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.