Publications

Tertiary mathematics has a central place in teacher education, yet in recent years there is growing evidence that realizing its potential affordances in secondary mathematics teaching is far from trivial. Research suggests that utilizing tertiary mathematics in secondary teaching requires interweaving it with knowledge for teaching secondary mathematics. Little is known about the underlying processes, which are often tacit and highly personal. In this article we analyze affordances of tertiary mathematics for teaching secondary probability. A group of mathematicians and experienced secondary teachers jointly inquired into the mathematics that could be addressed in school when discussing a popular probability game - the River Crossing game (henceforth "the game"). This context was chosen as an extreme case, in the sense that the mathematics underlying the game is so nuanced and complex that applying tertiary knowledge to mathematize and understand it is generally not feasible for secondary teachers. Thus, it is not clear how tertiary mathematics can inform teachers about using the game in class. Our analysis shows how the conflicting perspectives of teachers and mathematicians on what mathematics students may learn by playing the game initially hindered utilization of tertiary mathematics. Nevertheless, rapprochement was achieved, highlighting four different trajectories for interweaving knowledge of tertiary mathematics with knowledge for teaching secondary mathematics towards using the game in ingenious ways that respect both mathematical and pedagogical concerns. Our findings suggest that tertiary mathematics may have affordances for secondary mathematics teaching even in situations where teachers lack tertiary-level understanding of the underlying subject-matter.

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.

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 growing availability of technology, teachers are becoming significant co-designers of the curriculum-resequencing, editing or supplementing a primary textbook with publicly available digital learning objects. This places a grave responsibility on teachers, who must have a clear view of the rationale and didactic intentions of the textbook in order to "re-source" it while maintaining its coherence. Yet, teachers are seldom provided with means to understand textbooks' structure and underlying design principles, particularly at the grain size of individual learning objects. We are developing tools to express the "voice" of a textbook-one tool for assigning specific metadata to learning objects (i.e. "tagging", sometimes referred to as "coding") and another for analysing the coherence of the textbook based on the tagged metadata. In the current study, the 3 pre-calculus chapters of a high school textbook were tagged. Based on the tagged metadata, we used the second tool to reveal the relative prominence of different "types" of tasks, analysed by the categories of metadata. We focused on commonalities across taggers to explore the nature of the textbook. Triangulating our findings in an interview with the textbook author, we show that some patterns in the metadata reflect explicit design principles, such as avoiding symbolic representations in tasks that open a topic, while others reflect tacit principles-making sense to the author though not explicitly intended-such as eliciting non-technological student justifications in the topic of derivative. We conclude that this methodology offers novel opportunities for the study of textbook design.

Teachers and mathematicians hold different perspectives on the teaching and learning of whole number arithmetic. Though these perspectives may be complementary, sharing them across communities is challenging. An unusual professional development course for primary school teachers, initiated and taught by research mathematicians, provided a setting for productive sharing of both mathematical and pedagogical perspectives. Drawing on the theory of commognition, I analyze two lesson segments, one, designed by one of the mathematician-instructors, included a discussion of an alternate method for performing vertical subtraction; the other, initiated by a 3rd grade teacher, is a discussion of an authentic classroom activity. Through these analyses, and drawing on the notion of boundary as sociocultural differences between communities, I reveal some mechanisms of the perspective-sharing that took place, and discuss what and how the parties learned from and with each other. I also highlight the role of a participant-observer researcher as a broker in this process, supporting events of boundary-crossing in which the parties came to explicate, and sometimes change, their own perspectives on teaching and learning mathematics with respect to the perspectives of others. I propose this PD setting as a model for sharing perspectives across these communities.

Reform committees typically comprise members from diverse communities of practice. Research has shown that achieving productive cross-community collaboration in mathematics education is far from straightforward. The work of reform committees is typically confidential, yet circumstantial evidence suggests that cross-community interactions are less productive than they could be. Due to the crucial influence of such committees on mathematics education at the national level, we suggest that cross-community interactions in reform committees should be an explicit topic of research. We propose boundary-crossing as a framework, and apply it to analyze cases of collaboration, including a setting that simulates the work of committees-a mathematics education forum where mathematicians, educational researchers and teachers (including past and present members of reform committees) meet to discuss issues of common interest. We identify the crucial role played by brokers in facilitating cross-boundary learning, and propose that suitable committee members should be designated as brokers. We call on the community to intentionally study and teach the role of broker in this context.

In this paper we investigate task design in an unusual professional development course for elementary school teachers, conceived and taught by research mathematicians. Prior analysis singled out relevance for teaching as a critical design issue for engaging teachers in effective learning. The aim of the current research is to uncover how relevance for teaching was achieved without compromising mathematical rigor and depth. Findings are based on an analysis of three representative cases of task designing in which the authors where involved – one as instructor and task designer, the other as participant observer. Our analysis reveals a designing model that first addresses purely mathematical concerns and then refines tasks, taking into consideration a series of constraints imposed by the requirement of relevance for teaching. Using Schoenfeld’s Resources-Orientations-Goals framework for decision-making, we show how the mathematicians drew on their special knowledge of mathematical content to achieve such relevance in ingenious ways. We find that tasks were best aligned with Knowles’s principles of Adult Learning in cases where the designers appropriated the teachers’ point of view, no longer seeing the need for relevance as a constraining imposition, but rather as an opportunity to combine and merge knowledge specific for teaching and purely mathematical knowledge.

University mathematicians and elementary school teachers have their own particular perspectives on whole number arithmetic. The study reported herein investigates how these different perspectives may interact productively in a professional development scenario, observed by a mathematics-education researcher, to achieve new insights. A discursive analysis of a professional development lesson on Division with Remainder, taught by a university mathematician, shows how the differences between the parties' perspectives led to the growth of new insights on the topic, its role in the curriculum, and its teaching. This meeting of communities is offered as a model for achieving rich expertise - an emergent discourse that is more than the sum of its parts.

In this paper we analyze teachers' beliefs about the knowledge needed for teaching elementary school mathematics. Eliciting such beliefs is important for designing and evaluating teacher education. We find indications of these beliefs in anonymous feedback questionnaires the teachers submitted in an in-service professional development course. This indirect approach avoids discrepancies between teachers' declared (conscious) beliefs and tacit beliefs that actually influence their learning. We found that beliefs changed during the course, at first favouring pedagogical content knowledge (PCK) as a learning goal, and shifting toward subject matter content knowledge (SMCK). The significance of this research is not only its findings but also its method, which avoids some issues inherent in traditional methods.