Computing Square Roots by Hand
After writing a post about Multiplication, I was going to go off and write a post about the method I’d learned in middle school for computing a square root by hand. It looked a lot like long division, except the digits were considered in pairs, and to be honest, it’s been so long since I used it that I forgot all the details until I looked it up. And that’s when I realized I didn’t have to write it up at all, since Maria of the Homeschool Math Blog had already written it up wonderfully on her main site: both how to do it and why it works (which I was never taught when I originally learned the algorithm).
The how-to post is followed by a great discussion of the relative merits of the long-division-like algorithm versus using the Babylonian method (of iterative approximations) which a commenter presents. You should definitely go read it!
But does computing square roots by hand have any relevance today?
When I was in middle school, you couldn’t buy a calculator at Walmart for $1 that would compute square roots for you. It was useful to actually be able to compute a square root by hand on occasion (though the main instance I remember using it for was to compute the square root of my student ID number, to appease the seniors who would ask for it during freshman initiation at my high school).
But today when you can buy a calculator for $1 that does the four basic operations plus square roots, is there any point in knowing how to do it by hand? Well, most (but by no means all) people still think it’s useful to learn to add, subtract, multiply and divide by hand, even when calculators are readily available that can do all this for you. But square roots?
Maria argues that:
…studying and practicing these algorithms, just like the long division algorithm, will give your child exercise in simple mental math calculations. It is important to let children get lots of mental addition, subtraction, multiplication, and division exercise. Why? Because that helps them to get familiar with numbers and to develop a ‘number sense’.
Now I’m all for number sense. It seems to be sorely lacking in many (most) students today, at all levels. I’m happy to blame most of that on the overuse of calculators in the classroom. Others feel that it is due to the use of “Reform” math curricula such as Everyday Math, that aim to supplement competence with comprehension, but in many cases (particularly when poorly taught) seem to replace competence with, well, not much of anything.
But even if we accept that number sense is important, and critically lacking in so many students, is there something to be gained by teaching an obscure square root formula, in particular? I agree that carrying out the algorithm provides practice in basic operations. But would more practice on those operations be just as good?
Then again, does it do any harm to practice the basic operations in a way that puts another tool in a student’s toolbox?
I find that I don’t have strong feelings on this. The algorithm is highly forgettable, in my opinion, so I’m not convinced that learning it really adds a tool to a student’s toolbox. And why it works is beyond the comprehension of most kids at the upper-elementary and middle school levels, where it seems most likely to be taught. But at an age (maybe 5th or 6th grade) when many students find square roots interesting (dare I say “sexy”) it may be a good motivator for practice that doesn’t seem so mundane.
What do you think? And do you think that the successive approximations method, which is much easier to understand, is a better choice for students, despite being arguably messier to carry out by hand?