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Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving 总被引：1，自引：0，他引：1
This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in,
mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about
proof is likely to be important for teachers to have as they engage students in the activity of proving. We explain that existing
research informs the knowledge about the logico-linguistic aspects of proof that teachers might need, and we argue that this knowledge should be complemented by what we call knowledge of situations for proving. This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms.
We identify two sub-components of the knowledge of situations for proving: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity. In order to promote understanding of the former type of knowledge, we develop and illustrate a classification of proving
tasks based on two mathematical criteria: (1) the number of cases involved in a task (a single case, multiple but finitely
many cases, or infinitely many cases), and (2) the purpose of the task (to verify or to refute statements). In order to promote
understanding of the latter type of knowledge, we develop a framework for the relationship between different proving tasks
and anticipated proving activity when these tasks are implemented in classrooms, and we exemplify the components of the framework
using data from third grade. We also discuss possible directions for future research into teachers’ knowledge about proof.
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| Andreas J. StylianidesEmail: |
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Gabriel J. Stylianides Andreas J. Stylianides George N. Philippou 《Journal of Mathematics Teacher Education》2007,10(3):145-166
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.
相似文献
| George N. PhilippouEmail: |
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In this article we elaborate a conceptualisation of mathematics for teaching as a form of applied mathematics (using Bass's idea of characterising mathematics education as a form of applied mathematics) and we examine implications of this conceptualisation for the mathematical preparation of teachers. Specifically, we focus on issues of design and implementation of a special kind of mathematics tasks whose use in mathematics teacher education can support the development of knowledge of mathematics for teaching. Also, we discuss broader implications of the article for mathematics teacher education, including implications for mathematics teacher educators' knowledge for promoting mathematics for teaching. 相似文献
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J. Sandefur J. Mason G. J. Stylianides A. Watson 《Educational Studies in Mathematics》2013,83(3):323-340
We report on our analysis of data from a dataset of 26 videotapes of university students working in groups of 2 and 3 on different proving problems. Our aim is to understand the role of example generation in the proving process, focusing on deliberate changes in representation and symbol manipulation. We suggest and illustrate four aspects of situations in which example generation seems to play a positive role in proving. These aspects integrate qualities of students and of problems: experience of utility of examples in proving, personal example spaces and technical tools, formulation of the problem, and relational necessity. Our analysis led to integrating two theoretical ideas: the alignment of conceptual insight and technical handle when trying to prove; and manipulating, getting-a-sense-of, and articulating as phases of work associated with example construction. 相似文献
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Andreas J. Stylianides Gabriel J. Stylianides 《Educational Studies in Mathematics》2009,72(2):237-253
In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof,
and we examine their ability to construct proofs and evaluate their own constructions. We claim that the combined “construction–evaluation”
activity helps illuminate certain aspects of prospective teachers’ and presumably other individuals’ understanding of proof
that tend to defy scrutiny when individuals are asked to evaluate given arguments. For example, some prospective teachers
in our study provided empirical arguments to mathematical statements, while being aware that their constructions were invalid.
Thus, although these constructions considered alone could have been taken as evidence of an empirical conception of proof,
the additional consideration of prospective teachers’ evaluations of their own constructions overruled this interpretation
and suggested a good understanding of the distinction between proofs and empirical arguments. We offer a possible account
of our findings, and we discuss implications for research and instruction. 相似文献
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Street Karin E. S. Malmberg Lars-Erik Stylianides Gabriel J. 《Educational Studies in Mathematics》2022,111(3):515-541
Educational Studies in Mathematics - Self-efficacy in mathematics is related to engagement, persistence, and academic performance. Prior research focused mostly on examining changes to... 相似文献
9.
Mathematical tasks embedded in real-life contexts have received increased attention by educators, in part due to the considerable levels of student engagement often triggered by their motivational features. Nevertheless, it is often challenging for teachers to implement high-level (i.e., cognitively demanding), real-life tasks in ways that exploit their motivational features without overshadowing the mathematics involved. This paper proposes an analytic framework for describing and explaining the classroom implementation of different kinds of tasks, and uses this framework to analyse a classroom episode where a secondary teacher implemented with low fidelity a high-level, real-life mathematical task. Implications for research are discussed. 相似文献
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Gabriel J. Stylianides Andreas J. Stylianides Leah N. Shilling-Traina 《International Journal of Science and Mathematics Education》2013,11(6):1463-1490
The activity of reasoning-and-proving is at the heart of mathematical sense making and is important for all students’ learning as early as the elementary grades. Yet, reasoning-and-proving tends to have a marginal place in elementary school classrooms. This situation can be partly attributed to the fact that many (prospective) elementary teachers have (1) weak mathematical (subject matter) knowledge about reasoning-and-proving and (2) counterproductive beliefs about its teaching. Following up on an intervention study that helped a group of prospective elementary teachers make significant progress in overcoming these two major obstacles to teaching reasoning-and-proving, we examined the challenges that three of them identified that they faced as they planned and taught lessons related to reasoning-and-proving in their mentor teachers’ classrooms. Our findings contribute to research knowledge about major factors (other than the well-known factors related to teachers’ mathematical knowledge and beliefs) that deserve attention by teacher education programs in preparing prospective teachers to teach reasoning-and-proving. 相似文献