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Abstract and Figures

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, in particular, knowledge at the mathematical horizon.
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From disturbance to task design, or a story
of a rectangular lake
Rina Zazkis
1
Ami Mamolo
2
Published online: 12 November 2016
Springer Science+Business Media Dordrecht 2016
Abstract 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, in particular, knowledge at the mathematical horizon.
Keywords Teacher educator Knowledge at the mathematical horizon Awareness
Dilation Account of Accounting for
What triggers a new phase of personal development? Most frequently there is some
form of disturbance which starts things off. It may be a surprise remark in a lesson, a
particularly poor showing on a test, something said by a colleague, something
asserted in a journal or book, or a moment of insight (Mason 2002, p. 10).
How may a teacher educator’s personal mathematical knowledge influence her teaching?
Numerous studies on mathematics teacher development have demonstrated that math-
ematics teachers learn through their teaching experiences (e.g., Leikin and Zazkis 2010).
This learning is multifaceted and includes personal pedagogical growth, gaining further
insights into students’ thinking, learning about the feasibility of various instructional
approaches, about implementing new curricula or technological tools, as well as enhancing
personal mathematics. We extend this research on teachers’ ‘‘learning through teaching’
by focusing on a teacher educator, a teacher of teachers.
&Rina Zazkis
zazkis@sfu.ca
1
Simon Fraser University, Vancouver, Canada
2
University of Ontario Institute of Technology, Toronto, Canada
123
J Math Teacher Educ (2018) 21:501–516
https://doi.org/10.1007/s10857-016-9361-z
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... According to Sherin (2002), LTT exists as a critical and at times intractable aspect of developing as a teacher. Previous studies have uncovered a wealth of value within LTT as teachers have developed both discipline specific content knowledge (CK) (Elmendorf, 2006;Leikin, 2006;) and pedagogical content knowledge (PCK) (Perkins et al., 2015;Zazkis & Mamolo, 2018) through this learning process. In terms of the mechanisms through which LTT occurs, Schön (1987) describes a simple three-part framework that encompasses LTT: planning some sort of lesson or learning experience, implementing that plan with students, and then reflecting on the results. ...
Conference Paper
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While previous research has examined what students learn as they agentically solve mathematics problems, less is known about what teachers learn through this process. We use this paper to address this oversight by analyzing interviews, lesson plans, and video observations of five novice elementary and middle school math teachers as they taught between six and eight lessons each. Findings reveal that teachers who rely on prescripted approaches more regularly developed knowledge related to classroom management and climate. Alternately, teachers who allowed students to agentically solve problems more routinely developed a deep understanding of students' mathematical thinking. This study therefore contributes to extant learning through teaching literature by illustrating how pedagogical choices directly affect what teachers learn within these approaches.
Chapter
This chapter explores how competing influences from K-12 schooling and teacher education, respectively, can be examined through the lens of actor-oriented transfer (AOT). A form of scripted role-play is used to evoke an experience of contingency in the form of an unexpected mathematical approach. Of particular interest are the possible tensions and incongruences among prospective teachers’ expressed intentions, the pedagogical and content knowledge being developed in their teacher education program, and the generalization of attitudes, approaches, and assumptions from their experiences as K-12 students. This study extends the applicability of the AOT lens to the case of mathematical knowledge for teaching and highlights implications for research and instruction in teacher education.
Chapter
We stated previously that the tasks described in this book are triggered by our previous interactions with our students, many of whom are prospective teachers. There is a repeated “experience of disturbance” when students draw erroneous conclusions or offer suggestions which lead to erroneous conclusions. However, at times, ideas that appear at first as “obviously wrong” lead to a correct answer. Exploration of these ideas leads to new insights and some surprises. It further also leads to new insights into mathematics and deeper appreciation of mathematical relationships.
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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 characters attempted to convince each other of their point of view. They had to imagine and articulate fictional characters' reasoning, as well as design a potential pedagogical intervention. We outline what script writing revealed about the participants' understanding of the structure of natural and rational numbers and of mathematical argumentation, and discuss the affordances of this methodological tool in teacher education.
Book
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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 refining our use of this tool for a number of years and using it in a variety of situations involving mathematics thinking and learning. The goal of this proposed book is to offer a comprehensive survey of the affordances of the tool, the results of our studies-particularly in the area of pre-service teacher education, and the reasons that the tool offers such productive possibilities for both researchers and teacher educators. © 2013 Springer Science+Business Media New York. All rights are reserved.
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This article examines pre-service secondary school teachers’ responses to a learning situation that presented a student's struggle with determining the area of an irregular hexagon. Responses were analyzed in terms of participants’ evoked concept images as related to their knowledge at the mathematical horizon, with attention paid toward the influence of one on the other. Specifically, our analysis attends to common features in participants’ understanding of the mathematical task, and explores the interplay between participants’ personal solving strategies and approaches and their identified preferences when advising a student. We conclude with implications for mathematics teacher education research and pedagogy.
Article
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We describe and analyze three episodes from mathematics classrooms. In each case, the teacher was confronted by a "contingent" situation that they had not anticipated or planned for yet that offered interesting and fruitful learning possibilities if pursued. In two cases, we analyze the teacher's response; in the third, we speculate how they might have responded. In each case, we propose that the teacher's ability to capitalize on these contingent situations is underpinned by their knowledge and awareness of the mathematical potential of the unexpected opportunity and by an interest in, and commitment to, mathematical enquiry.
Book
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 refining our use of this tool for a number of years and using it in a variety of situations involving mathematics thinking and learning. The goal of this proposed book is to offer a comprehensive survey of the affordances of the tool, the results of our studies�particularly in the area of pre-service teacher education, and the reasons that the tool offers such productive possibilities for both researchers and teacher educators.
Following Star (2005, 2007), we continue to problematize the entangling of type and quality in the use of conceptual knowledge and procedural knowledge. Although those whose work is guided by types of knowledge and those whose work is guided by qualities of knowledge seem to be referring to the same phenomena, actually they are not. This lack of mutual understanding of both the nature of the questions being asked and the results being generated causes difficulties for the continued exploration of questions of interest in mathematics teaching and learning, such as issues of teachers' knowledge.