Square roots and logarithms without a calculator (Part 5)

I’m in the middle of a series of posts concerning the elementary operation of computing a square root. This is such an elementary operation because nearly every calculator has a \sqrt{~~} button, and so students today are accustomed to quickly getting an answer without giving much thought to (1) what the answer means or (2) what magic the calculator uses to find square roots. I like to show my future secondary teachers a brief history on this topic… partially to deepen their knowledge about what they likely think is a simple concept, but also to give them a little appreciation for their elders.

One way that square roots can be computed without a calculator is by using log tables. This was a common computational device before pocket scientific calculators were commonly affordable… say, the 1920s.

As many readers may be unfamiliar with this blast from the past, Parts 3 and 4 of this series discussed the mechanics of how to use a log table. In Part 6, I’ll discuss how square roots (and other operations) can be computed with using log tables.

In this post, I consider the modern pedagogical usefulness of log tables, even if logarithms can be computed more easily with scientific calculators.

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A personal story: In either 1981 or 1982, my parents bought me my first scientific calculator. It was a thing of beauty… maybe about 25% larger than today’s TI-83s, with an LED screen that tilted upward. When it calculated something like \log_{10} 4213, the screen would go blank for a couple of seconds as it struggled to calculate the answer. I’m surprised that smoke didn’t come out of both sides as it struggled. It must have cost my parents a small fortune, maybe over $1000 after adjusting for inflation. Naturally, being an irresponsible kid in the early 1980s, it didn’t last but a couple of years. (It’s a wonder that my parents didn’t kill me when I broke it.)

So I imagine that requiring all students to use log tables fell out of favor at some point during the 1980s, as technology improved and the prices of scientific calculators became more reasonable.

I regularly teach the use of log tables to senior math majors who aspire to become secondary math teachers. These students who have taken three semesters of calculus, linear algebra, and several courses emphasizing rigorous theorem proving. In other words, they’re no dummies. But when I show this blast from the past to them, they often find the use of a log table to be absolutely mystifying, even though it relies on principles — the laws of logarithms and the point-slope form of a line — that they think they’ve mastered.

So why do really smart students, who after all are math majors about to graduate from college, struggle with mastering log tables, a concept that was expected of 15- and 16-year-olds a generation ago? I personally think that a lot of their struggles come from the fact that they don’t really know logarithms in the way that students of previous generation had to know them in order to survive precalculus. For today’s students, a logarithm is computed so easily that, when my math majors were in high school, they were not expected to really think about its meaning.

For example, it’s no longer automatic for today’s math majors to realize that \log_{10} 4213 has to be between 3 and 4 someplace. They’ll just punch the numbers in the calculators to get an answer, and the process happens so quickly that the answer loses its meaning.

They know by heart that \log_{b} xy = \log_{b} x + \log_{b} y and that \log_b b^x = x. But it doesn’t reflexively occur to them that these laws can be used to rewrite \log_{10} 4213 as 3 + \log_{10} 4.213.

When encountering 10^{1.8123}, their first thought is to plug into a calculator to get the answer, not to reflect and realize that the answer, whatever it is, has to be between 10 and 100 someplace.

Today’s math majors can be taught these approximation principles, of course, but there’s unfortunately no reason to expect that they received the same training with logarithms that students received a generation ago. So none of this discussion should be considered as criticism of today’s math majors; it’s merely an observation about the training that they received as younger students versus the training that previous generations received.

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So, do I think that all students today should exclusively learn how to use log tables? Absolutely not.If college students who have received excellent mathematical training can be daunted by log tables, you can imagine how the high school students of generations past must have felt — especially the high school students who were not particularly predisposed to math in the first place.

People like me that made it through the math education system of the 1980s (and before) received great insight into the meaning of logarithms. However, a lot of students back then found these tables as mystifying as today’s college students, and perhaps they did not survive the system because they found the use of the table to be exceedingly complex. In other words, while they were necessary for an era that pre-dated pocket calculators, log tables (and trig tables) were an unfortunate conceptual roadblock to a lot of students who might have had a chance at majoring in a STEM field. By contrast, logarithms are found easily today so that the steps above are not a hindrance to today’s students.

That said, I do argue that there is pedagogical value (as well as historical value) in showing students how to use log tables, even though calculators can accomplish this task much quicker. In other words, I wouldn’t expect students to master the art of performing the above steps to compute logarithms on the homework assignments and exams. But if they can’t perform the above steps, then there’s room for their knowledge of logarithms to grow.

And it will hopefully give today’s students a little more respect for their elders.

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