ABSTRACT
The most central, recurrent tension in mathematics teaching is between the presumed conflicting goals of ‘understanding’ and teaching for understanding on the one hand, and fluency or automaticity of performance (and teaching for that) on the other. There are also entrenched views about which has to come first in a teaching context, or indeed whether they may be developed relatively independently of one another. Geoffrey Howson (1982, p. 21) summarises the views of sixteenth-century mathematics teacher Robert Recorde: “Understanding is vital therefore, but mastery of a technique may well precede understanding.” What are ‘rote’ methods (somehow always connected
with ‘memory’) and why are they always to be shunned? Can you be too fluent, can you have too much understanding? Can an
excess of one actually detract from the other? Understanding in mathematics is automatically assumed to be a universal good, and the more the better. Yet it is recognised in other arenas that too much understanding can inhibit action, as the complexity perceived can inhibit decisions. And to be too understanding can prevent suitable boundaries from being maintained.
