# Slide rules and calculators

Several of my math teacher colleagues are of the opinion that calculators have destroyed math sense. I am not convinced that this is directly true: Calculators are a tool, nothing more.

A few months ago, I saw a video by the mythically amazing Vi Hart which led me to an epiphany: Perhaps the problem isn’t the birth of the calculator, but rather the death of the slide rule. Video killed the radio star (oh-a-oh-a), and Hewlett-Packard killed the slide rule.

In the video, Vi talks about how multiplication can be put on the same sort of number line that addition is on, but with the gaps representing consistently increasing values. To a younger generation, this might seem like an epiphany once it’s understood: But it’s precisely how a slide rule works.

At its heart, a slide rule consists of two logarithmic scales. Let’s say you want to multiply 25 by 32. Slide the top bar along until the marker at 1 is aligned with the marker at 2.5 on the bottom bar. Then look at the marker at 3.2, which will be aligned with 8 on the bottom bar. $$2.5 \times 3.2 = 8.00$$, and so $$25 \times 32 = 800$$. (Image courtesy of Wikipedia)

The slide rule reinforces several important mathematical concepts. First, it relates multiplication/division to addition/subtraction the same way that Vi Hart’s video does.

In modern education, the standard number line hammers away at the consistent distribution of numbers, but it doesn’t allow for the notion that the number line is not the only useful way of scaling values. Polar coordinates are sometimes introduced in trigonometry, but this seems to be fading. So student get a monochromatic way of seeing numbers: In 1D, there’s the number line; in 2D, there’s the Cartesian plane; in 3D, if they get so far, there’s simply a third coordinate.

Slide rules force the perspective that there are different ways of distributing numeric values in a meaningful way. Indeed, exponential distributions are very useful in a variety of scientific fields (including engineering and banking), and a poor understanding of them thus leads to problems.

Also, slide rules force an understanding of place value and scientific notation. It’s impossible to directly multiply 25 and 32 on a slide rule: You must multiply 2.5 by 3.2 and then adjust the decimal point appropriately. To effectively use a slide rule, you need to keep in mind how you’re adjusting the decimal point to set up a problem, as well as how that decimal point is affected by multiplication and division.

I am not saying that people who don’t learn slide rules can’t get the same understanding. I was not trained on a slide rule: I was among the first to have a calculator freely available to me as a high school student. However, all of my math teachers would have been trained on a slide rule, since these were standard tools until the 1970s.

I am also not saying that the slide rule is better than the calculator. Pragmatically speaking, the calculator is far better. $$4132 \times 5121$$ on a slide rule is going to give an approximation, and will result in some amount of squinting and cursing for the “best approximation”. It is no trouble at all on a calculator. There’s a reason why a centuries-old technology died out within a few decades of the introduction of a Better Mouse Trap.

What I am saying is that the slide rule taught certain useful number sense concepts that seem to be lacking in a lot of modern students. From that standpoint, it’s not that the calculator is directly responsible for the loss of these skills, but rather that the loss of the slide rule created an educational void that hasn’t yet been effectively filled.