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"Koichu, Boris"
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Implementation of mathematics education research as crossing the boundary between disciplined inquiry and teacher inquiry
Inquiry into the teaching and learning of mathematics is a central aspect of the work of both the mathematics teacher and the mathematics education researcher, yet there are profound differences between the practices and processes underlying educational inquiries within each community. These differences are known to hinder implementation of research, yet they can also become a valuable resource. This paper explores a particular model of implementation of research: mathematics teachers adopt and adapt practices and processes of disciplined educational inquiry in a co-learning partnership with mathematics education researchers. We present a narrative inquiry of a teacher–researcher community that designed and studied classroom activities aimed at encouraging students to ask meaningful mathematical questions. The data analysis, informed by the literature on boundary objects and boundary crossing, highlights how the teachers and researchers leveraged their different processes and practices of educational inquiry as a resource in their collaborative inquiry. We conclude by suggesting that informed and thoughtful attention to the differences between teacher inquiry and disciplined inquiry may support and enhance mathematics education research implementation.
Journal Article
Dissecting success stories on mathematical problem posing: a case of the Billiard Task
\"Success stories,\" i.e., cases in which mathematical problems posed in a controlled setting are perceived by the problem posers or other individuals as interesting, cognitively demanding, or surprising, are essential for understanding the nature of problem posing. This paper analyzes two success stories that occurred with individuals of different mathematical backgrounds and experience in the context of a problem-posing task known from past research as the Billiard Task. The analysis focuses on understanding the ways the participants develop their initial ideas into problems they evaluate as interesting ones. Three common traits were inferred from the participants' problem-posing actions, despite individual differences. First, the participants relied on particular sets of prototypical problems, but strived to make new problems not too similar to the prototypes. Second, exploration and problem solving were involved in posing the most interesting problems. Third, the participants' problem posing involved similar stages: warming-up, searching for an interesting mathematical phenomenon, hiding the problem-posing process in the problem's formulation, and reviewing. The paper concludes with remarks about possible implications of the findings for research and practice.
Journal Article
Tensions as springboards to actions in a partnership between mathematics teachers and mathematics education researchers
Research–practice partnerships around educational research may have beneficial outcomes but also present tensions. By considering the dynamics of manifested tensions, our study aims to understand how teachers engage with the various stages of the research in an inquiry-based professional development community consisting of eleven in-service teachers and three mathematics education researchers. In light of Heider's Balance Theory, we identify and analyze tensions expressed by teachers in the community discourse. Findings indicate that epistemic tensions related to teachers' and researchers' different cultural orientations act as powerful generators of inclusionary and exclusionary actions shaping community members' participation paths. While downplaying epistemic tensions can evoke individual actions detrimental to learning and destructive to the community's existence, awareness of and well-timed coping with tensions can become a springboard for community development.
Journal Article
Implementation-related research in mathematics education: the search for identity
Implementation has always been a paramount concern of mathematics education, but only recently has the conceptualizing and theorizing work on implementation as a phenomenon begun in our field. In this survey paper, we conduct a hermeneutic review of mathematics education research identified as related to the implementation problematics. The first cycle of the review is based on examples of studies published in mathematics education journals during the last 40 years. It is organized according to five reasons for developing implementation research. The second cycle concerns 15 papers included in this special issue and is organized by four themes, as follows: objects of implementation, stakeholders in implementation, implementation vs. scaling up, and implementability of mathematics education research. The paper is concluded with a refined glossary of implementation-related terms and suggestions for future research.
Journal Article
Sense making in the context of algebraic activities
This article concerns student sense making in the context of algebraic activities. We present a case in which a pair of middle-school students attempts to make sense of a previously obtained by them position formula for a particular numerical sequence. The exploration of the sequence occurred in the context of two-month-long student research project. The data were collected from the students' drafts, audiotaped meetings of the students with the teacher and a follow-up interview. The data analysis was aimed at identification and characterization of the algebraic activities in which the students were engaged and the processes involved in the students' sense-making quest. We found that sense-making process consisted of a sequence of generational and transformational algebraic activities in the overarching context of a global, meta-level activity, long-term problem solving. In this sense-making process, the students: (1) formulated and justified claims; (2) made generalizations, (3) found the mechanisms behind the algebraic objects (i.e., answered why-questions); and (4) established coherence among the explored objects. The findings are summarized as a suggestion for a four component decomposition of algebraic sense making.
Journal Article
Who-Is-Right tasks as a means for supporting collective looking-back practices
The looking-back stage is rarely observed in students’ problem solving in spite of its recognized importance. The importance of this stage is attributed to practices of engagement with queries on verification of the obtained solution(s), comparative consideration of alternative solutions, and formulation of implications for future problem solving. We refer to such practices as
looking-back practices
. In the present study we explored the hypothesis that the looking-back practices can be evoked in small-group classroom discussions of controversial worked-out solutions to word problems. Such tasks are known as Who-Is-Right tasks. The data consisted of audio- and videotapes of six small groups of high-school students working on a Who-Is-Right task in the context of percentage. The data analysis, informed by a discursively-oriented perspective on problem solving, attended to strategies, dialogical moves and mathematical resources enacted by the students towards attempted agreement as to which of the solutions should be endorsed and why. The findings imply that Who-Is-Right tasks have undeniable potential for supporting collective looking-back practices. In addition, the study contributes to the literature on enactment of mathematical resources in problem-solving discourse and on patterns of students’ dialogic participation in small-group problem solving.
Journal Article
A fictional dialogue on infinitude of primes: introducing virtual duoethnography
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 inquiry and exemplify our approach in a dialogue related to various proofs of infinitude of primes. We view Lakatos' (1976) dialogue in the seminal Proofs and Refutations as an example of virtual duoethnography. We discuss the affordances of this approach as an alternative to the formal ways of presenting research in mathematics education.
Journal Article
An A Priori Measure of Visual Difficulty of 2-D Sketches Depicting 3-D Objects
Aiming to enhance understanding of visual obstacles inherent in two-dimensional (2-D) sketches used in high school spatial geometry instruction, we propose a measure of visual difficulty based on the ratio between 2 attributes of the sketches: potentially misleading geometrical information (PMI) and potentially helpful geometrical information (PHI). Practical, theoretical, and methodological implications are inspected and discussed.
Journal Article