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Cook, John Paul; Dawkins, Paul; Reed, Zackery – For the Learning of Mathematics, 2021
In this paper we analyze common solutions that students often produce to isomorphic tasks involving proportional situations. We highlight some key distinctions across the tasks and between the different equations students write within each task to help elaborate the different interpretations of equivalence at play: numerical, transformational, and…
Descriptors: Equations (Mathematics), Mathematical Concepts, Measurement, Concept Formation
Melhuish, Kathleen; Czocher, Jennifer A. – For the Learning of Mathematics, 2020
Within a study of student reasoning in abstract algebra, we encountered the claim "division and multiplication are the same operation." What might prompt a student to make this claim? What kind of influence might believing it have on their mathematical development? We explored the philosophical roots of "sameness" claims to…
Descriptors: Mathematics Instruction, Elementary Secondary Education, Algebra, Multiplication
Komatsu, Kotaro; Fujita, Taro; Jones, Keith; Naoki, Sue – For the Learning of Mathematics, 2018
Kitcher's idea of 'explanatory unification', while originally proposed in the philosophy of science, may also be relevant to mathematics education, as a way of enhancing student thinking and achieving classroom activity that is closer to authentic mathematical practice. There is, however, no mathematics education research treating explanatory…
Descriptors: Mathematics Education, Grade 8, Mathematical Concepts, Thinking Skills
Cooper, Jason; Pinto, Alon – For the Learning of Mathematics, 2017
"The root of 18 is closer to 4 than it is to 5 because 18 is closer to 16 than it is to 25". Is this statement, voiced in an 8th grade class, valid? We suggest hypothetical arguments upon which this statement might be based, and analyze them from two complementary perspectives--epistemic and pedagogical--drawing on Toulmin's notion of…
Descriptors: Grade 8, Secondary School Mathematics, Inquiry, Mathematical Logic
Wasserman, Nicholas; Weber, Keith – For the Learning of Mathematics, 2017
In this article, we consider the potential influences of the study of proofs in advanced mathematics on secondary mathematics teaching. Thus far, the literature has highlighted the benefits of applying the conclusions of particular proofs to secondary content and of developing a more general sense of disciplinary practices in mathematics in…
Descriptors: Mathematics Instruction, Secondary School Mathematics, Mathematical Concepts, Teaching Methods
Weber, Keith; Mejia-Ramos, Juan Pablo – For the Learning of Mathematics, 2015
Conviction is a central construct in mathematics education research on justification and proof. In this paper, we claim that it is important to distinguish between absolute conviction and relative conviction. We argue that researchers in mathematics education frequently have not done so and this has lead to researchers making unwarranted claims…
Descriptors: Mathematics Education, Educational Research, Mathematical Concepts, Mathematical Logic
Hanna, Gila; Mason, John – For the Learning of Mathematics, 2014
This article discusses the concepts of "key ideas" and "memorability" and how they relate to the metric "width of a proof" put forward by the Fields medalist Timothy Gowers (2007) in a recent essay entitled "Mathematics, memory and mental arithmetic". The paper looks at the meaning of these concepts and…
Descriptors: Mathematics Education, Mathematical Logic, Mathematical Concepts, Teaching Methods
Dawkins, Paul Christian – For the Learning of Mathematics, 2014
This paper demonstrates how questions of "provability" can help students engaged in reinvention of mathematical theory to understand the axiomatic game. While proof demonstrates how conclusions follow from assumptions, "provability" characterizes the dual relation that assumptions are "justified" when they afford…
Descriptors: Mathematical Logic, Teaching Methods, College Mathematics, Mathematical Concepts
Davis, Brent – For the Learning of Mathematics, 2014
An argument is presented for including exponentiation as a "basic" of school mathematics. Key elements of the argument include the topic's increasing relevance in a rapidly changing world and its utility for supporting understandings of role of analogy in mathematical thought. The discussion is illustrated with an account of a…
Descriptors: Mathematics Instruction, Mathematical Concepts, Grade 8, Inquiry
Leron, Uri; Zaslavsky, Orit – For the Learning of Mathematics, 2013
We analyze the role of generic proofs in helping students access difficult proofs more easily and naturally. We present three examples of generic proving--an elementary one on numbers, a more advanced one on permutations, and yet more advanced one on groups--and consider the affordances and pitfalls of the method by reflecting on these examples. A…
Descriptors: Mathematics Instruction, Mathematical Logic, Validity, Teaching Methods
Zwicky, Jan – For the Learning of Mathematics, 2010
How are we to understand the power of certain literary metaphors? The author argues that the apprehension of good metaphors is importantly similar to the apprehension of fruitful mathematical analogies: both involve a structural realignment of vision. The author then explores consequences of this claim, drawing conceptually significant parallels…
Descriptors: Figurative Language, Mathematical Concepts, Mathematics Instruction, Mathematical Logic
Weber, Keith – For the Learning of Mathematics, 2010
Many mathematics educators have noted that mathematicians do not only read proofs to gain conviction but also to obtain insight. The goal of this article is to discuss what this insight is from mathematicians' perspective. Based on interviews with nine research-active mathematicians, two sources of insight are discussed. The first is reading a…
Descriptors: Mathematical Concepts, Mathematics Instruction, Mathematics Education, Mathematical Logic
Peer reviewed
Alcock, Lara; Simpson, Adrian – For the Learning of Mathematics, 2002
Explores differences between mathematical definitions and non-mathematical definitions by contrasting three approaches to mathematical reasoning. Examines the consequences of the use of those approaches in a first course in real analysis. Analyzes students' reasoning behavior during analysis and suggests reasons for students' difficulties with the…
Descriptors: Higher Education, Learning Problems, Mathematical Concepts, Mathematical Logic
Peer reviewed
Ball, Deborah Loewenberg – For the Learning of Mathematics, 1990
The role of methods courses to help prospective elementary teachers learn to teach mathematics was investigated. The influence of past experience on preservice teachers, their lack of subject knowledge, and the lack of time set aside for a methods course are discussed. (KR)
Descriptors: College Mathematics, Higher Education, Mathematical Concepts, Mathematical Logic
Peer reviewed
Steiner, Hans-Georg – For the Learning of Mathematics, 1988
Describes two kinds of elements in mathematics: Euclid's and Bourbaki's. Discusses some criticisms on the two concepts of elements from a philosophical, methodological, and didactical point of view. Suggests a complementarist view and several implications for mathematics education. (YP)
Descriptors: Epistemology, Mathematical Concepts, Mathematical Logic, Mathematicians