6 1 A Taste of Things to Come 1.2 Switches and flows: some questions for cognitive psychologists I could not fail to disagree with you less. Anonym One naive notion of function—the one which can eventually be con- ceptualized as analytic function—describes a function as a depen- dency between two quantities which can be expressed by a formula. Historically, this understanding of functional dependence led to the development of the concept of analytic function.2 However, computations with y = |x| require use of two different formulae for y 0 and y 0 if you think a few seconds about how you manipulate functions like y = ||||x| 1| 1| 1|, you will see that this is not a symbolic manipulation of the kind we do with analytic expressions, but something quite the opposite, very discrete and consisting almost entirely of flipping, as in bipolar switches LEFT—RIGHT, UP—DOWN. The difference between the “switch” and “flow” modes of computation is felt and recognized by almost every mathemati- cian. We humans are apparently quite good at flipping mental switches (when the number of switches is reasonably small the limitations are possibly of the same nature as in the subitizing/counting threshold see Section 4.1). Graphic manipula- tion with compound functions built from y = |x| is so efficient because they appear to engage some small but efficient switchboards in our brains. In contrast, most procedures of formula-based undergrad- uate calculus obviously follow some smooth, “choiceless” pattern. I would make a wild guess that the choiceless, rearrangement-of- formulae routines of elementary calculus and algebra invoke a type of brain activity which is ruled by rhythm and flow, as in music or reciting of chants. I am more confident in suggesting that, in any case, it is very different from the switch-flipping of discrete math- ematics. I base this milder conjecture on the anecdotal evidence that problems which require one to combine the two activities (for example, where the calculations should follow different routes de- pending on whether the discriminant = b2 4ac of a quadratic equation ax2 + bx + c = 0 is positive or negative) cause substan- tial trouble to beginning learners of mathematics, especially if they have not been warned in advance about the hard choices they will face.
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