Thinking About Mathematics: Calculators

For my philosophy of mathematics course I am reading the book Thinking About Mathematics, by Stewart Shapiro. It is an excellent read and I highly recommend it. I hope to give this book a fuller treatment on this blog sometime over the summer.

In the meantime, I just finished reading Shapiro’s chapter on Formalism and he makes a telling editorial comment on the use of calculators in math education. Below are a few excerpts from the chapter (emphasis added).

The various philosophies that go by the name of ‘formalism’ pursue a claim that the essence of mathematics is the manipulation of characters. A list of the allowed characters and allowed rules all but exhausts what there is to say about a given branch of mathematics. According to the formalist then, mathematics is not, or need not be, about anything, or anything beyond typographical characters and rules for manipulating them.

For better or worse, much elementary arithmetic is taught as a series of blind techniques, with little or no indication of what the techniques do, or why they work. How many schoolteachers could explain the rules for long division, let alone the algorithm for taking square roots, in terms other than the execution of a routine?

The advent of calculators may increase the tendency toward formalism. If there is a question of justifying or making sense of, the workings of the calculator, it is for an engineer (or a physicist), not a teacher or student of elementary mathematics. Is there a real need to assign ‘meaning’ to the button-pushing?

We hear (or used to hear) complaints that calculators ruin the younger generation’s ability to think, or at least their ability to do mathematics. It seems to me that if the basic algorithms and routines are taught by rote, with no attempt to explain what they do or why they work, then the children might as well use calculators.

Food for thought.