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Fostering pre-service teachers’ attention to mathematical objects: The realization tree mediator as a teaching representation

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Abstract

Using teaching representations is one of the common means for preparing pre-service mathematics teachers (PSTs) to learn how to teach mathematics meaningfully. However, turning PSTs’ attention to the mathematical objects that students engage with during the lesson is a challenging pursuit. This paper suggests using the Realization Tree Mediator (RTM) as a visual teaching representation for PST learning to implement cognitively demanding tasks in their classrooms. The RTM depicts the mathematical object at the core of a task and its different realizations while illustrating imaginary classroom discussions in which different realizations and the links between them are produced. By analyzing PSTs’ conversations around RTM images, we show how contrasting pairs of RTMs can be used for eliciting conversations about various aspects of cognitively demanding instruction. We discuss the affordances of the RTM as a teaching representation and how it can assist in making the mathematical content and meanings visible, public, and communicable among teachers and teacher educators.

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Notes

  1. As can be seen in Fig. 1, the RTM includes realizations that were not mentioned during the lesson (the unshaded cells). One may wonder how this is possible since there are infinitely realizations that can be depicted as the unshaded realizations. Indeed, the construction of the RTM was an iterative process in which 10 Hexagon lessons in secondary schools were analyzed, and from these lessons, the various possible (unshaded) realizations were taken (Weingarden et al., 2019). That said, the RTM’s developers did not aim to include any possible realization that may be authored in the lesson, only those that are common or reasonable to expect in middle-school lessons. Moreover, discussions of RTMs with PSTs or in professional development settings often include ideas about which additional realizations may be added to the tree.

  2. The RTM, as it was originally designed, does not provide specific information about individual students’ participation, such as which students said what and how many students participate in the discussion. However, the RTM is not inherently devoid of individual student information and adding this information can be easily done (e.g., Knox & Kontorovich, 2022). It is thus a flexible decision–whether to include student information or not and it is a matter of choice–what one wishes to highlight and attend to, vs. what one wishes to background for a particular PST discussion. In this study, we chose to background the individual level of student participation because we wished to discuss with PSTs the mathematical objects at the core of the task and the main mathematical narratives that can be produced by engaging with this task. We believed that providing information about individual students would deter from this focus, leading the discussion, for example, to focus on how many students participated or why specific students were more dominant than others.

  3. The Israeli speaking norms are more often characterized by directness and blunt, certain informality and a willingness to express opinions openly and freely (Blum-kulka et al., 2002; Katriel, 2004). This, in contrast to American teachers, for example, who are expected to interact in a more restrained manner. We are aware that in other cultures, such speech can be viewed as relatively disrespectful, but in the Israeli culture it is acceptable and normative. However, in the discussion, we refer to the possibility of the RTM as tacitly provoking deficit-talk about teachers and teachers’ actions, and we discuss the different ways by which teacher educators can handle it.

  4. The expression \(10n:2+2-n\) is akin to the expression \(\frac{10n}{2}+2-n\). Although it is less common to describe division in algebraic expressions with the notation of a colon, this was the notation used in the lesson. The notations used in the RTM aim to precisely replicate the notations used in the observed lessons.

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Acknowledgment

This article is based on a dissertation completed at the Technion – Israel Institute of Technology, Haifa, Israel. We would like to thank Miriam Wallach, Naama Ben-Dor, Gil Schwartz, Nadav Marco, and the anonymous reviewers for their careful reading and thoughtful critique of the earlier versions of this article.

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    Merav Weingarden & Einat Heyd-Metzuyanim

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Weingarden, M., Heyd-Metzuyanim, E. Fostering pre-service teachers’ attention to mathematical objects: The realization tree mediator as a teaching representation. J Math Teacher Educ (2024). https://doi.org/10.1007/s10857-024-09622-w

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