**The Way We Used To Multiply**

The old way to multiply required a student to add the products of 36 x 4 and 36 x 2. The trick is to add that 0 at the end of the second product.

### How Kids Learn To Multiply Now

These days, students add four products to get the answer.

You cannot become good at algebra without a mastery of arithmetic but arithmetic itself is no longer the ultimate goal. Thus the emphasis in teaching mathematics today is on getting people to be sophisticated, algebraic thinkers.

That doesn’t mean that kids can skip learning their multiplications tables. But the way it’s taught now is you get to the multiplication tables by understanding the number system and understanding what numbers mean.

If it sounds too complex for adults, let alone 7-year-olds. With all due respect, there’s nothing elementary about elementary mathematics education.

Consider, then, a teacher who tells her students what a “ratio” is, expecting them to remember the definition. Now imagine a teacher who has first graders figure out how many plastic links placed on one side of a balance are equivalent to one metal washer on the other side. Then, after discovering that the same number of links must be added again to balance an additional washer, the children come to make sense of the concept of ratio for themselves. Which approach do you suppose will lead to a deeper understanding?

To my mind, the most important feature of good teachers is that they put themselves in the student’s position. Rather than giving clear and accurate instructions and grading tests; the main objective is to help the students understand the material. Whether one is delivering a lesson or talking with one’s students, one has to remember that what seems perfectly obvious and transparent to one may be mysterious and opaque to those who have not encountered the ideas before. There is room for input from the teacher myself, and there is a delicate balance between helping the students by putting my own stamp on the material, and confusing them by introducing too many extraneous ideas or strategies.

To say that teaching from a constructivist perspective is characterized by a paradox – don’t give young children more than they can handle, but do give them a chance to show you what they can do – is to put this positively. The flip side is that the Old School manages to screw up on both counts, simultaneously failing to understand children’s developmental limitations (“Just drill ‘em until they get it”) and failing to appreciate their minds (“Use the technique I showed you”). This double fault reflects on educators who “overestimate children academically and underestimate them intellectually.” Unconventional teachers dedicate themselves to avoiding both traps.