Rina Zazkis

Learning Through Teaching Mathematics

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Teachers’ Opportunities to Learn Mathematics Through Teaching

In this chapter we discuss theoretical and empirical grounds for teachers’ learning through teaching (LTT). We review several theories on teachers’ knowledge and the potential changes in this knowledge, and then focus on learning mathematics. We consider several specific examples of teachers’ learning in a variety of instructional situations and identify the types of learning that have occurred. We further identify the sources of LTT, the types of knowledge acquired through teaching, and consider factors that support teachers’ learning.

Contextual Considerations in Probabilistic Situations: An Aid or a Hindrance?

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 the theoretical lens of reducing abstraction. The results show a strong dependence on a contextualized interpretation of the task, even when formal mathematical knowledge is evidenced in the responses.

“It Is Very, Very Random Because It Doesn’t Happen Very Often”: Examining Learners’ Discourse on Randomness

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. In the second study, we focus in depth on participants’ perceptions of randomness in a clinical interview setting. Particular attention is given to the participants’ ways of communicating the idea of randomness, as featured in the gestures that accompanied their discourse. We conclude that particular consideration of the notion of randomness—as intended in statistics and probability versus everyday uses of the term—deserves attention of instructors and instructional materials.

What Have I Learned: Mathematical Insights and Pedagogical Implications

In this chapter, I share three stories of my personal learning that was triggered by interaction with students. I analyze these stories using disaggregated perspective on learning, considering how the instructor’s mathematics and pedagogy is implemented to consider mathematical and pedagogical issues with prospective teachers. In the first story, I present a novel pedagogical approach in order to enhance teachers’ mathematics related to a translation of a parabola. In the second story, I share my learning of mathematics related to Affine transformations triggered by students’ errors that lead to correct results. In the third story, I explore several examples of numerical variation, presenting it as a powerful pedagogical tool.

Dialogues on Dialogues: The Use of Classical Dialogues in Mathematics Teacher Education

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 highlight meta-mathematical issues, such as definitions and proofs. We share several scripting assignments used with teachers and several student-written scripts produced in response to such assignments. We elaborate on the benefits of bringing classical dialogues for discussion in classes of mathematics teachers. These include, but are not limited to, enculturation by exposure to historical context, reinforcement of mathematical ideas and concepts, introduction to subsequent readings and assignments, and extended variety of tasks for the use in mathematics teacher education.

“I Understand” Talk in Script Writing: A Case from Euclid’s Elements

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 knowledgeable individuals who interact on mathematical matters. Our study is on one of such opportunities: a task of writing a script for a dialogue between fictional characters of historically different backgrounds, who discuss a particular proof from Euclid’s Elements, in the context of number theory. Twenty-three prospective secondary teachers responded to the task. We enquired what do the script writers identify as presumed points of difficulty in the given proof and what explanations are constructed by the participants to result in “I understand” claims. Three types of potential difficulty were attended to: number theory concepts and terms, claims within the proof, and the central idea of the proof. The points of difficulty were explained in various ways, such as producing procedural definitions and translating Euclidian terms into the modern terms. Implications for research on script writing and on the use of historical sources in mathematics education are drawn.

Dialogues on Numbers: Script-Writing as Approximation of Practice

Script-writing is a novel pedagogical approach and research tool in mathematics education. The goal of this chapter is to introduce the approach and exemplify its implementation. A script-writing task presents a prompt, which usually includes an incomplete argument or erroneous claim of a student. Prospective teachers address the prompt by creating a script for a dialogue—presenting an imaginary interaction between a teacher and her students, or among different students. In this chapter I exemplify several results of implementing script-writing tasks and discuss advantages of this approach. In particular, I focus on the concepts related to elementary number theory, prime numbers and factors of a number, and demonstrate how the understanding of these concepts can be explored and refined, as script-writers create characters who discuss particular claims. I suggest that engaging prospective teachers in script-writing is one possible way to support and improve preparation of mathematics teachers.

On the Use of Dialogues: Looking Back and Looking Forward

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.

Ceci n’est pas une Pratique: A Commentary

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.