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The Educational Value of Slide RulesEvery year, 160 to 180 new students enter into one of my 6 high school Precalculus classes. And before I learn their names, I already know what they feel about math. As a subject, there is nothing more polarizing. It's the only subject taught in school where a student can be excused for carrying the label "bad at math," a label often passed down from parent to child. And as a teacher, my first task is to overcome their preconceived attitude toward math.
Compounding the problem is an educational system that reinforces the "bad at math" idea. This is because children are ability-grouped in their formative years, classified to an ability "level," and then never given a chance later when their development supports it, to bridge back up to where they should be. The student is left with no opportunity because it is deemed by "educators" that it's no longer worth the effort. |
So when a third grader struggles with multiplication tables or a sixth grader lacks algebraic reasoning, they are started on an educational path that will not allow them the opportunity of others when they are actually ready to take it, only to be left with a wrong idea of what mathematics is.
Because this is the situation of roughly 70% of my students when I first get them, I consider myself less a teacher and more of a redeemer of math. I figure if I do the latter, then the former will be accomplished. So I work hard to make sure math is what its supposed to be...confidence-building, enlightening, and enjoyable.
Such is my job of a math apologist...to give them positive experiences in mathematics that they were never given the opportunity to have.
If you enter my classroom, you will be surrounded by mathematical history. Over 200 calculating devices, primarily slide rules, hang on my walls to elicit curiosity and questioning with my students. The depth and breadth of what could be accomplished with these slide rules goes far beyond computational mathematics, but also into a
Over 50 years ago, the slide rule was introduced to students as early as the third grade, continuing with them throughout their scholastic years. It was a way of life during a time when the slide rule was part and parcel to the educational process, much like calculator is today. However, the difference is that using a slide rule always requires certain math disciplines that a calculator does not require. It also provides an insight into math that simple keystrokes with a calculator does not.
What follows is not about a data-driven - or even an anecdotal - attempt to show how students are worse off from using calculators, nor is it an indictment on modern educational methods. Rather, it is to show insight into what education with a slide rule did well and why it may be worthwhile for today's educators to consider a return to the slide rule, for at least a short while, to enhance classroom learning.
Because this is the situation of roughly 70% of my students when I first get them, I consider myself less a teacher and more of a redeemer of math. I figure if I do the latter, then the former will be accomplished. So I work hard to make sure math is what its supposed to be...confidence-building, enlightening, and enjoyable.
Such is my job of a math apologist...to give them positive experiences in mathematics that they were never given the opportunity to have.
If you enter my classroom, you will be surrounded by mathematical history. Over 200 calculating devices, primarily slide rules, hang on my walls to elicit curiosity and questioning with my students. The depth and breadth of what could be accomplished with these slide rules goes far beyond computational mathematics, but also into a
Over 50 years ago, the slide rule was introduced to students as early as the third grade, continuing with them throughout their scholastic years. It was a way of life during a time when the slide rule was part and parcel to the educational process, much like calculator is today. However, the difference is that using a slide rule always requires certain math disciplines that a calculator does not require. It also provides an insight into math that simple keystrokes with a calculator does not.
What follows is not about a data-driven - or even an anecdotal - attempt to show how students are worse off from using calculators, nor is it an indictment on modern educational methods. Rather, it is to show insight into what education with a slide rule did well and why it may be worthwhile for today's educators to consider a return to the slide rule, for at least a short while, to enhance classroom learning.
Introduction |
There is no wikipedia-like definition of "slide rule" that doesn't use the word "obsolete" in the first sentence. And when the first affordable hand-held calculator was introduced in the 1970s, we could have measured the rate of obsolescence with a stopwatch. A college student could have entered as a freshman with a slide rule in hand and would have graduated with the calculator that his professor required for a junior-level course. But what was not asked, by anybody, was what would be missed from education by following the technology train?
Technology, from the greek root would "technes," literally means "young or child-like." As such, anything progressive enough as a tool is " technology." The first shovel was technology; the first slide rule. And the slide rule shouldn't have been replaced anymore than the irreplaceable shovel, at least not completely. While the assumption today is that any digital method should supersede any analog analogue, there will always be something missed in the transference. Ask an electrical engineer to toss out his multimeter "with the needle" for the sake of the his digital meter on his desktop and he will look at you as if you were crazy. This is because he knows that the analog device will reveal something about a measurement that the digital version will not. |
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Not to argue with the virtues of a calculator; nor am I advocating against any advancements. But when superior technology comes along, there will still be pedagogical value to that which it replaces, especially when there is 300+ years of using it. As a person who is one generation removed from using a slide rule scholastically, I find learning it today to be revolutionary. It has reminded me that something doesn't have to be practical in order to have value. Make no mistake, a slide rule is no longer practical, and this is the reason for its obsolescence.
But the slide rule is a tangible, visual way to see the way numbers work - and it's this aspect that warrants a reevaluation of its practicality today. In the least, it's akin to an abacus or counting beads in the primary years. Or it's a more sophisticated number line in the secondary years. As educators, we are always looking for models that assist us with abstract principles. While the abstractions make math challenging, it's also that aspect which makes it rewarding. And therein lies the polarizing nature of math. How many student give up before the reward? |
Sidebar: Are Calculators also Obsolete?An argument can be made that the replacement for the slide rule has become obsolete as well.
The classroom set of Texas Instrument calculators (TI-84 Plus) in my classroom likely cost my school around $150 each. Yet it's based on a technology/chip set that is more than 25 years old - and I shouldn't have to tell you how long that is in "computer years." Low resolution, non-touch, black-and-white screens; non-intuitive, extended-menu interface for functions beyond the simple operations; and no Internet connectivity. Not only has a calculator's technology been usurped by phone apps and computers, but it is also the least convincing visual or pedagogical way to understand math. Students could now be using, for free, Desmos or Wolfram Alpha on the smartphone they already own. Such "apps" can not only give you solutions like the calculator, but can show you step-by-step how to do it, as well as showing you extra information and extensions to the learning. Moreover, I can use Desmos to quickly and easily set-up a slider on a value to show visually its effect on a function. They, and more powerful PC solutions like a programming language (e.g. Python or MatLab) and even an Excel Spreadsheet, cannot only solve problems, but they also let students create their own problems. The idea for this is simple - if you give people a new tool, they will seek opportunities to use the tool. The only time I ever saw students use our calculators in this way was before smartphones when a few enterprising students would dive the phone's programmability, but anybody who has done that on a TI calculator can tell you how unintuitive and complicated that is. Simply put, the calculator is not a good tool. So why use them? The answer is more political than we need to be here, but let's just say teaching math has come down to standards-based instruction. In Texas, our standards require us to use a "graphing calculator" at the high school level. Texas schools using Texas calculators that were made obsolete by the first iPhone 15 years ago. And there is no sign of this changing anytime soon, no matter how much assessing we do, which is another debate for another article! So we continue to advocate the use of a device that is not only obsolete by 15 years, but provides remarkably little value beyond letting us do computations. The slide rule was given no such leash!
Shown above - Various ways to piggyback a camera. Piggyback DSLR with large lens atop a fork & wedge mounted SCT (upper left), a DSLR fixed atop a refractor with GEM (lower left), and a variety of cameras mounted in both piggyback and side-by-side configurations (right). |
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"It's the abstractions that make math challenging, but it's also that aspect which makes it rewarding. And therein lies the polarizing nature of math." What follows are 5 different ways in which our math lives are worse from no longer having slide rules. Some of these aspects are a direct reflection of the device itself; what it does and how it can help us in the classroom. But some of these are an indictment on the way we do math education today, and I hope that by showing how "old technology" can be used in a modern classroom, that educators might find some enlightenment about our current pedagogy.
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As a Manipulative |
Look up any state standards for a math class and you will see the idea of a "math manipulative" expressed. Defined, a math manipulative would constitute any concrete, hands-on tool for learning. As an educator earlier in my career, this idea, often coupled with the learning "center," became thematic with good education. And then the technology-initiative (calculators, then smartphones) happened, which said that using technology is better than not using it - and the idea of a manipulative went by the wayside. It too was considered a concrete, hands-on approach to learning.
But this was an error in judgment, as the focus of an manipulative was to master a specific learning objective. Technology is seldom that specific, nor does it necessarily promote a "center" of learning - at least not the way we use our devices. It's more of a passive participant to our learning, sometimes a resource for research, and always for computations. It's a jeopardy-like game, a self-grading assignment system, a video conferencing application, or an online curriculum. It's there when we need it, but often NOT an active, integral role in the mastery of a learning objective. One of the early uses of the smartphone in education, which spawned the BYOD (bring your own device) initiatives, was that we could trick students into learning through educational video games. In fact, I remember being rewarded with a trip to the elementary school library where I could play a Star War's themed text-based adventure game on the school's only Apple II Computer. To be honest, it didn't teach me to read, it just encouraged it. Note: I learned to read before I was 4 years old from store signs and billboards, which takes on the idea of a manipulative in that words and letters are supported with visual cues of shapes and colors. Call it early multimedia, which has value in education from a technology perspective as long as the objective is well-defined. Multimedia (videos or video games) begins to tap into the power of today's devices, but its effectiveness can be strongly debated, as can any use of technology on a platform that also doubles as entertainment; as a student's diversion from learning. So perhaps it's time to bring back slide rules as a manipulative? If the objective is fractions and proportions, then then setting the C and D scales for a single fraction gives all equivalences across the scales. It gives a subtle feel for the continuity or closeness of values. It's builds an intuition, especially when connected to the active feel of the slide rule. |
Reasonableness of Solution |
On ever test I give my Precalculus students, somebody will ask if they can use a calculator. That question, though asked by one person verbally, is actually one that half the class has. It's almost as if the class got together to find the volunteer brave enough to ask it. They feel that if they only had a calculator, then everything will be miraculously better. Worse, when the calculator spits out the wrong answer, they have no understanding, no warning that it is incorrect. The idea of "reasonableness of solution" eludes them.
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Numeracy |
Numbers are fascinating in and of themselves. They do not exist just to be useful or functional. They aren't merely to give a language for math or to facilitate the solution to a problem. They have their own delight; their own magic. There are patterns, behaviors, and curiosities that are mostly lost on the generations since the end of the slide rule era.
Consider the pattern: log (3) = 0.477 log (30) = 1.477 log (300) = 2.477 log (3000) = 3.477 Or, did you know this? sin (3 degrees) = .052 tan (3 degrees) = .052 3 degrees = .052 radians How about this? sin (.0225 degrees) = .000392... sin (.225 degrees) = .00392... sin (2.25 degrees) = .0392... And finally... log 1 = 0 log 2 = .301 log 2 + log 5 = 1 = log 10 log 3 = .477 log 3 + log 3 = .954 = log 9 log 4 = .602 log 4 + log 2 = .903 = log 8 log 5 = .699 log 6 = .778 log 7 = .845 log 8 = .903 log 9 = .954 log 10 = 1 log 1 + log 2 + log 3 + log 4 + log 5 = 2.079 These numerical relationships are obvious to those who grew up with a slide rule. |
Proportions |
Fractions, ratios, percentages, and proportional relationships are difficult concepts for today's students to learn. It should not be. One of the more powerful aspects of the slide rule is that, because the scales are logarithmic, it is easy to align the rule to where one number is placed over the another, and the rest of the numbers across those scales will be in exactly proportion to each other. It's like a table of equivalent fractions, an adjustable one at that.
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Cross-Curricular Application |
One of the things that is not readily apparent when a student looks at the slide rules on my classroom wall is that they do more than "just" math. Slide rules were once ubiquitous, not only for math computations, but also for applications in all of todays STEM subjects.
I have slide rules that give the molecular weight of elements and compounds. Others offer scales to compute inductance and capacitance of electronic circuits. Another gives the margin as a percentage of profit on a selling price. Several of my slide rules allow for loading capacities of military planes, and even some are custom-designed for airplane manufacturing - not to mention the E6-B flight computer still required to be used by pilots today. |
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Many slide rules include formulas for unit conversions in a variety of fields, which likely also includes scale markings to coincide with said formulas.
Efficiency in Solving |
A calculator doesn't care how you type in the numbers. If you type it in wrong, it will happily give you the wrong answer. Garbage in; garbage out. But the slide rule is a bit more particular. It will require you to order your computations logically, or to do so in a way that demands efficiency. Likewise, it will put a premium into doing a little mental math along the way.
A really basic example of this is what happens when you multiply 4 x 6. It requires using the right index of the rule (C scale) to get the correct result; however, the inclination is to use the left index first, which would put the solution off the slide rule. To fix this, you either have to remember to use the right index OR you use an inverse (CI) scale and treat the problem like it's a division problem, because multiply two numbers is the same as dividing by its inverse, something that would have been common knowledge. While this also improves numeracy, it teaches students to be more efficient in the operations, especially when "chaining" together multiple products and quotients. |
Math History |
A practical use of math, that mathematics is both passive and active in reality, is lost upon students and teachers today. The question, "When am I ever gonna use this?" At one point in history, the math was absolutely intertwined with productive living. In fact, mathematical ideas, like anything else, is often born our of need. "Necessity breeds Invention." We forget that the logarithm was invented because an easier way to think about multiplying two numbers was necessary.
Or perhaps, more accurately, our kids never knew that? Pythagorean, Descartes, Newton, Liebniz. There is a rich history of mathematics that colors the actual math that we do. Euler. Riemann. Gauss. Pascal. Euclid. When first learning the slide rule today, we cross paths with guys like John Napier, William Oughtred, and Peter Mark Roget. The first guy invented logarithms, the second guy invented the slide rule, and the third guy wrote a thesaurus - oh, and he created the first log log scale on a slide rule. This is because the guys who created math things were doing other things as well...philosophers, ministers, scientists, poets, and mathematicians. We call them "renaissance men," but I choose to call them "learners." Today's learners need to understand the origins of the math they study. Because that math is birthed out of problems and questions that were, and still are, important. Math was developed out of need and curiosity. But today it's learned because it's required, which infuses a resentment within students that they never grow past. |
Encourages Discovery |
One of the aspects of slide rules that blindsided me was the thrill of discovery, both about the mathematics on which slide rules work AND the mathematics that they allowed me to do.
Puzzles do this same thing, BTW. In fact, if you put the slide rule into the hands of somebody who's never seen one, and then asked them to solve the puzzle about what all the scales do, it would be the beginning of a very satisfying journey of discovery. Now, this is what math is...discovery. I've known this for quite some time. But students seldom have that opportunity to do this as discovery in math often happens too late in the average student's journey. This is primarily because they errantly believe the math is arithmetic. Math is about the algorithm...the pattern...the process. By putting a calculator in the hands of students, we deny them the experience of learning the algorithm that makes it possible. |
