and elaborated upon. These number sets have been
known to mathematicians for thousands of years and the student now have enough
theoretical foundation to prove that the numbers sets meet the theorem’s criteria.
These lessons can also make use of direct measurements made by students from
around their classroom and school environment. The students can make two
measurements of right angle triangles that occur in these environments and make
predictions of what the third number should be. The students can then make the final

79
measurement and compare predicted and measured values. Reasons for any discrepancies
can then be discussed or hypothesized. In this way, all students see the real-world
application of the theorem and also apply the scientific method of prediction,
measurement, and critical reflection.
Project: Pythagorean art assignment. The project can be completed in a variety
of ways and is very student driven. Completing the project has the student essentially
creating their own practice problems by creating a series of right angle triangles on graph
paper. The process of creating overlapping right angle triangles has the effect of creating
an abstract picture of triangles with the associated math being shown on a separate page.
Students may also choose to create an image using right-angle triangles as the building
blocks of the image. This imposes a limitation that sparks creative and novel use of
triangles that students may have never otherwise encountered. The idea behind the project
is to have a student driven art project that links the concepts of mathematics and artistic
ability. Examples of this project can be found in Appendix D.
Algebraic Expressions Unit
The key concept of the algebraic expression unit is the representation of numbers
in multiple ways. A number can be represented symbolically by variables or numbers
may be represented through expanded expression with operations that give that number
as a simplified result. A number being represented by a variable is given by a=3. When
the student sees an “a” in the expression, then that “a” represents the value “3”. A number
being expressed as a series of operations is 3 + 4 – 5. When solved, this operation
represents the value of 2. The number 2 can also be represented by 10 / 5, which gives a
result of 2. Numbers have both a simplified and expanded expression and this unit

80
focuses on the manipulation of mathematical concepts to either simplify or expand the
representation of a given value.
Symbolic representations are a difficulty for many students with MD and this unit
will be a test of this cognitive domain. The emphasis is not only simplifying expressions
but on recognizing that a mathematical expressions is an expanded representation of
simpler term. Assistance that is given to learners with a specified problem in the area of
symbolic representation should emphasize the direct correspondence between the given
variable value (m = 5) and the expression (3m + 2). It is important to clearly show these
students that the “m” is really a “5” in this question. The added complexity of introducing
variables for substitution that must be solved by order of operation will make these
expressions more difficult than normal but explicit instruction will help to alleviate these
problems.
The KeyMath-3 assessment makes extensive use of questions that have their
origins in the concept of algebraic expressions. Students are often asked to simplify
operations and substitutions during the assessment and the concept finds its way into
several of the test categories. This highlights the fundamental nature of algebraic
expression in understanding more complex mathematics. The ability to compute and
simplify for algebraic expressions is responsible for a significant portion of a student’s
score on the KeyMath-3 assessment.
The follow unit was created using the curriculum goals of Grade 8 Math in the BC
Ministry of Education New Curriculum. The following table describes the key ideas of
each lesson and the potential math difficulties that can be encountered when teaching the
lesson.

81
Table 3
Algebraic Expressions Unit Summary
Lesson

Lesson 1: Basic
Expressions and
Simplification

Lesson 2: Order of
Operations

Lesson 3:
Substitution into
Expressions Using
Variables
Lesson 4:
Simplifying
Algebraic
Expressions with
Variables

Lesson 5: The
Distributive
Property

Lesson 6: The
Steps of
Simplifying
Expressions
Project: The Math

Key Idea(s)

Math Difficulties Approached by the
Lesson
This lesson begins to show students a new
Mathematical
way of approaching mathematics and how
expression are
expressions represent the number that is
complicated
their “answer”. This will be a very difficult
representations of
task for MD learners who have difficulty
single number
with symbolic representation.
The difficulty that many student will have
The order in which with this lesson is seeing how an
we solve problems expression can be solved multiple way but
can effect the
that there is a correct way. Continued
outcome
practice and use of manipulative examples
where order matters will help MD students.
Variables in a
This lesson will be difficult for those
question simply
learners who have trouble with symbolic
stand for a given
representation. Explicit instruction on the
value and are then variable becoming a number will hopefully
solved as normal
lessen the difficulty of this topic.
Parts of an equation There is a specific approach that can be
can be grouped
used to treat math as a language just like
together to create a any other. This concept is particularly
simplified
useful for algebraic terms and will help
expression that may MD learners with their understanding of
not be a single
like terms using algebra tiles or other
number
manipulatives.
This concept should be taught carefully to
A term multiplying MD learner and it should be made
a group of terms in a explicitly clear that the multiplying term is
bracket, applies to how many sets you have of what is in the
all terms inside the brackets. MD learners will have trouble
brackets equally
with the abstract and multistep nature of
the concept.
By using the logical Applying logic to the steps of the problem
sequence of simple such as knowing which step has more
to complicated, it is terms will help students to see that they can
possible to put the place a solved equation in order without the
steps of already
need to directly simplify each step. The
solved expressions pattern recognition may be difficult for MD
in order
learners.
A single term is
Some student with MD may have trouble

82
Map

represented in a
multitude of ways
with project criteria
guiding how the
term is expressed to
ensure that all
methods from the
unit are represented

with the amount of freedom that this
project gives and will need clearer and
more direct instruction to produce a final
product. Having an EA or teachers help to
create examples within the students
assignment and then having the learner
create a similar expression may alleviate
this problem

Lesson 1: Basic expressions and simplification. Student will initially be shown
a binder that has been organized into 4 sections using dividers. Each section has a number
of pieces of paper that may be blank or may be worksheets relevant to the topic of the
section (Math, Science, Socials, etc.). Students are asked what the object is. Students will
answer binder and some may answer book. The binder will then be opened and the
sections taken out in their entirety. The binder is now described as 4 sections with each
having a label. Now, rather than calling the binder a binder can we not then name if for
its four sections (A Math-Science-Socials-English)? While more complicated, it is still
factually correct and we are still naming the same object. Now each of the sections has its
papers taken out with lined paper going into a pile and handout going into another. Can
the initial binder now be called lined paper and worksheet from this section, lined paper
and worksheets from that section, lined paper and worksheets from that section and lined
paper and worksheets from that section and finally lined paper and worksheets from that
section. This is a yet more complication way of describing the same initial binder. Finally
the individual papers are counted in a final and overly complicated fashion. On the board
or projector, this pattern can be shown as an expanding pyramid with either the binder on
top expanding towards the bottom (the reverse of a typical problems) or just as easily in
the reverse (a typical pyramid shape).

83
In this example, we have highlighted both the overall pattern of how mathematical
expressions are simplified and the idea of representing objects by their constituents in an
increasingly complicated way. This is meant to show students that both simple and
complicated description are accurate and still represent the same value. The same can be
done for numerical values. This will be demonstrated concretely for students through the
use of math manipulatives.
In this lesson students are shown basic expressions ranging from simple (3 + 7) to
the more complex (3+5-7+12). These expressions will have varying operations but will
not yet be complicated to the point of needing order of operations knowledge, which is
the subject of subsequent lessons. The expression will then be simplified by solving the
various operations to reduce the amount of numbers. Students are encouraged to come up
with 3 term expressions that have a real world application. An example of this might be
the term 4 – 3 + 12 which might represent a student having 4 pencils, giving 3 to her
friends and then taking 12 from Mr. Stovel’s extra pencil tin. Students will be given
manipulatives for each expression and then be asked to demonstrate other ways that the
same amount could be represented with those same manipulatives. The expression for
each representation will be written down to further enforce the idea of written expressions
representing real world values. Such values are the focus of several curricular
competencies such as representations, justification, modelling, and estimation. These
terms can be directly referenced in the subsequent activities of the unit to give clear
examples of student competency in these areas.
Lesson 2: Order of operations. In this lesson student will be using concrete
manipulatives to show that operations done in various orders can result in different

84
answers. A key aspect of this lesson is how do we write or ask a question that tells
someone to do the steps in the order we intended or in the order that reflects what actually
happens when those numbers and operations are applied to real world scenarios.
The demonstration for this lesson involves an initial amount of math counters. For
this example we can say there are three counters. We can then say we start with a group
of 3, subtract two of them and then have five copies of the group (subtracting 2 and
multiplying by 5). We then do the reverse order by subtracting 5 copies of 2, which is
simply subtracting 10. This results in two different answers that are written on the board.
One of the answers cannot be illustrated with math counters and provides a brief
introduction to the idea of negative integers. The students are then asked which one is
correct. We have to then explain the order of what happened showing the ways of writing
the operations. When writing mathematical expression, we need to examine the
interpretations of the expression. For 3 x 5 + 2, is this three times five and then we add
two resulting in 17, or is this three times the five and two added together (7), which gives
7 copies of 3 resulting in 21. Communicating which operation should be done first is the
basis of understanding the order of operations and that there is a correct way to interpret a
given set of operations.
Students will be placing their math counters into stacks of a given number, asked
to split that stack that represents the initial stack, and then repeat this several times. This
will allow the student to show the original stack in multiple ways. As a wrap up activity
for the students, they can be given personal whiteboard and placed into groups. A number
between 10 and 20 is placed on the front board and students must come up with multiple
means of expressing that number. As an added piece of competition within the group,

85
students receive a point for each novel expression they have that no other member of their
group managed to create.
Lesson 3: Substitution into expressions using variables. The key idea of this
lesson is that a missing value of an expression is given and then “plugged-in”, or
substituted, into an expression that allows the expression to be simplified.
From a concrete standpoint this is shown using math counters with the inclusion
of a container. The counters are placed into stacks and the container is left empty. The
only action that can be taken is adding counters to or removing counters from the
container. By adding or removing counters to the container, we can then change the total
number of counters in there are in the container. This provides the student with the
essence of what a variable is. A variable is the portion of an expression that can be
changed or varied to give a different result.
Students will be given counters and containers to represent expressions. They will
set up the stacks of counters and then put the container next to the stacks. Students will
then write the corresponding mathematical operation of the counters and containers in
front of them with a variable being represented by the container. This will represent a
series of addition substitutions. An example of this is 5 + c. A stack of 5 counters is
placed next to an empty container.
In the case of subtraction, a separate type of counter can be used (this would help
to introduce the idea of negative values). An example of this might be 5 – c. Because this
“c” represents a subtraction, it might be a different colour (red if the initial container was
blue) than the container used for the previous example of addition. For subtraction, the
students could also physically remove counter. This second option may be less abstract

86
and appealing to those students who have difficulty with understanding that adding
negative value object counts as subtraction.
In the case of multiplication questions, the number of counters in the container
might represent the number times a particular stack is being copied. In this way, the
number of counters in the container are almost like a set of instructions, telling the learner
how many sets of the stack they need to make. It could also represent the number of rows
or groups of variables that are being created. An example of this might be a stack of 5
with 3 counters in the container. This would be giving the student instructions that there
need to be 3 stacks of the initial 5 counters.
For division, the number of counters in the container could represent the counters
that will make up each new stack. An example of this might be a stack of 10 with 5
counters placed in a container. This would mean that the 10 needs be split into stacks of
5. Students would count out the stacks and determine that they now have 2 even groups
with no counters left over. The number of stacks that can be made represents the
simplified expression. Extra care should be taken to make sure number work out well and
that situations that do not solve evenly (12/5) unless students have an understanding of
remainders and these situations should be avoided for simplicity. Even if students are not
comfortable with remainders, this method gives them an idea on what remainders signify
and could lead to further discussion about what could be done (breaking the counters into
pieces). This may lead to increased understanding of decimals and fractions.
This lesson focuses on giving students the understanding that a variable is the part
of an equation that can change and that the amount of change is determined by the
amount of the variable. This concrete basis for the operations gives a reference point for

87
all student and helps to make the concept less abstract for those students with difficult
with symbolic representation. See Appendix C for examples of each scenario.
Lesson 4: Simplifying algebraic expressions with variables. In this lesson, the
main idea for students is that certain terms can be combined to make the expression
simpler. The math counters provided should have some sort of overall organizing
characteristic, such as size, shape, or colour. The amounts should initially be grouped
inefficiently so that students can practice organizing, or simplifying, the groups. This will
allow students to learn that variables can be grouped together as well as numbers
(constants). An example of this would be the expression 3a + 2 – 2a + 5 – 6. Those terms
with variables (3a and – 2a) can be combined and might be represented by blue and red
squares. Blue representing positive integers with the “a” variable and red squares
representing negative integers with the “a” variable. The constants (+2, +5 and -6) can be
combined and would be represented with red and blue circles with red being negative and
blue being positive. A review of integers, especially the use of integer tiles, would be
necessary for this unit to succeed. A review could be done as part of the unit or this
algebra unit to follow a unit on integers. Given that integers are now mainly found at the
Grade 7 level, a standalone review of integers may be necessary to ensure student success
and learning. The final simplified expression will be a + 1 and would be represented by a
single blue square and a single blue circle.
When asking students to set up the initial conditions (groupings), it will be
beneficial to have pictures or drawings of the initial setup on the board for students to
copy and then simplify. If numbers are used on the board, part of the exercise is made
redundant by not having students determine on their own what number are being

88
represented and therefore not showing understanding of what the visual representations
are meant to express. The use of pictures rather than numbers also makes the lesson more
accessible to those students with MD who may have difficulty with symbolic
representations.
Once students have shown competency in setting up given scenarios, students
may be given the opportunity to set up their counters and containers and have other
students determine the initial expression. This will test students’ ability to understand the
mathematical expression of a given situation and will provide opportunity for students to
challenge one another.
Lesson 5: The distributive property. This lesson focuses on the concept of
multiplication affecting the terms inside of a set of brackets. The key concept to gain
from this lesson is that the outside term influences both inside terms equally. This can be
proven to student using the concrete manipulatives of the earlier lessons. Using the idea
that one type of counter indicates the number of copies being made (multiplying
coefficient outside of the brackets) and two separate groups of counters can be used to
represent those values that are being copied (the terms inside of the brackets). Students
should be shown that the outside term or the number of copies does not change the final
result if the inside terms are combined before applying the multiplier. An example of this
is 2( 3 + 4 ). The 2 can be applied to the inside terms first to produce 6 + 8 which gives
14. The inside terms could be combined first to gives 2(7), which also gives 14. This idea
illustrates that there is often more than a single way to solve a problem and determine a
correct solution. There are more efficient ways of solving problems, such as order of
operations, and this fact should be directly stated to students.

89
Lesson 6: The steps of simplifying expressions. In this lesson, several long
expressions have been simplified over several steps to give a single correct answer. The
entire simplification process is then cut into strips and jumbled up and combined with
other expressions and their steps. Students have to then correctly assemble the strips into
a correct order. Rather than directly testing mathematical calculation knowledge, this
activity allows students to use problem-solving skills and reason to determine what order
the steps should occur in. The overall pattern of increasing complexity from single term
to complex term is what will allow the students to place the steps into the correct order.
Some mathematical knowledge will be necessary to determine which numbers
correspond to previous or subsequent steps but there is never an over reliance on
computing the equation from beginning to end.
Project: The math map. This project is a variation of an earlier activity in the
unit where students had to create multiple ways of expressing a single number that was
placed on the board. In the project, students will be creating a mind-map on a piece of 11
x 14 paper bubbles emanating from a central number. This initial central number will
range from 10 to 20 depending on student ability but a minimum number of expressions
must be met, such as needing 2 additions, subtractions, multiplications, additions,
distributions, etc. Some of the expressions must also be visual representations, like those
used earlier in the unit. Students must also make use of variables in at least some of their
expressions and give the substituted values for those variables that will give the
simplified term in the central bubble.
This project is an extensive open-ended question that has students displaying their
understanding according to their ability. Learners with math disabilities will need

90
guidance in creating their expression and may need to be able to express themselves
orally or with manipulatives to show that they can produce expression that are
representative of their chosen number.
Surface Area Unit
This unit makes extensive use of concrete models that will give students a
reference point for the majority of the work in the unit. The unit starts with the use of nets
to create 3-dimensional shapes and these shapes will serve as the basis of concrete
examples for the subsequent lessons. The vocabulary lesson is an overview of many of
the terms that will be encountered in the unit with both formal and informal definitions to
give students a clear understanding of what is meant by each term. The area and surface
area lesson make use of the models built at the beginning of the unit to break down the
nets and shapes that constitute the 3-D shapes and will help students to understand what
formulas will need to be applied to calculate total surface area. The project that sums up
the unit is actually a selection of 4 projects that students may select from based on their
own interest or ability.
The readily apparent practical nature of area calculations and the use of hands-on
models will help to develop and maintain student interest. Student interest and the
recognition of the importance of a topic are factors that can improve student performance
on mathematical tasks. Learners with math disabilities can have particular troubles in
attending to tasks in math and this unit serves to remove this obstacle through direct
application of the math to student’s lives and the use of concrete models to improve
students understanding. The unit’s models and projects also help to appeal to more

91
kinesthetic learner by allowing students to construct models that show their
understanding.
The follow unit was created using the curriculum goals of Grade 8 Math in the BC
Ministry of Education New Curriculum. The following table describes the key ideas of
each lesson and the potential math difficulties that can be encountered when teaching the
lesson.
Table 4
Surface Area Unit Summary
Lesson

Key Idea(s)

Lesson 1: Folding
of 2-dimensinoal
Nets Into 3dimensional
Shapes

3D shapes can be
represented as 2D
nets that are
comprised of
multiple 2D shapes

Math Difficulties Approached by the
Lesson
MD learners that require concrete examples
of concepts will be able to see the
relationship between 2D net and 3D object.
There are no calculation so MD learners
with dyscalculia will not be affected by this
lesson

MD learner can also have reading
There are a wide
disabilities (RD). These are addressed by
Lesson 2:
variety of terms that the use of informal definitions using
Geometry
exist to describe
language developed by the class to more
vocabulary
shapes in the surface clearly define certain terms. Cloze versions
area unit
of the definitions are available. Use of
labeled images helps to reinforce concepts.
Various shapes have
MD learners can struggle with pattern
differing numbers of
recognitions. The patterns are clearly
edges, faces and
Lesson 3: Anatomy
explained and written out using
vertices, There are
of a 3-dimensional
mathematical formulas. This also helps
patterns in these
Objects
with those MD students who have memory
numbers amongst
difficulties and removes the number of
certain types of
individual facts that need to be retained.
shapes
Lesson 4: Creating
There are formulas MD learners are repeating the formulas to
Pictures Using
for the areas of
improve memorization. There are no
Area Formulas of
shapes that apply to calculations so dyscalculia students are not
2-dimensional
total surface area
at a disadvantage.
Shapes

92

Lesson 5: Surface
Area of
Rectangular
Prisms

Lesson 6: Surface
Area of Triangular
Prisms and
Pyramids

Lesson 7: Surface
Area of Cylinders

3D shapes are
comprised of
multiple 2D shapes
with area formulas.
Marking scheme
emphasizes
displaying how to
solve a problem, not
solving the problem
3D shapes are
comprised of
multiple 2D shapes
with area formulas.
Marking scheme
emphasizes
displaying how to
solve a problem, not
solving the problem
3D shapes are
comprised of
multiple 2D shapes
with area formulas.
Marking scheme
emphasizes
displaying how to
solve a problem, not
solving the problem

Project: 3-floor
Students are given
House Plan, 1-floor
multiple means of
Apartment Model,
expressing
3-dimensional
themselves and their
Stellated Platonic
understanding of
Solid, Novel Cereal
surface area
Box
mathematics.

This is a multistep process that will strain
the working memory of learners. Choosing
to focus on one aspect, such as nets or
formulas, will limit demands on student
working memory. Receiving marks for
multiple pieces of information ensure that
students receive some credit for displaying
some of the knowledge learned in class.
This is a multistep process that will strain
the working memory of learners. Choosing
to focus on one aspect, such as nets or
formulas, will limit demands on student
working memory. Receiving marks for
multiple pieces of information ensure that
students receive some credit for displaying
some of the knowledge learned in class.
This is a multistep process that will strain
the working memory of learners. Choosing
to focus on one aspect, such as nets or
formulas, will limit demands on student
working memory. Receiving marks for
multiple pieces of information ensure that
students receive some credit for displaying
some of the knowledge learned in class.
MD learners can have difficulty with
calculations and spatial reasoning. The last
2 projects are meant to address this by
having student create basic or repetitive
objects that fit together easily with the
assistance of adults. The varied number of
projects going on ensure that no students
stands out as needing assistance.

Lesson 1: Folding of 2-dimensinoal nets into 3-dimensional shapes. The unit
begins with students selecting 3 nets from a selection of options. The nets fold into a
variety of 3-dimensional shapes that will be examined at various points throughout the
unit. The shapes include objects such as rectangular prisms and cylinders as well as more

93
complicated objects like dodecahedrons and octahedrons. Students are asked to colour
and decorate each of their models and then fold them into their 3-dimensional shape. It is
explicitly stated that the models will be used by themselves and other students as
examples, demonstrations, and tools to solve problems.
This lesson gives students the opportunity to be creative from the outset and also
allow them to create their own tools that will be used later in the unit. This connection to
later lessons gives the seemingly creative and fun lesson a sense of purpose that drives
students to complete their models.
Lesson 2: Geometry vocabulary. The geometry vocabulary sheet is a 3-column
handout that is given to students to go over the relevant vocabulary for the unit. There is a
cloze-sentence version for those students who may have writing difficulties so that they
can still participate and follow along with the notes. The goal of this lesson to give a list
of key vocabulary terms that will be used throughout the unit (including reference to the
models made in the previous lesson) and to give formal and informal language definitions
of the terms along with an image illustrating the concept.
The first column of the handout contains the term already typed out. The second
column is labeled “Definitions” and is where both the formal and informal definitions
will be written. The formal definition will contain jargon terms and language that not all
students will be familiar with. The informal definition that will be written in this space is
a simplified version of the language to give students a clearer picture of what the formal
language is saying. An example of this would be the definition of the term vertex on the
vocabulary sheet. The formal definition is “the point on the object where several edges
meet” and is itself a relatively simple definition. The informal definition would be written

94
based on student feedback and conversation about what this formal term means. An
example of the informal definition might be something like “the pointy and pokey part of
a pyramid”, which is an accurate description of vertex, even though it is not necessarily a
complete definition that uses jargon. In this way, students are participating in the
construction of the language that describes the objects and concepts that will be studied in
the unit.
The final column is for an image that describes the concept in as simple a way as
possible. This image may include labels or sketches of the objects that were made in the
first lesson as a means of describing the vocabulary term in question.
Lesson 3: Anatomy of a 3-dimensional objects. In this lesson students will be
using the objects construction in the first lesson and describing them according to the
vocabulary defined in the second lesson. Students will be given a sheet that lists 3D
objects made in the first lesson, along with several that were not created in the first lesson
(such as spheres or octagonal prisms). Students are asked to state how many faces, edges,
and vertices each of the objects possesses. Use of the objects that were made in the first
lesson, in addition to models that may be brought in from a math resource room, will give
students a physical object to interact with so they might better count the faces, edges, and
vertices.
Upon completing the activity in this lesson, the class then looks for patterns in the
faces, edges, and vertices for shapes that are related. Formulas are then produced to help
students who may not be able to remember the exact number of ease measurement but
can be applied to arrive at a correct answer. An example of this is the pattern seen in
prisms. The number of sides will vary between rectangular, triangular, and octagonal

95
prisms but the total number of sides will always be 2 (the number of bases) + the number
of sides of the base. This pattern will be developed from the information on the activity
sheet so that students are given the opportunity to create formulas that describe patterns
from a list of data. Other such shape patterns occur for edges and vertices and can be
applied to the different types of pyramids.
The IXL program provides activities that are similar to this for additional practice
to reinforce the ideas learned in class.
Lesson 4: Creating pictures using area formulas of 2-dimensional shapes.
This lesson is a review of the area formulas for triangles, squares, rectangles, and circles.
For circles, the formula for circumference is included as circumference is needed to solve
for the surface area of cylinders.
The activity of this lesson is writing each of the formulas multiple times in order
to create the shape of the object to which it applies. This will reinforce the relationship
between the formula and the shape to which it corresponds. The triangle formula (1/2 b x
h) is repeatedly written to fill the shape of a triangle, the square formula (s2) is written to
fill the shape of a square, etc. For circles, the circumference formula is used to write
along the circumference of the circle. Students are encouraged to use colours and to
arrange multiple shapes to create an image that is made of the various formulas. Most
learners will able to complete this assignment with modification being made for those
learners who have difficulty writing having the shapes already drawn for them and
repeating the formulas a minimal number of times.
The numerical practice of the formulas will make use of IXL as it gives
immediate feedback and can be completed at the student’s own pace.

96
Lesson 5: Surface area of rectangular prisms. This lesson focuses on
rectangular and square prisms and the shapes that comprise them. Students are shown
what the nets are for the prisms (including cubes) and the number of each shape that
needs to have its area determined. For examples a square prism has 2 squares and 4
rectangles that are all the same and a cube has 6 sides that are all the same. The
associated formulas for each shape are shown and students are given a clear process to go
through when they write out their calculations.
An introductory activity for this lesson is giving students several different
configurations of the nets of the shapes. Students are placed in small (2-3 members)
groups with several different nets for a given shape. Some of the nets will fold to create
rectangular prisms and cubes while others will not work. At the end of the activity, those
configurations that create the complete prism are placed together while those that did not
create prisms are placed together. For learners with math disabilities, emphasis should be
placed on those configurations that do work so as to limit confusion and different
configurations that are memorized.
Marks are assigned so that students will needs to draw out the nets, label the sides
with the correct lengths, give formulas for each different shape, substitute values into the
formulas, show work for completing the formulas, give the areas of each shape, and then
show a final step where they add all of the individual areas together to obtain a total
surface area. The fact that these questions are very long is explicitly stated but students
are told that many of the marks come from simply drawing the shape, labeling sides, and
giving formulas. It is possible to achieve sufficient marks to pass a question by simply
drawing nets, giving formulas, and substituting. The calculations simply add to the mark.

97
This is done to alleviate the stress of needing to obtain a final answer and instead
focusing on illustrating understanding of what is required to get a final answer. This
removes the barrier that some students with math disability have with calculating
correctly. Students are also reminded that, while the questions are long, each individual
step is a basic formula question that just happens to apply to a more complicated 3D
shape.
Clearly delineating the distribution of the marks for each question will help
students to see the different steps and will serve as a modified program for those learners
that warrant modification. Teachers will be able to set specific goals related to parts of
the question to determine student math competencies in determining surface area.
Lesson 6: Surface area of triangular prisms and pyramids. This lesson is
similar to the previous lesson but applies to triangular prisms and pyramids. The nets for
each are drawn out and the marking schemes are similar. These shapes may give students
more difficulty as these prisms are not encountered as often and the calculations involve
fractions.
Lesson 7: Surface area of cylinders. Cylinders are one of the most complicated
shapes to obtain surface area for because they make use of multiple formulas and the
numbers cannot always be made round because of the nature of pi. This situation is
partially accommodated for with the marking scheme used for previous shapes that
highlights the need to show how an answer can be obtained, not necessarily just the
ability to produce an answer.
Project: Applications of surface area. The project for this unit is actually a
selection of 4 different projects that students may choose from based on interest. The 3-

98
floor house plan is a drawing of the floor plans for 3 floors of a home that students
design. Each floor must have a minimum of 4 rooms. The cost of painting and flooring
the rooms must also be determined using a price list provided to the students. The 1-floor
apartment model is similar to the house plan but has the students actually construct the
model but still calculate costs to paint and floor the apartment. The 3-dimensional
stellated Platonic solid requires students to pick 2 of the following: octahedron,
dodecahedron, or dodecahedron. Students then add the appropriate pyramids to each
surface to create a stellated version of that shape. Students need state the number of faces,
edges and vertices of both the original and stellated shapes as well as calculating the total
surface area before and after the object has been stellated. The novel cereal box project
has students creating a 3-dimensional shape that can be used as a cereal box. Students
must decorate their cereal box and decide on what kind of cereal they think it should be
used for. The students must also calculate the cost of producing enough cereal boxes to
supply all of the grocery stores in their area.
When introducing the project, it will be helpful for students to be given examples
of projects and the math involved in each. A worksheet that has scaled down versions of
each project and the associated calculation will help guide the decisions of students to
select a project that both interests them and is within at their ability level.
Each of these projects makes use of the knowledge that students have acquired in
the unit but have significant variation in how they accomplish this. The projects also vary
significantly in how a student can give a creative expression of their work. Because
student project will vary greatly from one another, students that struggle in math will not
stand out as receiving additional assistance as each student is doing something quite

99
unique from the other students around them. Learners that have reached the substituteonly stage and are able to measure and count the relevant information can be given
adapted version of projects. The practice of taking measurements and then placing them
into formulas will be hands-on practice and is a skill that can be focused on as the main
educational goal that can be reported on with regards to student achievement.
IXL Reinforcement for Units
All of the above units will have reinforcement of their relevant ideas with
assigned IXL sections. The program will give immediate feedback and rote practice for
the basic skills necessary to fully comprehend the concepts being taught. It will be
emphasized that IXL be used as independent practice whenever possible because of its
immediate feedback for students.
It is encourage that educators preview the IXL activities they wish to use before
teaching a lesson so that notes may be tailored towards the language of IXL. This is
partially because teacher have no control over the types of questions asked by particular
IXL problem set and students may not be aware of how to answer a question that is asked
in an unfamiliar manner.
Chapter Summary
These units are constructed with the special needs learner in mind but still cover
the necessary material for all learners. The units make use of concrete examples to ensure
that learners with math disabilities may better access the material as part of the class.
Student choice was given in several instances to allow for multiple means of expressing
understanding. The emphasis on students expressing how to get a correct answer, rather
than just getting a correct answer, opens the door to having learners that struggle with

100
computation still being able to give an acceptable solution on how to solve a problem.
The lessons listed do not constitute the entirety of each unit and will need to be
supplemented with practice classes or additional sheets and materials. These units are
general guides that have taken into account the needs of learner with MD. Certain subsets
of math disability will require different considerations depending on the math lesson.
Certain lessons will be differentially affected by particular deficits and these have been
pointed out to educators in each of the lesson summaries found in the appendix. By using
these general guidelines, teachers can create lessons that take into account the needs of
learners with MD while still maintaining the integrity of the lesson and its concept.

Chapter 5: Reflection and Conclusion
Reflection
The intention of this project was to summarize and make use of the current
research on mathematical pedagogy with respect to special education and to use a
diagnostic tool, KeyMath-3, to guide construction of instructional math units. By building
units around possible problems that students with math disability might encounter, the
unit and its concepts can be made more accessible for all learners.
In my research, I encountered a wealth of information about the various ways that
students can exhibit mathematical learning difficulties and that students will not

101
necessarily exhibit all of these deficits. It is therefore important to provide varied
instruction that does not stress a single cognitive domain, as this will make mathematics
an overwhelmingly difficult task for those learners who have difficulty in that domain.
The varied approach to mathematical instruction gives students multiple viewpoints of
the same topic. This gives average and high achieving students a broader understanding
of the material while giving low achieving students or students with math disabilities an
access point for the material.
A major take-away from this research project was the separation of mathematical
operations from the scenarios that they describe in common math pedagogical practice.
By this I mean that I have begun to notice that the basic operations learned in the
elementary years are new and distinct from one another but after the middle school years
there is relatively little that is new in terms of operations. Students learn counting,
addition, subtraction, division and multiplication and it is these operations that form the
basis of all later mathematics. In many situations, the previous skills are only applied in
different scenarios and to describe different real-world situations. This has led me to
place a great deal of importance on the explicit instruction of the operations that are
necessary for a given concept and to describe the real-world scenario to which the math is
meant to apply. An example of this would be in the surface area unit that has formulas
involving multiplication and addition that are applied to multiple shapes and then
combined to give a total surface area. The overall amount of math can be overwhelming
for students but breaking down each step and showing the basic operations
(multiplication, addition, etc.) can help to alleviate anxiety in students. Giving students
these instructions will hopefully alleviate the confusion and anxiety that can accompany

102
learning a “new” type of math. This viewpoint also makes the mastery of, or at least
functional ability to complete, these basic operations of paramount importance to success
in math.
The ability to automatically mark a student’s work should not be underestimated.
The time it takes to mark an assignment can greatly impact the time spent discussing
errors or correcting student misunderstandings. By reducing time spent marking, not only
do students get immediate feedback as to whether they are correct, but teachers are freed
to directly interact with students for more time or to spend time on other activities that
can directly address the misunderstandings of students.
The end projects of each unit often involved some form of artistic expression of
the math concept being taught for that unit. Mathematics can become a one-dimensional
subject for students with its only expression being numbers and correct answers. By using
projects that encourage students to artistically represent math concepts, it was my hope
that these learners no longer hold on to the stigma that math class needs to be only for
those with number driven dispositions. Artistic expression has its place mathematics. By
expressing learning in this manner, I also hoped that students would be driven to create
more math expressions in order to create more art. This intrinsic drive could be a more
powerful motivator than simply completing more math problems that I, as the teacher,
had created. The intrinsic drive to create art projects around mathematical concepts would
also increase the relevancy of the mathematics for the student, which has been shown to
improve student performance in mathematics (Boyd & Bagerhuff, 2009; Sayeski &
Paulsen, 2010; Shin & Bryant, 2015).

103
Any methods taught should be taught to the class as a whole and any goals should
be made achievable by the class as a whole. This is not to say that students who have
higher or lower levels should all produce the same end project but that all students should
be able to access the same information and produce the same baseline product (Dieker &
Rodriguez, 2013; Sayeski & Paulsen, 2010). Equitable access to the material does not
mean there will be equitable production of material. The differentiating factor for the
learners should be what they produce, not what they are taught.
Students need to understand why they are learning a given topic and this is an
important question to consider not only from the perspective of students but also from the
perspective of the teacher. Teachers need to understand the why of the education being
administered and how that education will be useful to the student in the future.
Considering this question at the time of unit construction will ensure that there is also an
end-goal and purpose to each lesson with an overarching goal for the unit beyond simple
mastery of basic skills (Zheng, Flynn, & Swanson, 2013). Having the students know how
they will use the math later in real-life situations will provide more incentive to practice
the skills now or in the present learning situation. Lack of interest can lead to lack of
motivation which further leads to a lack of ability, as the student never practices the basic
skills.
Using the KeyMath-3 to guide the construction of the units and assignments has
allowed me to word questions and build the understanding of students that will directly
influence any future assessment they may be given. When these students are assessed
using a KeyMath diagnostic in the future, the questions will be reflective of material they
have encountered in the classroom using a variety of means that were designed to

104
improve their understanding of that specific concept. The learners will have encountered
concrete examples of the concepts that improve understanding (Agrawal & Morin, 2016;
Witzel, 2005) and will have been allowed to express understanding in a variety of ways at
various stages of solving the problem. When the KeyMath-3 assessment is administered
in the future to these students, it will be a more valid assessment because of the thorough
way in which these learners have been exposed to the material that are being tested on.
The goal of this project was successful in that it has allowed me to use the
knowledge and practices I have gained through my research to create entire math units
that will improve instruction to all learners. Research has informed my teaching practice
and I now have a greater understanding of not only math pedagogy but student
difficulties with mathematics and the assessment of student math ability. All of these
factors allow me to teach mathematics to all students effectively and work with other
professionals to advance student understanding of mathematics.

Conclusion
By using the KeyMath approach to unit construction, the language of the math
lessons was tailored towards the assessment that would lead to diagnosis of math
disability. This allowed the units to be tailored towards deficiencies in mathematical
understanding and ability while simultaneously considering the need for stronger math
learners to be able to express their level of mathematical understanding. The units
constructed tied together both the KeyMath-3 and the BC Ministry of Education Grade 8
Curriculum in a way that guided teaching practice towards preparing students for possible

105
assessment using the KeyMath-3 diagnostic assessment. This practice will hopefully
improve student performance on diagnostic tests and increase student confidence in their
own ability to be proficient in mathematics. Students should be encouraged to see special
education diagnoses as temporary and that they can improve their level of mathematical
understanding to the point where their designation may be removed. Mathematical
understanding is not a static personality trait and the KeyMath instructional approach is
one way of improving special needs learner’s mathematical understanding and
performance.

References
Anderson, L. W., & Krathwohl, D. R. (2001) A taxonomy for learning, teaching, and
assessing: A revision of Bloom’s taxonomy of educational objectives. New York:
Longman.
Bakker, M., van den Heuvel-Panhuizen, M., & Robitzsch, A. (2016). Effects of
mathematics computer games on special education students' multiplicative
reasoning ability: Mathematics computer games in special education. British
Journal of Educational Technology, 47(4), 633-648.

106
Björklund, C. (2012). One step back, two steps forward - an educator's experiences from
a learning study of basic mathematics in preschool special education. Scandinavian
Journal of Educational Research, 56(5), 497-517.
Bouck, E. C., & Flanagan, S. (2009). Assistive technology and mathematics: What is
there and where can we go in special education. Journal of Special Education
Technology, 24(2), 17-30.
Boyd, B., & Bagerhuff, M. E. (2009). Mathematics education and special education:
Searching for common ground and the implications for teacher education.
Mathematics Teacher Education and Development, 11(2009), 54-67.
Branum-Martin, L., Fletcher, J. M., & Stuebing, K. K. (2013). Classification and
identification of reading and math disabilities: The special case of comorbidity.
Journal of Learning Disabilities, 46(6), 490-499.
Breaux, K. C., Avitia, M., Koriakin, T., Bray, M. A., DeBiase, E., Courville, T., . . .
Grossman, S. (2017). Patterns of strengths and weaknesses on the WISC-V, DASII, and KABC-II and their relationship to students’ errors in oral language, reading,
writing, spelling, and math. Journal of Psychoeducational Assessment, 35(1-2),
168-185.
British Columbia Ministry of Education. (2016). BC’s New Curriculum. Retrieved from
https://curriculum.gov.bc.ca/curriculum/overview
Brownell M. T., Sindelar P. T., Kiely M. T., & Danielson L. C. (2010). Special education
teacher quality and preparation: Exposing foundations, constructing a new model.
Exceptional Children, 76, 357-377.

107
Burny, E., Valcke, M., & Desoete, A. (2012). Clock reading: An underestimated topic in
children with mathematics difficulties. Journal of Learning Disabilities, 45(4), 351360.
Chong, S. L., & Siegel, L. S. (2008). Stability of computational deficits in math learning
disability from second through fifth grades. Developmental Neuropsychology,
33(3), 300-317.
Coben, D. (2003). Adult numeracy: Review of research and related literature.
Nottingham, UK: University of Nottingham.
Connelly, A. J., (2008). KeyMath-3 diagnostic assessment: Canadian edition. Toronto.
Pearson
Daugherty, S. J., Campana, K. A., Kontos, R. A., Flores, M. K., Lockhart, R. S, & Shaw,
D. D. (1996). Supported education: A qualitative study of the student experience.
Psychiatric Rehabilitation Journal, 19(3) 59–71.
Dekker, M. C., Ziermans, T. B., & Swaab, H. (2016). The impact of behavioural
executive functioning and intelligence on math abilities in children with intellectual
disabilities: Impact EF and IQ on math in children with MBID. Journal of
Intellectual Disability Research, 60(11), 1086-1096.
Dieker, L. A., & Rodriguez, J. A. (2013). Enhancing secondary cotaught science and
mathematics classrooms through collaboration. Intervention in School and Clinic,
49(1), 46-53.
Doabler, C. T., Clarke, B., Stoolmiller, M., Kosty, D. B., Fien, H., Smolkowski, K., &
Baker, S. K. (2016). Explicit instructional interactions: Exploring the black box of a
tier 2 mathematics intervention. Remedial and Special Education, 38(2), 98-110.

108
Ekstam, U., Korhonen, J., Linnanmäki, K., & Aunio, P. (2017). Special education preservice teachers' interest, subject knowledge, and teacher efficacy beliefs in
mathematics. Teaching and Teacher Education, 63, 338-345.
Elliott, S. N., Kurz, A., Tindal, G., & Yel, N. (2016). Influence of opportunity to learn
indices and education status on students’ mathematics achievement growth.
Remedial and Special Education, 38(3), 145-158.
Erford, B. T., & Biddison, A. R. (2006). The math essential skills screener - upper
elementary version (MESS-U): Studies of reliability and validity. Assessment for
Effective Intervention, 31(4), 33-47.
Erford, B. T., & Klein, L. (2007). Technical analysis of the Slosson–Diagnostic math
screener (S-DMS). Educational and Psychological Measurement, 67(1), 132-153.
Esmonde, I. (2009). The Canadian journal for science, mathematics and technology
education: Special issue on equitable access to participation in mathematical
discussions: Looking at students' discourse, experiences, and perspectives.
Canadian Journal of Science, Mathematics and Technology Education, 9(4), 289292.
Faramarzi, S., & Sadri, S. (2014). The effect of basic neuropsychological interventions on
performance of students with dyscalculia. Neuropsychiatria i Neuropsychologia,
9(2), 48.
Furman, T., & Rubinsten, O. (2012). Symbolic and non-symbolic numerical
representation in adults with and without developmental dyscalculia. Behavioral
and Brain Functions : BBF, 8(1), 55-55.

109
Gal, I. (Ed.). (2000). Adult numeracy development: Theory, research, practice. Cresskill,
NJ: Hampton Press.
Garrett, A. J., Mazzocco, M. M. M., & Baker, L. (2006). Development of the
metacognitive skills of prediction and evaluation in children with or without math
disability. Learning Disabilities Research & Practice, 21(2), 77-88.
Griffin, C. C., Jitendra, A. K., & League, M. B. (2009). Novice special educators’
instructional practices, communication patterns, and content knowledge for
teaching mathematics. Teacher Education and Special Education, 32(4), 319-336.
Haggarty, L., & Pepin, B. (2002). An investigation of mathematics textbooks and their
use in english, french and german classrooms: Who gets an opportunity to learn
what? British Educational Research Journal, 28(4), 567-590.
Hallahan, P. D., Kauffman, M. J., & Pullen, C. P. (2015). Exceptional learners:
Introduction to special education. Toronto: Pearson
Hale, J. B., Fiorello, C. A., Dumont, R., Willis, J. O., Rackley, C., & Elliott, C. (2008).
Differential ability scales – second edition (neuro)psychological predictors of math
performance for typical children and children with math disabilities. Psychology in
the Schools, 45(9), 838-858.
Hoskyn M., Swanson H. L. (2000). Cognitive processing of low achievers and children
with reading disabilities: A selective meta-analytic review of the published
literature. School Psychology Review, 29, 102–119.
Hsieh, H.F., Shannon, S.E., (2005). Three approaches to qualitative content analysis.
Qualitative Health Research. 15(9), 1277-1288

110
Hutchinson, L. N. (2017). Inclusion of exceptional learners in Canadian schools. A
practical handbook for teachers. (5th ed.) Toronto: Pearson
Johnston, B. (2002). Numeracy in the making: Twenty years of Australian adult
numeracy: An investigation by the New South Wales Centre Adult Literacy and
Numeracy Australian Research Consortium (ALNARC). Sydney: University of
Technology.
Katz, V. J., Jankvist, U. T., Fried, M. N., & Rowlands, S. (2014;2013;). Special issue on
history and philosophy of mathematics in mathematics education. Science &
Education, 23(1), 1-6.
Kelley, M., & Legge, C. (2006). Teacher's challenge: seven days in a struggling school.
Retrieved from https://curio.ca/en/video/teachers-challenge-seven-days-in-astruggling-school-1034/
Ketterlin-Geller, L. R., Chard, D. J., & Fien, H. (2008). Making connections in
mathematics: Conceptual mathematics intervention for low-performing students.
Remedial and Special Education, 29(1), 33-45.
Kitchenham, G. A. (2011). Preparing 21st century teachers. Implications for teacher
education. NorthWest Passage 9(2), 7-10.
Krippendorf, K. (1969) Models of messages: three prototypes. In G. Gerbner, O. R.
Holsti, K. Krippendorf, G. J. Paisly & Ph.J. Stone (Eds.), The analysis of
communication content. New York: Wiley
Loong, E. Y. K. (2014). Fostering mathematical understanding through physical and
virtual manipulatives. Australian Mathematics Teacher, 70(4), 3-10.

111
Lipson, J. (2004). Indigo children and schools. Spiral Wisdom. Available at:
http://www.spiralwisdom.com/indigo-children-schools/
Markon K. E., Chmielewski M., Miller C. J. (2011). The reliability and validity of
discrete and continuous measures of psychopathology: A quantitative review.
Psychological Bulletin, 137(5), 856–879.
Mayring, P. (2000). Qualitative content analysis. Forum: Qualitative Social Research,
1(2)
Mogalakwe, M. (2006). The use of documentary research methods in social research.
Research report. African Sociological Review, 10(1), 221-230
Morgan, P. L., Frisco, M. L., Farkas, G., & Hibel, J. (2010). A propensity score matching
analysis of the effects of special education services. The Journal of Special
Education, 43(4), 236-254.
Murphy, M. M., Mazzocco, M. M. M., Gerner, G., & Henry, A. E. (2006). Mathematics
learning disability in girls with turner syndrome or fragile X syndrome. Brain and
Cognition, 61(2), 195-210.
National Numeracy Review Report Panel. (2008). National Numeracy Review Report.
Barton, ACT: Commonwealth of Australia. Human Capital Working Group
Council of Australian Governments.
National Research and Development Centre for Adult Literacy and Numeracy. (2005).
Does numeracy matter more? University of London: Parsons, S., & Bynner, J.

Northcote, M., & McIntosh, A. (1999). What mathematics do adults really do in
everyday life? The Australian Primary Mathematics Classroom, 4(1), 19–21.

112
Parkin, J. R., & Beaujean, A. A. (2012). The effects of wechsler intelligence scale for
Children—Fourth edition cognitive abilities on math achievement. Journal of
School Psychology, 50(1), 113-128.
Payne, G., & Payne, J. (2004). Key concepts in social research. London, UK: Sage
Pliner, S. M. & Johnson, J. R. (2004). Historical, theoretical, and foundational principles
of universal design in higher education. Equity & Excellence in Education, 37,
105–113.
Price, G. R., & Fuchs, L. S. (2016). The mediating relation between symbolic and
nonsymbolic foundations of math competence. PloS One, 11(2), 1-14.
Prior, L. (2003) Using documents in social research. London. Sage.
Powell, S. R. (2015). Connecting evidence-based practice with implementation
opportunities in special education mathematics preparation. Intervention in School
and Clinic, 51(2), 90-96.
Robinson, S. L., & Skinner, C. H. (2002). Interspersing additional easier items to enhance
mathematics performance on subtests requiring different task demands. School
Psychology Quarterly, 17(2), 191-205.
Rose, M. B., & Holmbeck, N. G. (2007) Attention and executive functions in adolescents
with spina bifida. Journal of Pediatric Psychology. 32(8) 983-994.
Sayeski, K. L., & Paulsen, K. J. (2010). Mathematics reform curricula and special
education: Identifying intersections and implications for practice. Intervention in
School and Clinic, 46(1), 13-21.
Schulte, A. C., & Stevens, J. J. (2015). Once, sometimes, or always in special education:
Mathematics growth and achievement gaps. Exceptional Children, 81(3), 370-387.

113
Scott, J. (2006). Documentary research. London, UK: Sage Publications
Scott, J. 1990. A matter of record, Documentary sources in social research. Cambridge:
Polity Press
Seo, Y., & Bryant, D. (2012). Multimedia CAI program for students with mathematics
difficulties. Remedial and Special Education, 33(4), 217-225.
Shin, M., & Bryant, D. P. (2015). Fraction interventions for students struggling to learn
mathematics. Remedial and Special Education, 36(6), 374-387.
Shin, M., & Bryant, D. P. (2016). Improving the fraction word problem solving of
students with mathematics learning disabilities: Interactive computer application.
Remedial and Special Education, 38(2), 76-86.
Speijer, J., Gray, C., Peirce, S., & Doherty, G. (2016). Mathematics and special
education: Two teachers in one classroom. Gazette - Ontario Association for
Mathematics, 54(3), 33.
Stanescu-Cosson, R., Pinel, P., van De Moortele, P F, Le Bihan, D., Cohen, L., &
Dehaene, S. (2000). Understanding dissociations in dyscalculia: A brain imaging
study of the impact of number size on the cerebral networks for exact and
approximate calculation. Brain : A Journal of Neurology, 123(11), 2240-2255.
Stegemann, C.K. & Aucion, A. (2018). Inclusive education. Stories of success and hope
in Canadian context. Don Mills, ON: Pearson
Stotsky, S. (2009). Licensure tests for special education teachers: How well they assess
knowledge of reading instruction and mathematics. Journal of Learning
Disabilities, 42(5), 464-474.

114
Stuebing K. K., Fletcher J. M., LeDoux J. M., Lyon G. R., Shaywitz S. E., Shaywitz B.
A. (2002). Validity of IQ-discrepancy classifications of reading disabilities: A
meta-analysis. American Educational Research Journal, 39, 469–518
Sullivan, A. L., & Field, S. (2013). Do preschool special education services make a
difference in kindergarten reading and mathematics skills?: A propensity score
weighting analysis. Journal of School Psychology, 51(2), 243.
Swanson, H. L., & Jerman, O. (2006). Math disabilities: A selective meta-analysis of the
literature. Review of Educational Research, 76(2), 249-274.
Swanson, H. L. (2015). Cognitive strategy interventions improve word problem solving
and working memory in children with math disabilities. Frontiers in Psychology, 6,
1-13.
Tezer, M., & Kanbul, S. (2009). Opinions of teachers about computer aided mathematics
education who work at special education centers. Procedia - Social and Behavioral
Sciences, 1(1), 390-394.
Tolar, T. D. (2016). Math performance in turner syndrome: Implications for math
disability research. Developmental Medicine & Child Neurology, 58(2), 110-110.
Van Garderen, D., Scheuermann, A., Poch, A., & Murray, M. M. (2016). Visual
representation in mathematics: Special education teachers’ knowledge and
emphasis for instruction. Teacher Education and Special Education, 41(1), 7-23
Vygotsky, L.S. (1978). Mind in society: The development of higher psychological
processes. Cambridge, Massachusetts. Harvard University Press.
Willcutt, E. G., Petrill, S. A., Wu, S., Boada, R., DeFries, J. C., Olson, R. K., &
Pennington, B. F. (2013). Comorbidity between reading disability and math

115
disability: Concurrent psychopathology, functional impairment, and
neuropsychological functioning. Journal of Learning Disabilities, 46(6), 500-516.
Williams, T. O., Fall, A., Eaves, R. C., Darch, C., & Woods-Groves, S. (2007). Factor
analysis of the KeyMath—Revised normative update form A. Assessment for
Effective Intervention, 32(2), 113-120.
Witzel, B. (2005). Using CRA to teach algebra to students with math difficulties in
inclusive settings. Learning Disabilities: A Contemporary Journal, 3(2), 49–60.
Xin, Y. P., & Tzur, R. (2016). Cross-disciplinary thematic special series: Special
education and mathematics education. Learning Disability Quarterly, 39(4), 196198.
Zheng, X., Flynn, L. J., & Swanson, H. L. (2013). Experimental intervention studies on
word problem solving and math disabilities: A selective analysis of the literature.
Learning Disability Quarterly, 36(2), 97-111.
Zurina, H., & Williams, J. (2011). Gesturing for oneself. Educational Studies in
Mathematics, 77(2/3), 175-188.

Appendix A: Sample Rainbow Squares Table
121
100
81
64
49
36
25
16
9
4
1

1

2

3

4

5

6

7

8

9

10

11

116

* This is just a sample of this assignment. On typical graph paper, students are able to
create a table that includes up to 25 x 25. This is often sufficient for most questions that
will be asked in this unit.

Appendix B: Samples of Algebraic Container Models
Addition (5 + c)

The five (5) counters outside are set in the question. The container is what can be varied
and has three (3) added to it. Students count the total number of counter and see that there
are eight (8).

117

Subtraction (5 – c)

The container now has three (3) red counters added which are removed from the initial
amount of 5, leaving two (2) counters.
Multiplication (5x)

The container now has three (3) counters that vary distinctly to show the different
operation and thought process. The counters of the container illustrate the number of
stacks of the initial value, which now give a total of 15 of the initial counter.

Division (12/x)

The initial amount is now nine (9) and the container has 3 counters indicating division,
which will be the number of even groups into which the initial set of counter is split
(divided). The number of counters under each “divide” counter is the answer to the
question.

118

Appendix C: Samples of Pythagorean Art Projects

119

120