by C.J. Uberroth

___

In teaching high school mathematics for the last seven years, I’ve begun to realize that there are some issues in our field that seem ubiquitous. The first is the death of estimation. Students faced with a simple arithmetic problem will sooner jump to their phone for the answer than think about what answer they should obtain. This presents a serious problem. Since students are not thinking ahead about what the approximate answer should be when multiplying 2.5 by 8, they are just as likely to answer 200 as they are 20. All it requires is missing the decimal button, something that would immediately be corrected, if the student had already estimated the answer.

Secondly, I find myself longing for students who actually understand the core concepts of arithmetic and algebra. By using the calculator, the concepts that relate the different parts of mathematics fall by the wayside. How can students see the linear connection between second differences of a quadratic function or the subtle relationship between the places of numbers and how they interact under multiplication? The logic and beauty of patterns that one discovers breeds a real hunger for further knowledge, and without this hunger, the quality of our mathematicians will decay.

Finally, I have a hard time taking seriously some of the issues on which my colleagues expend a significant amount of their energy. Some of them, like simplifying fractions by their common factors, I understand: reporting a value of ten fourths as opposed to five halves make a person look uneducated. Others seem needlessly obtuse in a technologically advanced world. In the history of mathematics, it was necessary to rationalize denominators of fractions when the denominator contained an irrational number, most often an un-simplified radical. This was important when division was done by hand. If students need to estimate the value of one divided by radical two, they would be faced with a near impossibility due to the never-ending nature of the square root of two. By rationalizing the denominator, you can turn the fraction to radical two divided by two, making the estimation a feasible process.

Today, we have the technology to evaluate these expressions without the hassle of rationalization. This is by my no means to say I don’t see value in learning the process required to achieve this rationalization, but I couldn’t believe that my colleagues would still lose their minds over un-rationalized fractions. The same goes for solving an equation. If a student’s method of solving an algebraic equation is different from the teacher’s, points will be marked off, regardless of the solution.

Recently, my mind wandered to my Grandfather’s slide rule, a fascinating piece of technology. Years ago, I found it in his garage and was immediately enthralled. I set out looking for any information I could find on how to solve problems with this beautiful, plastic rectangle. My labors paid off, and the complexity of device astounded me. Arithmetic on a slide rule is fairly simple: Align the initial number to the secondary number, slide the cursor to the secondary number’s other occurrence, and then read the solution. The important task comes when you realize that the slide rule returns the same answer for 120 * 6 as it does for 1200 * 60: 7.2. It is up to the user to know what this value really represents. The death of estimation that I referred to earlier would make this tool largely useless to students, if it were reinstated in the classroom today, without substantial, remedial education.

The slide rule also does a remarkable job of visually representing the concepts in arithmetic. Each scale is constructed in intervals, so one can visualize how each of the functions on the slide rule change. This visualization is exactly what I’ve noticed students now lack. Understanding that the gap between log(1) and log(2) is an exponential difference; understanding that distance between percentage growth and decay changes as the principal value changes; these concepts are not challenging, but finding students who can make these connections is rare, something I think is detrimental to the quality of math education today.

With all this said, the reality of the slide rule’s passing does not surprise me. After all, we as a culture have come to value expedience above all else. What did surprise me was the age and longevity of the slide rule. It’s origins can be traced back to the early *1600’s*. All through the 19th century, various slide rules were created and produced to serve as calculation tools for everything from simple arithmetic to radical estimation to . The power of the slide rule far exceeded its juvenile competitors in the mechanical calculator and early computers. The skill and speed of a trained slide rule user was unmatched. Ultimately, I feel like we’ve lost a great deal of our skill with numbers as this tool disappeared and we have become ignorant of its history.

Even if the sort of remediation required was provided, I don’t think that that replacing calculators with slide rules in math classes would constitute some sort of magical fix for our current educational woes, but I find myself wondering if the quality of our math education might significantly improve if we did. Would students have a larger appreciation for the history of mathematics? Would they be more capable of recognizing and analyzing unknown patterns? If slide rules were the only calculating devices we permitted in class, would the students’ capacity for mental arithmetic be improved? I am inclined to answer “Yes” to all of these questions.

Perhaps it is time to consider a more rigorous fundamental approach to the development of numeracy in our young people. Perhaps it is time to return to the age of the slide rule.

Several slide rules

My father was an accountant and every night after dinner he would sit at his desk working with a stack of papers he brought home from the office slide rule in hand. He had a Japanese colleague at work who used an abacus and was equally rapid. Just remembering..

I remember that slide rule, or “slip stick” as we used to call it. But it wasn’t very important in mathematics. I mostly used it in science lab classes.

Like it or not, the slide rule isn’t coming back. I kept my slide rule around for decades, but I never again used it. In a world of inexpensive calculators, nobody wants a slide rule.

Yes, estimation was fundamental to using a slide rule. And estimation is still very important today. It is such a pity that it is a lost art. Maybe schools should be teaching that.

I’m honestly not worried about that. My interest in mathematics came early, and I don’t think it would have been different with calculators. And I suspect that is true of other budding mathematicians.

Agreed but I do think I’d have a hard time making an argument that there wouldn’t be more interest in math if we had a stronger system for mathematical investigation.

> Perhaps it is time to consider a more rigorous fundamental approach to the development of numeracy in our young people. Perhaps it is time to return to the age of the slide rule.

It won’t happen (as you certainly know …).

There would be massive protests that education is “destroying the self-confidence” and the “inquisitiveness” of pupils. Educational researchers would point out that there’s no way “the natural curiosity” of young people could arrive at or be stimulated by something alien and artificial like slide rules etc.

‘T is a pity, because if I understand you correctly, you’re pleading for something essential to education: critical thinking. For pupils asking themselves after they’ve made a calculation: “Can this answer be correct?”

I’m glad my real consideration came through. Thank you for the comment. I definitely long for the expectation to change from the destination to the journey.

Yes, journey and destination. Any course has to have goals, but it would be nice if syllabi allowed some flexibility. The high school maths syllabus I was exposed to (calculus and so on) seemed to be too much focused on preparing us for university engineering courses. Anyone with a more philosophical (for want of a better word) cast of mind was not really being catered for. I remember we had a class in our final or penultimate year which I found really interesting and I asked the teacher (who was a young physics graduate) after the class why we couldn’t do more of this kind of thing. It was outside the syllabus.

They probably do. But I suspect the problem is that the teacher has to aim for the average student, and cannot take the time to inspire the really bright student because that would confuse the others.

Thinking back to my own high school days, I think the library was more important than the classroom.

There seems to be a pedagogy question here. Why not ask students to estimate the result before they do the calculation? Maybe get the class to do an estimate together and then get them to do the calculation individually.

Alan

Terrific piece, my friend. Full disclosure: the essay was hatched during a late night Scotch drinking session, when I queried CJ about Slide Rules. I had been re-reading some of Heinlein’s juvenile novels, and slide rules are all over them.

With regard to the ongoing discussion: CJ is absolutely right. The slide rule is remarkable in that it is a labor saving tool, whose use does not undermine the cultivation of calculating skills, because its use presupposes them, something that is true neither of the electronic calculator or the computer.

This is not specific to mathematics instruction. Spell and grammar check software have had a devastating effect on students’ ability to write. And we know what computers have done to handwriting. My college students’ writing looks like the writing of third graders.

During that drinking session, I told CJ I would order enough slide rules for him to teach all of his classes. He told me that there was no point. They wouldn’t be able to use them, unless they had some serious remedial education, and they weren’t about to get that anytime soon.

I should also call attention to the following. Labnut, in a comment on my last New Year’s Musings, lamented that he no longer had the opportunity to use his slide rule.

“The things I miss!

– using my slide rule. I loved the tactile feel of calculating. I still have it but nobody recognises this ancient artefact.”

https://theelectricagora.com/2018/01/01/new-year-musings/

Dan-K, thanks for that reference. I enjoyed rereading your post and I am looking forward to next year’s New Year Musings. I wonder how different it will be?

One more thing. Could everyone here please tell CJ to write more often? Thanks.

Terrific essay. I was looking for a couple of slide rules for my kids recently for similar reasons. I also have given my kids plenty of estimation exercises to ensure that they can see the sensible ball park answers straight away.

My brother was telling me that it has become quite a serious problem in engineering that some engineers can’t see that the results they have got from, for example, a finite element analysis are absurdly wrong.

And yes, CJ, please write more often!

Agree with Dan K. that CJ should write more frequently. I enjoyed the article above and found it thought provoking.

Looks like there’s an app for that, in fact several in the Apple app store. Touch screens might actually be better suited for sliders, rather than typing. Haven’t tried a virtual one though.

I also agree that the estimation is more related to physics or other applied maths, like the famous Fermi problems. Symbolic maths is done on chalkboards, which many maths professonials still insist on instead of the whiteboards that have replaced them elsewhere.

Dan-K,

“

One more thing. Could everyone here please tell CJ to write more often? Thanks.”Indeed.

Happily this post comes just after Mark’s post about experientialism(which I saw too late to comment). I took down my old circular slide rule from the shelf where I keep it as an exhibit, along with my book of five figure log tables. I played with them, re-activating forgotten memories, and once more enjoyed the tactile act of computing. It was, well, quite a magical experience. What made it so magical? I then picked up my smartphone and fondled the hand made leather case constructed from Kudu leather(the inedible remains of a hunting trip). I tapped out a birthday message to a friend, enjoying the feedback from beep and vibration. And then I started on this comment. As I type it my loudspeakers play back the recorded sounds of manual typewriter key presses(my Enter key sounds like a real carriage return!)

So what is going on here? We are grounded in reality by the physical experience of a feedback control loop. Every action we take is rewarded by a signal that a reaction has taken place. These signals can be tactile, auditory, visual or olfactory. The strength, immediacy and combination of these signals is what makes the experience real. Take away these signals and the ground is taken away from under out feet. We deeply need confirmation of the success of our actions and it is this that grounds us in reality.

If we weaken these signals or delay them we become anxious and uncertain. They are vital to our well being. Reality exists when we act and experience the reaction from the feedback control loop. The important thing, though, is that this is a future oriented experience. We act so that an anticipated or desired outcome will take place. It is our success in doing this that gives lives meaning and satisfaction. The gratification of experiencing the feedback control loop spurs us onward.

I couldn’t agree more. There is such a rich fulfillment from experiencing all we do in math and science in a tactile way. It actually drives my students crazy that I insist on doing everything I teach without an electronic calculator, at least once, for many reasons of course but especially for the sensation of it.

Thank you all so much for your kindness and comments. It’s great to be back.

Still have my old K & E. I remember when HP brought out the very first hand held calculators around 1973 and they cost around $450. That’s crazy.

My older brother got one of those.

It used RPN, Reverse Polish Notation.

You didn’t type / 3 / + / 4 / = / but / 3 / enter / 4 / + /.

He had to work the whole summer to pay for it.

I loved translating formulas into RPN.

That old HP makes me just as nostalgic as my brother’s slide rule, which I also liked to play with.

It was easy to get something wrong, going from the usual notation to RPN.

This left me with a serious character deficiency.

After having done a calculation with a mechanical device, I still often check with pencil & paper if the answer can be correct. Almost everyone I know finds this funny or weird.

And I always try to reduce a mathematical formula to it’s simplest form before I use mechanical device. This often gives me a good “feel” for the behavior of the formula, allows to check for consistency in extreme cases etc.

Some here have suggested that “slide rule thinking” is more important for applied mathematics and physics etc. I’m not so sure. All the mathematicians (and the physicists) I know are good at it. Mathematical thinking implies understanding what a formula tells you. If you describe some simple problem – a guy throwing a ball or so – and your calculator tells you that it hits the ground with a velocity 50,000 times lightspeed, you should know that it can’t be true, just by looking at the formula.

I’m bigly in favor of slide rule thinking, because it forces you to be more than an extension of a mechanical device. You have to be *human*, you have to try to *understand* what a formula or a bunch of numbers is telling you.

But I’m afraid it won’t happen.

I also teach High School math. Special Ed and Inclusion Geometry to be exact. My students do not come to me with a solid foundation in even the most rudimentary mental math. The times I have had a student grab a calculator to add zero to something are discouragingly numerous. And before you jump to the elephant in the room and conclude that SpEd means dumb and dumb means incapable, I gotta say, I just don’t think it is the case. My guys do need practice, and that is exactly what the calculator has stolen from them at the same time that it has given them access to math that they might never have been able to approach otherwise. But approach is all they do because nothing is ingrained without considerable practice. When I was in high school i remember that we regularly got 30-40 practice problems that were homework if we didn’t finish. Today, even in Gen Ed courses it is closer to 10-12, books never go home, and parents often complain if homework is excessive (Admin passively discourages homework.)

They are amazed that I am able to do simple algebraic equations in my head much faster than they can do them on paper. And by they I am referring to new teachers, not the students. But I’m not a math guy by a long shot, my background is in history. But in order to ride herd over a classroom of students I can only spend a minute or two at each desk before jumping to the next and that means I’m doing most everything in my head (have to have a pencil in hand, can’t think without one, even if I don’t scratch lead on paper). I guess where I’m going with all this is that students with calculator in hand spend all their time learning a formula, which buttons to push, when to zig and when to zag. What they don’t learn much of is what they are actually doing. Math is chore instead of what it should be, a puzzle and ultimately, a language.

I do worry that my students will never see mental math as a way to approach the world or make basic economic decisions. Jumping between fractions, decimals, and percents is a lost skill. Estimating the real world value of a s

square root, lost. Adding tax and then taking off the sale percentage, lost. Making a plan to build a deck out back, lost. And this doesn’t even begin to approach the wonders that were done with compass and straightedge. The pyramids were built using string and straightedge as their mathematical tools. What can you do with string and a stick?

I went to Amazon to look for a slide rule. They don’t sell them anymore.

http://www.sphere.bc.ca/test/sliderule.html

https://www.sliderule.ca/buy.htm

Interesting piece. Much in agreement over the diagnosis of the problem.

We used slide rules regularly from about 7th grade or so on into most of high school — talking late 60’s here. My dad gave me my first one: a leftover implement of intriguing obscurity from his work as a mechanical engineer. It had some precision to it, a fine red hairline on the sliding glass piece to match against nicely calibrated divisions underneath. In my education it was far more kosher in science classes, chemistry and physics, than in math. This was true at both school levels, although it was initially introduced to us as a math curio.

You’re right, it was very useful as a device to visualize logarithmic scaling. But we were supposed to know how to calculate mentally well before this. 3rd grade, etc. The estimation issue you’ve mentioned (basically: what is the order of magnitude?) was a skill we’d have either long ago picked up or still be floundering over. The slide rule would not influence the cultivation of this skill, per se. When I entered university, initially as a math major, electronic calculators (mostly from Texas Instruments) were coming into vogue. But they were still a no-no or pooh-pooh’d within any true math context. I had friends attending the engineering institute up the street though, and it was all the rage there. I remember being incredulous that these devices were allowed in exams for fluid dynamics and the like. I think the idea there was that pragmatic speed mattered, and by then you either knew how to calculate or were not enrolled in engineering.

The way I see the issue is this. Math and calculational reasoning, even in arithmetic, is a kind of vital basic cognitive skill that should be exposed early in grade school. True, not all will take to this or display interest or sympathy with the topic. That’s okay… I was pretty turned off by religion classes (memorization exclusively) in grade school. But the exposure should be there under those terms. No external tools. Cellphones? Ridiculous. The idea is to unveil a cognitive skill within the child, not reinforce how to look up stuff. As for slide rules, I would limit their use too. Maybe something interesting to introduce as a curio, but not as an everyday tool.

They are cool though, specially those intricate yellow ones.

For a million years we have been tool using animals. Tools have multiplied our abilities in a most marvellous manner. However the power of tools goes well beyond their utility. They have also given us the power to create and it is this creative power that we possess which is possibly the most important difference between ourselves and other higher order mammals. It is this combination of utility, which multiplies power, and creative possibility that results in the almost mystical bond between us and our tools.

As a result we are emotionally invested in our tools in a way that goes well beyond their utility. We touch, caress, admire and enjoy our tools. We take inordinate pleasure in constructing and perfecting them. We are made complete when we experience the synergy of holding, feeling and using our tools, whether for their utility or for their creative possibilities.

Of all the senses it is the tactile sense which most closely connects us with our tools and arouses the emotional energy that bonds us with our tools.

I like the idea of the Return of the Slide Rule.

It reinforces the materiality of math, and “the view that there are no such things as [abstract] mathematical objects.”

https://plato.stanford.edu/entries/fictionalism-mathematics/