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.
GOD & MATH: Thinking Christianly About Math Education 
Dang. I wish I’d had you as my math teacher growing up. Any chance you’ll come tutor Analie?
I have that same book in-hand and have read the quotes you posted in context. The comment about the use of calculators being just as helpful to a student’s understanding of the material as the use of the basic algorithms and routine tricks we teach our kids now a days was meant in a philosophical context. It is because we don’t teach our kids about the objects of math from anything other than a defaulted philosophical formalism (defaulted because it is never really considered from a curriculum or test making standpoint) that gives way to the idea that knowing the steps of a long division algorithm by memory is no different than punching the numbers and operations into a calculator. By this, i believe Shapiro and you have a point. How do we remedy this? Stop teaching brute factual procedures and replace instructional time with student-mentor question answering time and investigation. Our computational power is only going to get better, but the thing that computers cannot do very well or at all is reason. Someone please replace our 18th century math curriculum with a kind of interactive socratic reasoning.
Great points Jesse. I completely agree.