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Geometric Cognition

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Interdisciplinary Perspectives on Math Cognition

Part of the book series: Mathematics in Mind ((MATHMIN))

Abstract

Geometric Cognition, or the more inclusive Spatial/Visual Reasoning in mathematics and its applications are often overlooked in a focus on mathematical cognition in algebra, calculation and logic. Both school curriculum and pedagogy, and discussions of the cognitive work of the mathematical mind can miss the central role of spatial/visual reasoning in supporting learning and applications of mathematics. Gaps in spatial visual reasoning can develop during schooling which does not build 3-D spatial reasoning and become a barrier to many post-secondary areas of work and study. This chapter builds on reflections of decades as an active applied geometer and instructor of future teachers. It also draws on both historical examples and recent studies to document the central contribution of geometric cognition to developing the mathematical mind.

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References

  • Boaler, J. Chen L., Williams, C., Cordero, M. (2016). Seeing as understanding: The importance of visual mathematics for our brain and learning. Journal of Applied Computational Mathematics 5: 325.

    Article  Google Scholar 

  • Brosterman, N. (1997). Inventing kindergarten. New York: Harry N. Abrams.

    Google Scholar 

  • Burke, H., Gardony, A., Hutton, A., and Taylor, H.. Thinking 3d! Improving mathematics learning through embodied spatial training; Cognitive Research: Principles and Implications (2017), 2: 1-18.

    Google Scholar 

  • Burton, L. (2004). Mathematicians as enquirers: Learning about learning mathematics. New York: Springer.

    Book  Google Scholar 

  • Cánovas, C. P. and Monzanares, J. V. (2014). Conceptual mappings and neural reuse. Frontiers in Human Neuroscience 29 April 2014 doi: https://doi.org/10.3389/fnhum.2014.00261

  • Copelewicz, J. (2019). The brain maps out ideas and memories like spaces. Quanta Magazine. https://www.quantamagazine.org/the-brain-maps-out-ideas-and-memories-like-spaces-20190114/

  • Davis, B. et al (2015). Spatial reasoning in the early years: Principles, assertions, and speculations. New York: Routledge.

    Google Scholar 

  • Edwards, B. (1999). Drawing on the right side of the brain. New York: Tarcher.

    Google Scholar 

  • Engage (2019). Spatial reasoning for engineering (accessed January 2019) https://www.engageengineering.org/spatial/whyitworks/

  • Fauconnier, G. and Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. New York: Basic Books.

    Google Scholar 

  • Gardner, H. (1985). Frames of mind: The theory of multiple intelligences. New York: Basic Books.

    Google Scholar 

  • Gardner, H. (2006). Multiple intelligences: New horizons in theory and practice. New York: Basic Books.

    Google Scholar 

  • Geometry (1967). Geometry K-13 OISE Report 1967 www.math.yorku.ca/~whiteley/geometry.pdf Posted with permission of the Ontario Institute for Studies in Education of the University of Toronto.

  • George, W. (2017). Bringing van Hiele and Piaget together: A case for topology in early mathematics learning. Journal of Humanistic Mathematics 7: 105-116.

    Article  Google Scholar 

  • Goodings, D. C. (2006). From phenomenology to field theory: Faraday’s visual reasoning. Perspectives on Science 14: 40-65.

    Article  Google Scholar 

  • Grandin, T. (2006). Thinking with pictures: My life as an autistic. London: Bloomsbury.

    Google Scholar 

  • Hadamard, J. (1945). The psychology of invention in the mathematical field. New York: Dover.

    MATH  Google Scholar 

  • Halmos, P. (1955). Review of Polya. Bulletin of the American Mathematical Society 61: 243-245.

    Article  MathSciNet  Google Scholar 

  • Henderson, D. and Taiminia, D. (2004). Experiencing geometry: Euclidean and non-Euclidean geometry with history. Boston: Pearson.

    Google Scholar 

  • Hoffman, D. (2000). Visual intelligence: How we create what we see. New York: W. W. Norton and Co.

    Google Scholar 

  • Kosslyn, S., Thompson, W., Ganis, G. et al (2006). The case for mental imagery. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Klein, F. (1924) Elementary mathematics from an advanced standpoint: Geometry. New York: Dover.

    Google Scholar 

  • Klein’s Erlanger Program (1872) https://en.wikipedia.org/wiki/Erlangen_program.

  • Mamalo, A., Ruttenburg-Rozen, R., Whiteley, W. (2015). Developing a network of and for geometric reasoning. ZDM: International Journal on Mathematics Education 47: 483-496.

    Article  Google Scholar 

  • Mamalo, A. and Whiteley, W. (2012). The popcorn box activity and reasoning about optimization. Mathematics Teacher 105(6): 420-426.

    Article  Google Scholar 

  • Mann, R. (2005) Gifted students with spatial strengths and sequential weaknesses: An overlooked and under-identified population. Roeper Review 2005.

    Google Scholar 

  • Mason, J. and Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics 15: 277-289.

    Article  Google Scholar 

  • Maxwell, J. Clerk (2003). The scientific papers of James Clerk Maxwell. New York: Dover.

    MATH  Google Scholar 

  • Mix, K. and Battista, M. (eds.) (2018). Visualizing mathematics; The role of spatial reasoning in mathematical thought. New York: Springer.

    Google Scholar 

  • Polya, G. (1954a). Mathematics and plausible reasoning. Volume I: Induction and analogy in mathematics. Princeton: Princeton University Press.

    MATH  Google Scholar 

  • Polya, G. (1954b). Mathematics and plausible reasoning. Volume II: Patterns of plausible inference. Princeton: Princeton University Press.

    MATH  Google Scholar 

  • Polya, G. (1956). On picture writing. American Mathematical Monthly 63: 687-697.

    Article  MathSciNet  Google Scholar 

  • Possin, K. L. (2010). Visual spatial cognition in neurodegenerative disease. Neurocase 16(6): 466-487.

    Article  Google Scholar 

  • Rauscher, F. H. et al (1997). Music training causes long-term enhancement of preschool children’s spatial-temporal reasoning. Neurology Research 19(1): 2-8.

    Article  Google Scholar 

  • Schulze, B. and Whiteley, W. (2018). Rigidity and scene analysis. In: C. Toth, J. Goodman and J. O’Rourke (eds.), Handbook of Discrete and Computational Geometry.

    Google Scholar 

  • SIGGRAPH (2002). White paper: Visual learning for science and engineering. http://education.siggraph.org/conferences/other/visual-learning

  • Silverman, L. K. (1995) Effective techniques for teaching highly gifted visual-spatial learners. https://eric.ed.gov/?id=ED418535 (with full text).

  • Sorby, S. (2019). Higher education services: Visualizing success https://www.higheredservices.org/ (accessed Jan 2019). See also the TedX talk: https://www.youtube.com/watch?v=cJZIhl28HFI

  • Tall, D. et al (2012). Cognitive development and proof. Proof and Proving in Mathematics Education, Michael de Villiers and Gila Hanna (eds.), April 2012.

    Google Scholar 

  • Turner, M. (2014). The origin of ideas: Blending, creativity, and the human spark. New York: Oxford University Press.

    Google Scholar 

  • Uttal, D. H. et al (2013) The malleability of spatial skills: A meta-analysis of training studies. Psychological Bulletin 139: 352-402.

    Article  Google Scholar 

  • West, T. (2009). In the mind’s eye: Creative visual thinkers, gifted people with dyslexia, and the rise of visual technologies. New York: Random House.

    Google Scholar 

  • Wainer, H. (2007). Graphic discovery: A trout in the milk and other visual adventures. Princeton: Princeton University Press.

    MATH  Google Scholar 

  • Wainer, H. (2000) Visual revelations: Graphical tales of fate and deception from Napoleon Bonaparte to Ross Perot. New York: Psychology Press.

    Google Scholar 

  • Whiteley, W. (1999). The decline and rise of geometry in 20th century North America. Proceedings of the 1999 CMESG Conference. www.math.yorku.ca/~whiteley/cmesg.pdf.

  • Whiteley, W. (2002). Teaching to see like a mathematician www.math.yorku.ca/~whiteley/Teaching_to_see.pdf.

  • Whiteley, W. (2005). Learning to see Like a Mathematician. In: G. Malcom (ed.), Multidisciplinary approaches to visual representation and interpretation, pp. 279-292. Oxford: Elsevier.

    Chapter  Google Scholar 

  • Whiteley, W. (2010). As geometry is lost—What connections are lost? What reasoning is lost? What students are lost? Does it matter? Plenary Talks PIMS Changing the Culture 2010. https://www.pims.math.ca/files/AsGeometryIsLost_0.pdf.

  • Whiteley, W. (2012). Mathematical modeling as conceptual blending: Exploring an example within mathematics education. In: M. Bockarova, M. Danesi, and R. Núñez (eds.), Cognitive science and interdisciplinary approaches to mathematical cognition. München: Lincom Europa.

    Google Scholar 

  • Whiteley, W. (2014). Seeing like a mathematician: There is a diversity of ways to support the active learning of mathematics, Paper and Presentation at Forum for Action: Effective Practices in Mathematics Education’ https://mathforum1314.wordpress.com

  • Whiteley, W. (2019). Big ideas in geometry, http://wiki.math.yorku.ca/index.php/Big_Ideas_Concepts_Procedures (accessed January 2019).

  • Whiteley, W. and Paksu, A. D. (2015). Reasoning with quadrilaterals using hands, eyes, and technology, Workshop, Ontario Association of Mathematics Educators Annual Conference, May 2015 https://www.researchgate.net/publication/322505303_Reasoning_with_Quadrilaterals_using_hands_eyes_and_technology.

  • Yaglom, I. M. (I–IV): Geometric transformations MAA New Mathematical Library Vol 8, 21, 24, 44; 1962, 1968, 1973, 2009.

    Google Scholar 

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Authors and Affiliations

  1. Mathemtics and Statistics, York University, Toronto, ON, Canada

    Walter Whiteley

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  1. Walter Whiteley
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Correspondence to Walter Whiteley .

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Editors and Affiliations

  1. University of Toronto, Cognitive Science Network, Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada

    Marcel Danesi

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Whiteley, W. (2019). Geometric Cognition. In: Danesi, M. (eds) Interdisciplinary Perspectives on Math Cognition. Mathematics in Mind. Springer, Cham. https://doi.org/10.1007/978-3-030-22537-7_13

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