Steven Pinker -- How the Mind Works - Hampshire High Italian ...

Good Ideas 341objects other than the ones they were designed for. Ordinarily, lines andshapes are analyzed by imagery and other components of our spatialsense, and heaps of things are analyzed by our number faculty. But toaccomplish Mac Lane's ideal of disentangling the generic from theparochial (for example, disentangling the generic concept of quantityfrom the parochial concept of the number of rocks in a heap), peoplemight have to apply their sense of number to an entity that, at first, feelslike the wrong kind of subject matter. For example, people might have toanalyze a line in the sand not by the habitual imagery operations of continuousscanning and shifting, but by counting off imaginary segmentsfrom one end to the other.The second way to get to mathematical competence is similar to theway to get to Carnegie Hall: practice. Mathematical concepts come fromsnapping together old concepts in a useful new arrangement. But thoseold concepts are assemblies of still older concepts. Each subassemblyhangs together by the mental rivets called chunking and automaticity:with copious practice, concepts adhere into larger concepts, andsequences of steps are compiled into a single step. Just as bicycles areassembled out of frames and wheels, not tubes and spokes, and recipessay how to make sauces, not how to grasp spoons and open jars, mathematicsis learned by fitting together overlearned routines. Calculusteachers lament that students find the subject difficult not becausederivatives and integrals are abstruse concepts—they're just rate andaccumulation—but because you can't do calculus unless algebraic operationsare second nature, and most students enter the course withouthaving learned the algebra properly and need to concentrate every dropof mental energy on that. Mathematics is ruthlessly cumulative, all theway back to counting to ten.Evolutionary psychology has implications for pedagogy which are particularlyclear in the teaching of mathematics. American children areamong the worst performers in the industrialized world on tests of mathematicalachievement. They are not born dunces; the problem is that theeducational establishment is ignorant of evolution. The ascendant philosophyof mathematical education in the United States is constructivism,a mixture of Piaget's psychology with counterculture andpostmodernist ideology. Children must actively construct mathematicalknowledge for themselves in a social enterprise driven by disagreementsabout the meanings of concepts. The teacher provides the materials andthe social milieu but does not lecture or guide the discussion. Drill and