# Preservice teachers’ knowledge of proof by mathematical induction

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## Abstract

There is a growing effort to make *proof* central to all students’ mathematical experiences across all grades. Success in this goal depends highly on teachers’ knowledge of proof, but limited research has examined this knowledge. This paper contributes to this domain of research by investigating preservice elementary and secondary school mathematics teachers’ knowledge of proof by mathematical induction. This research can inform the *knowledge about preservice teachers* that mathematics teacher educators need in order to effectively teach proof to preservice teachers. Our analysis is based on written responses of 95 participants to specially developed tasks and on semi-structured interviews with 11 of them. The findings show that preservice teachers from both groups have difficulties that center around: (1) the essence of the base step of the induction method; (2) the meaning associated with the inductive step in proving the implication *P*(*k*) ⇒ *P*(*k* + 1) for an arbitrary *k* in the domain of discourse of *P*(*n*); and (3) the possibility of the truth set of a sentence in a statement proved by mathematical induction to include values outside its domain of discourse. The difficulties about the base and inductive steps are more salient among preservice elementary than secondary school teachers, but the difficulties about whether proofs by induction should be as encompassing as they could be are equally important for both groups. Implications for mathematics teacher education and future research are discussed in light of these findings.

## Keywords

Proof Mathematical induction Content knowledge Knowledge fragility Mathematical reasoning Tasks Teacher education Collegiate mathematics## Notes

### Acknowledgements

This paper uses data that were collected by the first two authors for their senior thesis, which was conducted at the University of Cyprus under the supervision of the third author; the first two authors have contributed equally to this paper. The authors wish to thank Dina Tirosh and anonymous reviewers for useful comments on an earlier version of the paper.

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