The terms ‘concept image’ and ’concept definition’ were formulated in 1980 by Shlomo Vinner. When he visited me in Warwick that year, I had a huge quantity of data gleaned from undergraduate mathematicians that I could not analyse from a mathematical viewpoint. Shlomo’s idea immediately clarified the issues and resulted in the joint paper on Concept Image and Concept Definition published in 1981. Here I should declare that this has led to two different meanings given to ‘concept image’ in the literature. Shlomo’s definition was philosophically based and was a thought experiment to analyse what happens when students focus in different ways on images and definitions. My perception was more humanly based, so that where Shlomo talked about ‘the mind’ and thought about it as separate from ‘the brain’ in a cartesian sense, I always thought of the mind as the way the brain works, so that it is an indivisible part of the structure of the brain. Shlomo has always written about‘concept image’ and ‘concept definition’ as being ‘two distinct cells’ which enables him to make subtle analyses of different ways of employing the two distinct ideas. As the concept definition is a form of words that can be written or spoken, I regard this as part and parcel of the total concept image in the mind/brain. It is up to you to choose which version you want. It is easy to accommodate both. However, when I say ‘concept image’ I refer to the definition given in Tall and Vinner 1981.

When I think about concept image, I think of the conundrum raised by the composer Mendelssohn, in response to a comment that music is too vague to be represented by musical notation. He replied, on the contrary, that music is too precise ever to be captured by notation. Speaking of concept image can sometimes be vague, but as Pat Thompson once told me, this is precisely what makes it so useful. It helps us to grasp that there are subtleties in mathematical thinking that cannot be precisely conveyed by the apparent precision of mathematics.

Selected papers involving Concept Image and Concept Definition

1981a (with Shlomo Vinner) Concept image and concept definition in mathematics, with special reference to limits and continuity, Educational Studies in Mathematics, 12 151-169.
1988e Concept Image and Concept Definition, Senior Secondary Mathematics Education, (ed. Jan de Lange, Michiel Doorman), OW&OC Utrecht, 37-41.
2000b (with Eddie Gray, Demetra Pitta), Objects, Actions And Images: A Perspective On Early Number Development. Journal of Mathematical Behavior,18, 4, 1–13.
1989e Concept Images, Generic Organizers, Computers & Curriculum Change, For the Learning of Mathematics, 9,3 37-42.
1993e (with Lan Li), Constructing Different Concept Images of Sequences and Limits by Programming, Proceedings of PME 17, Japan, 2, 41-48.
1998a (with Liz Bills), Operable Definitions in Advanced Mathematics: The case of the Least Upper Bound, Proceedings of PME 22, Stellenbosch, South Africa, 2, 104–111.

last modified: Tuesday, February 11, 2003