Publications

We substantiate the following claim: multi-variable narrative in qualitative research on problem posing bears promise for a better understanding of causality relationships between ways in which problem-posing activities are organized on the one hand, and characteristics of processes, products, and effects of problem posing on the other hand. Our notion of multi-variable narrative is first introduced by means of a hypothetical scenario. We then discuss relationships between different types of variables while adapting the terminology developed in mediation analysis literature to problem-posing situations and suggest heuristics for choosing problem-posing variables in research that aspires to inform practice. This is followed by an illustration in the context of a problem-posing activity by mathematics teachers. The illustration shows how features of the posed problems can be related to the problem-posing task organization, and how these relations may be mediated or moderated by particular features of the problem-posers, and by choices they make.

Professors in proof-based mathematics courses often intend that the feedback they provide on students' flawed proofs will promote proof comprehension. In this theoretical article, we investigate how such feedback can be formulated. Drawing on Lakatos's process of proof and refutation, we propose the notion of heuristic refutation feedback for feedback on a flawed proof that contains a mathematical argument, possibly incomplete, that logically implies that the student's proof is invalid. Such feedback is heuristic in the sense that interpreting and utilizing it invites mathematical reasoning that can contribute to development of proof comprehension. We build on Toulmin's model of argumentation to analyze possible variations in the formulation of feedback. Based on data from a Real-Analysis course, we highlight two key decisions entailed in formulating refutation feedback: deciding what flawed (possibly implicit) claim in the student's proof to refute, and deciding how explicitly to present the various elements of the refutation argument. We exemplify these decisions in two particular types of refutation feedback that we have identified: refutation by counter-example and refutation by false implication. We show how different formulations of refutation feedback may afford different opportunities for student-engagement with particular facets of proof comprehension. Our findings suggest how professors can purposefully tailor feedback in order to achieve particular didactic goals.

Situations where students encounter mathematical "impasses" instances where their current discourse is incommensurable with the discourse demanded to solve a task have the potential to stir emotional responses of different types. They can engender feelings of confusion and bafflement, or even embarrassment, or alternatively open students' curiosity to the ways by which more advanced mathematics can solve a problem. In this study, we closely examine the subjectifications (affective communication) of a group of graduate students encountering such an impasse in the form of a request to account for the area and perimeter of the Sierpiński triangle. We analyze these subjectifications in relation to an a priori mathematical analysis of the impasse inherent in the task. We show two major types of subjectifications, found to be communicated by distinct members of the classroom: "I'm baffled", and "this is not a problem". We show how the students subjectifying "this is not a problem" were those who avoided engagement with the impasse by attending only to the infinite process that defines the fractal, while those students who subjectified "I'm baffled" were those who engaged with the process as well as its outcome. Moreover, despite the lack of substantial contribution to the exploration of the impasse by those students who subjectified "this is not a problem", they were positioned as the "explainers", in a position of power relative to those students who expressed bafflement. We conclude by discussing the dialectical relationship between identity and engagement with mathematical impasses.

In their enactment of the curriculum, teachers have a substantial role as instructional designers. Accordingly, any evaluation of the progression of students learning should first be concerned with the pedagogical intentions of the teacher. In this article we present a method for reconstructing teachers implicit and tacit considerations in their selection, sequencing and enactment of tasks. Two 11^th grade teachers tagged all of the tasks that comprised a 5-week learning progression. Tagging consisted of assigning values to prescribed categories of metadata. Visual representations of the metadata revealed patterns in the tagged progressions, and allowed the teachers to reflect upon these patterns. Both teachers, though guided by very different didactical considerations, validated that many of their explicit and implicit intentions were revealed in the representations of the progressions. Furthermore, both teachers had the opportunity to reflect on tacit aspects of their instructional design that they were not previously aware of.

With the emergence of e-textbooks, along with expectations to integrate technology in instruction, teachers are becoming significant co-designers of the curriculum. Informed selection and sequencing of learning resources requires sensitivity to didactic nuance, and tools to support development and application of such sensitivity. Teachers practices are constrained by the aspects of resources that are \u201csearchable,\u201d which are limited using standard search engines, and the selection of tasks is influenced by the engines ranking of search results, reflecting their popularity. We are developing and researching a coupled pair of tools to support mathematics teachers in making informed curricular decisionsa tool for tagging learning resources with prescribed categories of didactic metadata and a dashboard for browsing collections of resources according to this tagged metadata. In this article, we investigate affordances of these tools for the professional development of mathematics teachersboth practicing and pre-service teacher candidates. Viewing the dashboard, along with the metadata that it encodes, as a boundary object between the teachers and the researchers perspectives on curricular design, we show how teachers learned through acts of boundary crossing, conceived as transitions and interactions between the two communities curricular discourses. We show how using the dashboard in a task-selection assignment encouraged teachers to reflect on their practicemaking explicit the tacit considerations that they apply to curricular decisions and articulating them from the researchers perspective. We also describe the emergence of \u201chybrid\u201d search strategies, integrating multiple perspectives to create practices that are both didactically informed and practically relevant for instruction.

Research mathematicians often play a central role in determining educational policies, yet the relevance of their mathematical expertise may be (and indeed often is) questioned. A mathematician who developed a game based on non-Euclidean geometry, and a high school teacher, participated in a group discussion on the game, and were interviewed separately to elicit their perspectives on enriching advanced-track students through inquiry. Though their perspectives appeared in many ways incompatible or incommensurable, we suggest ways in which many of their conflicting concerns can be reconciled. In this we are proposing a model of cooperation among communities, where mathematicians and teachers contribution to mathematics education is mediated by mathematics education researchers.

Is it possible that a meeting of mathematicians and primary school teachers will be productive? This question became intriguing when one professor of mathematics initiated a professional development course for practicing primary school teachers, which he taught alongside a group of mathematics Ph.D. students. This report scrutinizes the uncommon meeting of these two communities, who have very different perspectives on mathematics and its teaching. The instructors had no experience in primary school teaching, and their professed goal was to deepen the teachers' understanding of the mathematics they teach, while teachers were expecting the course to be pedagogically relevant for their teaching. Surprisingly, despite this mismatch in expectations, the course was considered a success by teachers and instructors alike. In our study, we analyzed a lesson on division with remainder for teachers of grades 3-6, taught by the professor. The framework used for the data analysis was mathematical discourse for teaching, a discursive adaptation of the well-known mathematical knowledge for teaching framework. Our analysis focuses on the nature of the interactions between the parties and the learning opportunities they afforded. We show how different concerns, which might have hindered communication, in fact fueled discussions, leading to understandings of the topic and its teaching that were new to all the parties involved. The findings point to a feasible model for professional development where mathematicians may contribute to the education of practicing teachers, while they are gaining new insights themselves.

Increasing globalization encourages assumptions of universalism in teaching and learning, in which cultural and contextual factors are perceived as nonessential. However, our teaching and learning are unavoidably embedded in history, language and culture, from which we draw to organize our educational systems. Such factors can remain hidden but can also provide us with opportunities to gain a deeper understanding of constraints that are taken for granted. This chapter provides a meta-level analysis and synthesis of the what and why of whole number arithmetic (WNA). The summary provides background for the whole volume, which identifies the historical, cultural and linguistic foundations upon which other aspects of learning, teaching and assessment are based. We begin with a historical survey of the development of pre-numeral and numeral systems. We then explore the epistemological and pedagogical insights and highlight the differences between linguistic practices and their links with the universal decimal features of WNA. We investigate inconsistencies between spoken and written numbers and the incompatibility of numeration and calculation and review a number of teaching interventions. Finally, we report the influence of economics and business, academic mathematics, science and technology and public and private stakeholders on WNA to understand how and why curriculum changes are made, with a focus on the fundamental losses and gains.

"The root of 18 is closer to 4 than it is to 5 because 18 is closer to 16 than it is to 25". Is this statement, voiced in an 8th grade class, valid? We suggest hypothetical arguments upon which this statement might be based, and analyze them from two complementary perspectives--epistemic and pedagogical--drawing on Toulmin's notion of field-dependence in argumentation and on Freeman's classification of warrants based on the type of underlying intuition, belief or prior understanding. We propose that our investigation of this statement may serve as a model for inquiry in teacher preparation.

The TWG14, University Mathematics Education (UME), was launched in CERME7 (Nardi, González-Martín, Gueudet, Iannone, & Winsløw, 2011) showcasing the fast growth of research in UME and also the specificity of the research context. In particular, the abstract, formal nature of a significant portion of the mathematical content; the absence of national curriculum guidelines, and therefore the great variations in organization and practices across institutions; the general lack of professional development or of systematic preparation for teaching; and, the volume of content to learn in a short period of time, and the degree of autonomy expected from the students and faculty.