© Authors, 2017
Petrarca, D., & Kitchen, J. (Eds.). (2017). Initial teacher education in Ontario: The first year of four-semester
teacher education programs. Ottawa, ON: Canadian Association for Teacher Education. ISBN 978-0-9947451-7-0.
Retrievable from http://cate-acfe.ca/polygraph-book-series/
REDESIGNING K-12 TEACHER EDUCATION: A FOCUS ON
COMPUTATIONAL AND MATHEMATICAL THINKING
George Gadanidis, Ann LeSage, Ami Mamolo and Immaculate Namukasa
Western University and University of Ontario Institute of Technology
The recent change in teacher education in Ontario, moving from a single year to a two-year
program, has offered us an opportunity to rethink and redesign our Kindergarten – Grade 12 (K-
12) teacher education programs. A major shift has been happening within and outside of education
due to a renewed focus on different mathematical ways of thinking, including computational
thinking (CT) (Grover & Pea, 2013; Wing, 2006, 2008, 2011; Yadav, Mayfield, Zhou, Hambrusch
& Korb, 2014). In this chapter we discuss how CT has been integrated into teacher education
programs at two Ontario universities and its connection to mathematics education.
Mathematical thinking in/for teaching has been widely discussed (e.g., Ball, Thames & Phelps,
2008; Gadanidis & Namukasa, 2009; Mason, 1989) with attention toward how ways of thinking
and being mathematical can inform teachers’ responses to students (e.g., Mason & Davis, 2013;
Zazkis & Mamolo, 2011). Through its publications, the Ontario Ministry of Education has
emphasized an importance in teachers’ understanding of students’ potentially disparate
mathematical thinking. What CT is and what it looks like in K-12 education is less emphasized
and not well-defined, as it has not yet been integrated in K-12 curricula (National Research Council
[NRC], 2010). Nevertheless, "computational thinking" is not a new focus, having been previously
proposed by Perlis—in the 1960's, as noted by (Guzdial, 2008)—by Seymour Papert (1980), and
by diSessa (2000). An important stimulus for current discussions of CT in K-12 education has
been Jeanette Wing's (2006; 2008; 2011) advocacy for the inclusion of CT in K-12 education.
Wing (2006) stated, “To reading, writing, and arithmetic, we should add CT to every child’s
analytical ability” (p. 33).
There are a variety of ways of defining mathematical and CT. For the purposes of this chapter, we
focus on (i) articulating our understanding of each within the context of teacher education, and (ii)
analysing connections and complements between mathematical and CT for teaching. Yadav et al.
(2014) note that "prominent features of computational thinking revolve around abstraction and
automation, indicating the ability to dissect problems, abstract the high-level rules, and use
technology to automate the problem-solving process" (p. 5:1). Similarly, Wing (2006) and Aho
(2012) point to formulating and solving problems, designing systems and algorithms as key.
Abstracting, formulating and solving problems, generalizing, and applying as well as objectifying
imagined objects are also recognized as key practices in mathematical thinking (e.g., Mason, et
al., 1981/2010; Mason, 1989; Radford 2003; Sfard, 1991). Grover and Pea (2013) note examples
of "children successfully designing LOGO software to teach fractions (Harel & Papert, 1990) and
science (Kafai, Ching, & Marshall, 1997) (p.42).
Notwithstanding the similarities, CT is not mathematical thinking and vice versa. The distinctions
are in many ways as important as the connections – while CT may offer powerful ways of
mathematical modelling (NRC, 2012) and common patterns in student conceptions have been
noted (Perkins & Simmons, 1998), "the approach to problem solving generally described as CT is
a recognizable and crucial omission from the expertise that children are expected to develop
through routine K-12 Science and Math education" (Grover & Pea, 2013, p. 40).
CASE 1: CT IN MATHEMATICS EDUCATION FOR ELEMENTARY PRESERVICE
In this section we present the case of developing a mandatory 18-hour CT in mathematics
education course for elementary school teacher candidates at Western University.
The course covers these themes:
1. A critical look at the role of computer coding and digital making in teaching mathematics
concepts and relationships to elementary school children.
2. The past, present and future possibilities of CT in elementary school mathematics
education are situated within the context of mathematics education.
The course has two assignments:
1. A reflection on a journal article or a personal experience; or a short paper to be submitted
to a mathematics education journal.
2. Design a CT + math task; or teach and reflect on a CT + math task.
Teacher candidates are also offered the option to propose their own assignments related on the
theme of the course, to be negotiated with and approved by the instructor. From past experience
using this option in other courses, although most teacher candidates tend to select to complete the
assignments in the course outline, the flexibility we offer is an opportunity for exercising agency
and allowing students to pursue personal and professional course goals, typically results in greater
effort and attention and a more immersive experience.
320 Chapter 17
The course is offered over nine weeks as nine two-hour classes, in the winter term of the first year
of our two-year program. It was offered for the first time in January-March of 2016. It is a blended
course, with the five odd numbered classes (e.g., Week1, Week 3) being in a regular classroom
and the four even numbered classes online. The online component serves a number of purposes: It
is a forum for discussing assigned course readings, such as: excerpts from Papert's (1980)
Mindstorms; Wing's (2008) paper on Computational Thinking and Thinking about Computing; a
video of a keynote address by Hoyles & Noss (2015) on The New Coding Curriculum in England
– The Maths Scratch Project (available at http://researchideas.ca/coding/proceedings.html); and
teacher interviews on math + CT (such as the one available at
http://researchideas.ca/wmt/c6b1b.html). The online component is sometimes used to offer a
flipped classroom experience, where students complete online modules while the face-to-face class
offers opportunities to consolidate, reflect and extend. At other times, it is used as a place to extend
classroom activities through related readings,
classroom documentaries and teacher interviews,
along with online discussion.
The face-to-face component is a place to explore
hands-on, in small groups, math related coding: in
Scratch or in Python (see example in Figure 1), using
programmable robots to model mathematical
relationships (see example in Figure 2), and sticker-
based circuits to perform mathematical relationships
(see example in Figure 3).
The face-to-face component also serves to reflect on
and discuss the affordances of CT in mathematics
teaching and learning and for the instructor to model
pedagogy that teacher candidates can use in their own
WHY FOCUS ON CT IN MATHEMATICS
A coding focus in mathematics education is not new:
it was an integral component of Papert's (1980) work
with Logo. "One key difference, compared to Logo’s
history, has been the serious consideration given by
those in education policy and decision-making
positions to include coding skills in mandated
(Gadanidis, 2014, p. 313).
For example, starting in
the Fall of 2014, the new
national curriculum of
England mandates that
children at all grades will
Figure 2. Sphero the robot is coded to
walk a circle.
Figure 3. Sticker-based circuits used to "perform" that the first 3
odd numbers fit in a 3x3 square.
Figure 1. Python Code for printing odd
numbers and their sums.
Gadanidis, LeSage, Mamolo & Namukasa 321
learn to code. Another key difference is that unlike Logo's coupling of CT and mathematics, the
current focus appears to be as an end in itself. The historical pairing of CT and mathematics is not
accidental or arbitrary: there are important conceptual links between the two fields of study.
Integrating CT with one or more curriculum areas also makes it easier to implement in an already
crowded school curriculum. In addition, as elaborated in greater detail in Gadanidis, Hughes,
Minniti and White (2016), there are important affordances of CT, which can be beneficial to
Low Floor, High Ceiling
Following Papert's (1980) lead with Logo,
there exist today several CT environments
that offer a low floor and a high ceiling
experience. That is, they allow even the
youngest children to engage with minimal
prerequisite knowledge, while providing
opportunities to explore more complex
concepts and representations. Some
examples of CT environments include:
programmable robots (Figure 2); sticker-
based electronic circuits (Figure 3); and
block-based programming languages such
as Scratch from MIT (Figure 4). The low floor, high ceiling affordance of CT complements the
focus of our mathematics teacher education program on offering teacher candidates models and
hands-on experiences with tasks that help them see how mathematical ideas in the early grades
connect with more complex concepts in higher grades. Sample tasks will be shared in the "CT
resources" section below.
The process of abstraction, which is at the heart of mathematics, is a prominent feature of CT
(Yadav et al., 2014). Wing (2008, p. 3717) states that "In computing, we abstract notions beyond
the physical dimensions of time and space. Our abstractions are extremely general because they
are symbolic, where numeric abstractions are just a special case." In Figures 1 and 4 we see how
code is used to abstract the processes of finding sums of odd numbers and drawing a square,
respectively. What is interesting about these abstractions is that they have a tangible feel. For
example, the code in Figures 1 and 4 makes the abstractions of “sum of odd numbers” and “draw
a square” feel tangible by turning them into code or algorithmic objects that can be manipulated,
listed, printed, drawn, graphed, and so forth (Gadanidis, 2015a). A physical aspect to this tangible
feel is added when using programmable robots and sticker-based circuits (Figures 2-3). This
objectification of abstractions (Hazzan, 1999) may help students experience a reduced level of
abstraction of mathematical concepts and relationships.
Wing (2008) notes that "Computing is the automation of our abstractions" (p. 3718). This
automation of abstractions makes dynamic modelling possible, offering opportunities to question
the roles and impacts of the various parameters, to make changes in the computer code and to see
Figure 4. Scratch block-based code for drawing a
322 Chapter 17
the mathematical reaction immediately. For example, changing the values of parameters in Figures
1 and 4 can instantly model variations, such as finding sums of even numbers or drawing a triangle,
respectively. Such "play" with mathematics relationships offers students opportunities to
experience the pleasure of mathematical surprise, such as “Odd numbers hide in squares!” (Figures
1 and 3). Surprise is an important part of mathematics learning (Movshovitz-Hadar, 1994; Watson
& Mason, 2007) and the related uncertainty and excitement is part of mathematicians' "world of
knowing" (Burton, 1999, p. 138). These are important mathematics teaching and learning
experiences that we seek to offer to our teacher candidates. Play is associated with student agency,
offering opportunities to pursue personally meaningful experiences. Burton (1999) suggests that
agentic control makes a substantial difference in achievement and attitude towards mathematics.
Papert (1993) adds, "I am convinced that the best learning takes place when the learner takes
charge" (p. 25).
Mathematics for Teachers
The CT in mathematics education course is connected to the mathematics-for-teacher component
of our mathematics teacher education program. Computational thinking in its various forms is in
part used to revisit the mathematics-for-teachers experiences, to investigate them more deeply, and
see them in a fresh light through the new representations available with CT. The low floor, high
ceiling of CT is also an integral component of the mathematics-for-teachers experiences we design.
Unlike Ball, Thames & Phelps (2008) (for example) we do not distinguish between mathematics-
for-teachers and mathematics-for-students (Gadanidis & Namukasa, 2007, 2009). Rather, we see
the mathematics-for-teachers experiences we create as opportunities to experience how math ideas
can be stretched across grades, and to model mathematical connections as well as innovative
teaching strategies (e.g., tool-based and hands-on strategies) for their own teaching practice. Hsieh
(2013) argues that from different curricular perspectives much of what has been identified as
mathematics-for-teachers is also knowledge required of students, such as in the Taiwanese context.
Research Informed Practice
Many of the activities we use in our mathematics teacher education courses come from our
research classrooms, where we work collaboratively with teachers to design experiences that
engage young mathematicians. Over the past five years, through funding by SSHRC, KNAER and
Western's Teaching Support Centre, some of these research classroom activities have been
documented and shared in the online resource, What will you do in math today? (WMT), freely
available at www.researchideas.ca/wmt. The mathematics content in WMT is organized in four
chapters: (1) number; (2) pattern & algebra; (3) measurement & geometry; and (4) data &
probability. These chapters contain activities, videos, animations, classroom documentaries,
interviews with mathematicians, as well as extensions to coding.
Gadanidis, LeSage, Mamolo & Namukasa 323
WMT also contains a chapter specifically focusing on CT and its connections to mathematics
teaching and learning. The introduction to the CT chapter includes an animation showing the
connection between math and coding, which can be
shared as an eCard (Figure 5). The CT chapter
contains a variety of content that we use in the CT
in mathematics education course at Western
University. The first section offers an overview of
the history and current state of CT and its connection
to mathematics education, as well as keynote
addresses on this theme by Celia Hoyle, Richard
Noss and Yasmin Kafai, from a June 2015
Symposium on Math + Coding at Western
University. The second section shares classroom-
tested activities along with teacher interviews. We
will be adding activities to this section as we
develop them. The third section offers math and
coding animations and games, where parameters in
the code can be changed to model different
situations. Figure 6 shows a simulation of rolling a
die to get the first number in __ + __ = 8,
calculating the second number, and then plotting
the pair of numbers on a grid to notice that the
points line up. The simulations are also coded in
Scratch and the code is available to use and edit to
create variations. Figure 7 shows the Scratch code
version for the simulation in Figure 6. The fourth
section contains a list of resources for CT and
Figure 5. Animation and math eCard on
math and coding.
Figure 6. Math & coding animation.
Figure 7. Scratch code version of simulation.
324 Chapter 17
CASE 2: CT IN MATHEMATICS AND SCIENCE EDUCATION FOR SECONDARY
SCHOOL PRESERVICE TEACHERS
In our secondary mathematics teacher education program at Western University, we have a
mandatory 36-hour CT in mathematics and science education course.
The course description is similar to the elementary CT course described above, except for the focus
on secondary school and the inclusion of science education.
The course has three assignments:
1. A reflection on a journal article or a personal experience; or a short paper to be submitted
to a mathematics education journal.
2. Design and present an assessment instrument for a CT tool.
3. Design a CT + math task; or teach and reflect on a CT + math task.
The course is offered over nine weeks as 18 two-hour classes, in the winter term of the first year
of our two-year program. It was offered for the first time in January-March of 2016. Similar to
our CT in mathematics education course for elementary teachers, the secondary course is also a
blended course, with the five odd numbered weeks (e.g., Week 1, Week 3) being in a regular
classroom and the four even numbered classes online. The secondary course is different from its
elementary counterpart in two ways: (1) it makes CT connections across both mathematics and
science education; and (2) it ladders to our Masters of Professional Teacher Education Program in
Mathematics Education (Mathematics MPED) as a similar course exists in that program (to
"ladder" means that students get graduate credit for this course if they are accepted to our
Mathematics MPED program). Otherwise the elementary and secondary CT counterparts are quite
similar in approach and focus, so we won't repeat what we have written above.
CASE 3: A CERTIFICATE COURSE IN CT AND MATHEMATICS EDUCATION
Over the last five years, Western Education has partnered with the Fields Institute for Research in
Mathematical Sciences to offer Certificate Courses for Mathematics Teachers (available at
http://researchideas.ca/wmt/courses.html). These are publicly available, self-serve online courses
which teacher candidates at Western University have the option to complete based on the
program’s course completion criteria, and receive a Certificate of Completion for their resume.
The five courses currently offered are listed below:
2. Pattern & Algebra
3. Measurement & Geometry
4. Data & Probability
5. Computational Thinking & Math
The first four Certificate Courses have a cost-recovery fee of $30/course. These courses can also
be completed without cost if a Certificate of Completion is not needed. The Computational
Thinking & Math Certificate course is the latest addition and is currently offered at no cost, as a
service to our students and to the wider mathematics education community.
Gadanidis, LeSage, Mamolo & Namukasa 325
As mentioned above, CT is not mathematical thinking and vice versa. Through the first three
cases, we have highlighted how, at their intersection, the respective disciplines can offer support
for one another. Through coding, learners can uncover mathematical structure, connections, and
new understandings. Through thinking mathematically, learners can appreciate structures,
techniques, and disciplinary values associated with CT. We now turn our attention toward two
cases that have de-coupled mathematical thinking from CT, addressing them separately in courses
that promote interdisciplinary approaches in STEAM (science, technology, engineering, arts, and
CASE 4: MATHEMATICAL THINKING FOR ELEMENTARY AND SECONDARY
In this section we present the case of developing two mandatory 36-hour courses in mathematical
thinking at the University of Ontario Institute of Technology (UOIT). The courses were designed
for teacher candidates preparing to teach mathematics at the (i) elementary and intermediate school
levels, and (ii) secondary and senior school levels. Although a similar 18-hour elective course was
previously offered to elementary teacher candidates in the 1 year program, moving to a two year
program provided an opportunity to redesign and extend courses aimed at enriching teacher
candidates’ understanding of K-12 mathematics education as it connects to STEAM (science,
technology, engineering, arts, mathematics) contexts and applications.
While the design is the same for both courses, the topics covered vary, as do the assignments.
Notwithstanding the differences, the courses were developed to be congruent in the themes and
values that are emphasized and explored, and as such, we focus on the commonalities in our
The course themes include:
1. Developing conceptual understanding, procedural skills, and confidence in the
mathematical knowledge required for teaching.
2. Challenging current perspectives of mathematics and mathematics pedagogy.
3. Exploring diverse ways of reasoning with and about mathematics, including mathematical
communication and connections to other subject areas.
4. Making connections amongst mathematical ideas, physical and virtual representations.
The assignments for both courses can be clustered around three main themes:
1. Reflective activities--include unpacking personal histories and experiences with
mathematics, de-contextualizing and then re-contextualizing mathematics, and exploring
how these experiences influence understanding of, and confidence with, mathematics;
2. Problem-solving teacher candidates’ content knowledge, as well as how it can be mobilized
in designing tasks, responding to student inquiries, conceptualizing and organizing the
3. Mathematical trajectories--include addressing a surprising mathematical result, common
misconceptions or errors, with attention to related prior and future knowledge, ways of
resolving the surprise or error, and modes of communicating with students.
326 Chapter 17
The actual assessment items include weekly learning modules, cumulative group projects and
presentations, independent mathematical investigations, as well as peer-to-peer teaching and
learning opportunities via mini-lessons and role-playing activities. Based on our experiences,
these activities help foster teachers’ personal mathematical knowledge as well as pedagogical
awareness of how to mobilize such knowledge in teaching, particularly when interacting during,
and responding to, unplannable teaching moments.
The Mathematical Thinking and Doing course is offered in the second term of a 16-month, four
term program, with the first offering from January-March, 2016. This nine-week blended delivery
course includes three face-to-face hours and one asynchronous (online) hour per week. The
instructional design of the course adheres to research rooted in effective professional development
in mathematics education. For example, the course provides opportunities for teacher candidates
to develop their mathematical knowledge for teaching by actively engaging in mathematical
thinking and learning (Hill, 2004; Manouchehri & Goodman, 2000; Ross, 1999; Spillane, 2000).
The course explores various math concepts including connections between them, with an emphasis
on analyzing multiple representations and abstracting, formulating and solving problems (e.g.,
Mason, et al., 1981/2010). Consequently, embedded in the course design are multiple opportunities
for teacher candidates to explore similar problem solving tasks as their students (Saxe, Gearhart,
& Nasir, 2001; Siegler et al., 2010), to reconceptualise mathematical representations, to discuss
the nature of the mathematics and mathematics pedagogy, and to reflect on their learning
experiences (Li & Kulm, 2008; Saxe et al., 2001; Tirosh, 2000).
WHY FOCUS ON MATHEMATICAL THINKING IN TEACHER EDUCATION?
While it is not uncommon (although we might argue it is nevertheless not common enough!) to
have mathematics content courses as part of pre-service teacher education, one distinction that we
make with these courses is that they are mandatory for all teacher candidates – elementary,
intermediate, secondary mathematics teachers, and secondary teachers in areas other than
mathematics. There are three important motivations behind this design: (1) every K-8 classroom
teacher is required to teach mathematics – including (for example) secondary school English
teachers, who will be accredited to teach grades 7-12, which includes mathematical content such
as integers, rational number arithmetic, and early algebra concepts; (2) secondary mathematics
teacher knowledge of mathematics tends to be mainly procedural with limited understanding of
the conceptual or structural facets (e.g., Mamolo & Pali, 2014); and (3) mathematical
understanding is necessary for “non-STEM related” areas, such as socio-political community
engagement and policy enactment, yet leaders in these areas struggle to interpret data trends and
their consequences (e.g., Hughes Hallett, 2015).
Mathematics for Teaching
Although a direct correlation between student achievement in mathematics and their teacher’s
understanding of mathematics has been well documented (Burton, Daane, & Giesen, 2008; Hill,
Rowan, & Ball, 2005; Ma, 1999); researchers have yet to reach a consensus on the nature, breadth
and depth of the mathematical knowledge required for teaching. Central to this discussion is the
complex dimension of teacher knowledge known as mathematical knowledge for teaching (Ball,
Hill, & Bass, 2005; Hill et al., 2008; Hill & Ball, 2009; Hill, Schilling, & Ball, 2004). At the core
of mathematical knowledge for teaching is a deep understanding of common content knowledge,
Gadanidis, LeSage, Mamolo & Namukasa 327
that is “the basic skills that a mathematically literate adult” possess combined with the “specialized
knowledge for teaching mathematics” (Ball et al., 2005, p. 45). Unfortunately, many elementary
and secondary teacher candidates lack common mathematical knowledge and, therefore struggle
to develop a foundation to build their specialized mathematical knowledge for teaching. In an
effort to address this challenge, we modified the structure and content of our program and math
courses to provide all teacher candidates with more opportunities to re-learn mathematics and build
their capacity for mathematical thinking.
In addition to the common and specialized mathematical knowledge for teaching, research has
pointed to the relevance of what has been called “horizon” knowledge (e.g., Ball & Bass, 2009;
Zazkis & Mamolo, 2011), and which includes understanding of mathematical structure,
disciplinary norms and values, and the interconnectedness of mathematical ideas and concepts.
Metaphorically speaking, a teacher’s horizon depends on his/her “location” in the landscape. The
“higher up” one climbs, the broader the view – that is, the more personal mathematical knowledge
acquired by teachers, the broader their view of student learning and future/past trajectories is. Ball
et al. (2008) point out that teachers “who do not themselves know a subject well are not likely to
have the [pedagogical content] knowledge they need to help students learn this content” (p.404).
Potari et al. (2007) observed that robust mathematical knowledge allowed teachers to interact with
students and their ideas more easily and effectively, and they suggested that awareness of the
connections amongst different mathematical areas contributed to teachers’ ability to create rich
mathematical learning environments. Similarly, Chinnappan and Lawson (2005) andBaturo &
Nason (1996) noted the importance of teachers’ understanding of the interconnectedness of
mathematical concepts and procedures, linking this understanding to opportunities fostered or
missed in supporting student learning.
Research Informed Practice
Similar to Western University, UOIT strives to create opportunities for teacher candidates that
encourage them to make mathematical connections across subjects and grade levels, to model
mathematical ideas, and to change their location in the landscape by broadening their personal
mathematical knowledge. One instructional strategy we use to foster content knowledge and
engage elementary teacher candidates in problem-solving and problem-posing is through
integrating mathematics and children’s literature.
Throughout the term, children’s books are introduced as a means for exploring the mathematics
content at hand. By providing a problem-solving context which integrates math and language, we
create a venue for teacher candidates to re-experience learning math from a new perspective or a
new “location” in the landscape. By the end of the term, teacher candidates in the elementary
division are required to create, solve and analyze rich problem solving tasks inspired by children’s
The resources used to support teacher candidates’ integration of mathematics and literature are
products of current and previous research projects. These resources include:
1. A database of approximately 1000 books that could be used to teach K-8 mathematics,
organized by grade level, strand and math concepts.
328 Chapter 17
2. A subset of my top recommended children’s books organized by grade level, strand and
3. A low inference checklist for assessing the quality of early counting picture books.
CASE 5: COMPUTATIONAL THINKING FOR ELEMENTARY AND SECONDARY
As mentioned, at UOIT we offer a separate mandatory course dedicated to fostering CT across
disciplines and grades, from K-12. The course, Coding and Communication, has two sections –
one for elementary and one for secondary teacher candidates. The first course offering was from
September to November 2016, in the final term of the four term program. It will include two hours
per week of face-to-face teaching and two hours per week of asynchronous (online) activities.
The description for the two sections of this course is the same, and appears below. As with the
Mathematical Thinking and Doing course, specific topics, foci, and assessments will vary.
Today’s youth are born into a technology rich environment vastly different from that experienced
by even quite recent generations. Students will increasingly need skills in coding and
communication to be active participants in a digital world and for the future workplace. This
course will introduce teacher candidates to leading-edge pedagogies and skills for learning and
teaching the foundations and fundamentals of programming. By exploring and analyzing an array
of child-friendly software geared at developing the basics of coding and digital communication for
(PJ/IS) students, teacher candidates will develop innovative pedagogies for STEM learning in the
21st century. Topics may include: coding educational games, developing mobile apps, LEGO
robotics, and digital storytelling.
In resonance with UOIT’s approach to the Mathematical Thinking and Doing course, Coding and
Computation will include content and assessment items that vary, respectively, across PJ and IS
sections, yet will offer congruent experiences, themes, and values for teacher candidates. The
assignments for both courses will be:
1. Project-based – with a focus on developing ‘computational thinking for teaching’, which
includes a “functional literacy” or “performative competency” in coding, familiarity with
multiple coding platforms, including block coding (e.g., Scratch, Alice) and coding
languages (e.g., Python), communicational possibilities and constraints for coding, and
related pedagogical knowledge for K-12.
2. Collaborative and hands-on – with an emphasis on peer-to-peer mentorship through
engagement in communities of practice in a “maker-space” environment.
3. Interdisciplinary – highlighting different practical purposes and applications for
computational thinking, coding, and communication in (e.g.) online participation and
production, digital story-telling, social justice and equity, and scientific exploration.
The course was offered in the final semester of a four-semester program, and teacher candidates
will have taken prerequisite courses in mathematics (PJ and IS), two STEM-focused 36-hour
Gadanidis, LeSage, Mamolo & Namukasa 329
methods courses (PJ), at least one integrated 36-hour STEAM methods course (IS), and one 36-
hour digital literacies course (PJ and IS).
Implementation of the course faces non-negligible challenges, including institutional constraints
regarding student-instructor ratios, physical lab space, and the timing of the semester (it is a
condensed 9-week semester). Pedagogical challenges include addressing affective and content-
knowledge issues similar to those experienced in mathematics courses – we expected high levels
of anxiety, related in particular to the high level of abstraction and precision required in coding, as
well as to the ‘newness’ of integrating coding into teaching practice (for example, it is unlikely
that teacher candidates will experience such initiatives during their practicum unless they have
opportunity to implement them). One approach we are taking to address and pre-empt some of
these issues is to involve teacher candidates in extracurricular opportunities to work with peers
and school children in coding environments. For instance, we offer summer and March Break
camps to which teacher candidates volunteer to work with K-12 students on activities with Lego
robotics and coding, as well as UOIT houses the STEAM-3D Maker Space Lab, which is a
collaborative, learner-entered and constructionist-pedagogy-focused environment for interested
learners and researchers (Hughes & Morrison, 2014).
WHY CODING IN TEACHER EDUCATION?
We will not repeat the arguments made earlier with respect to low-floor / high-ceiling learning
opportunities, abstraction and dynamic modeling except to say that we agree and see them as
broadly beneficial for the development of 21st century literacy needs.
To be literate in the 21st century, students need to both read critically and to write functionally
across a range of media forms and formats. In personal, civic, and professional discourse, multiple
modes of expression facilitated by the multimodal, multimedia nature of digital media are not
luxuries but essential components of knowing and communicating. (Hughes, Laffier, Mamolo,
Morrison, & Petrarca, 2015).
Critical Digital Literacies (CDL).
At its heart, CDL pedagogy emphasizes critical and equitable participation in democratic society
via the co-creation of knowledge, practices, skills and values. The approach is inquiry-based,
fostering analytic and performative skill in the uses of digital technologies for learning,
communication, and societal participation (Hughes & Morrison, 2014). An important goal is the
promotion of linguistic (including coding languages) knowledge, skills, and understanding that
may increase the competence and confidence of students persistently left out of the ‘digital native’
demographic. In thinking about coding as an element of CDL, with relevance beyond its
applications to mathematics and computing sciences, we note that fluency with digital technologies
requires competency in both literacy and (abstract) mathematical/deductive reasoning. This is
evidenced in the characterization of 21st century literacies provided for by the National Council
for the Teachers of English: to “pose and solve problems collaboratively and strengthen
independent thought” as well as “manage, analyze and synthesize multiple streams of simultaneous
information, create, critique, analyze, and evaluate multimedia texts” (NCTE, 2013).
330 Chapter 17
Reading and Writing the World.
Informed by Freire (1970/1993), we extend his notions of reading and writing the world to a
context of digitally-enhanced learning. Understanding the socio-political, cultural-historical
conditions of one’s life, community, and world (reading the world), and taking action to transform
one’s life, community, and world (writing the world) take on new meaning when contextualized
within the complexities of modern digitally-enhanced societies. These complexities relate to both
the scope of information required to read the world (e.g., an emphasis on highly condensed,
numeracy-embedded, dynamic images and information), as well as to how it is accessed,
distributed, vetted, and developed or refuted (e.g., via social media, special-interest online
publications, blackout censorship). As indicated above, there is a strong correlation between CT
and mathematical activity, and we see coding as one of the vehicles through which to enhance
access to important pillars of mathematics education. Inequitable access to mathematics education
has been acknowledged both to restrict student opportunities for broader academic success, and to
limit civic participation which depends on critical interpretation of statistics, numerical trends, and
their societal implications (e.g., Anderson & Tate, 2008; Skovsmose & Valero, 2008). In the
context of computer programming or coding, inequities exist with respect to both opportunities
with which to engage in CT or digital making (such as app design, robotics, or digital storytelling),
as well as access to current and affordable hardware which can serve as a sufficiently sophisticated
platform for programming in various languages. With respect to this latter point, we are piloting
a project that incorporates Raspberry Pi’s © (https://www.raspberrypi.org/), a fully functioning
computer that costs under $40USD (plus accessories). Figure 8 shows a Raspberry Pi next to more
traditional “educational tools”. The Pis offer an easy-access coding platform with Scratch and
Python pre-installed and ready to go, which avoids some of the complications associated with
navigating disparate shells, environments, and libraries
available for Python, particularly as the Faculty of
Education offers a BYOD environment. The Pis can also
be connected to different sensors (e.g., temperature,
humidity, cameras) that allow for scientific data
collection, experimentation, and inquiry.
NEXT STEPS: RESEARCHING CASES
The shift to a two year teacher education program in
Ontario created the opportunity to introduce and experiment with new ideas in teacher education.
In particular, our focus on CT in teacher education is one innovation that is both unique and timely.
In all five cases discussed in this chapter, we are embarking on new territory, and as such, we are
conducting respective, but related, research studies.
We will be conducting research studies on the (i) CT in mathematics education, (ii) mathematical
thinking and doing, and (iii) coding and communication courses. The analyses will be qualitative,
seeking to identify themes that answer the following questions:
a) What do teacher candidates learn about both computational thinking and mathematics and
b) What attitudes and identities do teacher candidates develop towards computational
(coding) thinking and mathematics?
c) What role do the online resources and experiences play in (a) and (b)?
Figure 8. Raspberry Pi
Gadanidis, LeSage, Mamolo & Namukasa 331
d) What role do the face-to-face experiences play in (a) and (b)?
Data will be collected in the following ways:
pre and post questionnaires
observations of face-to-face classes
individual and focus group interviews with 3-4 teacher candidates from each of the sections
of the course
We will be using a case study approach, which is suitable for collecting in-depth stories of teaching
and learning and studying a ‘bounded system’ (that is, the thoughts and actions of participants of
a particular education setting) so as to understand it as it functions under natural conditions (Stake,
2000a, 2000b; Yin, 1994). Each of the sections of teacher candidates will be treated as an
individual case and we will use content analysis (Berg, 2004) to identify themes related to our
research questions. We will use cross-case analysis to compare/contrast the 5 cases.
The research will be repeated annually for three years, to study three cohorts of teacher candidates.
Research data will help inform how we design the course in subsequent years.
The research will also inform what additional resources we develop to support future course
offerings. At present, we plan to: (1) develop additional classroom case studies to serve as teaching
ideas, teaching models, and objects for reflection and discussion; and (2) maintain a list of links to
Aho, A. V. (2012). Computation and computational thinking. The Computer Journal, 55, 832-835.
Anderson, C. R., & Tate, W. F. (2008). Still separate, still unequal: Democratic access to
mathematics in U.S. schools. Chapter 13. In Handbook of international research in
mathematics education, 2nd ed., 299–318. New York: Routledge. Retrieved from
Ball, D. L., & Bass, H. (2009). With an eye on the mathematical horizon: Knowing mathematics
for teaching to learners’ mathematical futures. Paper prepared based on keynote address
at the 43rd Jahrestagung für Didaktik der Mathematik held in Oldenburg, Germany, March
1–4, 2009. Retrieved from https://www.mathematik.tu-
Ball, D., Hill, H., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics
well enough to teach third grade and how can we decide? American Educator, 29(3), 14–
Ball, D.L., Thames, M.H. & Phelps, G. (2008). Content knowledge for teaching. Journal of
332 Chapter 17
Teacher Education, 59(5), 389-407.
Baturo, A., & Nason, R. (1996). Student Teachers’ Subject Matter Knowledge within the domain
of area measurement. Educational Studies in Mathematics, 31, 235-268.
Berg, B. L. (2004). Qualitative research methods for the social sciences. New York: Pearson
Burton, L. (1999). The practices of mathematicians: what do they tell us about coming to know
mathematics? Educational Studies in Mathematics, 37, 121-143.
Burton,M., Daane, C. J., & Giesen, J. (2008). Infusing mathematics content into a methods course:
Impacting content knowledge for teaching. IUMPST: The Journal, 1. Retrieved from
Chinnappan, M., & Lawson. J. M., (2005). A framework for analysis of teachers’ geometric
content knowledge and geometric knowledge for teaching. Journal of Mathematics
Teachers Education, 8, 197–221.
diSessa, A. A. (2000). Changing minds: Computers, learning, and literacy. Cambridge: MIT Press.
Freire, P. (1970 / 1993). Pedagogy of the oppressed. (M.B. Ramos, Trans.) New York : Continuum
Gadanidis, G. (2012). Why can't I be a mathematician? For the Learning of Mathematics, 32(2),
Gadanidis, G. (2014). Young children, mathematics and coding: A low floor, high ceiling, wide
walls learning environment. In D. Polly (Ed). Cases on Technology Integration in
Mathematics Education (pp. 312-344). Hersey, PA: IGI Global.
Gadanidis, G. (2015a). Coding as a Trojan Horse for mathematics education reform. Journal of
Computers in Mathematics and Science Teaching, 34(2), 155-173.
Gadanidis, G. (2015b). Coding for young mathematicians, K-8. London, ON: World Discoveries,
Gadanidis, G., Hughes, J., Minniti, L. & White, B. (2016). Computational thinking, grade 1
students and the Binomial Theorem. Digital Experience in Mathematics Education.
Gadanidis, G. & Namukasa, I. (2007). Mathematics-for-teachers (and students). Journal of
Teaching and Learning, 5(1), 13-22.
Gadanidis, G. & Namukasa, I. (2009). A critical reflection on mathematics-for-teaching. Acta
Scientiae, 11(1), 21-30.
Gattegno, C. (1974). The common sense of teaching mathematics. New York, NY: Educational
Grover, S. and Pea, R. (2013). Computational thinking in K-12: A review of the state of the field.
Educational Researcher, 42(1), 38-43.
Guzdial, M. (2008). Paving the way for computational thinking. Communications of the ACM,
Harel, I., & Papert, S. (1991). Software design as a learning environment. Constructionism.
Norwood, NJ: Ablex Publishing Corporation. pp. 51–52. ISBN 0-89391-785-0.
Hazzan, O. (1999). Reducing abstraction level when learning abstract algebra concepts.
Educational Studies in Mathematics, 40, 71-90.
Hill, H. (2004). Professional development standards and practices in elementary school
mathematics. The Elementary School Journal, 104(3), 215–231.
Hill, H., & Ball, D. L. (2009). The curious – And crucial – Case of mathematical knowledge for
teaching. Phi Delta Kappan, 91(2), 68–71.
Hill, H., Ball, D. L. & Schilling, S. G. (2008). Unpacking pedagogical content knowledge:
Gadanidis, LeSage, Mamolo & Namukasa 333
Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for
Research in Mathematics Education, 39(4), 372–400.
Hill, H., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for
teaching on student achievement. American Education Research Journal, 42(2), 371–406.
Hill, H., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers’ mathematics
knowledge for teaching. The Elementary School Journal, 105(1), 11–30.
Hoyles, C. & Noss, R. (2015). The New Coding Curriculum in England – The Maths Scratch
Project. Keynote address at the 2015 Mathematics and Coding Symposium, Western
University, London, ON. (available at http://researchideas.ca/coding/proceedings.html)
Hsieh, F-J. (2013). Strengthening the conceptualization of mathematics pedagogical content
knowledge for international studies: A Taiwanese perspective. International Journal of
Science and Mathematics Education, 11, 923-947.
Hughes, J. (2015). STEAM-3D Maker Space Lab: http://janettehughes.ca/lab/
Hughes, J., Laffier, J., Mamolo, A., Morrison, L. & Petrarca, D. (2015). Re-imagining pre-service
teacher education in Ontario, Canada – A journey in the making. Proceedings of the first
conference for HEIT, Dublin, Ireland.
Hughes Hallett, D. (2015, June). Connections: Mathematical interdisciplinary, personal, and
electronic. Plenary address at the 39th Annual Meeting of the Canadian Mathematics Study
Group, Moncton, NB, Canada.
Kafai, Y. B., Ching, C. C., & Marshall, S. (1997). Children as designers of educational multimedia
software. Computers & Education, 29, 117–126.
Li, Y., & Kulm, G. (2008). Knowledge and confidence of pre-service mathematics teachers: The
case of fraction division. ZDM, 40(5), 833–843.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understandings of
fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.
Mamolo, A. & Pali, R. (2014). Factors influencing prospective teachers’ recommendations to
students: Horizons, hexagons, and heed. Mathematical Thinking and Learning, 16(1), 32-
Manouchehri, A., & Goodman, T. (2000). Implementing mathematics reform: The challenge
within. Educational Studies in Mathematics, 42, 1–34.
Mason, J. (1989). Mathematical abstraction as the result of a delicate shift of attention. For the
Learning of Mathematics 9 (2), 2-8.
Mason, J., Burton, L. & Stacey, K. (1981/2010). Thinking mathematically. Bristol, UK:
Movshovitz-Hadar, N. (1994). Mathematics theorems: An endless source of surprise. For the
Learning of Mathematics, 8(3), 34-40
Maturana, H. R. (1988). Reality: The search for objectivity or the quest for a compelling argument.
The Irish Journal of Psychology, 9(1), 25–82.
National Council of Teachers of English (NCTE). (2013, February 15). The NCTE Definition of
21st Century Literacies. Retrieved from
National Research Council (NRC) (2010). Report of a workshop on the scope and nature of
computational thinking. Washington, DC: National Academies Press.
National Research Council (NRC) (2011). Committee for the workshops on computational
thinking: Report of a workshop of pedagogical aspects of computational thinking.
334 Chapter 17
Washington, DC: National Academies Press.
National Research Council (NRC) (2012). A framework for K–12 science education: Practices,
crosscutting concepts, and core ideas. Washington, DC: National Academies Press.
Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York, NY: Basic
Papert, S. (1993). The children's machine. Rethinking school in the age of the computer. New
York, NY: Basic Books.
Perkins, D. N., & Simmons, R. (1988). Patterns of misunderstanding: An integrative model for
science, math, and programming. Review of Educational Research, 58(3), 303–326.
Potari, D., Zachariades, T., Christou, C., Kyriazis, G., & Pitta-Pantazi, D. (2007). Teachers’
mathematical knowledge and pedagogical practices in the teaching of derivative. In D.
Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European
Society for Research in Mathematics Education (pp. 1955–1964).
Radford, L. (2003). Gesture, speech, and the sprouting of signs: A semiotic-cultural approach to
students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37-70.
Ross, J. (1999). Implementing mathematics education reform: What the research says. In D.
MacDougall (Ed.), Impact mathematics (pp. 48–86). Final Report to the Ontario Ministry
of Education & Training. Toronto: OISE/UT.
Saxe, G. B., Gearhart, M., & Nasir, N. S. (2001). Enhancing students’ understanding of
mathematics: A study of three contrasting approaches to professional support. Journal of
Mathematics Teacher Education, 4(1), 55–79.
Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., et al. (2010). Developing
effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE
#2010-4039). Washington, DC: National Center for Education Evaluation and Regional
Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from
Spillane, J. P. (2000). A fifth-grade teacher’s reconstruction of mathematics and literacy teaching:
Exploring interactions among identity, learning, and subject matter. The Elementary
School Journal, 100(4), 307–330.
Sengupta, P., Kinnebrew, J.S., Basu, S. Biswas, G. and Clark, D. (2013). Integrating computational
thinking with K-12 science education using agent-based computation: A theoretical
framework Educational Information Technology, 18, 351–380.
Sfard, A. (1991). Reification as the birth of metaphor. For the Learning of Mathematics, 14 (1),
Skovsmose, O., Valero, P., & Christensen, O. R. (Eds.). (2008). University science and
mathematics education in transition. Dordrecht: Springer-Verlag New York Inc.
Sneider, C., Stephenson, C., Schafer, B. and Flick, L. (2014). Exploring the science framework
and the NGSS: Computational thinking in elementary school classrooms. Science and
Children, 52(3), 10-15.
Stake, R. (2000a). Case Studies. In N. Denzin & Y. Lincoln (Eds.). Handbook of qualitative
research, 2nd Ed. (pp. 435-454). Thousand Oaks, CA: Sage Publications.
Stake, R. E. (2000b). The case study method in social inquiry. In R. Gomm, M. Hammersley & P.
Foster (Eds). Case study method: Key issues, key texts. London: Sage Publications.
Tirosh, D. (2000). Enhancing pre-service teachers’ knowledge of children’s conceptions: The case
of division of fractions. Journal for Research in Mathematics Education, 31, 5–25.
Watson, A. & Mason, J. (2007). Surprise and inspiration. Mathematics Teaching, 200, 4-7.
Gadanidis, LeSage, Mamolo & Namukasa 335
Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49(3), 33-35.
Wing, J. M. (2008). Computational thinking and thinking about computing. Philosophical
Transactions of the Royal Society A, 366(1881), 3717-3725.
Wing, J. (2011). Research notebook: Computational thinking—What and why? The Link
Magazine, Spring. Carnegie Mellon University, Pittsburgh. Retrieved from
Yadav, A., Mayfield, C., Zhou, N., Hambrusch, S. and Korb, J.T. (2014.). Computational thinking
in elementary and secondary teacher education. ACM Transactions on Computing
Education, Vol. 14(1), 5: 1-16.
Yin, R. (1994). Case study research: Design and methods (2nd ed.). Beverly Hills, CA: Sage
Zazkis, R. & Mamolo, A. (2011). Reconceptualizing knowledge at the mathematical horizon. For
the Learning of Mathematics, 31(2), 8 – 13.
336 Chapter 17