ABSTRACT
In this chapter, I offer a further contribution to a series of discussions about the nature and varieties of mathematical understanding which has taken place in the past (Backhouse, 1978; Buxton, 1978b; Byers & Herscovics, 1977; Skemp, 1976; Tall, 1978). By 1978 seven categories had been proposed, which I subsequently suggested (see Chapter 13) could be re-arranged into a table showing three kinds of understanding and two modes of mental activity. However, as I was aware at the time, my analysis of formal understanding was incomplete, since the words 'form' and 'formal' are used with two distinct meanings, of which I only dealt with the first. (i) There is 'form' as in 'formal proof.' This is the meaning used by Buxton (1978) and I have already suggested (see Chapter 13) that we distinguish this one by calling it 'logical understanding.' (ii) There is 'form' as used in statements such as, "This equation can be written in the form y = mx + c." This is the meaning used by Backhouse (1978), and is also that in the first part of the definition given by Byers and Herscovics (1977): "Formal understanding is the ability to connect mathematical symbolism and notation with relevant mathematical ideas ... '' I now suggest that we distinguish this meaning by calling it symbolic understanding.
