Visualisation and mathematical giftedness

Presmeg, Norma

doi:10.1007/bf00305075

Cited by 176 publications

(121 citation statements)

References 4 publications

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“…It might appear that the research on visualization (Krutetskii, 1969(Krutetskii, , 1976Presmeg, 1986Presmeg, , 1992Presmeg, , 2006) is related to Lana and Jesse's ways of thinking about graphs. For instance, one could consider Jesse's thinking about graphs to be related to Krutetskii's visual reasoning construct, and Lana's thinking about graphs to be related to Krutetskii's analytical reasoning construct.…”

Section: Relationship Of Results To Literature On Visualizationmentioning

confidence: 99%

“…For instance, one could consider Jesse's thinking about graphs to be related to Krutetskii's visual reasoning construct, and Lana's thinking about graphs to be related to Krutetskii's analytical reasoning construct. One could also view Lana's thinking as related to Presmeg's construct of prototypes (Presmeg, 1986(Presmeg, , 1992 because Lana used shapes to help her associate a face (the graph) with a name (the equation). On the other hand, the connection becomes less clear when we note that Jesse expressed his visual reasoning in symbolic notation, and Lana's interpretations of her symbolic formulations were in terms of associated shapes.…”

Section: Relationship Of Results To Literature On Visualizationmentioning

confidence: 99%

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Students’ images of two-variable functions and their graphs

Weber

Thompson

2014

Educ Stud Math

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This paper presents a conceptual analysis for students' images of graphs and their extension to graphs of two-variable functions. We use the conceptual analysis, based on quantitative and covariational reasoning, to construct a hypothetical learning trajectory for how students might generalize their understanding of graphs of one variable functions to graphs of two variable functions. To evaluate the viability of this learning trajectory, we use data from two teaching experiments based on tasks intended to support development of the schemes in the HLT. We focus on the schemes that two students developed in these teaching experiments and discuss their relationship to the original HLT. We close by considering the role of covariational reasoning in generalization, consider other ways in which students might come to conceptualize graphs of two-variable functions, and discuss implications for instruction.

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Section: Relationship Of Results To Literature On Visualizationmentioning

confidence: 99%

Section: Relationship Of Results To Literature On Visualizationmentioning

confidence: 99%

Students’ images of two-variable functions and their graphs

Weber

Thompson

2014

Educ Stud Math

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show abstract

“…Out of over a thousand students, the most gifted nine were classified with five analytic (verbal logical), one geometric (visual-pictorial), two combining both (one more visual, the other more verbal) and one who was not classified. Presmeg (1986) found that the most outstanding senior school mathematics students in her study (7 pupils out of 277) were almost always nonvisualizers. Of 27 'very good' students (10% of the sample), eighteen were nonvisualizers and five were visualizers.…”

Section: The Transition From Proof In Embodiment and Symbolism To Formentioning

confidence: 78%

Making Sense of Mathematical Reasoning and Proof

Tall

2014

Mathematics &Amp; Mathematics Education: Searching for Common Ground

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The sensori-motor language of mathematicsThe cognitive development of mathematical thinking and proof is based on fundamental human aspects that we all share: human perception, action and the use of language and symbolism that enables us to develop increasingly sophisticated thinkable concepts within increasingly sophisticated knowledge structures. It is based on what I term the sensori-motor language of mathematics, (Tall, forthcoming).Mathematical thinking develops in the child as perceptions are recognised and described using language and as actions become coherent operations to achieve a specific mathematical purpose. According to Bruner (1966), these may be communicated first through enactive gestures, then iconic images, then the use of symbolism, including not only written and spoken language but also the operational symbolism of arithmetic and the axiomatic formal symbolism of logical deduction.The theoretical framework proposed here follows a similar path enriched by the experience over time, building from conceptual embodiment that combines the enactive and iconic modes of human perception and action, developing into the mental world of perceptual and mental thought experiment. Embodied operations, such as counting, adding, sharing, are symbolised as manipulable concepts in arithmetic and algebra in a second mental world of operational symbolism. As the child matures, there is a further shift into a focus on the properties of mental objects as in Euclidean geometry, or the properties of arithmetic operations that are recast as 'rules' that underlie the generalized operations and expressions in algebra. Each of these leads to different forms of mathematical proof: Euclidean proof in geometry and symbolic proof, based on the 'rules of arithmetic' in arithmetic and algebra.

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“…Presmeg (1992) deems that the mode of representation, namely the visual-spatial mode, plays a key role in solving problems, because such presentation "abandons specific details and sketches out a pure relationship." [7] Paul Vickers&Joe Faith used the tools of category theory to provide a general framework for understanding visualization in practice [8] , Owen G. Scott have found greater demand for the professional use of visualization in the workplace, and the evaluation frameworks for effective visualization [9] . Delinda Van Garderen emphasized that using Visual-spatial representation can enhance the abilities of mathematical problem solving [10] .…”

Section: Literature Review On Visualizationmentioning

confidence: 99%

Literature Review of Applying Visual Method to Understand Mathematics

Yu¹

2015

MATEC Web of Conferences

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Visualisation and mathematical giftedness

Cited by 176 publications

References 4 publications

Students’ images of two-variable functions and their graphs

Students’ images of two-variable functions and their graphs

Making Sense of Mathematical Reasoning and Proof

Literature Review of Applying Visual Method to Understand Mathematics

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