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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/

CHAPTER 17

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

INTRODUCTION

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

TEACHERS

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.

OVERVIEW

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

classrooms.

WHY FOCUS ON CT IN MATHEMATICS

TEACHER EDUCATION?

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

curriculum documents"

(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

mathematics education.

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.

Abstraction

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.

Dynamic Modelling

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

square.

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

mathematics.

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.

OVERVIEW

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:

1. Number

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

mathematics) education.

CASE 4: MATHEMATICAL THINKING FOR ELEMENTARY AND SECONDARY

PRESERVICE TEACHERS

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.

OVERVIEW

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

discussion.

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

curriculum;

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.

Broadening Horizons

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

literature.

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

math concept.

3. A low inference checklist for assessing the quality of early counting picture books.

CASE 5: COMPUTATIONAL THINKING FOR ELEMENTARY AND SECONDARY

PRESERVICE TEACHERS

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.

OVERVIEW

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.

Research

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

pedagogy?

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

online postings

assignments

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.

Resources

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

external resources.

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