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.