## Abstract

Several researchers have documented prospective teachers’ (PTs’) conceptions of various mathematical topics. However, less is known about how PTs’ conceptions develop. To address this gap, I designed two tasks with the goals of addressing the PTs’ initial conceptions of multidigit whole numbers and helping them develop more sophisticated ones. I examined how PTs’ conceptions changed while working on these tasks in two settings (a teaching experiment with 6 PTs and a mathematics methods course with 33 PTs) and modified the tasks on the basis of the results. Consistent with prior findings, this study showed that PTs entered with limited conceptions. This study showed further that (a) well-designed tasks (addressing the PTs’ incoming conceptions as well as focusing on the desired conceptions) can help PTs develop content knowledge, (b) conceptual difficulties may persist even with well-designed tasks, and (c) artifacts of children’s mathematical thinking can be used to develop mathematical content knowledge. Instructional implications are discussed.

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## Notes

One of the 35 PTs in the class did not provide consent, and 1 did not plan to teach elementary school. Thus, I report on 33 PTs.

We considered the Mayan numbers as base-20 numbers. In the literature, the Mayan numeral

*dot shell shell*can be found as representing 18 × 20 rather than 20 × 20 as we would expect in a base-20 system. We disregarded this aspect in this lesson (Bennett and Nelson 2007).None of these 33 PTs had participated in the teaching experiment prior to their methods course.

In base ten, when one states, for example, “This is the 10’s column,” whether the reference is to a label or a quantity is unclear because in base-ten the labels and the quantities have the same names.

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## Acknowledgments

I would like to thank my mentor and advisor Randy Philipp for his continued interest in my work and for the walks during which we brainstorm ideas. I would also like to thank Bonnie Schappelle, who has also continuously supported my work by sharing my interest and giving me feedback. In addition, I would like to thank Signe Kastberg, who helped with the enactment of the teaching experiment, as well as Briana Mills, Krista Stand, and Jodi Fasteen, who work(ed) with me tirelessly to understand prospective teachers better.

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Thanheiser, E. Developing prospective teachers’ conceptions with well-designed tasks: explaining successes and analyzing conceptual difficulties.
*J Math Teacher Educ* **18, **141–172 (2015). https://doi.org/10.1007/s10857-014-9272-9

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DOI: https://doi.org/10.1007/s10857-014-9272-9

### Keywords

- Prospective teacher
- Teacher education
- Content knowledge
- Whole number
- Place value
- Task design