Understanding Square Roots vs. Computing Square Roots

Some time ago, I wrote a post about why it’s so hard to learn square roots. Several people left comments referencing the process of extracting square roots on paper, including someone this morning, who posted an entire description of Heron’s method for extracting square roots. I was rather surprised, because, when I wrote that post, I wasn’t thinking about the process of extracting square roots at all, I was thinking about how hard it is for my students to grasp what a square root means. The reality is, I learned to extract square roots on paper in 4th or 5th grade, thought it was really incredibly cool, did it a bunch of times, and then promptly forgot how. In the nearly 20 years since (during which I earned a master’s degree in math, so I really was using this sort of thing), I have not once found myself in a situation where I needed to extract a root by hand. If I’m dealing with roots, mostly I don’t care what the value of the root is, and if I do need an approximate value, I either estimate it (24 is just a bit less than 25, so sqrt(24) must be just a little smaller than 5) or use a calculator. I never find myself in a grocery store or in front of a class of students needing to know the exact value of a square root down to the decimal point on the fly.

At any rate, I think it’s important to distinguish between computation and understanding, because they are not the same thing (though sadly, tests and grading systems don’t distinguish between the two very well). Of course, the two are not totally unrelated. A good algorithm can lead to some insight into the underlying math (although this is only true if you understand enough fundamentals to be able to glean those insights) and someone who understands the math can usually use that knowledge to help remember (or reconstruct) the algorithm or to adjust the algorithm or skip steps in specific situations. But on the whole, computation and understanding are different things. A trained monkey (or at least a trained student, or a calculator) can learn to compute answers to arithmetic problems but have absolutely no idea what they are doing. And a mathematician (like me) can understand exactly what square roots are and how they work without having a clue how to compute one in the wild.

None of that gets at whether both understanding and computational skill are important, but I’ll leave that for another day.

P.S. I know for a fact that one of the people who left a comment understands this difference quite clearly, as she was my teacher in elementary school.  I remember her adamantly insisting that I demonstrate that I understood how fraction multiplication worked before she’d let me use the “trick” to do it on paper.

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