The Development of Control of Language in Mathematical Activity

Abstract

My chief preoccupation is with the difficulties experienced in the teaching and learning of mathematics. Many of these difficulties seem to grow out of the presumption that truth is embodied in the exactness both of statements made in mathematics and in the consequences arising from such statements. We are accustomed to the use of forms recognisable as mathematical and to the beliefs which relate our responses to the appearance of such forms. For example, though we are surprised by ‘He saw 2 women walking towards him’, preferring ‘two’, we see immediately the significance of 5 + 2 = 7, or 3 × 2 = 6. It is the conviction with which we meet those forms in mathematics with which we are familiar that allows us potential access to rich analytically complex systems. I wish to emphasise that the truths of mathematics and hence also the difficulties of understanding, lie in the convictions or beliefs and not in the forms themselves. In the example given above I asserted that you would be ‘surprised’ to see the form ‘2’ in a sentence but ‘see immediately’ the significance of that same form ‘2’ in the expression ‘5+2=7’. But my explicit assertion about the form ‘2’ is implicitly concerned with assumptions I am making about your beliefs in connection with the usage of the form ‘2’ together with the two higher forms ‘sentence’ and ‘expression’. My assertion is hence not simple about the form ‘2’ although that is how it was expressed.

Reprinted by permission of the R.P.A.