Algorithms are Recipes

At 6:30 this morning, I grabbed paper and pencils, and Eric and I figured out why the algorithm for extracting square roots in Katie's math book works. Why did we choose to do math during our early morning time together? Well, finding square roots was yesterday's math lesson in A Beka Pre Algebra. I followed the directions, walked her through a couple of problems, and was ready to set her working on her own when she said, "But I don't understand it. I can't do it if I don't understand it."

Just the day before my mother told me she skipped the section on extracting square roots with my niece. I never did it myself until yesterday. Eric also never learned how in school but taught himself some time along the way using Newton's Method. A calculator makes fast, efficient business of finding square roots. Why should we bother?

An algorithm is basically a recipe for solving a math problem. If you follow the procedure correctly, the right answer will be the result. In traditional math education, like A Beka PreAlgebra, the recipe or algorithm is what matters and helping the student understand why rarely merits mention. A recipe for pancakes may warn against overbeating because it makes them tough but usually lacks an explanation. Knowing that they become tough because the gluten develops and creates more structure in the batter doesn't change the recipe. Following directions and practice make good food and good math, not an understanding of the process.

Do I believe that? Absolutely not. My favorite biblical proverb has always been "Wisdom is the principal thing; therefore get wisdom: and with all thy getting get understanding." I do think knowing how gluten functions in doughs and batters and how I can make changes to vary the results makes me a better cook, even if it is in an intangible way. In cooking I manipulate ingredients to vary the results; in math I vary the results through manipulating numbers. Michael Ruhlman's Ratio: The Simple Codes Behind the Craft of Everyday Cooking explains the value of knowing the mathematical ratios for doughs and batters, imparting the craft of cooking with an elegant simplicity that made the cookbook a pageturner for me. Deriving square roots without a calculator will make Katie and me magicians performing a number trick, but understanding the algorithm will raise it above that.

First we turned to our supplemental math curriculum, Math-U-See. Steve Demme on the DVD usually explains everything comprehensively, but he lacked an explanation for finding square roots, so we decided to figure it out ourselves in the early morning light. I extracted a square root, then squared it to examine the process. Eric compared it to the squares of binomials and trinomials. Somewhere along the way, I found a way that made it clear for me, he found one that did the same for him, and we compared notes.

A couple of hours later I explained it to Katie. "Oh! It's like multiplying backwards!" she said and then completed the lesson. I wish understanding everything about life could be as simple as math and cooking.