# Visualization and Learning in Mathematics Education

**DOI:**https://doi.org/10.1007/978-3-319-77487-9_161-4

## Definitions and Background

Visualization in mathematics learning is not new. Because mathematics involves the use of signs such as symbols and diagrams to represent abstract notions, there is a spatial aspect involved, that is, visualization is implicated in its representation. However, in contrast with the millennia in which mathematics has existed as a discipline, research on the use of visual thinking in learning mathematics is relatively new. Such research has been growing in volume and depth since the 1970s, initiated by Bishop (1973, 1980) and later Clements (1981, 1982), who investigated preferences of individual learners with regard to visualization in mathematics and how spatial abilities interacted with these preferences. Visualization has internal and external forms (Goldin 1992), which may be designated as visual mental imagery and inscriptions, respectively (Presmeg 2006). Presmeg defined a visual image as a mental sign depicting visual or spatial information and...

## Keywords

Signs Symbols Diagrams Spatial aspect Representation Preferences of individual learners Spatial abilities Visual mental imagery Inscriptions Visual image Ana-vis scale Logic Strength of mathematical processing Type Verbal-logical Visual-pictorial Analytic geometric and harmonic types Reluctance to visualize Pedagogy Abstraction Generalization One-case concreteness Prototype Uncontrollable image Compartmentalization Dynamic imagery Pattern imagery Metaphor Mnemonic advantages Interactive dynamic geometry software Gestures Conversion processes Registers Connections Idiosyncratic visual imagery Reification Computer technology Overarching theory of visualization## References

- Arcavi A (2003) The role of visual representations in the learning of mathematics. Educ Stud Math 52:215–241CrossRefGoogle Scholar
- Battista MT (2009) Highlights of research on learning school geometry. In: Craine TV (ed) Understanding geometry for a changing world. Seventy-first yearbook. National Council of Teachers of Mathematics, Reston, pp 91–108Google Scholar
- Bishop AJ (1973) The use of structural apparatus and spatial ability – a possible relationship. Res Educ 9:43–49CrossRefGoogle Scholar
- Bishop AJ (1980) Spatial abilities and mathematics education–a review. Educ Stud Math 11:257–269CrossRefGoogle Scholar
- Clements MA (1981) Visual imagery and school mathematics. Part 1. Learn Math 2(2):2–9Google Scholar
- Clements MA (1982) Visual imagery and school mathematics. Part 2. Learn Math 2(3):33–38Google Scholar
- Dreyfus T (1991) On the status of visual reasoning in mathematics and mathematics education. In: Furinghetti F (ed) Proceedings of the 15th conference of the International Group for the Psychology of Mathematics Education (PME), vol 1, pp 33–48Google Scholar
- Duval R (1999) Representation, vision, and visualization: cognitive functions in mathematical thinking. Basic issues for learning. In: Hitt F, Santos M (eds) Proceedings of the 21st north American PME conference, vol 1, pp 3–26Google Scholar
- Eisenberg T (1994) On understanding the reluctance to visualize. Zentralbl Didakt Math 26(4):109–113Google Scholar
- Goldin GA (1992) On the developing of a unified model for the psychology of mathematics learning and problem solving. In: Geeslin W, Graham K (eds) Proceedings of the 16th PME international conference, vol 3, pp 235–261Google Scholar
- Hershkowitz R, Ben-Chaim D, Hoyles C, Lappan G, Mitchelmore M, Vinner S (1989) Psychological aspects of learning geometry. In: Nesher P, Kilpatrick J (eds) Mathematics and cognition. Cambridge University Press, Cambridge, pp 70–95Google Scholar
- Krutetskii VA (1976) The psychology of mathematical abilities in schoolchildren. University of Chicago Press, ChicagoGoogle Scholar
- Nardi E, Jaworski B, Hegedus S (2005) A spectrum of pedagogical awareness for undergraduate mathematics: from “tricks” to “techniques”. J Res Math Educ 36(4):284–316Google Scholar
- Owens K (1999) The role of visualization in young students’ learning. In: Zaslavsky O (ed) Proceedings of the 23rd PME international conference, vol 1, pp 220–234Google Scholar
- Piaget J, Inhelder B (1971) Mental imagery and the child. Routledge and Kegan Paul, New YorkGoogle Scholar
- Presmeg NC (1985) The role of visually mediated processes in high school mathematics: a classroom investigation. Unpublished PhD dissertation, University of CambridgeGoogle Scholar
- Presmeg NC (1986) Visualization in high school mathematics. Learn Math 6(3):42–46Google Scholar
- Presmeg NC (1992) Prototypes, metaphors, metonymies, and imaginative rationality in high school mathematics. Educ Stud Math 23:595–610CrossRefGoogle Scholar
- Presmeg NC (2006) Research on visualization in learning and teaching mathematics: emergence from psychology. In: Gutierrez A, Boero P (eds) Handbook of research on the psychology of mathematics education. Sense, Rotterdam, pp 205–235Google Scholar
- Presmeg NC (2008) An overarching theory for research on visualization in mathematics education. In: Plenary paper, proceedings of Topic Study Group 20: visualization in the teaching and learning of mathematics. 11th International Congress on Mathematics Education (ICME-11), Monterrey, 6–13 July 2008. Published on the ICME-11 web site: http://tsg.icme11.org (TSG20)
- Suwarsono S (1982) Visual imagery in the mathematical thinking of seventh grade students. Unpublished PhD dissertation, Monash UniversityGoogle Scholar
- Yerushalmy M, Shternberg G, Gilead S (1999) Visualization as a vehicle for meaningful problem solving in algebra. In: Zaslavsky O (ed) Proceedings of the 23rd PME international conference, vol 1, pp 197–211Google Scholar
- Yu P, Barrett J, Presmeg N (2009) Prototypes and categorical reasoning: a perspective to explain how children learn about interactive geometry. In: Craine TV (ed) Understanding geometry for a changing world. Seventy-first yearbook. National Council of Teachers of Mathematics, Reston, pp 109–126Google Scholar
- Zimmermann W, Cunningham S (1991) Visualization in teaching and learning mathematics. Mathematical Association of America, Washington, DCGoogle Scholar