Minus the Math Anxiety:Breaking Down the
Barriers Between Students and Mathematics
by
Elliott Freeman
A Thesis
Presented to the Department of Mathematics
and the Robert D. Clark Honors College
in partial fulllment of the requirements for the degree of
Bachelor of Science
June 2021
An Abstract of the Thesis of
Elliott Freeman for the degree of Bachelor of Science
in the Department of Mathematics to be taken June 2020
Title: Minus the Math Anxiety: Breaking Down the Barriers Between Students and Mathematics
Approved: Christopher D. Sinclair, Ph.D.
Christopher D. Sinclair
The public’s opinion on mathematics is overwhelmingly negative. Millions of individuals suer from a
diagnosable condition called math anxiety. There are, however, ways to combat this issue. Evidence-based
practices such as explicit instruction have been shown to decrease the load on working memory, something
math anxiety limits. By using explicit instruction and other evidence-based practices and high-leverage
practices we can reduce math anxiety and increase math achievement. Another compounding issue is the
public perceived lack of relevancy of mathematics in schools. This issue can be alleviated by teaching
discrete mathematics, a branch of mathematics that has close ties to computer science which is essential
in our current society. Teaching python coding concurrently with discrete mathematics only increases
the connection to "the real world". Additionally, teaching about diverse individuals in computer science
and mathematics, along with other methods of reducing math anxiety and alternative testing methods can
further decrease, or make more manageable, the prevalence of math-related anxiety.
ii
Acknowledgments
I would like to thank my thesis advisor Christopher Sinclair, Associate Professor for his continued belief
that my work on this project always had a positive rst derivative, even when I was doubting my progress.
His support and advice were an essential component of completing this project. I would also like to thank
the other members of my thesis committee Nancy Nelson, Research Associate Professor and Kathleen Free-
man Hennessy, Sr Instructor I/Dir UG Study for their patience and support. Nancy Nelson also deserves
recognition for being the rst person to teach me about special education in mathematics, without this
knowledge I would not have been able to complete this project. Kathleen Freeman Hennessy’s knowledge
and teaching skills also need acknowledgment; she is the rst person who taught my the Python program-
ming language. I would like to recognize the invaluable assistance of the Robert D. Clark Honors College as
well as the Mathematics Department at the University of Oregon for their support and all they taught me
over the past ve years. My thanks goes out to Andrew Schopieray and Boris Botvinnik for teaching me
discrete mathematics. I cannot begin to express my thanks to my family whose support and advice made
this project not seem impossible. I would like to especially extend my gratitude towards my wife who not
only supported me through every obstacle but whose patience when I brought home countless books on
teaching, discrete mathematics, and python is unrivaled.
iii
Table of Contents:
Introduction 1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
An Ode to Discrete Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Incorporating Python . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Educational Practices 6
Research to Practice Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
What are Evidence-Based and High Leverage Practices? . . . . . . . . . . . . . . . . . . . . . . . . 9
Explicit Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Logistics 20
Diverse Individuals in Mathematics & Computer Science . . . . . . . . . . . . . . . . . . . . . . . 20
Other Individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Methods of Reducing Math Anxiety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Content 29
Mathematics Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Computer Science Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Sample Lesson Plans 37
Mathematics Lessons: Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Computer Science Lessons: Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Connection Project: Truth Tables & Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Conclusion 51
iv
List of Figures
1 Bar Chart of Topics Covered in Example Syllabi . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Bar Chart of Topics Covered in Example Textbooks . . . . . . . . . . . . . . . . . . . . . . . 35
3 Review Question: Turning Statements to Symbolic Logic . . . . . . . . . . . . . . . . . . . . 38
4 Truth Table for Negation of p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Truth Table for Conjunction of p and q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Empty Truth Table for Disjunction of p and q . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7 Full Truth Table for Disjunction of p and q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
8 Empty Truth Table for Conditional If p Then q . . . . . . . . . . . . . . . . . . . . . . . . . . 40
9 Demonstration Question 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
10 Demonstration Question 1 Empty Truth Table . . . . . . . . . . . . . . . . . . . . . . . . . . 41
11 Demonstration Question 1 Full Truth Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
12 Demonstration Question 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
13 Demonstration Question 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
14 Demonstration Question 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
15 Demonstration Question 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
16 Complex Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
17 String Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
List of Tables
1 High-Leverage Practices for K-12 Special Education Teachers . . . . . . . . . . . . . . . . . . 10
2 Table of Topics Covered in Example Syllabi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Table of Topics Covered in Example Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Table of Topics Covered in Example Textbooks Part 2 . . . . . . . . . . . . . . . . . . . . . . 33
5 Comparison Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Logical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
v
Introduction
Background
The construct of math anxiety was rst introduced in 1957 by Dreger and Aiken who said that "Many
persons report in clinical sessions and in academic classes that they are emotionally disturbed in the pres-
ence of mathematics. It has been the writers’ experience that reactions of this nature to mathematical and
arithmetical materials constitute a major eld of emotionality in academic situations" [A1]. However, people
have been expressing math anxiety for centuries. A verse from the sixteenth century says "Multiplication is
vexation ... and practice drives me mad". Mathematics education has come a long way since then but math
anxiety still exists and is prevalent in our society. Math anxiety is a diagnosable condition characterized by
feelings of frustration and anxiety that limit the individual’s capability to interact with mathematics in a
meaningful way. It has been dened as "a feeling of tension and anxiety that interferes with the manipula-
tion of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic
situations" [A2].
Math anxiety seems to consist of multiple components. The two components are worry and emotion-
ality. "Worry is the cognitive component of anxiety, consisting of self-deprecatory thoughts about one’s
performance. Emotionality is the aective component of anxiety, including feelings of nervousness, ten-
sion, and unpleasant physiological reactions to testing situations" [A3]. While the aective part of math
anxiety eects the actual performance of the student with math anxiety, the cognitive part inuences their
daily life and career path; an individual with math anxiety is less likely to pursue careers that require ex-
tensive mathematics education. Furthermore, studies have shown that math anxiety increases with age and
that women are more likely to suer math anxiety than men [A4].
One consequence of math anxiety is its limiting eect on working memory. Working memory plays a
vital role in mathematical cognition. Working memory capacity is the range of information that an individ-
ual can process at the same time to perform complex tasks, and selectively shines on relevant information
from one moment to another. Learning new information requires a substantial amount of working memory,
something that, if you have math anxiety, is more dicult to procure. For individuals with disabilities or
conditions that further limit working memory, the process of learning more information is signicantly
harder. In addition to limiting working memory, the symptoms of math anxiety include panic, paranoia,
passive behavior, and a lack of condence. Studies have also found that there exists an inverse relationship
between math anxiety and self-ecacy [A5]. Here we dene self-ecacy as in individual’s condence in
their ability to perform mathematics, something which directly impacts motivation to pursue mathematics
[A6]. Math anxiety is pervasive, studies have shown it aects anywhere from 6% to 68% of the population
depending on the study [A7].
2
An Ode to Discrete Mathematics
Discrete mathematics emerged as a separate area of study in the 1960s. An essential part of the mathe-
matics behind computer science, it includes combinatorics, graph theory, counting, probability, and more.
Unlike other areas of mathematics discrete mathematics allows students to very quickly answer the question
"When am I going to use this in the real world?" by using non-trivial "real world" questions that are inter-
esting and challenging. While high school curriculums cover some discrete mathematics, its coverage is
limited and depends highly on which school district you are looking at, as the United States has no national
or even state-wide curricula. Elements of discrete mathematics are however in the NCTM standards. Dis-
crete mathematics is also featured in many middle school and high school mathematics competitions. While
other areas of mathematics like algebra and geometry are done by rote memorization, discrete mathematics
requires creative, exible thinking and problem-solving. In the words of Susan Picker:
An important aspect of discrete mathematics is, that it enables students to see mathematics
more accurately, as having many problems which are still unsolved. With only a small amount
of technical background students can understand enough about a class of problems such as graph
colouring to understand that no algorithm exists which will always guarantee the most ecient
colouring. It means that students can stand at the cutting edge of mathematical knowledge with
the real possibility of making a contribution; something they could not have previously believed.
There are a number of other student beliefs about mathematics that discrete mathematics se-
riously challenges. Such widely held beliefs as mathematics problems can only have one right
answer, that doing maths means memorizing formulas and giving them back in the right order
at the right time, that most math problems can be solved within minutes, and that mathematics
can only be created by geniuses, are disproved in a discrete mathematics class where there may be
many answers to a problem, there may be no formula to follow, students may work on a problem
over several days, and where students may see something that the teacher has never seen that way
before [A8].
3
Incorporating Python
Under normal circumstances, discrete mathematics is taught in college after students have had some
experience with programming. Discrete mathematics is, however, often taught abstractly and disjointed,
without the use of programming. This can make some concepts more dicult to grasp. By incorporating
a language such as Python, whose syntax is relatively intuitive and has its similarities to mathematical no-
tation and algorithmic pseudocode, we can teach students abstract concepts in practical ways. Python is a
simple but powerful object-oriented programming language used by novices and experts alike. Best of all,
it’s free. Using Python allows students to study the mathematics not only theoretically, but practically. It
also helps answer the question of "When will we use this in real life?" as knowledge of computers is consid-
ered essential in this day and age. This is not just a one way relationship, learning discrete mathematics can
help to improve coding skills as experts often use discrete mathematics knowledge to approach the design
of a problem.
4
Motivation
I have been a teacher’s assistant and tutor for 8 years and in that time I’ve seen a lot of students with math
anxiety. Being able to help them understand math is one of my greatest joys. I created this curriculum as a
way to connect to them as I believe that teaching discrete mathematics using eective educational practices
may be the key to turning attitudes towards math from fear to understanding. Furthermore, I believe that
discrete mathematics it in itself important to teach not only in college but in high school as well. There is a
whole other world of mathematics that a large majority of students never get to see because they don’t go
into computer science. However, discrete mathematics can be very disjoint and hard to understand. This is
why I believe it is important to enact these high-leverage and evidence-based practices. I also believe that
adding computer science to the curriculum can help the scaolding of the course to be clearer and more
dened.
5
Educational Practices
Research to Practice Gap
One of the systemic problems that exists in education is the gap between research and practice. While
a key goal of research is the improvement of practice, little of said research makes its way into teaching
practices in the classroom. This means that, despite the proven eectiveness of the evidence-based and
high-leverage practices mentioned here, the practices are not widely implemented. Four conceptual is-
sues, as identied by Broekkamp and van Hout-Wolters, constitute the gap between research and practice
[B9]. The rst is that educational research yields only few conclusive results. This can be due to a myriad
of factors including the complexity of education, the marginal amount of research compared to other ap-
plied disciplines, the narrow or one-sided focus of researchers, etc. The second problem is that educational
research yields only few practical results. This could be due to inconclusive results or, if unambiguous ev-
idence exists, the results often have no practical value. Other times evidence relates to problems that are
too insignicant or too remote from the context of interest. Thirdly, practitioners believe that educational
research is not conclusive or practical, an issue that is fed by the previous two issues mentioned. A study
by Gore and Gitlin found that teachers dismissed research as they found it to be not "practical, contextual,
credible, or accessible" [B10]. Finally, the fourth issue they found was that practitioners make only little
(appropriate) use of educational research. This could be partly due to their mistrust of research but could
also be caused by the limited fulllment of the following two conditions. First, that practitioners " - at least
some of them - should be trained to access information about research results (e.g., through reading re-
search articles), and to critically evaluate and test them. Furthermore, practitioners should receive support
from their organizations for actively using research in the form of time, money, assistance, and ultimately,
collaboration" [B9]. The limited fulllment can also explain the frequent sloppy and inappropriate use of
research by practitioners.
Another article views the gap dierently. Vanderlinde and van Braak pose that there are two opposing
types of knowledge at play here: research-based knowledge, i.e. what is published in scientic journals,
and pedagogical knowledge, i.e. the knowledge used in classrooms [B11]. The problem there is the tension
6
formed by the practitioner asking for solutions to current and ongoing issues while the research seeks
new knowledge. McIntyre expands on this, saying that another contrasting feature is that practitioners
are looking for practical knowledge whereas researchers are prioritizing other values such as clarity and
coherence. Furthermore, research must be, to have any value, abstract, theoretical, or generalized while
teachers need research that enables them to solve the unique and context-rich problems they are facing in
the classroom everyday [B12].
While it is beyond the scope of this thesis to solve this systemic of a problem, possible solutions can
be touched on. First, Broekkamp and van Hout-Wolters propose four possible solution models. One possi-
ble solution is the Research Development Diusion Model, where "practice-oriented research expands on
fundamental research" [B9]. The model assigns a central role to mediators who make decisions on what to
include, discard, or adapt from research for practitioners. The Evidence-Based Practice model describes the
"systematic application of research results in educational practice" [B9]. Unlike the RDD model which is
diverse in what it includes, the EBP model is concerned only with empirical evidence for eective teaching
methods. The model of Boundary-Crossing Practices concerns cooperation between dierent professional
domains towards a common task. Finally, the model of Knowledge Communities believes in intense collab-
oration between dierent professional domains, that they prot from each other’s expertise and generate
new knowledge.
McIntyre suggests that it would be helpful if research is intelligently planned from the beginning with
the intention of informing practitioners of ways to adjust their practices [B12]. One interpretation of this,
is the suggested use of high-leverage practices. High-leverage practices are "practices that are essential
to eective teaching and fundamental to supporting student learning" [B13] They are essential practices
designed provide a clear focus for teacher preparation. Despite their proven eectiveness, these practices
are not widely used. There are multiple possible explanations for why this is:
1. Teacher preparation largely occurs in settings that are removed from the practice of teaching,
and most coursework emphasizes reection, investigation, and knowing about practice rather than
directly focusing on the use of practices in classrooms (Ball et al., 2009)
2. Most teacher education programs have very limited connections between coursework and eld
7
experiences. When this fragmentation occurs between theory and practice, practice primarily fo-
cuses on the ’conceptual underpinnings of teaching as opposed to the concrete practices new teachers
may need to enact when they begin teaching – practice is not at the core of the curriculum’ (Gross-
man et al., 2009, p.275).
3. The specic practices that PSTs learn to use are largely left to chance. This occurs primarily
because teacher educators expect that practices will be learned during eld experiences, the com-
ponent of their programs over which they have the least control (Grossman & McDonald, 2008).
PSTs thus learn practices that they happen to encounter in eld settings, and these may or may not
be eective practices.
4. Eective instructional practices that are critical to the success of beginning teachers are seldom
systematically taught during teacher preparation through closely aligned coursework and eld
experiences (Forzani, 2014). [B14]
Changes to teacher preparation were proposed in 2010 by the National Council for Accreditation of Teacher
Education [B15]. A work group convened by the Council for Exceptional Children developed several high-
leverage practices intended to be paired with culturally responsive teaching [B16].
8
What are Evidence-Based and High Leverage Practices?
According to one article entitled Unraveling Evidence-Based Practices in Special Education by Brian G
Cook and Sara Cothren Cook, evidence-based practices are practices "that are supported by multiple, high-
quality studies that utilize research designs from which causality can be inferred and that demonstrate
meaningful eects on student outcomes" [B17]. Evidence-based practices are eective, they create a real
dierence in student life and performance. It is important to note that no one evidence-based practice will
work for everyone. Evidence-based practices originated in the 1990s in the eld of medicine. The iden-
tication of evidence-based practices soon spread to other eld such as nursing, psychology, agriculture,
and education. They represent what has the most potential for improving student performance, especially
for struggling students who require as eective of instruction as is possible to succeed. The problem with
evidence-based practice is a problem previously discussed, teachers have a noticeable lack of faith in re-
search which leads to a lack of implementation of evidence-based practices. What these teachers might not
necessarily know is that evidence-based practices go through an evidence-based review which examines
the research design, quality of research studies, quantity of research studies, and magnitude of eect [B17].
Teachers currently face higher demands to prepare a diverse range of students for the high levels of
performance required of them by colleges or careers. It is essential that whatever time teachers have is
spent using the most eective practices possible. However, teachers are often entering schools without
sucient initial preparation for them to have the skills to enact those practices [B18]. Therein lies the
problem that high-leverage practices are trying to resolve. Researchers believe that by incorporating these
22 educational practices across four areas (collaboration, assessment, social/emotional/behavioral, and in-
struction) into teacher preparation and education [B16]. 1-3 are collaboration, 4-6 are assessment, 7-10 are
social/emotional/behavioral, and 11-22 are instruction.
9
Table 1: High-Leverage Practices for K-12 Special Education Teachers
1. Collaborate with professionals to increase student success
2. Organize and facilitate eective meetings with professionals and families
3. Collaborate with families to support student learning and secure needed services
4. Use multiple sources of information to develop a comprehensive understanding of a student’s
strengths and needs
5. Interpret and communicate assessment information with stakeholders to collaboratively design and
implement educational programs
6. Use student assessment data, analyze instructional practices, and make necessary adjustments that
improve student outcomes
7. Establish a consistent, organized, and respectful learning environment
8. Teachers provide positive and constructive feedback to guide student’s learning and behavior (behav-
ior focus)
9. Teach social behaviors
10. Conduct functional behavioral assessments to develop individual student behavior support plans
11. Identify and prioritize long- and short-term learning goals
12. Systematically design instruction toward a specic learning goal
13. Adapt curriculum tasks and materials for specic learning goals
14. Teach cognitive and metacognitive strategies to support learning and independence
15. Provide scaolded supports
16. Use explicit instruction
17. Use exible grouping
18. Use strategies to promote active student engagement
19. Use assistive and instructional technologies
20. Teach students to maintain and generalize new learning across time and settings
21. Provide intensive instruction
22. Teachers provide positive and constructive feedback to guide students’ learning and behavior (learn-
ing focus)
10
Explicit Instruction
Explicit Instruction is a highly structured, systematic method that breaks instruction down into small
steps. It is direct and uses clear, concise, and consistent language both in designing and delivering instruc-
tion. Explicit instruction uses scaolds, a series of supports that are lessened over time as the students
gain independent mastery of the subject in question. Rosenshine (1987) characterized explicit instruction
as "a systematic method of teaching with emphasis on proceeding in small steps, checking for student
understanding, and achieving active and successful participation by all students" [B19]. Reviews on the
eectiveness of explicit instruction include Jere Brophy’s and Thomas Good’s chapter in the Handbook of
Research on Teaching entitled "Teacher Behavior and Student Achievment" which reviewed dozens of stud-
ies [B20], N.L Gages’ and Margaret C Needels’ article "Process-Product Research on Teaching: A Review
of Criticisms" in The Elementary School Journal [B21], and Barak Rosenshine’s and Robert Stevens’ chapter
in the Handbook of Research on Teaching entitled "Teaching Functions". Between 2008 and 2009, three gov-
ernment reports were released that recommended the use of explicit instruction: a report by the National
Mathematics Advisory Panel in 2008, a 2008 report by the Institute of Education Sciences, and a report in
2009 by the National Center for Education Evaluation and Regional Assistance. Meta-analyses have been
conducted in the eld of special education by Sharon Vaughn, Russell Gersten, and David J Chard [B22];
and H. Lee Swanson and Maureen Hoskyn [B23], among others.
There are 5 essential components of explicit instruction; explicit instruction should segment complex
skills, use modeling and "think-alouds" to draw attention to important content features, use systematically
faded supports and prompts, provide frequent opportunities for students to respond and receive feedback,
and create purposeful practice opportunities. Other important and commonly used components of explicit
instruction include selecting critical content, using logic to sequence skills, ensuring students have the
requisite prerequisites and background knowledge necessary for success, presenting a wide range of exam-
ples and non-examples, and maintaining a brisk pace [B24]. Though explicit instruction has been viewed
through many dierent lenses and talked about in many dierent forms, the essential components remain
constant. The reasoning on why ecient instruction is so eective is not constant. Researchers have argued
that explicit instruction is eective because it reduces the cognitive load and resulting stress on working
11
memory [B25], while others contend that the teaching behaviors align with several applied behavior anal-
ysis principles [B26]. Regardless of the reasoning, the truth remains evident, explicit instruction is eective
for both special and general education purposes.
Explicit instruction can be taught in many formats. One such format is the explicit instruction cy-
cle. It consists of seven phases: curriculum-based assessment, planning, advance organizer, demonstration,
guided practice, independent practice, and maintenance. CBA has evolved over the last few decades as a
measurement system that uses data regarding student performance and progress to make data-based deci-
sions which helps to design ecient and eective education. "CBA helps to pinpoint curricular objectives
that act as the criteria for the identication of instructional targets and for the assessment of students’ skills
upon entry into an academic curriculum." [B27] Curriculum-based assessment (or measurement) was orig-
inally developed for special education as a way of testing the eectiveness of a intervention model called
data-based program modication (DBPM). The University of Minnesota’s Institute for Research on Learn-
ing Disabilities ran a 6 year long study which ultimately generated a set of criteria which established the
technical adequacy, treatment validity, and logistic feasibility of the measures [B28].
CBA is technically adequate, time ecient, and easy to teach. It can be used to improve individual
instruction programs, predict performance, screen for at-risk students, enhance instructional planning, re-
duce bias in assessments, and more [B29]. There are also dierent kinds of CBAs which accomplish dierent
things. The three types of curriculum-based measurements are survey CBAs, untimed focused CBAs, and
timed focused CBAs (also called probes). A survey CBA’s purpose is to measure a wide range of concepts
and skills. This provides the instructor is a comprehensive view of the student’s mathematical ability. It can
also help pinpoint individual strengths and weaknesses. Focused CBAs are designed to test very specic
concepts or skills. They aim to represent all problem types of a specic skill area in order for the instructor
to understand a student’s level of understanding of a certain topic area. Untimed focused CBAs allow as
much time as needed to complete whereas timed focused CBAs are given a more limited time frame, usually
one minute. Probes (timed focused CBAs) aim to develop and measure declarative knowledge by testing
uency and recall [B30].
The next phase of the explicit teaching cycle is the planning phase. There are two key principles that
12
go into planning the lesson: data-based decision making and instructional alignment. Data-based decision
making entails using the information and data gathered during the curriculum-based assessment to inform
the decisions made in planning the lesson. There are multiple types that teachers can draw on to make
data-based decisions: input data, output data, process data, and context data. Input data is data on student
or teacher characteristics. Outcome data is the data regarding student achievement and well being. Process
data is data about instruction and types of assessments. Context data is on the school culture, building, or
the curriculum. Most commonly, teachers use outcome data to make decisions [B32]. Depending on the
type of CBA used, you will learn dierent outcome data. For example, survey CBAs help make decisions
about the appropriate place in the curriculum and how to group students according to instructional needs.
Regardless of the type or purpose of the CBA, an important step in data-based decision making is, when
looking at outcome data, determining whether or not the student in question has met the criteria for the
rate of accuracy.
The other key principle to take into consideration when planning lessons is instructional alignment.
"Instructional alignment exists when there is a match in the stimulus conditions of intended outcomes,
instructional processes, and instructional assessment" [B31]. This means that what a teacher is wanting
to teach must match the activities or worksheets the students complete. There are three components of
instructional alignment: objectives, instruction, and evaluation. Teachers should decide, when planning
the lesson, what the core objectives of that lesson are. This lets them choose worksheets and activities that
follow the same objectives. In keeping instruction in mind, they should take care in choosing the correct
instructional strategies to match the context of the lessons. Teachers should also share the objectives with
their students as students can gain autonomy by knowing the rationale of a particular activity or lesson.
The nal component of instructional alignment is evaluation. Evaluations are important for a myriad of
reasons, one of which is enhancing student motivation by letting students track their progress. It is essential
that evaluations cover material and objectives explicitly learned in class [B33].
13
The next step in the cycle to consider is the advance organizer. The purpose of an advance organizer is
to link new ideas to existing knowledge and thereby enable students to learn new ideas and information. "It
is designed to cue the relevant prior knowledge of a learner and it is usually presented at a higher level of
abstraction, generality, and inclusiveness than that of the planned lesson" [B34]. This manifests as teachers
asking questions, presenting an outline, or linking new knowledge to previously learned material. Advance
organizers can take many forms, whether they be charts and diagrams, or concept maps. There are two types
of advance organizers: expository and comparative. "Expository organisers function to provide the learner
a conceptual framework for unfamiliar material, and comparative organisers are used when the knowledge
to be acquired is relatively familiar to the learner" [B34]. Additionally, there are three phases of advance
organizer teaching. The rst stage is presentation of the advance organizer and consists of introducing the
goals of the lesson, the topic, and the materials required. It is important that it provides the framework
for the learning that later takes place. The next step is the presentation of the actual material, paying
special attention to the ordering of the material and what it means to the students. Student participation is
important at this juncture. The last stage consists of the teacher asking questions to cement the knowledge
and extend the thinking of the students.
Also included in advance organizer teaching are some necessary components. One such component
is review. An advance organizer should include review because it allows teachers to check that the stu-
dents know the necessary prerequisite knowledge and prepares the students for success in the oncoming
lesson. An important question to ask is what skills, knowledge, or concepts are needed for the students
to obtain success in the lesson. It is helpful to consult a scope and sequence chart to determine the con-
cepts necessary to review. The review should be brief and engage all students in order to get an accurate
view of each student’s preparedness for the lesson. The teacher should be continuously checking student
progress, providing feedback, and assessing whether to move onto the lesson or continue review. Another
such component to include in advance organizer teaching is stating the lesson objective and linking it to
prior knowledge. The statement of the lesson objective should be brief and contain simple language. One
last component to include is the development of relevance. It has been shown that when students know
why they are learning a new skill, motivation and learned are elevated [B30].
14
There are several strategies involved in the next phase of the explicit teaching cycle, demonstration.
During demonstration, it is important to incorporate teacher modeling. Teacher models should use clear,
consistent, and concise language and be unambiguous in their explanations. They should show the students
what they are learning and how to apply it [B35]. Critical aspects of teacher modeling includes the use
of clear language, inclusion of student involvement, and sucient number of examples to promote real
understanding. When determining the number of examples, teachers should examine the diculty of the
task, the knowledge brought to the classroom by students, and student response to instruction. One way
to implement teacher modeling is by using think-alouds. "The think-aloud is a technique by which the
individual voices her or his thoughts during the performance of a task" [B36]. It is as simple an idea as it
sounds, someone explaining their actions out loud while doing a task. Think-alouds help guide students
through making critical connections in their thinking and generalizing the information learned to other
applications. They "provide students with an opportunity to witness the expert thinking that is typically
hidden and thus an abstract concept to the students" [B37].
Another important strategy is maximizing student engagement. One denition of student engagement
says that student engagement is multifaceted and is dened under three aspects: behavioral engagement,
emotional engagement, and cognitive engagement. Behavioral engagement "draws on the idea of partic-
ipation" [B38]; it is considered essential for obtaining a positive educational outcomes and includes the
participation in academic, social, and extracurricular activities. Emotional engagement is all about creating
ties to the school and supplying the motivation to complete academic tasks. It encompasses reactions, both
positive and negative, to a myriad of factors including teachers, academics, classmates and the school itself.
Finally, cognitive engagement "draws on the idea of investment" [B38] and incorporates the motivation to
exert the necessary eort to master complex skills and ideas. The three types of engagement are interrelated
and constitute a dynamic process that has its inuences on learning and achievement. Another denition
of student engagement is as follows: "the student’s psychological investment in and eort directed toward
learning, understanding, or mastering the knowledge, skills, or crafts that academic work is intended to
promote" [B39].
An article on student engagement identied four motivational constructs which represent major con-
15
structs in the eld of achievement motivation. The rst construct, belongingness, is about the necessity
of establishing meaningful connections and relationships to others. Belongingness theory says that hu-
mans have a "pervasive drive to form and maintain at least a minimum of lasting, positive, and signicant
interpersonal relationships" [B40]. When applied to education and school achievement, the premise of be-
longingness theory is that "caring and supportive relationships facilitate student engagement and other
adaptive school behaviors" [B41]. Instructional practices that oer opportunities to foster belongingness
are demonstrating and encouraging mutual respect and teaching students to work together in a productive
manner. The next construct is competence, the necessity for humans to meet one’s goals successfully and
be successful in their interaction with their environment. One theory of competence is self-ecacy. Self-
ecacy means a person’s belief that they can be successful when completing a certain task or achieving
a certain goal. Studies show that "students who feel ecacious about learning tend to be competent and
engaged and are likely to set learning goals, use eective learning strategies, monitor comprehension, eval-
uate goal progress, and create supportive environments" [B42]. Teachers can help increase competence by
showcasing that mistakes provide information, thusly encouraging student persistence. The use of scaf-
folding, feedback, and appropriately challenging assignments can also help foster feelings of competence
[B43].
Autonomy, the third construct, is a psychological need. It is a need to behave and make decisions and
choices according to one’s own values and interests. As for the eectiveness of nurturing autonomy to
increase student engagement, one study showed that "teachers’ autonomy support was an even better pre-
dictor of students’ classroom engagement than was students’ own engagement during an earlier class" [B44].
Strategies important for nurturing autonomy include using language that isn’t controlling, providing ratio-
nales for tasks, acknowledging and respecting student perspectives, and allowing students to debate and
ask questions freely. One study said that teachers who were autonomy supportive "(a) adopt the students’
perspective; (b) welcome students’ thoughts, feelings, and behaviors; and (c) support students’ motivational
development and capacity for autonomous self-regulation" [B45]. The nal construct, meaningfulness, is
the only one of the constructs that is not considered a basic human need. When learning is meaningful, it
involves the development of interest or the appreciation of the topic being learned. To foster meaningful-
16
ness, one study recommends building on the prior knowledge of students, emphasizing your own interest
in the topic, invoke universal human experiences, and oer opportunities for students to think complexly
and participate in conversations with fellow peers to achieve shared understandings. [B43]
One additional strategy to promote student engagement is to provide opportunities to respond (OTR).
Eective delivery is characterized by three aspects: type of delivery, rate of presentation, and method of
response. OTRs are a subset of teacher questioning which features two types of questions: fact and higher
order (cognitive). Fact questions require mere regurgitation of information, while higher order questions
require independent thinking. Various modes of responding can also be provided. Students could respond
individually or by choral responding (as a group). Another strategy that promotes student engagement
is providing feedback. This has the added benet of also increasing student achievement. Feedback can
be positive or negative and timely feedback helps to cement concepts in student minds and correct any
errors that arise. Feedback can also help regulate student behavior. While feedback can be positive or
negative, the ratio between the two should heavily favor positive feedback. Increasing positive feedback
has been shown to increase productivity and accuracy. A myriad studies concluded that the ratio of positive
to negative feedback should be at least 4:1 [B37]. Additional eective strategies for questioning include
creating a classroom culture open to dialogue; using both preplanned and emerging questions; selecting an
appropriate level of questions based on learners’ need; avoiding trick questions and those that require only
a yes or no response; phrasing questions carefully, concisely, and clearly; addressing questions to the group
or to individuals randomly; using sucient wait time; responding to answers given by students; deliberately
framing questions to promote student interest; and using questions to identify learning objectives for follow-
up self-study. [B46]
The next stage in the explicit instruction cycle is guided practice. Here we make use of a practice
called scaolding. Scaolding supports means to "provide temporary assistance to students so they can
successfully complete tasks that they cannot yet do independently and with a high rate of success" [B47].
The concept behind guided practice is to start with a high level of support and to slowly decrease support as
the students become more capable of independently solving the problems until you can withdraw support
completely without a loss of success. Worked solutions is a primary example of this strategy. They are
17
"matched carefully to students’ current understanding of learning objectives and are systematically faded
to include fewer supports as students learn to solve the problems independently" [B48]. In addition to
algebraic problems and skills, scaolding can also be used to teach higher level cognitive strategies. Above
mentioned think-alouds are a scaold that can be used to teach such skills and strategies. It is essential
that during this process of scaolding, that feedback, both positive and constructive, is frequently given.
Student engagement is also crucial at this stage.
Especially during the instruction of higher level cognitive skills, but also generally, it is important to
consider the load on students’ working memory. Working memory is a "processing resource of limited ca-
pacity, involved in the preservation of information while processing the same or other information. Tasks
that measure [working memory] assess an individual’s ability to maintain task-relevant information in an
active state and to regulate controlled processing" [B49]. The capacity of working memory is typically de-
scribed as the amount of information that an individual can process while also performing complex tasks.
The functioning of working memory is akin to a "spotlight" that shines selectively on relevant information
from one moment to the next. Ineective functioning of working memory can lead to overloading of infor-
mation and forgetfulness. The use of clear, concise, and consistent language as well as activation of prior
knowledge, scaolding instructional supports, frequent review, and sucient practice time with visual aids
and feedback, when well integrated, are ideal for facilitating working memory [B50]. Guided practice, with
adequate use of visual aids, is one way to facilitate working memory.
As students reach independency in their ability to complete problems using a certain skill or strategy,
you can transition into independent practice. The goal of independent practice is to "develop unitization
of the strategy, that is, the blending of elements of the strategy into a single, unied whole" [B51]. It also
aims to decontextualize learning so the strategy can be applied easily and with little working memory. You
want to build and maintain skill prociency. It is important at this stage to try to limit prompting, with
the intention of using the students’ performance at this stage to assess their understanding of the topic or
skill. Independent practice can take many forms including homework, in-class worksheets, quizzes, etc..
It is important that independent practice, regardless of form be distributed, i.e. take place over a longer
period of time. This helps to cement skills, entering them into long-term memories. The practice that takes
18
place when learning is occurring, during demonstrations and guided practice, is called massed practice and
is equally as important. The goal for independent practice is to move on when 80% of students are accurate
80+% of the time. If students are not achieving that accuracy level, it may be necessary to reteach the
specied skill.
The last stage in the explicit instruction cycle is maintenance, which refers to the student’s continued
ability to accurately complete mathematics problems related to a certain skill or strategy. It is the stage in
which distributed practice takes place. It is also the stage where you make data-based decisions to change,
adjust the curriculum, or reteach a certain skill or strategy. Data-based decision making should also occur
at signicant stages during instruction. These stages include when to move from review to demonstration,
when to move from demonstrations to guided practice, and when to move from guide practice to indepen-
dent practice. Data collected should not be only acquired from tests but rather come from four dierent
places: input data is data about demographics of the student population, outcome data is data such as stu-
dent test scores and well-being, process data is data about the quality of instruction, and context data is
about policies and resources. Good data-based decision making should use data from all four types.
Data on their own, however, do not provide judgement, interpretation, or basis for action; they need to
be transformed to be useful. This process of transforming data should always begin with a clear purpose
as to why it’s being sought. This makes it possible to know what data should be collected. After data is
collected, the analyzing of the data can begin. Analyzing consists of "contextualizing, categorizing, calcu-
lating, connecting, and/or summarizing the data in a way that meets the purpose"[B52]. The next step is to
interpret the data. This consists of trying to make sense of what the data means and their implications for
the future and any action that is taken. Once all of that is clear, then you can take action on the data. Over-
all, the steps in data-based decision making should be to 1.) Using data, determine the instructional needs
of individual students, 2.) Dene the SMART (Specic, Measurable, Attainable, Realistic, Time-bound) and
challenging goals, 3.) Based on the information in Steps 1 and 2, decide which instruction approach is most
promising to achieve those goals, and 4.) Execute the strategy proposed in Step 3. [B53]. For data-based
decision making that takes place you can just use data to decide whether or not to move on. This four-step
process is designed for the maintenance stage of the explicit instruction teaching cycle.
19
Logistics
Diverse Individuals in Mathematics & Computer Science
"Despite decades of eort to increase the participation of women and people from underrepresented
minority groups (URM) in STEM majors and careers, and despite the increasing diversication of the US as
a whole, participation in STEM majors and careers remains stubbornly male and white" [C56]. There are a
myriad of possible reasons as to why this is the case but one possible reason is that they don’t see women
and URM in STEM careers. One potential solution is to provide students with a list of women and URM in
the eld of mathematics now and in the past. The following are examples of some diverse individuals in
computer science and mathematics.
Muhammad ibn Musa al-Khwarizimi
The words algebra and algorithm stem from one man, Muhammad ibn Musa al-Khwarizimi, an Per-
sian polymath born around A.D.780 in what is now Uzbekistan. Al-Khwarizimi accomplished most of his
work between 813 and 833, part of which was spent working in Baghdad in the House of Wisdom. He
was commissioned by the caliph (spiritual head of Islam) to do research and translate the treatises of Greek
mathematicians. Al-Khwarizimi’s work on elementary algebra in 820 included the writing of a book en-
titled "The Compendious Book on Calculation by Completion and Balancing", which was where the term
algebra derives from. The term algorithm originates from the title of another of his books which was on
Hindu-Arabic numerals and arithmetic entitled "Al-Khwarizimi Concerning the Hindu Art of Reckoning"
and written in 820. Al-Khwarizimi also wrote a another book entitled "The Image of the Earth" which is
based on Geography of Ptolemy but used improved values. In his book "The Compendious Book on Calcu-
lation by Completion and Balancing" Al-Khwarizimi talks about two mathematic operations: al-jabr which
means restoring or completion and al-muqabala which means balancing. Al-jabr is the process that consists
of removing negatives (whether they be units, roots, or squares) by adding the same quantity to each side.
Ex.
x2 = 40x − 4x2
20
becomes
5x2 = 40x
Al-muqabala is about taking all of the terms of the same type and bringing them to one side of the equation.
Ex.
50 + 3x + x2 = 29 + 10x
becomes
21 + x2 = 7x
"The Compendious Book on Calculation by Completion and Balancing" also features lattice multiplication,
something Al-Khwarizimi is credited with developing. [C57, C58]
Winifred "Tim" Alice Asprey
Asprey was born on April 8th, 1917 and died on October 19th, 2007. She was an American mathematician
and computer scientist. Asprey was one of only 200 or so women to earn a PHD from an American university
during the 1940’s. She was responsible for bringing Vassar College its rst computer in 1967 making it only
the second college in the United States to acquire a IBM System/360 computer. As an undergraduate, Asprey
attended Vassar College, where she met Grace Hopper also known as the "First Lady of Computing". She
then taught at a myriad of dierent private schools in New York City and Chicago before obtaining her
masters in 1942 and doctorate in 1945 from the University of Iowa. After which, Asprey returned to Vassar
college as a professor, where she stayed for 38 years. There, she campaigned tirelessly for Vassar College
to have its own computer, succeeding in 1967. Alongside her work with Vassar college, Asprey conducted
research at IBM. Asprey retired from Vassar in 1982 and in 1989 Vassar’s Asprey Advanced Computation
Laboratory was named after her. [C59, C60]
Enriqueta González Baz y de la Vega
González Baz was born on September 22nd in Mexico City. She studied to be a teacher at Escuela número
8 for women before being sent by her father to Escuela Doméstica for domestic studies. There, González
21
Baz caught the attention of one of her teachers, Elena Picazo de Murray, who encouraged González Baz to
pursue higher education. In 1994, at age 29, she became the rst women to earn a degree in mathematics
from the National Autonomous University of Mexico where she wrote her thesis on Special Functions (i.e.
Bessel, Gama, and Legender). González Baz completed her postgraduate work in Philadelphia at Bryn Mawr
College. González Baz went on to teach in various secondary schools including the National Preparatory
School. Additionally, she was a researcher at the Institute and taught mathematics at the Faculty of Sciences.
González Baz was also one of ve founding women who founded the Mexican Mathematical society. in 1943
She died on December 22nd, 2002. [C61, C62]
Farida Bedwei
Farida Nana Efua Bedwei is a software engineer from Ghana. She was originally born in Lagos, Nigeria
where she was diagnosed with cerebral palsy a day after she was born. Farida Bedwei is the cofounder of
a nancial technology company called Logiciel. She is also known for her accomplishments in the elds
of software architecture and deploying mobile services. She has won several awards including the Winner
of Most Inuential Women in Business and Government Award Financial Sector. She is also a member of
The Girls in ICT Committee, a group dedicated to encourage the increase of women in IT careers. Farida
Bedwei also created a superhero named Karmzah who has cerebral palsy and gets her power from her
crutches. [C63]
Manjul Bhargava
Born to an Indian family, his mother Mira Bhargava was his mathematics rst teacher. Manjul Bhargava
was valedictorian of his high school before earning his BA at Harvard. He completed his doctoral disserta-
tion at Princeton where he teaches today. Manjul Bhargava has earned over 10 awards and prizes including
the Field’s Medal which is one of the highest prizes a mathematician can earn; it is often described as the
Nobel Prize of Mathematics. Manjul Bhargava is also the third youngest full professor in the history of
Princeton. [C64]
22
Anita Borg
American-born computer scientist Anita Borg got her rst programming job in 1969 at age 20. She was
awarded her PHD in 1981 by New York University. Borg founded the Institute for Women and Technology
and the Grace Hopper Celebration of Women in Computing. Her goal was to have 50% representation for
women in computing by 2020. The rst Grace Hopper Celebration of Women in Computing took place in
Washington DC in June of 1994 and brought together over 500 women in computer science. Anita Borg
won many awards and wa recognized by Bill Clinton who appointed her to the Presidential Commission
on the Advancement of Women and Minorities in Science, Engineering, and Technology. [C65]
Kimberly Bryant
Kimberly Bryant was born in Memphis, Tennessee in 1967. A scholarship student to Vanderbilt Univer-
sity she majored in Electrical Engineering with minors in Computer Science and Mathematics. Bryant is
best known for her founding of Black Girls Code in 2011. Inspired to create a better life for her daughter
who was interested in coding Bryant hopes that Black Girls Code will help young women from minority
populations to stay in STEM. Black Girls code teaches computer programming to young women via after
school and summer programs. The nonprot organization has a goal of teaching one million black girls to
code by the year 2040. [C66]
Stephanie Castillo
Stephanie Castillo is the executive director of Latin Girls Code. Latin Girls Code is a Chicago based
program which seeks to provide education and resources on technology to interested young Latina women,
from ages 7-17. Seeking to teach not only technology but also entrepreneurial skills, the group holds work-
shops, hackathons, and week-long programs. In addition to teaching, Castillo also provides advice as an
immigration advisor as many of the young women who are interested in technology are also undocumented.
[C67]
23
Lynn Conway
Lynn Conway was born in Mount Vernon, New York in 1938. As a child she was shy and experienced
gender dysphoria. Conway entered MIT in 1955 but dropped out due to a failed gender transition in 1957-58.
Lynn Conway worked for several years as an electronics technician before attending Columbia University’s
School of Engineering and Applied Science, earning degrees in 1963. Recruited by IBM, Conway worked
on an architecture team designing an advanced supercomputer. On hearing of Harry Benjamin’s work
on gender transitions Conway sought out transitioning for the second time. On hearing of the intent to
transition, Lynn Conway was red from IBM. Upon completing her transition in 1968, Lynn Conway began
a new life working at several dierent tech companies including Computer Applications, Inc, Memorex,
Xerox PARC, and DARPA. Conway even served as a visiting associate professor at MIT. After going public
with her story in 2000, Conway became a transgender activist. 52 years after being red from IBM, IBM
apologized and warded Conway with the rare IBM Lifetime Achievement Award. [C68]
Marian Croak
Marian Croak grew up in New York City. She attended Princeton University before obtaining her doc-
toral degree from University of Southern California in 1982. Directly after graduating from USC she began
work at Bells Labs, part of AT&T. After over 30 years working at AT&T, Marian Croak left AT&T to join
Google as their Vice President for Engineering. One remarkable thing about her is that she holds more
than 200 patents. More than 100 of those patents have to do with Voice over IP which has to do with the
movement of voice communications and other multimedia sessions over the internet. Marian Croak was
inducted into the Women in Technology Hall of Fame in 2013. [C69]
Shakuntala Devi
Shakuntala Devi is an Indian mathematician also known as the "Human Computer". Her father worked
in the circus but quit when he realized what she could do. Devi could memorize large numbers, something
her father discovered when teaching her a card trick at the tender age of three. Her father took her on road
shows to exhibit her unusual gift for calculation. Devi toured Europe in 1950 and showcased her abilities
24
in New York in 1976. In 1977 Devi calculated the 23rd root of a 201-digit number in 50 seconds at Southern
Methodist University. The answer was 546,372,891 which was conrmed by the UNIVAC 1101 in the US
Bureau of Standards. A special program had to be made in order to perform such a large calculation, and the
speed at which the calculation was done was less than the time it took for Devi to do the same. In 1980 she
made the Guinness Book of World Records when she calculated the multiplication of two 13-digit numbers,
7,686,369,774,870 and 2,465,099,745,779. The numbers, which were picked at random by the Department of
Computing at Imperial College London multiply to 18,947,668,177,995,426,462,773,730, an answer she gave
in 28 seconds. In 1988 Devi agreed to be studied by Arthur Jensen, a professor of educational psychology
in California. Shakuntala Devi calculated the seventh root of 170,859,375 and cube root of 61,629,875 before
Jensen could even write down the problems in his notebook. Note: the answers are 15 and 395 respectively.
Devi also wrote a book on homosexuality despite not being one herself in which she advocates for total
acceptance of homosexuals in India. [C70]
Annie Easley
Annie Easley was born before the Civil Rights Movement which means that options for African-Americans
were extremely limited. Despite this her mother encouraged her to get an education. Easley graduated
valedictorian from Holy Family High School then went on to major in pharmacy at Xavier University. In
1955 Annie Easley applied for and was hired by the NACA (National Advisory Committee for Aeronautics)
as a "computer". Of the 2500 employees at NACA, she was one of only four African-American women. In
the 1970s she helped other female and minority students to pursue careers in STEM during college career
days and in 1977 received her Bachelors in Mathematics from Cleveland State. At NASA she was also held a
position as an Equal Employment Opportunity counselor where she fought discrimination. Easley worked
at NASA for 34 years. In 2015, after her death in 2011 she was inducted into the Glenn Research Hall of
Fame. A crater on the moon was named after Easley in February of the year 2021.
25
Other Individuals
• Elizebeth Friedman • Sal Khan • Christoper Strachey
• Jorge Galicia Urrutia • Soa Kovalevskaya • Audrey Tang
• Sophie Germaine • Peter Landin
• Terence Tao
• Laura Gomez I. • Ada Lovelace
• Valerie Thomas
• Evelyn Granville • Yukihiro Matsumoto
• Alan Turing
• Frank Greene • Maryam Mirzakhani
• John Urschel
• Jon Hall • Victor Neumann-Lara
• Pamela Harris • Emmy Noether • Dorothy Vaughan
• Piper Herron • Maria Ong • Gladys West
• Grace Hopper • Radia Perlman
• Sophie Wilson
• Mary Horton Ann • Srinivasa Ramanujan
• Edith Windsor
• Emma Iwao Haruka • Mina Rees
• Lixia Zhang
• Mary Jackson • John Resig
• Miguel de Icaza
• Katherine Johnson G. • Adriana Salerno
• Brian Katz • Nicholas Saunderson • Jesus de Loera A.
• Autumn Kent • Mary Somerville • Juana de la Cruz Ines
26
Methods of Reducing Math Anxiety
It has been shown in several studies that there exists a direct relationship between math anxiety and math
performance. One way to alleviate math anxiety is by expressive writing. Expressive writing is the process
where students write freely for 10-15 minutes about their thoughts and feelings regarding an upcoming
mathematics exam (or other important stressor). Studies have shown it reduces negative thoughts and
feelings in depressed individuals as well as having other physical and psychological benets for all parties,
whether they be depressed individuals, college students, or both. Studies have also shown that expressive
writing can increase the availability of working memory, something which persons with math anxiety have
a limited amount of [C71]. Expressive writing may also help with emotion regulation and distancing of
oneself from their sources of stress.
Other studies have shown that "psychological techniques emphasizing self regulation, emotional con-
trol, and reappraisal of physiological threat responses hold promise" [C72]. Retrieval practice has also been
found to decrease math anxiety. Retrieval practice is frequent, occurring at least once a week, quizzes and
practice tests that accounted for little or no change to a student’s grade. An important component of re-
trieval practice is automatic feedback. In one study, 72% of students surveyed said that they experienced
less anxiety towards tests because of retrieval practice, 22% said that it made no dierence and 6% said that
they were more nervous [C73].
Another study advocated for digital game-based learning (DGBL) as a method of alleviating math anx-
iety. An article entitled Eects of anxiety levels on learning performance and gaming performance in digital
game-based learning said that "Unlike learning performance, where learners with high anxiety demonstrated
worse learning performance than those with low anxiety, the former and later demonstrated similar gam-
ing performance [C74]. One such game that been shown to help reduce anxiety is the game Kahoot!. A
literature review done of over 70 studies noted that 70% of studies that mentioned student anxiety noticed
a reduction in anxiety due to playing Kahoot!. Only one study reported that it caused anxiety [C75].
27
Testing
An alternative to conventional testing which can cause test anxiety as well as math anxiety, is dynamic
testing. Dynamic testing is designed to "assess developing or yet-to-develop abilities that are the products
of underlying, but often unrecognized, cognitive capacities" [C76]. Dynamic testing consists of, according
to article entitled Learning Potential and Anxious Tendency, allowing students to ask for and receive hints
on their tests with the caveat of being docked points for doing so [C77]. In this way, the bias in educational
testing towards individuals with anxious tendencies can be at least partially alleviated. This is one option
for decreasing math anxiety.
Another option to decrease math anxiety is collaborative testing as shown by multiple studies [C78].
Collaborative assessments have been shown to promote critical thinking, improve test performance, reduce
test anxiety, support long-term retention of the material, and more. There are, of course, drawbacks to
collaborative testing, it is dicult to assess individual performance as well as cheating. The benets are
however are an increase in understanding, performance, and condence.
28
Content
Mathematics Topics
Table 2: Table of Topics Covered in Example Syllabi
HU[D79] FTU[D80] LU[D81] UV[D82] KAU[D83] DLSU[D84] ACC[D85] AU[D86] IVC[D87] DSC[D88] NYU[D89] VTU[D90]
algebraic systems x
algorithms x x x x
boolean algebra x x x x
combinatorics x x x x x x x
counting x x x x
functions x x x x x x x x x
graph theory x x x x x x x x x
lattices x
logic x x x x x x x x x x x
matrices x x x
modular arithmetic x x x x
number systems x
pigeonhole principle x x x x x
probability theory x x x
proofs x x x x x x x x x x x
recursion x x x
relations x x x x x x x x x
sequences x x x x x
set theory x x x x x x x
state machines x x x
summations x x
turing machines x
In an eort to understand the importance of individual topics covered under the broad category of dis-
crete mathematics I surveyed 12 dierent college syllabi to see in what frequency select topics were covered.
The result is the table pictured above. To better illustrate my ndings I also created a bar chart that lists the
percentage coverage of each topic by the syllabi aforementioned. The result is illustrated on the next page.
The colleges that I chose were Harvard University, Florida Tech University, Liberty University, University
of Virginia, King Abdulaziz University, De La Salle University, Austin Community College, Anna Univer-
sity, Imperial Valley College, Daytona State College, New York University, and Visvesvaraya Technological
University. These colleges and universities were chosen by virtue of their discovery whilst conducting
research.
29
Figure 1: Bar Chart of Topics Covered in Example Syllabi
Algebraic Systems 8.33
Algorithms 33.33
Boolean Algebra 33.33
Combinatorics 58.33
Counting 33.33
Functions 75
Graph Theory 75
Lattices 8.33
Logic 91.66
Matrices 25
Modular Arithmetic 33.33
Number Systems 8.33
Pigeonhole Principle 41.66
Probability Theory 25
Proofs 91.66
Recursion 25
Relations 75
Sequences 41.66
Set Theory 58.33
State Machines 25
Summations 16.66
Turing Machines 8.33
The topics that are covered here are, for the most part, broad categories. They give useful information
as to what broad categories should be covered but are of little help in deciding what individual concepts or
ideas should be addressed in the curriculum. To get a better idea of what topics are covered in textbooks I
obtained and extracted data from 32 dierent discrete mathematics textbooks and noted what topics were
covered from a list of more specic topics. The result is shown on the next two pages. The list of textbooks
is as follows:
· Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications by
Gerard O’Regan
· Discrete Mathematics and Its Applications by Kenneth H. Rosen
· Discrete and Combinatorial Mathematics: an Applied Approach by Grimaldi
· Discrete Mathematics with Applications by Susanna S. Epp
·DiscreteMathematical Structures with Applications to Computer Science by Jean-Paul Tremblay and R Manohar
30
· Applied Discrete Mathematics by William Shoa
· Introduction to Discrete Mathematics via Logic and Proof by Calvin Jongsma
· Mathematics for Computer Science by Eric Lehman, et al
· Theory and Problems of Discrete Mathematics by Seymour Lipschutz and Marc Lipson
· A Beginner’s Guide to Discrete Mathematics by W. D. Wallis
· Discrete Mathematics Using a Computer by John O’donnell
· DISCRETE MATHEMATICS : An Open Introduction by Oscar Levin
· Discrete Mathematics by Jean H. Gallier
· DISCRETE MATH WORKBOOK : A Companion Manual Using Python by Sergei Kurgalin
· Essential Discrete Mathematics for Computer Science by Todd Feil and Joan Krone
· Handbook of Discrete and Combinatorial Mathematics by Kenneth H. Rosen
· Foundations of Discrete Mathematics by K.D. Joshi
· Discrete Mathematics with Graph Theory by Edgar G Goodaire and Micheal M Parmenter
· Discrete Mathematics by Martin Aigner
· Foundations of Discrete Mathematics by Albert D Polimeni and H Joseph Straight
· Discrete Mathematics by K. A. Ross and C. R. B. Wright
· Mathematics : A Discrete Introduction by Edward R. Schneinerman
· Discrete Mathematics by James Hein
· Discrete Mathematics by Nomran L. Biggs
· An Introduction to Discrete Mathematics by Steven Roman
· Discrete Mathematics for New Technology by Rowan Garnier and John Taylor
· Discrete Mathematics by R. K. Bisht and H. S. Dhami
· Discrete Mathematics for Computer Science by Kenneth P. Bogart, et al
· Discrete Mathematical Structures : Theory and Applications by D. S. Malik and M. K. Sen
· Discrete Mathematical Structures by Bernard Kolman, et al
· Logic and Discrete Mathematics by Winfried K. Grassmann and Jean-Paul Tremblay
31
Table 3: Table of Topics Covered in Example Textbooks
Topic [D91] [D92] [D93] [D94] [D95] [D96] [D97] [D98] [D99] [D100] [D101] [D102] [D103] [D104] [D105] [D106]
Algebraic Structures x x
Algorithm Eciency x x
Binomial Theorem & Coecients x x x x x x x
Boolean Algebra x x x x x x x x x x x x
Circuits x x x x x x
Coding Theory x x x
Combinatorics x
Computability & Decidability x x
Congruences x x x x
Counting x x x x x x x x x x
Cryptography x x x x x x x
Financial Math x x x
Determinants x x
Directed Graphs x x x x x
Discrete Structures x x x
Equivalences x x x x x x x x x
Euler Trails & Circuits x x x x x x x x
Fields x x x
Finite-State Machines x x x x x x x
Functions x x x x x x x x x x x x x x x
Gaussian Elimination x x
Generating Functions x x x x
Graph Coloring x x x x x x x
Groups & Semigroups x x x x
Hamiltonian Paths & Cycles x x x x x x x x x
Inclusion & Exclusion x x x x
Induction x x x x x x x x x x
Innite Sets x x x
Isomorphisms x x x x
Lattices x x x x x
Mathematics History x x
Matrices x x x x x
Modular Arithmetic x x x x x x
Networks x x
Permutations & Combinations x x x x x x
Pigeonhole Principle x x x x x x x
Planar Graphs x x x x x x x x
Posets x x x x x
Predicate Calculus x x
Prime Numbers x x x x
Probability x x x x x x x
Proofs x x x x x x x x x x x
Propositional Calculus x x
Recurrence Relations x x x x x x x x
Recursion x x x x x x x x x
Relations x x x x x x x x x x x x
Rings x x x
RSA Encryption x x x x x x
Sequences x x x x x
Set Theory x x x x x x x x x x x x x x x x
Sums x x x x
Trees x x x x x x x x x x
Truth Tables x x x x x
Turing Machines x x x x x
Vector Spaces x x
Venn Diagrams x x x x x
Well-Ordering Principle x x x
Note: The top row of the table lists the citations of the textbooks.
32
Table 4: Table of Topics Covered in Example Textbooks Part 2
Topic [D107] [D108] [D109] [D110] [D111] [D112] [D113] [D114] [D115] [D116] [D117] [D118] [D119] [D120] [D121] [D122]
Algebraic Structures x x x x x
Algorithm Eciency x x x x
Binomial Theorem & Coecients x x x x x x x
Boolean Algebra x x x x x x x x x x
Circuits x x x x x
Coding Theory
Combinatorics x
Computability & Decidability
Congruences x x
Counting x x x x x x x x x x x
Cryptography x x x
Financial Math
Determinants x
Directed Graphs x x x x
Discrete Structures x x x
Equivalences x x x x x x x x x x
Euler Trails & Circuits x x x x x x x
Fields x x x x
Finite-State Machines x x x x x
Functions x x x x x x x x x x x x x
Gaussian Elimination x
Generating Functions x x x
Graph Coloring x x x x x x x x
Groups & Semigroups x x x x x x x
Hamiltonian Paths & Cycles x x x x x
Inclusion & Exclusion x x x x
Induction x x x x x x x x x
Innite Sets x x x
Isomorphisms x x x x x x x x
Lattices x x x
Mathematics History x
Matrices x x x x x
Modular Arithmetic x x x x x
Networks x x x x x x x x
Permutations & Combinations x x x x x x x x x x x
Pigeonhole Principle x x x x x x x
Planar Graphs x x x x x x x x
Posets x x x x x
Predicate Calculus x x x
Prime Numbers x x
Probability x x x x x
Proofs x x x x x x x x x x x
Propositional Calculus x x x
Recurrence Relations x x x x x x x x x x
Recursion x x x x x x
Relations x x x x x x x x x x
Rings x x x x
RSA Encryption x x
Sequences x x x x
Set Theory x x x x x x x x x x x x x x
Sums x
Trees x x x x x x x x x x x x x
Truth Tables x x x x x
Turing Machines
Vector Spaces x
Venn Diagrams x x
Well-Ordering Principle x
As with the data regarding the discrete mathematics syllabi, I constructed a bar chart that lists the
percentage coverage of each topic by the 32 textbooks I collected. The result is on the next page. The
next logical step after conducting research is analyzing and deciding how to use the data. After much
deliberation I made the decision to include a topic covered by the syllabi if at least 30% of the syllabi cover
it. Likewise I will include the more specic topics that pass the 30% coverage by discrete mathematics
textbooks. This means that the broad categories I will include in my curriculum are algorithms, boolean
algebra, combinatorics, counting, functions, graph theory, logic, modular arithmetic, pigeonhole principle,
proofs, sequences, and set theory. The more specic topics I plan on covering include:
· Binomial Theorem & Coecients
· Boolean Algebra 33
· Circuits
· Counting
· Cryptography
· Equivalences
· Euler Trails & Circuits
· Finite-State Machines
· Functions
· Graph Coloring
· Groups & Semigroups
· Hamiltonian Paths & Cycles
· Induction
· Isomorphisms
· Modular Arithmetic
· Networks
· Permutations & Combinations
· Pigeonhole Principle
· Planar Graphs
· Posets
· Probability
· Proofs
· Recurrence Relations
· Recursion
· Relations
· Set Theory
· Trees
· Truth Tables
34
Figure 2: Bar Chart of Topics Covered in Example Textbooks
Algebraic Structures 21.87
Algorithm Eciency 18.75
Binomial Theorem & Coecients 43.75
Boolean Algebra 68.75
Circuits 34.38
Coding Theory 9.38
Combinatorics 6.25
Computability & Decidability 6.25
Congruences 18.75
Counting 65.63
Cryptography 31.25
Determinants 9.38
Directed Graphs 21.13
Discrete Structures 18.75
Equivalences 59.38
Euler Trails & Circuits 46.88
Fields 21.87
Financial Math 9.38
Finite-State Machines 37.5
Functions 87.5
Gaussian Elimination 9.38
Generating Functions 21.87
Graph Coloring 46.88
Groups & Semigroups 34.38
Hamiltonian Paths & Cycles 43.75
Inclusion & Exclusion 25
Induction 43.75
Innite Sets 18.75
Isomorphisms 37.5
Lattices 25
Mathematics History 9.38
Matrices 31.25
Modular Arithmetic 37.5
Networks 31.25
Permutations & Combinations 53.13
Pigeonhole Principle 43.75
Planar Graphs 50
Posets 31.25
Predicate Calculus 15.63
Prime Numbers 18.75
Probability 37.5
Proofs 68.75
Propositional Calculus 15.63
Recurrence Relations 56.25
Recursion 46.88
Relations 68.75
Rings 21.87
RSA Encryption 25
Sequences 21.13
Set Theory 93.75
Sums 15.63
Trees 71.88
Truth Tables 31.25
Turing Machines 15.63
Vector Spaces 9.38
Venn Diagrams 21.3857
Well-Ordering Principle 12.5
Computer Science Topics
Based on reviews, the computer science textbook I chose was Bite-Size Python: An Introduction to Python
Programming by April Speight. I chose this book because it was marketed towards beginners and used clear,
consistent, and concise language to explain topics. It also was written by a female person of color and the
graphics in the book are of diverse individuals. The chapters in the book are listed below. While I want to
use this book as my main textbook to work out of, in order to cover more complex topics, it can’t be my only
textbook. The second textbook I chose was also designed to be easy for beginners to grasp as well as being
open sourced. The textbook is Think Python: How to Think Like a Computer Scientist by Allen Downey. In
addition to the topics covered by Bite-Size Python it also covers interface design, recursion, iteration, tuples,
data structures, les, classes and objects, inheritance, more debugging, and analysis of algorithms.
1. What is Python? 8. Lists
2. Install Python 9. for Loops
3. IDLE 10. while Loops
4. Variables 11. Functions
5. Numbers 12. Dictionaries
6. Strings 13. Modules
7. Conditionals and Control Flow 14. Next Steps
36
Sample Lesson Plans
Mathematics Lessons: Truth Tables
Lesson objective: Given a statement in symbolic form, students will create a truth table with a success
rate of 4/5 problems or 80%.
Advance Organizer:
Professional of the Day: (Display individual in mathematics and computer science eld who is from an
underrepresented minority ex. Winifred "Tim" Alice Asprey)
Review: Let’s begin today’s lesson by reviewing some vocabulary. Can anyone give me the denition of
the word negation? Student Response That’s right, a negation of a statement is the denial of a statement
and is represented by the tilde symbol. If I were to say the statement "Math is hard", what’s the negation
of that statement? Student Response That’s correct, the negation of the statement "Math is hard" is the
statement "Math is easy". Now, what does the word conjunction mean? Student Response You are correct!
Conjunction means the joining of two statements by the word and. What does the word disjunction mean?
Student Response That’s great! The word disjunction means the joining of two statements by the word
or. Now, we can interpret the word or in two ways, inclusive or and exclusive or. If I said the compound
statement "[Student Name] is at school or at home" would that be an example of an inclusive or exclusive
or? That’s right, the statement "[Student Name] is at school or at home" is an example of an exclusive or.
Final vocabulary word to review. What is a conditional statement? Student Response You are correct, a
conditional statement is a statement that can be written in the form "if p then q". [Student Name], can you
give me an example of a conditional statement? Student Response Good job! [Statement] is a great example
of a conditional statement.
Let’s review turning statements into symbolic logic which is what we covered in class last week. (Display
gure 3) Please translate the following compound statement into symbolic logic and write the answer on
your whiteboard and hold it up.
37
Figure 3: Review Question: Turning Statements to Symbolic Logic
p: The grass is brown
q: It is summer
r: It is raining
If it is summer and the grass is brown then it is not raining.
Nice job! I see most of you got it correct. The correct answer is (Display following)
(q ∧ p)→∼ r .
Objective and Link: You already learned how to turn statements into symbolic logic. Now we are going to
something dierent. We are going to take those statements written in symbolic logic and construct truth
tables using them.
Relevance: Truth tables are used to determine if two logical statements are equivalent or if the structure of
a logic statement is valid.
Demonstration: Truth tables are a way of telling if a compound statement is true of false. The entries in
a truth table are called truth values. Truth values represent either true or false. Often times a T is used to
denote true and a F is used to denote false but you could also use a 0 to denote false and a 1 to denote true. A
truth table list all possible combinations of the individual statements that make up a compound statement
to determine if the statement is true or not. Let’s start with the negation. Take a look at the following truth
table (Display gure 4)
38
Figure 4: Truth Table for Negation of p
p ~p
T F
F T
Figure 5: Truth Table for Conjunction of p and q
p q p ∧ q
T T T
T F F
F T F
F F F
Look at the rst column, the truth values T and F represent all the possible values that p can be. The
statement p can either be true or false. The next column indicates the possible values of the negation of p.
When p is true, p̃ is false and when p is false, p̃ is true. Does that make sense? Let’s look at another example
of a truth table. Let’s look at the truth table for the conjunction of p and q. (Display gure 5)
Now, there are more rows here because there are more possible combinations of what values p and q
can be. You could have p be true and q be true, p be true and q be false, p be false and q be true, and p be
false and q be false. If you have n simple statements p, q, r, ect., there will be 2n rows. So here we have
two simple statements and so we have 22 = 4 rows. Now let’s look at the last column. The last column
represents the values the conjunction of p and q. As you can see, only in the case where both p and q are
true is the conjunction true.
Now let’s construct a truth table for a disjunction, p or q. How many simple statements does the com-
pound statement p or q have? Student Response That’s correct, the compound statement p or q has two
simple statements p and q. So how many rows does our table need to have? Student Response That’s right!
Our table should have four rows. Let’s construct our truth table. I’ll set up the possible values for p and q
for you. (Display gure 6)
Our rst two columns are going to be the same, it’s just our last column that is going to be dierent.
Let’s ll that column. If p is true and q is true is p or q going to be true or false? Student Response That’s
correct. If p is true and q is true then p or q is true also. What about if p is true and q is false? Student
39
Figure 6: Empty Truth Table for Disjunction of p and q
p q p ∨ q
T T -
T F -
F T -
F F -
Figure 7: Full Truth Table for Disjunction of p and q
p q p ∨ q
T T T
T F T
F T T
F F F
Response That’s right! If p is true and q is false then p or q is true. If p is false and q is true is p or q going to
be true or false? Student Response You all are doing great! If p is false and q is true then p or q is going to be
true. Last one, if both p and q are false is p or q going to be true or false? Student Response Great job y’all.
If both p and q are false then p or q is going to be false. Here’s the nished truth table. (Display gure 7)
Before we get into some more complicated examples, let’s go over the truth table for a conditional
statement if p then q. How many rows are we going to have? Student Response That’s correct, we are going
to have four rows because we have two simple statements and 22 = 4. Now let’s ll in our table (Display
gure 8)
If p and q are both true, what is the value of our truth value? Remember those are the T and F values.
Student Response That’s right, if p and q are both true then the conditional if p then q is true. What about if
p is true and q is false? Student Response That’s correct. If p is true and q is false, if p then q is false. Now,
this is a tricky one, if p is false and q is true, what is the value of our truth value? Student Response Great
Figure 8: Empty Truth Table for Conditional If p Then q
p q p → q
T T -
T F -
F T -
F F -
40
Figure 9: Demonstration Question 1
Statement: I have a diploma or I have a full-time job and no diploma.
p: I have a high school diploma
q: I have a full-time job
Figure 10: Demonstration Question 1 Empty Truth Table
p q ~p q∧~p p ∨ (q∧~p)
T T - - -
T F - - -
F T - - -
F F - - -
job! Yeah, if p is false and q is true then the conditional if p then q is what is called vacuously true. It’s sort
of like the expression "when pigs y", it doesn’t matter what you promise to do when pigs y because pigs
are never going to y. What about our last case, if p and q are both false? Student Response Correct! If p
and q are both false then the conditional if p then q is going to be false. Are you ready to try something
trickier? Let’s try this problem: (Display gure 9)
First let’s write the statement in symbolic logic. Can I have a volunteer to come write the symbolic logic
on the board? Thank you [Student Name]! Student writes Thank you, you may sit down now. What do
you all think abut what [Student Name] wrote? Give a thumbs up if you agree and a thumbs down if you
disagree. Student Response I see most of you with your thumbs up. In symbolic logic the statement "I have a
diploma or I have a full-time job and no diploma" is p ∨ (q∧~p). Now let’s construct our truth table. When
we construct our truth table we want to break the statement into its smallest parts and build up. Let me
show you what I mean. (Display gure 10)
So, what I did was use something very similar to PEMDAS. I started at the innermost operation, the
negation of p and work my way out. That way, at each step I am only doing one operation. Let’s start
with column three. What is going to be the rst entry in column three? Remember negation means do the
opposite. Student Response That’s correct! If p is true then the negation of p is false. Now let’s do the second
entry. Put your thumb up if you think the truth value that goes in the second row of column 3 should be
true and put your thumb down if you think it should be false. Student Response Good job! It should be false
41
Figure 11: Demonstration Question 1 Full Truth Table
p q ~p q∧~p p ∨ (q∧~p)
T T F F T
T F F F T
F T T T T
F F T F F
Figure 12: Demonstration Question 2
Statement: I own a handgun and it is not the case that I am a criminal or a police ocer.
p: I own a handgun
q: I am a criminal
r: I am a police ocer
Answer:
p q r q ∨ r ~(q ∨ r ) p∧~(q ∨ r )
T T T T F F
T T F T F F
T F T T F F
T F F F T T
F T T T F F
F T F T F F
F F T T F F
F F F F T F
again. Now the next entry, if p is false the negation of p is... Student Response Correct! It should be true.
What about the last entry? Put your thumb up for true and thumb up for false. Student Response That’s
right, it should be true. Now let’s move to the next column. if q is true and the negation of p is false, what
is conjunction of q and not p? Student Response That’s right, it should be false. What about the next entry
in column 4? If q is false and the negation of p is false what should our truth value be? Student Response
That’s right! It should be false. Repeat for rest of truth table (See gure 11). As indicated in the truth table,
the statement "I have a diploma or I have a full-time job and no diploma" is true under all conditions except
one. The statement is false if you don’t have a high school diploma or a full-time job.
Repeat for gure 12, gure 13, gure 14, and gure 15. Then repeat with new problem and decreased
prompts. Decrease prompts gradually until students can complete problems without assistance.
The problems provided are sourced from a textbook entitled Mathematics: A Practical Odyssey [E123].
42
Figure 13: Demonstration Question 3
Statement: I walk up the stairs if I want to exercise or the elevator isn’t working.
p: I walk up the stairs
q: I want to exercise
r: The elevator is working
Answer:
p q r ~q p∨ ~q (p∨ ~q)→ r
T T T F T T
T T F F T F
T F T T T T
T F F T T F
F T T F F T
F T F F F T
F F T T T T
F F F T T F
Figure 14: Demonstration Question 4
[(p → q)∧ ~q]→~p
p q p → q ~q (p → q)∧ ~q ~p [(p → q)∧ ~q]→ ~p
T T T F F F T
Answer: T F F T F F T
F T T F F T T
F F T T T T T
Figure 15: Demonstration Question 5
[(p →~q)∧ ~q]→ p
p q ~q p →~q (p →~q)∧ ~q [(p →~q)∧ ~q]→ p
T T F F F T
Answer: T F T T T T
F T F T F T
F F T T T F
43
Computer Science Lessons: Conditionals
Lesson objective: Given a statement students will create a if statement with a success rate of 4/5 problems
or 80%.
Advance Organizer:
Professional of the Day: (Display individual in mathematics and computer science eld who is from an
underrepresented minority ex. Annie Easley)
Review: Let’s begin today with some review. Last week we talked about functions. I want you to write down
on your paper a function that takes no arguments but prints out your name. I’ll give you a minute do do
that then I’ll come around and check. Wait one minute then check answers Good job everyone, you all wrote
excellent functions. Now can I have a volunteer come up to the board and write a function that take two
arguments a and b, and returns what a mod b is. Thank you, [Student Name] for volunteering! Wait What
does everyone think of this function? Does the function return a mod b? Student Response That’s correct,
that is a great example of a function that returns a mod b.
Objective and Link: You already learned how to write functions. Now we are going to learn how to write
functions using conditionals or if statements.
Relevance: If statements are really useful if you only want a block of code to run under certain conditions.
Demonstration: First let’s talk about comparison operators. Comparison operators are similar to those
used in mathematics. One example of a comparison operator is "greater than". Can anyone give me another
example of a comparison operator? Student answer Yes, that’s a good example of a comparison operator.
Can anyone give me another? Continue until you have most of the operators then display table 5
Table 5: Comparison Operators
Operator Name Example
== Equal 5 == 5
!= Not Equal 26 != 3
> Greater Than 100 > 67
< Less Than 89 < 216
>= Greater Than or Equal To 90 >= 54
<= Less Than or Equal To 23 <= 77
[E124]
44
When you use a comparison operator like the ones shown, python will return a Boolean value of either
True or False. In all of the examples shown, python will return True because they are all true statements.
Will you all please write on your boards one example of a statement that is False using a comparison
operator? Give feedback You can also have more complex comparisons. Some examples include: Display
gure 16
Figure 16: Complex Comparisons
34 - 3 < 24 * 2$
13 % 3 == 17 - (8*2)
15 - (2*3 + 4) >= 36/4 + (5**2 - 4*7)
I will give you a minute to calculate whether each of these lines of code will return a True or False. Wait
Okay, thumbs up if you think the rst line of code will return the Boolean value True, thumbs down if you
think it will return False. Student Response That’s correct. The rst line will return True because 31 is not
equal to 48. Second line: thumbs up if it will return True. Student Response That’s right, the second line
will return True because 1 is equal to 1. What about the third line? Student Response Good job! The third
line will return False because 5 is not greater than or equal to 6. You can also use comparison operators to
compare strings. For example: Display gure 17
Figure 17: String Comparisons
sentence1 = "This string is short"
sentence2 = "This string is shorter"
print(len(sentence1) < len(sentence2))
favColor = "Blue"
color = "blue"
print(favColor == color)
What will that rst block of code print? [Student Name] what do you think? Will the rst block of code
print True or False? Student Response That’s correct. The rst block of code will print True because the
rst sentence has fewer characters than the second sentence. What about the second block of code? Will
it print True or False? Can I have a volunteer? Yes, [Student Name]? Student response That is right. That
45
was a tricky question wasn’t it? The second block of code will print False because a capital "B" is not the
same as a lowercase "b". Comparison operators are not the only type of operators that exist. Another type
of operator is a logical operator. These are very similar to what you’ve seen with symbolic logic. There are
three logical operators They are "and", "or" and "not". Display table 6
Table 6: Logical Operators
Operator Description Example
and Returns True if both statements are true 4 > 3 and 2 * 3 < 6
or Return True if either statements are true 3 + 2 == 2 ** 2 + 1 or 22 < 5
not Returns the opposite Boolean value of the statement not 64 >= 4 ** 3
Please write on your boards what each of the examples will return. Student response Good job everyone.
The rst example will return False because the rst statement is True and the second statement is False.
Remember, 6 is not less than 6. The second example will return True because the rst statement is True and
the second statement is False. Finally, the third example will return False because the statement is True.
You can use logical operators with strings as well. Now let’s move on to if statements. An if statement lets
you write code that will only run if certain conditions are met. If statements look like this: Display following
if some condition:
action
The formatting is similar to functions, anything you want in your if statement needs to be indented.
Let’s look at some examples. Display following
a = 12
b = 23
if a < b:
print("a is less than b")
dayOfTheWeek = "Tuesday"
temperature = 80
if dayOfTheWeek == "Saturday" and temperature >= 80:
print("Let’s go to the beach!")
46
For the rst block of code, will the line inside the if statement run? Put your hand up if you think it
will run. Student response That’s correct. Because a is less than b, i.e. 12 is less than 23, the code inside the
if statement will run and "a is less than b" will be printed. How about the second block of code? Will the
code inside the if statement run or not? Hands up if you think yes. Student response That’s correct, it will
not run because the dayOfTheWeek variable is not equal to "Saturday". Now, what if we want some code to
run under a certain condition but if that condition is not met we want dierent code to run? Here is where
we use if-else statements. If-else statements look like this: Display following
if some condition:
action
else:
different action
Now let’s write our own if-else statement. Say we wanted to know if a number, say, 27378 is divisible by
three. If it is, we want to print out "This number is divisible by 3" and if it is not we print out "This number
is not divisible by 3". How might we do that? Let’s start with the if statement. How do we tell if a number
is divisible by three? What code can we use to gure it out? [Student Name]? What do you think? Student
response That’s correct. A number is divisible by three if the number modulo 3 is equal to 0. Let’s put that
into code: Display following
x = 27378
if x % 3 == 0:
What do we need to put in the if statement? Student response That’s correct. The code inside the if
statement needs to say print("This number is divisible by 3"). Display following
x = 27378
if x % 3 == 0:
print("This number is divisible by 3")
Now we need to write the else condition. With the else condition you don’t have to specify a condition
because the condition is the negation of the if statement condition so all we have to do is write the code
that will go in the else condition. And that code is what? Student response That’s correct, the code that goes
in the else condition is print("This number is not div4is7ible by 3") Display following
x = 27378
if x % 3 == 0:
print("This number is divisible by 3")
else:
print("This number is not divisible by 3")
Which line will print, the one inside the if statement or the one inside the else condition? Thumbs up
if you think the code inside the if statement will run and thumbs down if you think the code inside the
else statement will run. Student response That’s correct, since 27378 is divisible by three, the line inside the
if statement will run and "This number is divisible by 3" will print. Sometimes you need the code to have
more than one condition. This is where if-elif-else statements come in. If-elif-else statements are similar to
if statements and if-else statements in the way they function. The formatting of if-elif-else statements look
like this: Display following
if some condition:
action 1
elif some other condition:
action 2
else:
action 3
Python rst looks at the if statement and determines whether or not to run the code inside based on
whether the condition is True or False. If the condition is False, Python looks at the elif condition. If that
condition is True, Python runs the code inside. If not, Python runs the code inside the else condition. Let’s
look at an example. Say that you want to print if a number is bigger than, equal to, or less than another
number. You would use an if-elif-else statement. You have two expressions x and y. Display following
x = 12*14 - 7(8**2 + 34*3) + 1000
y = (42%13) + 14(5-13*2) + 11*3 **3
You want to print whether x is less than, greater than, or equal to y. Let’s write the if statement rst. If
x is less than y we want to print "x is less than y" Display following
48
x = 12*14 - 7(8**2 + 34*3) + 1000
y = (42%13) + 14(5-13*2) + 11*3 **3
if x < y:
print("x is less than y")
If x is greater than y we want to print "x is greater than y". This will be our elif condition. Display
following
x = 12*14 - 7(8**2 + 34*3) + 1000
y = (42%13) + 14(5-13*2) + 11*3 **3
if x < y:
print("x is less than y")
elif x > y:
print("x is greater than y")
If neither is True, i.e. x == y, we want to print "x equals y". This will be our else condition. Display
following
x = 12*14 - 7(8**2 + 34*3) + 1000
y = (42%13) + 14(5-13*2) + 11*3 **3
if x < y:
print("x is less than y")
elif x > y:
print("x is greater than y")
else:
print("x equals y")
Which print statement will run? I’ll give you a minute to calculate. When you are ready, write 1 on your
board if you think the rst print statement will run, 2 if you think the second print statement will run, and
3 if you think the last print statement will run. Student response The answer is that the third print statement
will run because both x and y equal 6.
Repeat with new problem and decreased prompts. Decrease prompts gradually until students can complete
problems without assistance.
49
Connection Project: Truth Tables & Conditionals
Project objective: Given sample code, students will create functions that evaluate the truth of logical
operators for Boolean variables.
Project Instructions: Attached to this thesis is code I found while doing research on coding truth tables.
While not perfect, it serves as the basis for a project that combines the use of truth tables with if statements
and conditionals. If you were to use this project you might provide the code and have them write the
conditional, biconditional, and_func, or_func, and negate functions themselves. The source for this code is:
https://codereview.stackexchange.com/questions/145465/creating-truth-table-from-a-logical-statement
50
Conclusion
Math anxiety is a pervasive problem that spans centuries. There are several ways to combat this issue.
Some ways to combat the issue are evidence-based practices such as explicit instruction or high leverage-
practices, practices that are intended to be easily implemented and have been shown to be eective. It
also helps to teach mathematics where the relevancy is clearly dened. One such branch of mathematics is
called discrete mathematics. Discrete mathematics largely consists of the mathematics needed for computer
science. By implementing discrete mathematics alongside computer coding lessons the mathematics is
made more accessible and less anxiety inducing. Incorporating expressive writing, retrieval practice, digital
game-based learning, alternative testing methods and examples of diverse individuals in the mathematics
and computer science eld, are further methods of reducing math anxiety. All of these methods culminated
in a lesson plan on truth tables as an example of what this kind of teaching might look like. Future research
would include testing to determine the eectiveness of the practices not already evidence-based.
51
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