This study investigates how four prospective teachers interpret and use textbooks while learning to teach mathematics during university coursework and practicum teaching. Results indicate that prospective teachers had varied approaches to using textbooks ranging from adherence, elaboration, and creation. Factors influencing how they engaged with texts include their practicum classroom setting, access to resources, and their understanding of mathematics. Preservice teachers' attempts to modify textbook lessons raised pedagogical, curricular, and mathematical questions for them that were not easily answered by reference to the textbooks or teacher's guides. Findings indicate that the practicum can, however, challenge preservice teachers to be creative and flexible users of curriculum materials.

84Today we showed our elementary preservice teachers a video where an interviewer was asking young students what the answer to 5 ÷ 0 was. The students in the video consistently responded that the answer was 0, and when asked how they knew this, many responded: "my teacher taught me that." We asked our preservice teachers to imagine that these were students in their 5th grade class, and to discuss in their groups how they would respond to and challenge such students' thinking. The response was a complete silence for what seemed to be an eternity, until one preservice teacher sheepishly asked: "Do you mean to say that the answer is not 0?" (Instructor journal, Fall '93). Our interest in preservice teachers' understanding of division by 0 began over a decade ago while coteaching a mathematics methods course for elementary teacher candidates. The episode in the quote above relates what happened the day we came to class with the question of division by 0 as a context for exploring the challenges of listening and responding to students' mathematical ideas. We had planned to engage the class in thinking about how they might respond to children's mathematical ideas in ways that respect but still challenge and extend their reasoning.Preservice elementary school teachers' fragmented understanding of mathematics is widely documented in the research literature. Their understanding of division by 0 is no exception. This article reports on two teacher education tasks and experiences designed to challenge and extend preservice teachers' understanding of division by 0. These tasks asked preservice teachers to investigate division by 0 in the context of responding to students' erroneous mathematical ideas and were respectively structured so that the question was investigated through discussion with peers and through independent investigation. Results revealed that preservice teachers gained new mathematical (what the answer is and why it is so) and pedagogical (how they might explain it to students) insights through both experiences. However, the quality of these insights were related to the participants' disposition to justify their thinking and (or) to investigate mathematics they did not understand. The study's results highlight the value of using teacher learning tasks that situate mathematical inquiry in teaching practice but also highlight the challenge for teacher educators to design experiences that help preservice teachers see the importance of, and develop the tools and inclination for, mathematical inquiry that is needed for teaching mathematics with understanding.As the journal excerpt suggests, this particular class did not turn out at all as we had expected.We had imagined that the preservice teachers would be challenged to think of ways of explaining to young students why the answer to dividing by 0 is not 0 using mathematical arguments rather than appeal to authority. We thought they could find it difficult to explain why division by 0 might be "undefined," "impossible," or "not allowed" without using language or ...

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This paper explores what happens when students engage with mathematical tasks that make no attempt to be connected with students' everyday life experiences. The investigation draws on the work of educators who call for a broader view of what might count as real and relevant contexts for studying mathematics. It investigates students' experiences with two imaginative tasks and reports on the students' intellectual and emotional engagement. This engagement is examined and described in terms of the character and quality of the class and group discussions generated. Findings suggest that students can indeed engage productively with mathematics when it is explored in imaginative settings and that such contexts can help students support and sustain their engagement with the mathematics in the task.

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