Preservice teachers' knowledge of proof by mathematical induction

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. © Springer Science+Business Media B.V. 2007

Preservice teachers' knowledge of proof by mathematical induction

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. © Springer Science+Business Media B.V. 2007

Seeking research-grounded solutions to problems of practice: classroom-based interventions in mathematics education

Research on classroom-based interventions in mathematics education has two core aims: (a) to improve classroom practice by engineering ways to act upon problems of practice; and (b) to deepen theoretical understanding of classroom phenomena that relate to these problems. Although there are notable examples of classroom-based intervention studies in mathematics education research since at least the 1930s, the number of such studies is small and acutely disproportionate to the number of studies that have documented problems of classroom practice for which solutions are sorely needed. In this paper we first make a case for the importance of research on classroom-based interventions and identify three important features of this research, which we then use to review the papers in this special issue. We also consider the issue of 'scaling up' promising classroom-based interventions in mathematics education, and we discuss a major obstacle that most such interventions find on the way to scaling up. This obstacle relates to their long duration, which means that possible adoption of these interventions would require practitioners to do major reorganizations of the mathematics curricula they follow in order to accommodate the time demands of the interventions. We argue that it is important, and conjecture that it is possible, to design interventions of short duration in mathematics education to alleviate major problems of classroom practice. Such interventions would be more amenable to scaling up, for they would allow more control over confounding variables and would make more practicable their incorporation into existing curriculum structures. © 2013 FIZ Karlsruhe