Iconicity in mathematical notation: Commutativity and symmetry

Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects – those which visually resemble in some way the concepts they represent – offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd’s hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance.

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Can Authority be Sustained while Balancing Accessibility and Formality?

HERMES - Journal of Language and Communication in Business ◽

10.7146/hjlcb.v27i52.25132 ◽

2017 ◽

Vol 27 (52) ◽

pp. 11 ◽

Cited By ~ 1

Author(s):

Nigar Hashimzade ◽

Georgina A. Myles ◽

Gareth D. Myles

Keyword(s):

Mathematical Concept ◽

Verbal Description ◽

Mathematical Notation ◽

Formal Definition ◽

Mathematical Symbols ◽

Formal Definitions ◽

Oxford Dictionary

Economics has developed into a quantitative discipline that makes extensive use of mathematical and statistical concepts. When writing a dictionary for economics undergraduates it has to be recognised that many users will not have sufficient training in mathematics to benefi t from formal definitions of mathematical and statistical concepts. In fact, it is more than likely that the user will want the dictionary to provide an accessible version of a definition that avoids mathematical notation. Providing a verbal description of a mathematical concept has the risk that the outcome is both verbose (compared to a definition using appropriate mathematical symbols) and imprecise. For the author of a dictionary this raises the question of how to resolve this conflict between accessibility and formal correctness. We use a range of examples from the Oxford Dictionary of Economics to illustrate this conflict and to assess the extent to which a non-formal definition can be viewed as authoritative.

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The role of gestural polysigns and gestural sequences in teaching mathematical concepts

Gesture ◽

10.1075/gest.00013.ove ◽

2018 ◽

Vol 17 (1) ◽

pp. 128-157 ◽

Cited By ~ 1

Author(s):

Alice Ovendale ◽

Heather Brookes ◽

Jean-Marc Colletta ◽

Zain Davis

Keyword(s):

Learning Process ◽

First Graders ◽

Mathematical Concept ◽

Analytical Tool ◽

Mathematical Concepts ◽

Conceptual Role ◽

Symbolic Processes ◽

Representational Gestures ◽

Mathematical Signs

Abstract In this paper, we examine the conceptual pedagogical value of representational gestures in the context of teaching halving to first graders. We use the concept of the ‘polysign’ as an analytical tool and introduce the notion of a ‘mathematics gesture sequence’ to assess the conceptual role gestures play in explicating mathematical concepts. In our study of four teachers each teaching a lesson on halving, they produced representational polysign gestures that provided multiple layers of information, and chained these gestures in mathematical gestural sequences to spatially represent the operation of halving. Their use of gestures and their ability to use gestures accurately to convey mathematical concepts varied. During the lesson, learners, whose teachers used few representational gestures or used gestures that were conceptually incongruent with the mathematical concept, expressed more confusion than learners whose teachers used conceptually appropriate gestures. While confusion can be a productive part of the learning process, our analysis shows that producing conceptually appropriate gestures may be important in mediating concepts and the transition from concrete and personal symbolic processes to institutional mathematical signs.

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Making mathematical meaning: From preconcepts to pseudoconcepts to concepts

Pythagoras ◽

10.4102/pythagoras.v0i63.104 ◽

2006 ◽

Vol 0 (63) ◽

Author(s):

Margot Berger

Keyword(s):

Mathematical Knowledge ◽

Concept Formation ◽

Mathematical Object ◽

Mathematical Concept ◽

The Body ◽

Mathematical Meaning ◽

University Level ◽

Mathematical Signs

I argue that Vygotsky’s theory of concept formation (1934/1986) is a powerful framework within which to explore how an individual at university level constructs a new mathematical concept. In particular I argue that this theory can be used to explain how idiosyncratic usages of mathematical signs by students (particularly when just introduced to a new mathematical object) get transformed into mathematically acceptable and personally meaningful usages. Related to this, I argue that this theory is able to bridge the divide between an individual’s mathematical knowledge and the body of socially sanctioned mathematical knowledge. I also demonstrate an application of the theory to an analysis of a student’s activities with a ‘new’ mathematical object.

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