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Dawkins, Paul Christian – For the Learning of Mathematics, 2019

This paper sets forth a construct that describes how many undergraduate students understand mathematical terms to refer to mathematical objects, namely that they only refer to those objects that satisfy the term. I call this students' pronominal sense of reference (PSR) because it means they treat terms as pronouns that point to objects, like…

Descriptors: Mathematics Instruction, Calculus, College Mathematics, Undergraduate Students

Brown, Stacy – For the Learning of Mathematics, 2019

Recognizing identity not only as an important educational outcome but also as being inter-related to students' knowledge and practice, this paper explores an affordance of proof scripts; exploring students' identities. Specifically, drawing on data from teaching experiments and the construct of perceptual ambiguity, this paper presents an analysis…

Descriptors: Mathematical Logic, Validity, Mathematics Instruction, College Mathematics

Staats, Susan – For the Learning of Mathematics, 2018

A poetic structure occurs when a speaker's comment repeats some of the syntax and words of a previous comment. During a collaborative algebra task, a student explained a property five times over a few minutes, in slightly different ways. He consistently used poetic structures that were marked elaborately through discursive modes such as pause,…

Descriptors: Algebra, Mathematics Activities, Persuasive Discourse, Poetry

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

Shinno, Yusuke; Miyakawa, Takeshi; Iwasaki, Hideki; Kunimune, Susumu; Mizoguchi, Tatsuya; Ishii, Terumasa; Abe, Yoshitaka – For the Learning of Mathematics, 2018

The aims of the present study are two-fold. The first aim is to reveal the cultural and linguistic issues that need to be considered in the development of curricular content and sequencing for teaching mathematical proof in secondary schools in Japan. The second aim is to elaborate an epistemological perspective that may allow us to understand…

Descriptors: Mathematics Instruction, Foreign Countries, Cultural Influences, Language Usage

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

Whitacre, Ian; Bouhjar, Khalid; Bishop, Jessica Pierson; Philipp, Randolph; Schappelle, Bonnie P. – For the Learning of Mathematics, 2016

Rather than describing the challenges of integer learning in terms of a transition from positive to negative numbers, we have arrived at a different perspective: We view students as inhabiting distinct mathematical worlds consisting of particular types of numbers (as construed by the students). These worlds distinguish and illuminate students'…

Descriptors: Mathematics Instruction, Numbers, Number Concepts, Mathematical Logic

Tillema, Erik; Gatza, Andrew – For the Learning of Mathematics, 2016

We provide a conceptual analysis of how combinatorics problems have the potential to support students to establish non-linear meanings of multiplication (NLMM). The problems we analyze we have used in a series of studies with 6th, 8th, and 10th grade students. We situate the analysis in prior work on students' quantitative and multiplicative…

Descriptors: Mathematics Instruction, Multiplication, Mathematics Skills, Thinking Skills

Maciejewski, Wes; Barton, Bill – For the Learning of Mathematics, 2016

Originating from interviews with mathematics colleagues, written accounts of mathematicians engaging with mathematics, and Wes's reflections on his own mathematical work, we describe a process that we call mathematical foresight: the imagining of a resolution to a mathematical situation and a path to that resolution. In a sense, mathematical…

Descriptors: Mathematics Education, Mathematical Logic, Problem Solving, Imagination

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

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

Komatsu, Kotaro; Tsujiyama, Yosuke; Sakamaki, Aruta; Koike, Norio – For the Learning of Mathematics, 2014

It has become gradually accepted that proof and proving are essential at all grades of mathematical learning. Among the various aspects of proof and proving, this study addresses proofs and refutations described by Lakatos, in particular a part of increasing content by deductive guessing, to introduce an authentic process into mathematics…

Descriptors: Mathematics Instruction, Validity, Mathematical Logic, Guessing (Tests)

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