# Rina ZazkisSimon Fraser University · Faculty of Education

Rina Zazkis

PhD

## About

233

Publications

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3,590

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Citations since 2017

Introduction

Additional affiliations

September 1991 - present

## Publications

Publications (233)

We attend to the composition of even and odd functions, as featured in imagined dialogues between a teacher and students, composed by sixteen teachers in a professional development
program. Data were analyzed as aimed at addressing students’ intellectual needs, with particular attention to the need for causality and the need for certainty. The resu...

Knowing how best to respond to students’ mathematical inquiries is an important skill for all teachers to develop. A class of pre-service teachers (PSTs) was presented with a scripting task in which a student conjectured that 1/6.5 was “exactly in between” fractions 1/6 and 1/7. However, instead of addressing the student’s inquiry directly, many of...

This study aims to explore the notion of the density of the set of rational numbers in the set of real numbers, as interpreted by undergraduate mathematics students. The data comprise 95 responses to a scripting task, in which participants were asked to extend a hypothetical dialog between two student characters, who argue about the existence of on...

In this paper, we examine mathematicians’ views on the value of advanced mathematics for secondary mathematics teachers. The data comprise semi-structured interviews with 24 mathematicians from 10 universities. The findings indicate that the value of advanced mathematics courses for prospective secondary mathematics teachers lies in their potential...

We attend to the composition of even and odd functions, as featured in imagined dialogues between a teacher and students, composed by sixteen teachers in a professional development program. Data were analyzed as aimed at addressing students’ intellectual needs, with particular attention to the need for causality and the need for certainty. The resu...

We explore the responses of 26 prospective elementary-school teachers to the claim “1/6.5 is not a fraction” asserted by a hypothetical classroom student. The data comprise scripted dialogues that depict how the participants envisioned a classroom discussion of this claim to evolve, as well as an accompanying commentary where they described their p...

The original version of this chapter was revised due to some errors (listed below) in the text at page numbers 98 and 99.

We investigate how students make sense of irrational exponents. The data comprise 32 interviews with university students, which revolved around a task designed to examine students' sensemaking processes involved in predicting and subsequently interpreting the shape of the graph of f(x)=x^√2. The task design and data analysis relied on the concept o...

The work of teacher educators juxtaposes mathematics and pedagogy and often involves the use of pedagogically oriented tasks to help teachers extend their understanding of the underlying mathematics. However, in teacher education programmes, where students come from different mathematical backgrounds, the nature and extent of the participants’ prio...

Is the natural number 7 rational? Is it complex? We argue that the answers to these questions relate to the ways numbers are taught. Commonly, a new kind of numbers is presented as an expansion of a previously familiar kind of numbers, which results in a nested image of the relations between number sets. In this article, we introduce an alternative...

What is the connection between advanced mathematics and teaching secondary mathematics? I address this question, drawing on insights from the Special Issue on “Personal Mathematical Knowledge in the Work of Teaching” (CJSMTE 13(2), 2013) and reflecting on my personal teaching experience.

Windmill images and shapes have a long history in geometry and can be found in problems in different mathematical contexts. In this paper, we share and discuss various problems involving windmill shapes and solutions from geometry, algebra, to elementary number theory. These problems can be used, separately or together, for students to explore rota...

This study focuses on connections between linear functions and their graphs that were made by tertiary remedial algebra students. In particular, we describe students’ work on a Task designed to examine the connection between points on a graph and the equation of a line. The data consist of 63 responses to a written questionnaire and individual inte...

In the section titled On the Worth of Calculus for Calculus Per Se in the original article.

Acknowledging the contribution of mathematicians to the mathematical education of teachers, we explore mathematicians’ perspective on an envisioned Calculus course for prospective teachers. We analyzed semi-structured interviews with 24 mathematicians using the EDW (Essence-Doing-Worth) framework (Hoffmann & Even, 2018, 2019); and subsequently, we...

We explore prospective elementary-school teachers’ attempts to provide signs and symbols with mathematical meaning in a case involving non-integers as numerator or denominator. The data comprises 33 responses to a task inquiring about the existence of numbers between 1/6 and 1/7, in which the participants were asked to compose a hypothetical classr...

Acknowledging the significant contribution of mathematicians to the mathematical education of teachers, we explore the views of mathematicians on an envisioned Calculus course for prospective teachers. We analyzed the semi-structured interviews with 24 mathematicians, using the EDW (Essence-Doing-Worth) framework (Hoffmann & Even, 2018, 2019); and...

We investigate how students make sense of irrational exponents. The data is comprised of 32 interviews with university students, which revolved around a task designed to examine students sensemaking processes involved in the understanding of the concept. Both the task design and data analysis relied on the concept of sensemaking trajectories, blend...

Availability of easily accessible computational tools undoubtedly affects teaching and learning of mathematics. However, with technological advances often comes a blind trust in the reliability and accuracy of the digital information and unquestioned dependence on it. I focus on pitfalls in understanding mathematical ideas associated with the use o...

We explore prospective teachers’ attempts to explain atypical fraction representations, particularly when non-integers appear in the numerator or denominator. As practical implication, we advocate the use of the set model to interpret atypical representations and expand learners’ understanding of fractions.

For any convex quadrilateral, joining each vertex to the mid-point of the next-but-one edge in a clockwise direction produces an inner quadrilateral (as does doing so in a counter-clockwise direction). In many cases, a dynamic geometry measurement of the ratio of the area of the outer quadrilateral to the area of the inner one appears to be 5:1. It...

This paper reports a preliminary analysis of mathematicians' detailed accounts on why and how advanced mathematical knowledge matters for secondary mathematics teaching, and what specific connections can be made between advanced and secondary mathematics. Drawn from interviews with 18 mathematicians from 9 universities, the findings indicate that h...

We explore prospective elementary-school teachers' attempts to provide signs and symbols with mathematical meaning. The data comprises responses to a task inquiring about the existence of numbers between 1/6 and 1/7, in which the participants were asked to compose a hypothetical dialog between a teacher and students addressing this issue. We focus...

This study is concerned with the first experiences of in-service mathematics teachers in consuming scholarly mathematics education literature. Growing from the meta-didactical transposition model, we were interested in the praxeologies that may develop from teachers' engagement with research articles. The data were collected from a cohort of 13 tea...

In this study we focus on example spaces for the concept of a function provided by prospective secondary school teachers in an undergraduate program. This is examined via responses to a scripting task-a task in which participants are presented with the beginning of a dialogue between a teacher and students, and are asked to write a script in which...

We explore the notion of density of the set of rational numbers in the set of real numbers, as interpreted by undergraduate mathematics students. Participants' responses to a scripting task, in which characters argue about the existence of one or infinitely many rational numbers in a real number interval, comprise the data for our study. The framew...

Prospective secondary mathematics teachers frequently take as many (or more) mathematics courses from a mathematics department as they do methods courses from an education department. Sadly, however, prospective secondary teachers frequently view their mathematical experiences in such courses as unrelated to their future teaching (e.g., Zazkis & Le...

Do we need to insist on real numbers? We present our discussion of this question in
the style of duoethnography.

We extend the argument on the importance of teachers’ knowledge of advanced mathematics, group theory in particular. We present several examples in which familiarity with properties of groups guided the teacher’s responses in unexpected instructional situations, to which we refer as situations of contingency. We further describe how the situations...

We investigate responses of prospective secondary school teachers to a task related to the inverse function concept. We utilize ideas of fuzzy logic as a theoretical lens in analyzing the participants’ demonstrated avoidance of refuting the existence of an inverse to a quadratic function. The analysis is based on 29 responses to a scripting task, i...

In Canadian schools the acronym BEDMAS is used as a mnemonic, which is supposed to assist students in remembering the order of operations: Brackets, Exponents, Division, Multiplication, Addition, and Subtraction. In the USA schools the prevailing mnemonic is PEMDAS, where P denotes parentheses, and it further assists memory with the phrase “Please...

We present a story of a teacher educator’s response to a situation of contingency and describe how her experience enhanced her personal mathematical knowledge and influenced her teaching. In our analysis, we attend to different levels of awareness that support a teacher educator’s work and illuminate the qualities of a teacher educator’s knowledge,...

We believe that professional mathematicians who teach undergraduate mathematics courses to prospective teachers play an important role in the education of secondary school mathematics teachers. Thus, we explored the views of research mathematicians on the mathematics that should be taught to prospective mathematics teachers, on how the courses they...

In this paper, we highlight a further use of scripting tasks in the work of researchers-teacher- educators, which we present in two parts. In Part 1, we study prospective secondary school mathematics teachers’ responses to a scripting task on the topic of functions. The generated examples for a function that satisfies certain constraints provide a...

In our exploration of the order of operations we focus on the following claim: “In the conventional order of operations, division should be performed before multiplication.” This initially surprising claim is based on the acronym BEDMAS, a popular mnemonic used in Canada to assist students in remembering the order of operations. The claim was voice...

I highlight the main issues discussed in the chapters and wonder about the effect of engaging with representations of practice on actual teaching practice. I offer avenues for future studies in which representations of practice are designed by teachers, rather than researchers and teacher educators.

We illustrate how the classical dialogues – Galileo’s Dialogue on Infinity from Dialogues Concerning Two New Sciences, Plato’s Meno, and Lakatos’ Proofs and Refutations – can be used in teacher education. By re-capturing our conversation, we demonstrate the use of the classical dialogues to revisit mathematical notions, such as infinity, or to high...

In this chapter I outline special affordances of dialogue as a literary genre used by story writers, and demonstrate how these affordances are featured in the use of scripting in mathematics education. I conclude by exemplifying several ideas for follow up research.

The pursuit of understanding in communication between interlocutors of different knowledge is one of the most distinct characteristics of teaching as a professional occupation. Consequently, mathematics teacher preparation programs should provide opportunities for the prospective teachers to put themselves in the shoes of more and less knowledgeabl...

This book shows how the practice of script writing can be used both as a pedagogical approach and as a research tool in mathematics education. It provides an opportunity for script-writers to articulate their mathematical arguments and/or their pedagogical approaches. It further provides researchers with a corpus of narratives that can be analyzed...

We identify several dichotomies that emerge in the chapters of this section around the teaching and learning of mathematics.

TouchCounts is an open-ended multi-touch App, which provides unconventional opportunities for engagement with the concept of a number, counting, and number operations. We describe a series of tasks designed for use in TouchCounts, which take advantage of the affordances of this environment. We elaborate on various aspects of the tasks as related to...

This paper is focused on mathematical conventions, which account for the decisions of the mathematics community regarding definitions, names and symbols of concepts. We argue that tasks that request learners to create and discuss not necessarily historically valid, but convincing explanations of mathematical conventions, provide them with opportuni...

Mathematical conventions -- which account for the choices of mathematics community regarding definitions of concepts, their names and symbols -- are the focus of this paper. We introduce the task of unpacking, that is, offering plausible explanations and arguments for the choice of conventions. Using responses of four prospective teachers to the ta...

This study is concerned with tensions between the two different perspectives on the concept of angle: angle as a static shape and angle as a dynamic turn. The goal of the study is to explore how teachers cope with these tensions. We analyze scripts of 16 in-service secondary mathematics teachers, which feature a dialogue between a teacher and stude...

This study focuses on instances of creativity in the design of Lesson Play tasks and in prospective teachers’ responses to the tasks. A Lesson Play task assumes a theatrical interpretation of the word ‘play’ and requires teachers to write a script for an imaginary interaction between a teacher-character and student-characters, attending to a partic...

This study is concerned with tensions between the two different perspectives on the concept of angle: angle as a static shape and angle as a dynamic turn. The goal of the study was to explore how teachers cope with these tensions. We analyse scripts of 16 in-service secondary mathematics teachers, which feature a virtual dialogue between a teacher...

By focusing on a particular alteration of the comparative likelihood task, this study contributes to research on teachers' understanding of probability. Our novel task presented prospective teachers with multinomial, contextualized sequences and asked them to identify which was least likely. Results demonstrate that determinants of representativene...

Mathematics as well as mathematics education research has long progressed beyond the study of number. Nevertheless, numbers and understanding numbers by learners, continue to fascinate researchers and bring new insights about these fundamental notions of mathematics.

This paper is focused on mathematical conventions, which account for the decisions of the mathematics community regarding definitions, names and symbols of concepts. We argue that tasks that request learners to create and discuss not necessarily historically valid, but convincing explanations of mathematical conventions, provide them with opportuni...

We present a framework of researcher knowledge development in conducting a study in mathematics education. The key components of the framework are: knowledge germane to conducting a particular study, processes of knowledge accumulation, and catalyzing filters that influence a researcher decision making. The components of the framework originated fr...

In mathematics education research paradoxes of infinity have been used in the investigation of students’ conceptions of infinity. We analyze one such paradox - the Painter’s Paradox - and examine the struggles of a group of Calculus students in an attempt to resolve it. The Painter’s Paradox is based on the fact that Gabriel’s horn has infinite sur...

In order to explore the sum of sinusoids we extend the notion of least common multiple to rational numbers.

We introduce virtual duoethnography as a novel research approach in mathematics education, in which researchers produce a text of a dialogic format in the voices of fictional characters, who present and contrast different perspectives on the nature of a particular mathematical phenomenon. We use fiction as a form of research linked to narrative inq...

A significant body of research literature in mathematics education attends to mathematical proofs. However, scant research attends to proof comprehension, which is the focus of this study. We examine perspective secondary teachers’ conceptions of what constitutes comprehension of a given proof and their ideas of how students’ comprehension can be e...

This paper presents a new twist on a familiar paradox, linking seemingly disparate ideas under one roof. Hilbert's Grand Hotel, a paradox which addresses infinite set comparisons is adapted and extended to incorporate ideas from calculus – namely infinite series. We present and resolve several variations, and invite the reader to explore his or her...

Lesson play is a novel construct in research and teachers' professional development in mathematics education. Lesson play refers to a lesson or part of a lesson presented in dialogue form-inspired in part by Lakatos's evocative Proofs and Refutations-featuring imagined interactions between a teacher and her/his students. We have been using and refi...

Script writing by learners has been used as a valuable pedagogical strategy and a research tool in several contexts. We adopted this strategy in the context of a mathematics course for prospective teachers. Participants were presented with opposing viewpoints with respect to a mathematical claim, and were asked to write a dialogue in which the char...

We examine the responses of secondary school teachers to a probability task with an infinite sample space. Specifically, the participants were asked to comment on a potential disagreement between two students when evaluating the probability of picking a particular real number from a given interval of real numbers. Their responses were analyzed via...

We provide an overview of how the notion of randomness is treated in mathematics and in mathematics education research. We then report on two studies that investigated students’ perceptions of random situations. In the first study, we analyze responses of prospective secondary school teachers who were asked to provide examples of random situations....

The lesson play idea that we have been exploring in this book is an elaboration on a lesson or part of a lesson in a form of a script that presents interaction between a teacher and students, and among students. As a construct in teacher education, it provides various opportunities for the prospective (or practicing) teachers who write the plays, a...

In this chapter, we discuss lesson plays composed by prospective elementary school teachers that related to linear measurement. The particular focus is on using a ruler in determining a length of an object.

In this chapter, we discuss lesson plays composed by prospective elementary school teachers that are related to number sense and elementary number theory. The particular focus on the property of divisibility and divisibility tests.

This chapter concerns a variation of the lesson play assignment that focuses on the critiquing rather than the writing of a classroom interaction. In particular, we discuss reactions of a group of experienced elementary and middle school teachers to part of a play written by a prospective elementary school teacher. They were asked to consider what...

In this chapter, we develop in detail the construct of '‘lesson play’' as an imagined interaction between a teacher and his/her students. We provide an example of a lesson play and describe the affordances of the method.

In this chapter we describe the evolution of the Lesson Play Task for prospective teachers, as a regular assignment in our ‘methods’ courses.

In this chapter we discuss lesson plays composed by prospective elementary school teachers that are related to understanding fractions. The particular focus is on different methods of comparing fractions and on testing conjectures.

## Projects

Project (1)

Mathematics is saturated with conventions that may seem arbitrary. However, suggesting possible reasons for establishing some of them can lead to active engagement with mathematical concepts and untrivial ideas. The goal of this project is to explore learning that occurs as a result of such unpacking.