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The Educational Value of Slide RulesEvery year, 160 to 180 new students enter into one of my six 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 to "bridge" back up to their potential once their developmental growth supports it. The student is left with no further 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 another opportunity when they are actually ready for it, with no change to catch up to their peers and no positive experience to be gained.
Because this is the situation of roughly 70% of my students when I first get them, I consider myself both a teacher AND a redeemer of math. I've learned that I must do the latter before I can reach my own potential as the former. 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 300 calculating devices, primarily slide rules, hang on my walls to elicit curiosity and questioning within my students. The depth and breadth of what could be accomplished with these slide rules goes far beyond computational mathematics, but also into applications, history, number sense, and real-life models. These aspects have little value in today's math education, which values doing pseudo-applications (phony word problems that act like real-life math), computations and passing state-level standardized tests. The richness of mathematics is sacrificed for the achievement of all the wrong goals, which have more to do with reading comprehension than it does real mathematics.
So, has it always been this way?
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 both a teacher AND a redeemer of math. I've learned that I must do the latter before I can reach my own potential as the former. 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 300 calculating devices, primarily slide rules, hang on my walls to elicit curiosity and questioning within my students. The depth and breadth of what could be accomplished with these slide rules goes far beyond computational mathematics, but also into applications, history, number sense, and real-life models. These aspects have little value in today's math education, which values doing pseudo-applications (phony word problems that act like real-life math), computations and passing state-level standardized tests. The richness of mathematics is sacrificed for the achievement of all the wrong goals, which have more to do with reading comprehension than it does real mathematics.
So, has it always been this way?
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 pocket 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?
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Technology, from the Greek root word "technes," literally means "young or child-like." As such, anything progressive enough as a tool is "technology." The first shovel was technology; as was 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 historical use. 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 more like a 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 students give up before the reward? "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 8 different ways in which math education is 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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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 this 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" cannot 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 to solve. The idea for this is simple - if you give productive people a new tool, then 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 into the calculator's programmability, but anybody who has done that on a TI calculator can tell you how unintuitive and complicated that is. Simply put, the calculators in my classroom are not a good tool by today's standards. So why use them? In Texas, our standards require us to use a "graphing calculator" at the high school level. As such, Texas students will not be reaching for a "Casio" when your state's name is on education's most popular calculator. No matter that it was made obsolete by the first iPhone 15 years ago. Not to bury the lede, but we'll refrain from asking "why" here. Let us just say that a lot of money changes hands in education, with teachers not seeing much of it. Note: While writing this, my district has supplied me with new TI-84 Plus CE calculators, with color LCD, self-charging, and even with Python programming functionality. It's much better than what I had, and they are appreciated, but they are still not up to the capabilities of a good, free phone app! |
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. The proper use of a manipulative would have students match a specific visualization or analogy to a learning goal, such as using a ruler or number line to visualize the addition and subtraction of integers. A calculator does not replace the purpose of the ruler in that case, because the virtue of the manipulative is the hands-on, visual-reinforcement of an IDEA that takes place.
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Dig into your state's course objectives and you'll see process standards as required for student success. In Texas, this is what the "TEKS" are for Precalculus, which includes not only the concept of the manipulative, but also "mental math, estimation, and number sense" as techniques. (Note as well "reasonableness of the solution" in P.1.B above it.) The slide rule a perfect "manipulative" because it requires those techniques when using it. I would advocate the use of slide rules in modern education as a way to not only meet all of these requirements, but to help improve this skill deficiency in our students.
But replacing manipulatives with calculators is exactly what happened!
The idea of a manipulative was tossed by the wayside in favor of, first calculators, and then smart-devices. In this new paradigm, technology BECAME the manipulative. And it continues to this day. In fact, when my administrators assess my teaching formally, they only care about seeing that I'm using a form of technology, and not even as a part of a specific lesson plan. As long as I have the specific learning objective written on the board; good enough, it seems!
Technology replacing manipulatives; it all started with good intentions. In fact, I remember being rewarded with a trip to the elementary school library in the late 1970s 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, but it did encourage it. But this is quickly what technology became...let's trick students into learning through educational video games!
Note: I learned to read before I was 4 years old from store signs and billboards. While it's not something I could place in my hands, it's functioned like a manipulative in that words and letters are supported with visual cues of shapes and colors.
Certainly, one of the early uses of the smartphone in education, which spawned the BYOD (bring your own device) initiatives, was to use multimedia "apps" to help students learn in a fun way. But because their use was seldom specific in focus, the effectiveness of these devices can be strongly debated. Not only this, because this technology use takes place on a platform that also doubles as a student's entertainment, it also becomes a diversion FROM learning. This is technology use at its worse.
At its best, technology is still an almost entirely passive participant in our learning; it's either a resource for research or its used to do computations so we no longer have to. It's a jeopardy-like game or a self-grading assignment system or a video conferencing application or an online curriculum.
What it is NOT - if we are honest - is an active, integral role in the mastery of a learning objective. And in the spirit of using manipulatives as a pedagogical practice, technology has failed. While it could be used to function as a manipulative toward meeting specific educational objectives, we seldom require it to do so.
Education is far worse today because teachers are not held accountable to give students hands-on, visual modelling via manipulatives - though, judging from my state's requirements at right, it appears that it is most certainly required.* My guess is that either we've assumed that technology is providing that or we are given the choice to neglect it.
So perhaps it's time to bring back slide rules as a manipulative? This is because, unlike a calculator, slide rules gives students a very specific intuition about the computations they are performing. It is a number line, a conversion table, and a Vernier scale, all wrapped up into one. If the objective is to understand fractions and proportions, then setting the C and D scales for a single fraction gives all equivalences across the scales.
If the objective is to understand continuity, then the slide rule can be used to give a subtle feel for either the continuity or the closeness of values. It's builds an intuition, especially when connected to the active feel of the slide rule.
And importantly, as we will see in examples that follow, the slide rule helps support students in the development of mental math, estimation, and number sense skills that most teachers believe to be lacking in today's students.
*Note: Manipulatives are differentiated from technology here, as we can see by reading the 4th bullet point of the Precalculus standards below. So as indicated, manipulatives (and real objects) are intended to be used whether, or not, technology is used.
The idea of a manipulative was tossed by the wayside in favor of, first calculators, and then smart-devices. In this new paradigm, technology BECAME the manipulative. And it continues to this day. In fact, when my administrators assess my teaching formally, they only care about seeing that I'm using a form of technology, and not even as a part of a specific lesson plan. As long as I have the specific learning objective written on the board; good enough, it seems!
Technology replacing manipulatives; it all started with good intentions. In fact, I remember being rewarded with a trip to the elementary school library in the late 1970s 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, but it did encourage it. But this is quickly what technology became...let's trick students into learning through educational video games!
Note: I learned to read before I was 4 years old from store signs and billboards. While it's not something I could place in my hands, it's functioned like a manipulative in that words and letters are supported with visual cues of shapes and colors.
Certainly, one of the early uses of the smartphone in education, which spawned the BYOD (bring your own device) initiatives, was to use multimedia "apps" to help students learn in a fun way. But because their use was seldom specific in focus, the effectiveness of these devices can be strongly debated. Not only this, because this technology use takes place on a platform that also doubles as a student's entertainment, it also becomes a diversion FROM learning. This is technology use at its worse.
At its best, technology is still an almost entirely passive participant in our learning; it's either a resource for research or its used to do computations so we no longer have to. It's a jeopardy-like game or a self-grading assignment system or a video conferencing application or an online curriculum.
What it is NOT - if we are honest - is an active, integral role in the mastery of a learning objective. And in the spirit of using manipulatives as a pedagogical practice, technology has failed. While it could be used to function as a manipulative toward meeting specific educational objectives, we seldom require it to do so.
Education is far worse today because teachers are not held accountable to give students hands-on, visual modelling via manipulatives - though, judging from my state's requirements at right, it appears that it is most certainly required.* My guess is that either we've assumed that technology is providing that or we are given the choice to neglect it.
So perhaps it's time to bring back slide rules as a manipulative? This is because, unlike a calculator, slide rules gives students a very specific intuition about the computations they are performing. It is a number line, a conversion table, and a Vernier scale, all wrapped up into one. If the objective is to understand fractions and proportions, then setting the C and D scales for a single fraction gives all equivalences across the scales.
If the objective is to understand continuity, then the slide rule can be used to give a subtle feel for either the continuity or the closeness of values. It's builds an intuition, especially when connected to the active feel of the slide rule.
And importantly, as we will see in examples that follow, the slide rule helps support students in the development of mental math, estimation, and number sense skills that most teachers believe to be lacking in today's students.
*Note: Manipulatives are differentiated from technology here, as we can see by reading the 4th bullet point of the Precalculus standards below. So as indicated, manipulatives (and real objects) are intended to be used whether, or not, technology is used.
Reasonableness of Solution |
On every 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 wants to ask. It's almost as if the class got together to find the volunteer brave enough to ask it!
Students today feel that if they only had a calculator, then everything will be miraculously better. But after each test, when we go over commonly missed problems, it's almost always the calculator problems that they get wrong. This is largely because students assume no responsibility for such problems; they believe that the calculator would do everything right for them. When the calculator spits out the wrong answer, they have no understanding, no warning that it is incorrect. This is because students are either not taught to provide estimations in advance of the solution or because they simply lack the ability to do so. The prophylactical idea of a "reasonable" solution eludes them. |
With the 8 on the D scale aligned with the 7 on the C scale, the result can be read at the D scale, as shown here on the hairline of the cursor. This reading of 114 (and a little more) are the significant figures for any of the computations at the left. It becomes YOUR job to supply the necessary decimal.
It's different with a slide rule. As an example, on a calculator, you get the following outputs...
8 / 700 = .0114286
8 / 70 = .114286
8 / 7 = 1.14286
80 / 7 = 11.4286
800 / 7 - 114.286
But when doing this computation on a slide rule, all 5 computations are performed with the identical setting of the rule (see right). The user must know where the decimal place belongs.
Now, in a list of pros and cons for slide rules, having to supply your own decimal is usually listed among the disadvantages, especially in a calculator society that freely yields decimal precision for each unique problem. However, a math teacher can quickly understand the pedagogical advantages to placing your own decimal. Quite simply, the slide rule requires students to have a sense of magnitude and scale with any problem they are trying to compute.
It was quite commonplace for school children to estimate a problem such as "80 divided by 7" as being "a little more than 10, simply because if 70 / 7 = 10, then 80 / 7 would equal a "little bit more." This, of course, encourages the practice of better "numeracy" (see next topic point), but for the student it simply requires that they be constantly engaged with each step of a problem, having an idea of what they should expect. With a calculator, it's all too commonplace for a student to incorrectly input an operation of numbers, and because he or she had no idea of the problem's magnitude, the error is not often detected before moving to the next step.
As a teacher of any level of math, it is important to have students evaluate their solutions for reasonableness. In many cases now, as opposed to earlier in my career, this has become a missing skill for an alarming number of my students. I wish I didn't have to keep reminding them to watch their mode in the calculator when doing trigonometry, but alas, unless they know the answer in advance, most students don't catch the mistake. So, in my world of Precalculus, having some understanding of what a reasonable solution looks like is a skill that you do not want to be without!
8 / 700 = .0114286
8 / 70 = .114286
8 / 7 = 1.14286
80 / 7 = 11.4286
800 / 7 - 114.286
But when doing this computation on a slide rule, all 5 computations are performed with the identical setting of the rule (see right). The user must know where the decimal place belongs.
Now, in a list of pros and cons for slide rules, having to supply your own decimal is usually listed among the disadvantages, especially in a calculator society that freely yields decimal precision for each unique problem. However, a math teacher can quickly understand the pedagogical advantages to placing your own decimal. Quite simply, the slide rule requires students to have a sense of magnitude and scale with any problem they are trying to compute.
It was quite commonplace for school children to estimate a problem such as "80 divided by 7" as being "a little more than 10, simply because if 70 / 7 = 10, then 80 / 7 would equal a "little bit more." This, of course, encourages the practice of better "numeracy" (see next topic point), but for the student it simply requires that they be constantly engaged with each step of a problem, having an idea of what they should expect. With a calculator, it's all too commonplace for a student to incorrectly input an operation of numbers, and because he or she had no idea of the problem's magnitude, the error is not often detected before moving to the next step.
As a teacher of any level of math, it is important to have students evaluate their solutions for reasonableness. In many cases now, as opposed to earlier in my career, this has become a missing skill for an alarming number of my students. I wish I didn't have to keep reminding them to watch their mode in the calculator when doing trigonometry, but alas, unless they know the answer in advance, most students don't catch the mistake. So, in my world of Precalculus, having some understanding of what a reasonable solution looks like is a skill that you do not want to be without!
Improved 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 revealed with improved number sense. But I believe most of today's high school math teachers would agree that our student's sense about numbers is lacking. I believe this skill has progressively worsened on every generation since the end of the slide rule era; an inverse function of the reliance on technology to do it for us.
To support this statement, let's do some exploration about numbers and then show ways in which only the slide rules would have provided the intuition for it. Let's start with these facts... |
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sin (3 degrees) = .052
tan (3 degrees) = .052 3 degrees = .052 radians First, despite teaching trig graphs to kids, I NEVER made the association, in 28 years of teaching, that sine and tangent curves are so nearly identical when near the origin. Similar, yes, but not so exacting as this. But as you can see from the figures at right, both function curves look almost identical over their first 6 degrees. When I first discovered this, it took me a while to forgive myself for not knowing this fact. Quite simply, there has never been a context to make such a comparison for me. Why would I, when not only do we represent trig graphs in radians, but we never have a reason to show the sine and tangent curves on the same graph? Perhaps I could have had a teacher show it to me somewhere along the way, but that would have required them to a) know about it themselves and b) value you it enough to actually mention it to me. But these numerical relationships are obvious to those who grew up with a slide rule. In fact, many slide rules had an "ST" scale, the purpose of which was to compute the sine or tangent of small angles (up to ~5.7 degrees). This works well because the output of sine and tangent for angles in this domain are equivalent within 3 significant figures of resolution in which a typical 10" slide rule is capable. So it didn't matter if taking the sine or tangent of an angle like 5 degrees...they knew the output would be identical within a few decimal places. |
The sine graph zoomed to an x-window of 7 degrees. Compare to the tangent graph pictured below.
Comparing tangent to sine in this interval, the curves are approximate to within three significant figures.
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But for today's high-schooler? He or she can earn an entire credit in Precalculus and never realize that insight.
But notice as well that I slipped in one other observation, that the radian measure of 3 degrees is ALSO .052, and while this can easily go unnoticed, the implication is that the sin and tangent of small angles are also equivalent to the radian equivalent of that angle. This means that to find the sine or tangent of an angle like 2 degrees, a calculator or slide rule is not required. All one needs to do is divide by 57.2 (because there are 57.2 degrees in one radian).
But notice as well that I slipped in one other observation, that the radian measure of 3 degrees is ALSO .052, and while this can easily go unnoticed, the implication is that the sin and tangent of small angles are also equivalent to the radian equivalent of that angle. This means that to find the sine or tangent of an angle like 2 degrees, a calculator or slide rule is not required. All one needs to do is divide by 57.2 (because there are 57.2 degrees in one radian).
Here on this K&E Deci-Lon 10 slide rule, I've circled 3 on the D scale and have aligned it with the "R" gauge mark on the C scale. This gauge mark represents 57.2, which here is being divided into the 3. Note under the hairline on the cursor, 524 can be read. These significant figures require the user to know that .0524 would be the proper reading for the division. Students of the slide rule era would have known that sine and tangent at these angle sizes would require this decimal placement.
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Armed with this knowledge, even a 5th grader can compute the sine or tangent of an angle 6 degree or less, equipped only with long division! While it's doubtful any 5th grader would need to do this, high school students back in the day most certainly did. In some cases, if their slide rules did not have trig scales, or, more likely, the rule only had S and T scales with no provision for higher resolution for small angles, then dividing by 57.2 was indeed the method used. In fact, on many slide rules, 57.2 was included as a gauge mark on the C or D scale, for division of such small angles. And example of this is shown at left.
Let's look at another intuition about numbers, this time as it concerns logarithms. Note: For those needing a quick reminder, the log of a number is the value required to raise a base, in this case base 10, to yield the argument (the number in parentheses). Said as an exponential, 10 ^ 0.477 = 3 in the following example. How about this pattern of values? log (3) = 0.477 log (30) = 1.477 log (300) = 2.477 log (3000) = 3.477 |
An aspect of logarithms that my students are not required to learn is knowledge of the characteristic and mantissa of a log. For any number between 1 and 10, the log of that number will be a decimal value between 0 and 1. This result is known as the mantissa. To find the log of numbers greater than 10, a whole number value (the characteristic) is added to the mantissa depending on the number of digits of the input. This can be easily seen using a tool for logarithms that my students ARE required to know, the Product Rule for Logs, as applied to the above pattern:
log (3) = 0.477
log (30) = log (3 x 10) = log 3 + log 10 = 0.477 (mantissa) + 1 (characteristic) = 1.477
log (300) = log (3 x 100) = log 3 + log 100 = 0.477 (mantissa) + 2 (characteristic) = 2.477
log (3000) = log (3 x 1000) = log 3 + log 1000 = 0.477 (mantissa) + 3 (characteristic) = 3.477
Of course, the calculator inputs this value directly, assuming we are using the correct log base on the calculator (not always a given with students). However, the association between these values, the pattern, will not be obvious to students unless I have a reason as a teacher to show it to them. Doing so helps the students with a perspective of the magnitude of the logs themselves, giving them expectations for which they can check for reasonableness of solution. Likewise, the pattern reinforces this very important Product Rule for Logs, which my students are required to know. This gives students a visual and practical application for this rule, and logarithms themselves.
With a slide rule, the understanding is seamless with usage. For a base 10 logs, most slide rules have an "L" scale. Ironically, this scale for finding the base 10 log of a number is the only scale on a slide rule that's NOT itself logarithmic. Instead, it is a linear scale running from 0.0 to 1.0 on most rules (a scale of equal measures). See the images below for explanation...
log (3) = 0.477
log (30) = log (3 x 10) = log 3 + log 10 = 0.477 (mantissa) + 1 (characteristic) = 1.477
log (300) = log (3 x 100) = log 3 + log 100 = 0.477 (mantissa) + 2 (characteristic) = 2.477
log (3000) = log (3 x 1000) = log 3 + log 1000 = 0.477 (mantissa) + 3 (characteristic) = 3.477
Of course, the calculator inputs this value directly, assuming we are using the correct log base on the calculator (not always a given with students). However, the association between these values, the pattern, will not be obvious to students unless I have a reason as a teacher to show it to them. Doing so helps the students with a perspective of the magnitude of the logs themselves, giving them expectations for which they can check for reasonableness of solution. Likewise, the pattern reinforces this very important Product Rule for Logs, which my students are required to know. This gives students a visual and practical application for this rule, and logarithms themselves.
With a slide rule, the understanding is seamless with usage. For a base 10 logs, most slide rules have an "L" scale. Ironically, this scale for finding the base 10 log of a number is the only scale on a slide rule that's NOT itself logarithmic. Instead, it is a linear scale running from 0.0 to 1.0 on most rules (a scale of equal measures). See the images below for explanation...
This is a special K&E (Keuffel & Esser) slide rule that demonstrates the use of a logarithmic scale (top) and a linear scale (bottom) to discover the mantissa of a log. This rule aligns the input 3 on the top scale, with the logarithm output on the bottom scale, reading approximately 0.477 (note the use of 0.0 to 1.0, equally spaced). To compute log (30), the same setting is used, except that the user knows to add 1 to this 0.477 value.
Almost all slide rules have a base 10 log scale, often labelled "L." Note we see it here on one of K&E's more basic slide rules. The scale runs left to right in the middle of the slide rule, with the L label on the right. The input would be set on the logarithmic "D" scale in this case. Note again, reading at the hairline of the cursor, the value of ~0.477.
This interplay between linear and logarithmic scales has further importance to graphs, and so these concepts transition well to interpreting data. For example, exponential growth can be modeled with either linear or logarithmic axes. Witness the following charts representing the spread of the Coronavirus since the beginning of the Covid pandemic (courtesy of www.worldometers.info)...
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When an exponential graph - representing an exponentially growing virus - is modeled using a linear y-axis (on the left), the rapid growth of the event is felt (less so in this case when the axis labels jump 200 million each tick). But when that axis is logarithmic (on the right), in this case each tick advances 100x, we are able to see the rapid spread for the first 6 months of the pandemic until the exponential growth started to slow down. Graphically, we see this as a "flattening of the curve," which is what world leaders hoped that their citizens would work together to accomplish. Therefore, the logarithmic data here indicates that efforts to get the virus transmission under control were succeeding. So the next time somebody on social media says they haven't needed Algebra 2 since high school, you can let them know that since January, 2020, it's been a daily occurrence!
With school students, displaying data in multiple representations like this tells a more complete story. And it all starts with improving numeracy with our students. In this case, it's about discovering what logarithms are and what they can do.
With school students, displaying data in multiple representations like this tells a more complete story. And it all starts with improving numeracy with our students. In this case, it's about discovering what logarithms are and what they can do.
Encourages Discovery |
Speaking of which, one of the aspects of slide rules that blindsided me was the thrill of discovering how slide rules work.
In fact, if you put the slide rule into the hands of somebody who's never seen one, and then asked them to figure out what all the scales do, it could be the beginning of a very satisfying journey of discovery. |
Figure 1 - Scales created from the log table.
Figure 2 - Showing the multiplication of 2 x 3 = 6. The left index of the upper scale lines up on the 2 with the result read under the 3 on the bottom scale. For that matter, it also shows all multiples of 2 (up to 5) as well. You may also notice it shows fractions/proportions equal to 1/2 across the scales, something we'll talk about in the next section.
Figure 3 - This setting of the scales shows 6 x 4 = 24. Note that the right index of the upper rule is aligned with the 6, and the answer is read under the 4 on the bottom scale. The result, 2.4, would be understood as 24. You are responsible for your own decimal! The right index is used like this if using the left decimal takes you off the scale.
Figure 4 - How about division? It's essentially the inverse of multiplication, meaning that here you align the numbers to be divided and read the answer off of the index. The scales are showing to do 9 / 4 = 2.25. Here I have divided up the logs between 2 and 3 to provide more resolution for the decimal quotient.
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We lack the time for such an investigation here, so let's give a quick primer instead. Actually, let's BUILD one from paper, starting with the table of logarithmic values at right...
With this information, you will construct two logarithmic scales (or number lines). See Figure 1 at left. The standard size of most slide rules are 10" (or 25 cm) scale lengths, so let's start with a sheet of paper, pencil, and ruler. Draw a 10" line. Mark the ends of the line with tick marks (like on a ruler) and label them 1 and 10 respectively. These will represent the values of the log (1) and log (10) in the table above. In slide rule terms, they are known as the left and right indices (indexes). Moving on to log (2), change the .301 decimal to a percentage (30.1%) and mark the number line with a tick mark at 30% of the distance from the left index. That would be 3" if measuring with a ruler. Mark this tick mark with the number 2, which represents log(2). Do the same with the remainder of the table, yielding a number line from 1 to 10. Get as close as possible with your percentage approximations for more accuracy. For example, if log 3 = .477, then 47.7% would be a skosh shy of 4 3/4" on the ruler. Note that the gap between 0 and 1 is far greater than the gap between 9 and 10. This is as you should expect with a logarithmic scale. Use the first scale to create another similar to it in Figure 1. |
Yet another discovery, this from using a circular slide rule. Here, it is set to compute sin (1 degree), with 1 placed on the arc notated with the lowest red "S". The answer, 0.00174, is read off of the arc labeled with the red "C". Again, you must supply your own decimal. But this same setting also computes sin (0.1) on the red ST scale and sin (10) on the top red S scale. Would you have known that 0.1, 1, and 10 computed for sine all have the same significant digits of 174? Yeah, me neither!
For additional precision, feel free to mark the finer gradations. For example, adding ticks for log (1.1), while tedious, can give you better precision if the time is taken to do it accurately. I have done such between the numbers 2 and 3 on both scales to illustrate the point.
(Historical note: Slide rule makers used a machine to etch the tick marks. These machines were known as "dividing engines." With the technology, makers could produce scales representing log (1.01) with great accuracy. Doing so, they would promote the accuracy of their rules using the term "engine-divided.")
Now, for the MAGIC!
Put the left index of one of the scales onto the 2 of the other scale, as in Figure 2. Then, measure 3 on the top rule and look at where that value falls on the bottom rule. Observe that the result is 6. Congratulations, you just performed the computation 2 x 3 = 6. In addition, observe that you just performed the following computations as well:
20 x 3 = 60
2 x 30 = 60
200 x 3 = 600
20 x 300 = 6000
2000 x 30 = 60000
The same setting of your "slide" rule gives you all such computations when, as is expected, you provide your own decimal!
So, how does this work? Why does this yield the product of those numbers? Keep in mind that on your scales, 2 and 3 represent the length of log (2) on one scale and log (3) on the other. So you are literally measuring out a distance of log (2) and adding it to the distance of log (3). And if you recall the Product Rule of Logs discussed earlier...
log (2) + log (3) = log (2*3) = log (6)
Therefore, a slide rule is the physical manifestation of the Product Rule of Logs.
On a typical slide rule these two scales are typically labeled "C" and "D", and they are used to do both multiplication and division operations.
This exciting discovery inevitably leads us down a much larger rabbit hole, as other scales can be created using the the C and D scales as a reference. For example, building another scale, called the "A" scale on the slide rule, is easy to do simply by squaring the numbers on the D scale (see Figure 5). This gives a scale that can compute squares (read off of the A scale) or square roots (when read backwards off of the D scale).
Do the same with the cubic to yield a traditional "K" scale on many slide rules. Key sine ("S" scale) or tangent ("T" scale) off of the D scale and you can naturally compute trig values of any angle.
Now, THIS is discovery!
Students seldom have that opportunity, and when it does happen, it's often too late in the average student's journey. Once they've learned how about numbers, we put a calculator in their hand and the investigation about those numbers usually ends. And because students no longer need to use long division or multiplication on paper, they no longer remember those processes once they are needed later to factor via polynomial division or to find slant asymptotes or to multiply binomials.
Math is not arithmetic. It's not finding the answer. Math is the algorithm...the pattern...the process. By putting a calculator in the hands of students, we deny them the experience of learning or practicing the algorithm that makes it possible. And those algorithms will be used again in the future.
Almost every mathematical understanding about numbers can be more deeply understood by using a slide rule. It prompts the question, "How does this work?" And in my experience, students enjoy the discovery of it (see below).
Too bad the calculator doesn't prompt the same question. If it did, students would begin a wonderful deep-dive into the world of of electronics!
(Historical note: Slide rule makers used a machine to etch the tick marks. These machines were known as "dividing engines." With the technology, makers could produce scales representing log (1.01) with great accuracy. Doing so, they would promote the accuracy of their rules using the term "engine-divided.")
Now, for the MAGIC!
Put the left index of one of the scales onto the 2 of the other scale, as in Figure 2. Then, measure 3 on the top rule and look at where that value falls on the bottom rule. Observe that the result is 6. Congratulations, you just performed the computation 2 x 3 = 6. In addition, observe that you just performed the following computations as well:
20 x 3 = 60
2 x 30 = 60
200 x 3 = 600
20 x 300 = 6000
2000 x 30 = 60000
The same setting of your "slide" rule gives you all such computations when, as is expected, you provide your own decimal!
So, how does this work? Why does this yield the product of those numbers? Keep in mind that on your scales, 2 and 3 represent the length of log (2) on one scale and log (3) on the other. So you are literally measuring out a distance of log (2) and adding it to the distance of log (3). And if you recall the Product Rule of Logs discussed earlier...
log (2) + log (3) = log (2*3) = log (6)
Therefore, a slide rule is the physical manifestation of the Product Rule of Logs.
On a typical slide rule these two scales are typically labeled "C" and "D", and they are used to do both multiplication and division operations.
This exciting discovery inevitably leads us down a much larger rabbit hole, as other scales can be created using the the C and D scales as a reference. For example, building another scale, called the "A" scale on the slide rule, is easy to do simply by squaring the numbers on the D scale (see Figure 5). This gives a scale that can compute squares (read off of the A scale) or square roots (when read backwards off of the D scale).
Do the same with the cubic to yield a traditional "K" scale on many slide rules. Key sine ("S" scale) or tangent ("T" scale) off of the D scale and you can naturally compute trig values of any angle.
Now, THIS is discovery!
Students seldom have that opportunity, and when it does happen, it's often too late in the average student's journey. Once they've learned how about numbers, we put a calculator in their hand and the investigation about those numbers usually ends. And because students no longer need to use long division or multiplication on paper, they no longer remember those processes once they are needed later to factor via polynomial division or to find slant asymptotes or to multiply binomials.
Math is not arithmetic. It's not finding the answer. Math is the algorithm...the pattern...the process. By putting a calculator in the hands of students, we deny them the experience of learning or practicing the algorithm that makes it possible. And those algorithms will be used again in the future.
Almost every mathematical understanding about numbers can be more deeply understood by using a slide rule. It prompts the question, "How does this work?" And in my experience, students enjoy the discovery of it (see below).
Too bad the calculator doesn't prompt the same question. If it did, students would begin a wonderful deep-dive into the world of of electronics!
Proportions |
In terms of prerequisite skills, it's fractions, along with factoring, that continue to be a stumbling block for my Precalculus students.
I would love to know the answer to why my kids still can't get common denominators when adding fractions, six years since they should have learned it. They stare all day at their phones and have no problem knowing how much battery power they have remaining when expressed as a percentage. Yet, they are hopeless when it comes to tipping 15% for a meal. Say the word "ratio," and they act like you stole their souls. But proportions, they seem to understand those. One of the more powerful aspects of the slide rule is that, because the scales are logarithmic, it is easy to align the rule so when 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. |
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. My mother used a cardboard slide rule known as the EZ Grader, which computed the grade as a percentage given the total number correct against the total number of problems on an assignment. When I was in 2nd grade, I remember using the EZ Grader to help my mother grade her 2nd grade papers. It gave me a practical understanding of ratios before my school covered them. Oh, and I learned fractions by making biscuits from scratch when I was 5 years old.
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 even if you choose the most inefficient order of operations, the speed at which keystrokes can be made still makes it rather efficient. This, assuming you know what you are doing with it in the first place!
But the slide rule is a bit more particular in this regard. Efficiency depends on how you use it, particularly when multiple operations are required back-to-back, called "chaining" with a slide rule. It will require you to order your computations logically, otherwise you find yourself writing down intermediate answers for the next operation. |
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The classic example is doing the computation (3 x 4 x 5) / (6 x 7), simplified to whole numbers for demonstration purposes. There is a danger in doing this computation with a calculator, as students will typically multiply the entire numerator, followed by dividing the denominator. But as we know, if the parentheses are not honored in the denominator, then the 7 can easily end up being multiplied rather than being divided with the 6.
With the slide rule, if the numerator is multiplied first, followed by the denominator, then those intermediate values will need to be written down prior to finding the final quotient. But there's a trick...students of yesteryear would have known to work the problem as 3 / 6 x 4 / 7 x 5, in that order. This chaining of operations, which is totally permissible, allows the results of the each step to be used directly into the next operation, all without ever needing to use paper. Important for us, and today's students, is that you would never have reason to handle the problem that way, and so students would likely not ever know that doing so was possible. In this case, the parentheses "service" the problem, yet I find that students will almost always perform the parentheses quite literally. As such, they think of the computation as 60 / 42. |
Pictured is the Faber-Castell 2/83 N Novo-Biplex, regarded as one of the most beautiful and powerful slide rules ever produced. Note here the right hand side of the scales are labeled with the functions they perform. This feature was known as "self-documenting." It was mostly a European feature. Click on the image to see a close-up.
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With the slide removed because it's not needed here, this rule is set to find the outputs of 2.2. Reading closely, the square of 2.2 is on the A scale, the cube of 2.2 is on the K scale, and the log of 2.2 is on the L scale. These values can be read to three significant figures as 4.84, 10.6, and .342 respectively. Moving the hairline lets you quickly read-off these values for any number you desire.
Fact of the matter is, with fractions composed entirely of monomials, numbers (or factors) on top can be multiplied and numbers on bottom can be divided. While this sounds insignificant, in practice it means something quite powerful to students. I discovered this long ago when teaching transformations of trig graphs, watching students struggle when they are instructed to divide "B" values to the function's inputs. So when they are faced with "pi/2 divided by 2", they almost always fail to understand that all they need to do is place the 2 on the bottom - anything on the bottom is divided - leaving pi/4. Instead, I watch students reciprocate the 2 and then multiply, almost without exception.
While this speaks equally to poor numeracy, when students only understand order of operations in terms of "PEMDAS," then they will always be trapped by that rule. While rules are important, sometimes breaking the rules "legally" leads to more efficient math practices. The slide rule is full of a sorts of ways to rethink math.
Another example of when the slide rule makes you rethink math is what happens when you multiply something like 4.2 x 6.7. It requires using the right index on the 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 multiplying two numbers is the same as dividing by its inverse, something that would have been common knowledge in the slide rule era.
Likewise, there are also what is called "folded" scales on a slide rule. Such scales, usually labeled "CF" and "DF," are essentially the same as the C and D scales, yet the beginning of the scales through pi (or sqrt(10) depending on the manufacturer), is lifted off the scale and reapplied at the end. This yields a single index at the center of the rule. As such, if folded at pi, it can either be used to automatically multiply a number by pi (making circle computations a breeze), but also can be used instead of the C and D scales for multiplication operations to assure an index is always on the rule. This promotes the idea that there's more that one way to do things, and some ways are more efficient than others. This is something my students today seem to struggle with.
Speaking of efficiency, a slide rule can also be used as a function table to do multiple operations at once. For example, with the slide rule at left, when the middle slide is removed inputs on the D scale can be read as outputs on the L, K, and A scales by just moving the hairline. This slide rule, as are many of the European slide rules, are also "self-documenting," putting the functions they perform on the right side of the scale (see also above). As such, logs, cubes, and squares can be quickly read almost simultaneously, all without really needing to decipher what the L, K, and A scales actually mean.
Now this is something a calculator can do, but it requires a little preparation...type something like the squaring function into Y1 in the graph screen. Select the table for the graph. And read your results for multiple inputs. Of course this works well for integer inputs by default. To read off decimal inputs, you would need to customize the table for tenth or hundredths, and then scroll the table up and down all day long to get where you need it. Too bad the TI-84 doesn't have a manual cursor like a slide rule!
Likewise, slide rules will put a premium on doing a little mental math along the way. If efficiency is the goal, then any computation that can be done in one's head makes fewer computations required of the slide rule. This is a practice that calculator users should do as well, as far too many students within my classes will use the calculator for even the more simple of computations. Using a slide rule, for even a week, reinforces the idea that a calculator is not needed every time a computation is required. That there are mental tricks to arithmetic that slide rule era students did naturally that could also be used during today's age of the calculator.
Perhaps the most important aspect to communicate among students is that precision to a ridiculous number of decimal digits, like with a calculator, is mostly unnecessary to real-world applications. Conversely, the inability of a slide rule to work beyond 3 or 4 significant figures is less of a problem than it was historically made out to be. Strictly speaking, very seldom does anything built, designed, or modelled in this world require mathematical representations beyond a few significant figures. In fact, we never actually use the calculator's ability for this, since we round almost every decimal output we get from the calculator to 2 or 3 digits anyway. So from that point of view, the slide rule yields all the precision that is required for 99% of every application for which it's needed. Now, this isn't an advantage of a slide rule, per say, but when my students act as if it's a disadvantage, or that giving answers to ten decimals on a calculator is an advantage, I am quick to inform them of the important point that real-life math seldom requires any precision beyond a few digits. I remind them that most of America was built using a slide rule, from the Brooklyn Bridge to the Hoover Dam. And that has not changed even in the calculator era. Slide rules were an important part of efficiently building the world for more than 300 years. That's a lot of human history.
And speaking of history...
While this speaks equally to poor numeracy, when students only understand order of operations in terms of "PEMDAS," then they will always be trapped by that rule. While rules are important, sometimes breaking the rules "legally" leads to more efficient math practices. The slide rule is full of a sorts of ways to rethink math.
Another example of when the slide rule makes you rethink math is what happens when you multiply something like 4.2 x 6.7. It requires using the right index on the 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 multiplying two numbers is the same as dividing by its inverse, something that would have been common knowledge in the slide rule era.
Likewise, there are also what is called "folded" scales on a slide rule. Such scales, usually labeled "CF" and "DF," are essentially the same as the C and D scales, yet the beginning of the scales through pi (or sqrt(10) depending on the manufacturer), is lifted off the scale and reapplied at the end. This yields a single index at the center of the rule. As such, if folded at pi, it can either be used to automatically multiply a number by pi (making circle computations a breeze), but also can be used instead of the C and D scales for multiplication operations to assure an index is always on the rule. This promotes the idea that there's more that one way to do things, and some ways are more efficient than others. This is something my students today seem to struggle with.
Speaking of efficiency, a slide rule can also be used as a function table to do multiple operations at once. For example, with the slide rule at left, when the middle slide is removed inputs on the D scale can be read as outputs on the L, K, and A scales by just moving the hairline. This slide rule, as are many of the European slide rules, are also "self-documenting," putting the functions they perform on the right side of the scale (see also above). As such, logs, cubes, and squares can be quickly read almost simultaneously, all without really needing to decipher what the L, K, and A scales actually mean.
Now this is something a calculator can do, but it requires a little preparation...type something like the squaring function into Y1 in the graph screen. Select the table for the graph. And read your results for multiple inputs. Of course this works well for integer inputs by default. To read off decimal inputs, you would need to customize the table for tenth or hundredths, and then scroll the table up and down all day long to get where you need it. Too bad the TI-84 doesn't have a manual cursor like a slide rule!
Likewise, slide rules will put a premium on doing a little mental math along the way. If efficiency is the goal, then any computation that can be done in one's head makes fewer computations required of the slide rule. This is a practice that calculator users should do as well, as far too many students within my classes will use the calculator for even the more simple of computations. Using a slide rule, for even a week, reinforces the idea that a calculator is not needed every time a computation is required. That there are mental tricks to arithmetic that slide rule era students did naturally that could also be used during today's age of the calculator.
Perhaps the most important aspect to communicate among students is that precision to a ridiculous number of decimal digits, like with a calculator, is mostly unnecessary to real-world applications. Conversely, the inability of a slide rule to work beyond 3 or 4 significant figures is less of a problem than it was historically made out to be. Strictly speaking, very seldom does anything built, designed, or modelled in this world require mathematical representations beyond a few significant figures. In fact, we never actually use the calculator's ability for this, since we round almost every decimal output we get from the calculator to 2 or 3 digits anyway. So from that point of view, the slide rule yields all the precision that is required for 99% of every application for which it's needed. Now, this isn't an advantage of a slide rule, per say, but when my students act as if it's a disadvantage, or that giving answers to ten decimals on a calculator is an advantage, I am quick to inform them of the important point that real-life math seldom requires any precision beyond a few digits. I remind them that most of America was built using a slide rule, from the Brooklyn Bridge to the Hoover Dam. And that has not changed even in the calculator era. Slide rules were an important part of efficiently building the world for more than 300 years. That's a lot of human history.
And speaking of history...
Math History |
As we saw earlier with the Covid graphs, Math is hidden in so much of what we do that we often take it for granted. But it's important remember that it wasn't always this way. At one point in history, mathematics was absolutely intertwined with everyday life. People didn't always have computers programmed to do the math for us. So to accomplish things, people had to think and reason solutions to life's problems. Math wasn't just for Mathematicians, and in fact there wasn't even an occupation with the title, "Mathematician." The people who created all the math had other vocations...philosophers, ministers, scientists, and poets.
For them, these mathematical ideas were born out of the needs of the times. They created the math as they needed it. Necessity being the mother of invention. |
Pythagorean, Descartes, Newton, Liebniz, and Euclid. There is a rich history of mathematics that colors the actual math that we do. Euler. Riemann. Gauss. Pascal.
We teach Euclidean Geometry represented on a Cartesian coordinate plane and using the Pythagorean Theorem. We teach Precalculus by doing binomial expansion using Pascal's Triangle, systems of equations with Gaussian Elimination, and we expose our students to Euler's number for the first time. We study Calculus created by Newton and Liebniz by doing Reimann sums.
All this, yet, we are not required to teach kids about these math creators, nor the situations in which they felt compelled to create it. Our own academic language, which we use freely in class, is like a foreign language to those we teach...and for whatever reason, we do not view that as a problem.
So instead of "discovering" math as our ancestors did, today's students learn mathematics because it is required, and this often causes resentment within students who only wonder, "When am I ever gonna use this?"
Hopefully, as a teacher, you can answer that question for them. But if you can't, then even a shallow dive into math history can provide enrichment for students. In the very least, we should be supporting our academic language with vignettes of the people who gave it a name.
But if you are looking for something more hands-on, then I can think of nothing better than to show students a slide rule and how it works. It gives a perfect illustration for how a tool was created out of a need for the math that real people needed to do. In fact, 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.
So, if your students cannot discover math for themselves, then at least the slide rule can put them into the shoes of the people who did. It's a perfect example as to how mathematics was birthed out of problems and questions that were, and still are, important. What results is an appreciation of that perspective.
I provide opportunities for my own students to learn and use a slide rule as extra credit. While you might believe they do so reluctantly, I think you might be surprised how many students just think they are "cool." The hook is actually quite simple...I tell them I can do "X" with a slide rule and that sparks a curiosity. I provide a research project where they can earn points and I give them a slide rule for their efforts. This academic year, to date, I have given away 25 slide rules.
We teach Euclidean Geometry represented on a Cartesian coordinate plane and using the Pythagorean Theorem. We teach Precalculus by doing binomial expansion using Pascal's Triangle, systems of equations with Gaussian Elimination, and we expose our students to Euler's number for the first time. We study Calculus created by Newton and Liebniz by doing Reimann sums.
All this, yet, we are not required to teach kids about these math creators, nor the situations in which they felt compelled to create it. Our own academic language, which we use freely in class, is like a foreign language to those we teach...and for whatever reason, we do not view that as a problem.
So instead of "discovering" math as our ancestors did, today's students learn mathematics because it is required, and this often causes resentment within students who only wonder, "When am I ever gonna use this?"
Hopefully, as a teacher, you can answer that question for them. But if you can't, then even a shallow dive into math history can provide enrichment for students. In the very least, we should be supporting our academic language with vignettes of the people who gave it a name.
But if you are looking for something more hands-on, then I can think of nothing better than to show students a slide rule and how it works. It gives a perfect illustration for how a tool was created out of a need for the math that real people needed to do. In fact, 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.
So, if your students cannot discover math for themselves, then at least the slide rule can put them into the shoes of the people who did. It's a perfect example as to how mathematics was birthed out of problems and questions that were, and still are, important. What results is an appreciation of that perspective.
I provide opportunities for my own students to learn and use a slide rule as extra credit. While you might believe they do so reluctantly, I think you might be surprised how many students just think they are "cool." The hook is actually quite simple...I tell them I can do "X" with a slide rule and that sparks a curiosity. I provide a research project where they can earn points and I give them a slide rule for their efforts. This academic year, to date, I have given away 25 slide rules.
