Real Math Doesn't Use Calculators

Thank you for this post. I am a high school science teacher who does not allow my students to use calculators and am forever in conflict with our math department over their allowing the use (from 6th grade on up). My students often remark to me that they learned more math in my class than they did in their math course. I will be passing this article on to my colleagues.

I don't see how banning the calculator helps anything, though. As you note, math is a weird mix of two skills -- arithmetic and abstract symbol manipulation.

If you don't get the symbol part of it, it doesn't matter whether or not you have a calculator, because you can't do it. If you do get the symbol part of it, then calculators are mostly just a time-saving device on the arithmetic -- I mean, I don't think there are really people out there who can do integration by parts but get stymied by division.

(The exception to this, I suppose, is maybe linear algebra, where a fancy calculator can essentially do your problem sets for you.)

I see what you're saying about the psychological reason to ban calculators, but I think it's served just as well by allowing them and letting students notice that they're basically never necessary.

"The reason for restricting the use of calculators in introductory math and science classes is that calculators will be no help in higher math classes, and can be an active impediment to learning."

Could you elaborate on that last part? I get that higher math classes push students from number manipulation to completing abstract operations, but I don't understand how making students do long division helps that transition.

Full disclosure: as a journalism major who never got past differential equations in college, my experience with higher math is limited. And as someone who is embarrassingly bad with simple math, I'll give you my calculator when you take it from my cold, dead hands.

I was quite fond of what my high school pre-calculus/calculus teacher would do. Every test she gave had a first page (single or double sided) which was meant to be done calculator-free*. Once you turned it in to her, you received the rest of the exam, and were free to take out your calculator.

I can see some pitfalls in this method. For one, it forces you to finish the test first, rather than skipping around and doing the hard problems last. But it also could be used to introduce the concept of only taking out your calculator when you reach the stage where you've gotten the problem to its simplest state, and need only put in the numbers.

* It was mostly setting up problems, with a bit of simple math -- like knowing what the sine of 45 degrees was, or finding the tangent and cotangent of a 3-4-5 right triangle.

A famous Czech professor of maths once said that mathematics is the only academic subject that has no use for numbers.

The "impediment to learning" part comes from things like the tendency of students to immediately plug all the numbers into a formula as the first step of a problem, and then spend the rest of the problem manipulating nine-digit decimal numbers. This is the bane of many an introductory physics instructor, as it makes it almost impossible to find mistakes, particularly when those mistakes come from things like transposing two digits while entering them into the calculator. It also helps convince many students that physics is unremitting drudgery, as any time they make a mistake, they need to go all the way back to the start, and re-do a whole mess of nine-digit decimal operations.

The proper way to solve intro problems is to work them algebraically as far as possible, and only put numbers in as the very last step. Not only does this reduce the number of chances for silly arithmetic errors, but it also can lead to useful insights about the real dependence of things-- lots of problems in introductory mechanics will end up giving results that don't depend on the mass, for example, which you can see by writing things down algebraically and then cancelling out all the "m"s. Many students will see "m=37.2 kg" in the statement of the problem, though, and immediately plug that in, then spend a bunch of time manipulating numbers that are 37.2 times larger than they need to be. And they'll finish the problem still thinking that the final answer depends on the mass, because their reliance on calculators has obscured what's really important.

That makes a lot of sense. I totally remember doing the exact same thing in my high school algebra class a decade ago. And with the prevalence of souped-up graphing calculators with the younger sort (and their dancing and their rock music!)I imagine this problem only gets worse.

I always enjoyed bringing and using my slide rule whenever a professor would specify, "no electronic devices allowed on the exam". This was just a few years ago, so I certainly got my share of strange looks.

I totally agree with you Chad. I write this in 2006.

http://cyclequark.wordpress.com/2006/11/03/a-small-teaching-breakthroug…

Helping my daughter with her high school physics homework is a much different activity than lecturing to 200 students. I had a real insight last night into the kind of mistakes that students make. The problem was a classic block on a turntable problem. The turntable accelerates at a constant rate and the question is how long before the block slides off.
In this problem the mass of the block was not given, and that was a real stumbling block for my daughter. I told her to just write down m for the mass and keep working. Later in the same problem she needed the angular velocity and was given the angular acceleration. I said just find the angular velocity using the acceleration. I assumed that she would write
Ï = αt
but she said that she did not know the time. I pointed out that she was looking for the time. I got very excited at this point. I had discovered a fundamental problem she was having. Her bad habit of plugging in numbers immediately actually prevented her from doing this problem. I probably got too excited. She started to cry.
Since the beginning of the year I have encouraged her to solve the problem symbolically as far as possible and substitute in numbers as late as possible. In general she has ignored my well thought out fatherly advice and tried to plug in numbers as soon as possible. After she calmed down I explained how her way of doing things prevented her from solving the problem. I hope this lesson sticks.

I fear I disagree, taking more of an evolutionary approach. My rule was that any student who was foolish enough to over rely on a calculator was welcome to his (few women in physics grad shul in those days) failure. I would however, explain that the purpose of the test was to gather some insight into what they had learned rather than what they could compute so the manipulation of the maths via physics knowledge, with the occasional explanation, contained a lot more information than a umpteen digit number.

To me, the primary advantage (goal?) of limiting calculator use--even pre-college--is to help (force?) students to develop basic number sense and intuition about calculations. I have had students set up and need to solve a ratio that involved dividing by 10 and not being able to do it without a calculator. These were kids in 9th and 10th grade in a well-to-do high school, not young kids or remedial kids. There is zero excuse for a 14-year-old not being able to divide by 10 in his/her head--except they were so used to using a calculator that they had never learned any of the patterns that become very obvious when one does the math by hand.

Relying on a calculator from the time you learn to add (or divide) prevents you from learning skills such as estimation, rounding, etc. Learn those skills and you'll most likely only need a calculator once in a very rare while anyway--and why rely on something that can break, get lost, or run out of batteries?

Pre-calc in high school? What about actual calculus? In the UK, I did (differential) calculus at age 15, and then had three more years of mathematical education after that before going to university. If your high school pupils aren't studying calculus for several years before starting a mathematically-based subject at university, you have much worse problems than calculators.

Right on, Chad.

The joke among my friends who study math is, "I can't do multiplication! I'm a math major!" You see, multiplication of large numbers just never comes up, and a calculator does it better anyway.

"The reason for restricting the use of calculators in introductory math and science classes is that calculators will be no help in higher math classes, and can be an active impediment to learning."

If this is true, then the entire argument in favor of banning calculators is false. If calculators are of no help in higher level math, then how is teaching the person do to the tasks assigned to a calculator beneficial? A calculator is the mathematical equivalent of spell check. To teach without it is paying homage to an antiquated demagogy.

If your high school pupils aren't studying calculus for several years before starting a mathematically-based subject at university, you have much worse problems than calculators.

And yet, America hasn't run out of decent physicists. Fancy that.

With the over reliance on calculators, our kids stop thinking. They stop visualizing what the numbers do ... they stop using their intuition as Kate said. And I also agree with Kate that relying on calculators prevents you from estimating and rounding.

Have you ever stopped to think how much you use math on a daily basis? I didn't until my son entered Elementary school. He was allowed to use a calculator at a pretty young age, and it was then that I started to notice how much math I do in my head. I don't think I noticed before that because it's just so automatic. But because he has relied on calculators for so long, he has a limited ability to, say, estimate how much a few things at the grocery store is going to cost us. And he certainly doesn't do it automatically - if I force him to pay attention to it, he really struggles.

How many times have you gone someplace to get something you know the price of and looked in your wallet to see if you have enough cash to cover it? (Yeah, I know, most of use plastic which does essentially the same thing as a calculator - no math required!) I don't know if my son will ever think like that. I fear that he will not.

Even the simple concept of multiplying something by 10. Because he doesn't visualize numbers, he completely misses that all he has to do is add a zero to the end of whatever number he is multiplying. Or 20, simply double the multiplier(?) and add a zero. No calculator necessary. But for him, he can't see it.

And fractions - forget it. I am still stunned that my son can't look at, say, a window that is divided into four panes and see that one pane is one fourth of the window. He counts the panes and goes from there. I think it's his over reliance on calculators that has limited his ability to visualize things like that.

I think we're doing kids a huge disservice by letting them use calculators in elementary, middle and high school. I think kids should graduate from high school never having used them, and then use them in college on a limited scale. Or at the very least, let them use them infrequently in high school, when they've already developed the automatic responses to calculating. I can still remember the first time I was allowed to use a calculator on a test in college. Not that I couldn't have done it without one, but it made it go a lot quicker.

In the UK, I did (differential) calculus at age 15, and then had three more years of mathematical education after that before going to university. If your high school pupils aren't studying calculus for several years before starting a mathematically-based subject at university, you have much worse problems than calculators.

International comparisons of this type are not as straightforward as you might think. There are significant structural differences between the way the American public education system operates and the way most European systems operate. In particular, the American system does very little "tracking," in the sense of shifting students to different tracks of study based on their test scores or early grades. As a result, our high school math classes are taken by, and must therefore be pitched to, students who in many European systems would have stopped taking math a couple of years earlier.

In general, the best math students in US schools get at least one year of calculus in high school; this varies from one school district to another, depending on local resources, etc., but we see very few students in intro physics who haven't seen at least a little differential calculus; most have some idea of what an integral is before getting to college.

"The reason for restricting the use of calculators in introductory math and science classes is that calculators will be no help in higher math classes, and can be an active impediment to learning."

If this is true, then the entire argument in favor of banning calculators is false. If calculators are of no help in higher level math, then how is teaching the person do to the tasks assigned to a calculator beneficial?

I have bolded the relevant clause in the bit you quoted, to make it more obvious.

Calculators-- particularly the fancy graphing sort-- can be helpful for doing problems in introductory classes. They are helpful, though, at the expense of developing the real understanding necessary for higher-level classes, where the problems generally do not involve numbers at all, and in some cases may not even involve symbols that are directly related to numbers. As a result, while teaching a student to do math using a calculator may help them pass a high-school math test, or do simple introductory physics problems, over-reliance on calculators at an early stage can put them at a disadvantage when and if they move on to classes where the problems don't easily resolve into simple arithmetic.

I would never ban calculators outright-- I don't expect students to know how to evaluate square roots or trig functions in their heads, beyond the really simple cases (sine of 30 degrees is 1/2, etc.). I'm thinking of instituting an algebra requirement in future introductory classes, though-- docking them points for plugging non-zero numbers in before the very last step-- because I think their refusal to do algebra is incredibly damaging to their development as scientists or engineers.

docking them points for plugging non-zero numbers in before the very last step

I can't recommend that idea highly enough. My freshman physics professor did that very thing, and it was that process that pushed me into finally understanding the usefulness of symbolic manipulation (something I'd avoided in high school). Now the notion of plugging in a number before I get to the answer is...creepy.

I agree with Kate and Wendy that too-generous policies regarding calculator use in elementary school are actually harmful.

I suspect that the original reason for allowing calculators in math class was to alleviate the math phobia that observationally sets in for many American students during elementary school. But I think it actually has the opposite effect--people become so accustomed to using the calculator that they are even more afraid of doing simple arithmetic themselves.

It's not just in science and math education at the college level where we have suffered from this problem. Our current financial crisis arose in no small part because of a host of financial practices--such as always making only the minimum payment on a credit card, or taking out a mortgage where your monthly payment fails to cover the interest (much less the principal) on the loan--which on their face are absurd to anyone with even minimal number sense but nonetheless became routine. Since one of our major political parties is devoted to the concept of making government unable and/or unwilling to protect the unwary from scams (and too often to take the side of the perpetrators when these kinds of loans inevitably go bad), we have to teach students the number sense needed to avoid these pitfalls.

I am a US enginnering student who has finished my calculus requirements (up through diff eq). In my calculus 1 class, we were banned from ever using calculators. It made no sense to me then and it still doesn't make sense to me now. If students are able to place into calculus after a no-calculator math placement test, it should be assumed that their algebraic skills are sound. If a student wants to go into a math-intensive major, they probably have a sound understanding of algebra anyway. If not, they are in the wrong place.

When instructors get frustrated when their students use their calculators before symbolic manipulation, it isn't the calculator's fault, it is an instructor's fault. Students can't be expected to do math "the right way" when we are often led blindly into the dark alleyway of math and told that we have to do math a certain way "just because". It is the resposibilty of the instructors to pass on the understanding of "why" we need to do algebra first before plugging in numbers. Show some examples of problems in future math courses. Show the students how this will help them in the future.

Not allowing calulators is also un-realisitc. If a student figures out a way to do advanced math with a calulator, then by all means they should be allowed to work problems the same way they would outside of a classroom setting. Some of my later enginnering classes even let us use laptops and the internet on tests! I learned many patterns by using my calculator over and over again on trivial operations to the point where I would skip the calulator on my own accord.

How is having a scientific calculator different from having a slide rule?

Andrew - you blame the instructor. Ok, but how is the instructor supposed to train the students to use symbols first? Believe me, telling them to do it & showing them how to do it aren't enough for most students. You have to force them to do it. One possibility for forcing them to do it is to remove the calculator altogether.

I'm that old (35 years) math/physics teacher who allowed calculator use but gave minimal points for correct numerical answers. Problems had to be set up properly, manipulated properly, and then the calcs could be used. Most points were lost on "illegal alg!" errors.

Also, if a picture was going to be helpful (as it would be most of the time) and you didn't use one and made an error, the infamous "where pic?" label showed up. After a short time I ceased having arguments about solutions.

I always told my students that calculators would slow them down. However, we did program the law of cosines for SAS and SSS triangles into the calculators, and that was a real time saver. We did not do the law of sines, since my students knew that the law of sines would lie to you every time you looked for an obtuse angle.

Andrew, you really think that showing students who barely understand calculus a serious multi-variable calc E&M problem from Jackson will be at all useful in letting them understand why basic calculus is necessary? That's beyond absurd. The only result will be glazed eyes.

If you don't think what you're learning will be useful and can't trust your instructor word on it, there won't be anything he/she could do to change your mind. It's taken me years to realize the worth or lack thereof of various classes that I've taken, and certainly there isn't any speech or slide that could've just given me that understanding.

And of course, nobody bothers banning calculators from exams in physics (or I'm sure math) grad school - there is no point, since nobody uses them in any case. The problem is that if young students don't understand how to do simple things without a calculator they won't ever get to the more difficult problems. And these kids won't understand how their lack of basic math skills is hobbling them.

The only thing I'll say though is that if you do ban calculators in a physics class you should remove the annoying math too - i.e. no stupid games with units that aren't multiples of 10, leave constants in the answers without having to multiply them out, etc. Frankly I'd rather this sort of silliness was removed even if you don't remove calculators.

If you're going to do physics beyond the level of Giancoli, or math beyond the level of introductory calculus, you will absolutely need to be able to do abstract operations that a calculator will not help with.

Or worse, that they will help with. I confess to being fogey enough that Maple was totally unavailable when I was doing symbolic math in school, but it's progressed to the point where there are calculators with more power than my old computing center and they do symbolic math that I learned to do by hand.

IMHO allowing them in classes is much worse than the slide rule I had to argue for the right to use.

Oh, and get off of my lawn.

I sort of disagree. For instance, I'm terrible at arithmetic (like completely horrible) and I always reach for a calculator to do even the simplest computations. Nonetheless, I excelled at upper division math coursework and I am currently a grad student studying applied math.

Hell, I'd even go so far as to say I don't do most of my abstract symbolic manipulation by hand anymore, preferring to use Mathematica to do the grunt work for me.

So I think your objection is with calculators in physics classes, not in math classes? Because in math classes, you simply don't have the nine-digit decimals to plug in. Most math problems are solved with super-trivial arithmetic (realizing that you need to divide 37 by 89 is usually a clue that you've done something wrong earlier in the problem), and if they've got variables sitting around, they're rarely bound at all.

Ping!

Chad, one thing that bothers me about leaving calculators out altogether is that I found when I was TAing that students do not have a good sense of the relative scale of different quantities. In physics and engineering many approximatations depend critically on having a sense of which terms are big or small, and I thought calculators sometimes helped make it clear. (Also, I grew up using my calculator for symbolic integration, so this discussion shouldn't assume calculators are only for arithmetic.) On the other hand, people have no sense of how many digits past the decimal point are meaningful and my older profs assured me that it wasn't like that in the slide rule days.

The "impediment to learning" part comes from things like the tendency of students to immediately plug all the numbers into a formula as the first step of a problem, and then spend the rest of the problem manipulating nine-digit decimal numbers

Sadly, this tendency does not go away, once firmly established. Just last Wednesday I sat through a design review with every important quantity calculated out to five (5) significant digits. Even in the best of circumstances, that's ludicrous, and these were far from the best of circumstances. But that's what the modelling software spat out so that's what was on the slide.

I am about one promotion away from being a big enough bastard to quash that sort of nonsense in my presence, but not yet.

That said, I'd draw this line: Math class is where you work the math out by hand (or, if it's a numerical methods class, program it by hand) because math is what you're learning. You learn by drill.

Science class is where you work with a calculator, because math is just a tool in science classes, and science is what you're learning, whether it's chemistry, physics, biology, etc.

And a special post script to Andrew: Andrew, you're a student. By definition, you don't know it all, yet, and more than likely you don't even know some of the things you don't know. You might want to consider that your professors have been roughly in your position, before, but you haven't been in theirs.

Speaking as someone who got a (general) engineering degree 20 years ago, I'm all for "no calculators" while learning technical subjects. Unit conversion (or lack of conversion) because all the values were simply typed into the calculator and trusted blindly can be a real issue.

For example, in one lab we did a test reaction with a few grams of each reactant in a 1-liter chamber, and we had to figure out how large a vessel would be required to make a few kg of product in one batch. I was shocked at how many people forgot that grams-> millimoles, kg -> moles when dealing with atomic weights. That factor of 1.0e3 (easy to miss when using a calculator in scientific notation mode) ended up getting cubed (cranking through the formulas when calculating volume) and several groups ended up with a reaction chamber 1e9 meters on a side.

If you know algebra and only use the calculator (or a slide rule, for that matter) at the very end it's much harder to end up with an answer wrong by 9 orders of magnitude - and not realize it.

I think banning any tool is counterproductive. What one needs to do is teach the reason for doing arithmetic mentally. If your problems are "Compute this" well, you are going to get plug-n-chug, with or without calculators. Ask them instead, what happens when the mass is doubled? Ask them for qualitative behavior if you want them to think that way. And get them to guesstimate the answer before doing any detailed computations. Always.

I teach Environmental Masters students that did algebra 20 years ago and never saw math again. In one semester, we go through algebra, precalculus, probability, statistics, and calculus (differential, integral, Taylor polynomials, and differential equations). I do this by getting them to use calculators/GeoGebra/Excel to get past the arithmetic to learn the math concepts as well as do in-depth explorations. And they get to the point of where a typical good student is after 3 months post-math.

But at the same time, I have them working through Guesstimation so that they not only learn arithmetic, but also have fun with numbers and learn that by sitting back and thinking, one can answer quite a number of interesting questions. It gives them much more competency than any silly ban on calculator use would give.

I crashed and burned in basic arithmetic long before there were pocket calculators, only to learn in my '30's that I am dyslexic and couldn't (still can't) read the numbers reliably. Words have internal context for correction but numbers do not.

I understand multiplication just fine but couldn't get the same answer twice except by accident. And most of my (pointless) homework assignments were 25 problems or more. Countless nights, I went to bed in tears, and despite my fascination with science they assumed I was too stupid for algebra.

Despite multiple attempts I have never been able to memorize the multiplication tables that were so damn important for some reason. Today I call such things; "math trivia" while education standards seem to refer to them as "basics".

Yeah - every time I write about this issue, I have to control my anger. All these years later.

Here's my suggestion then for grade school arithmetic, somewhat based on joemac53's suggestion about giving minimal points for correct answers. Have fewer problems, and focus on understanding what the heck we're doing. Here's an example:

"A 16-oz smoothie is $2.60, while a 22-oz smoothie is $3.20. Write out the formula you would use to calculate the price per ounce for each smoothie."

I realize this kind of thing is difficult to grade in standardized tests.. But if a student can tick four or five of those without any trouble, they've got it. Doesn't matter if they use a calculator or count on their toes.

BTW I still use a slide rule. It's only good to three sigs, but is less confusing. Calculators are nearly useless to me, and spreadsheets are a nightmare, though I can accurately input numbers if someone will read them aloud to me as I type. Something breaks when I read numbers visually.

The only thing more disconcerting than giving them the value of a mass when it isn't really needed (but can be used) is to either give them a mass valeu when it has no role in the solution at all (does not appear in any of the relevant equations) or fail to give a value when it isn't needed because it cancels out. Panic ensues.

"I don't think there are really people out there who can do integration by parts but get stymied by division." But there are, just as there are quite a few who cannot deal with rational fractions because they can't divide polynomials.

And if you look at how the "Everyday Math" folks teach division (or, more precisely, at how the don't teach it) you will anticipate this problem getting much worse in the near future.

BTW, the thing that bothers me the most about calculators is that the students have been using them for a decade and don't know how to use them. I have watched students evaluate a complicated numerical expression by doing one operation at a time and then re-entering the intermediate result after rounding it. Utter crap, and these students have taken several math classes that allegedly require the use of a calculator!

I know plenty of folks,
including myself at times, who,
when given a calculator,
will come up with at least three
different answers to the same problem.

The ability to do math with a calculator is the absolute rock-bottom minimum skill required for modern society--

I disagree. An essential, and more basic skill is to know what the correct answer ought to look like. That involves an internalization of number sense that just won't come if the kids are using calculators.

I teach a low level chemistry class. The kids don't know which number to put into the calculator first in a division problem. When they calculate percent composition, they are likely to tell me that some compound is 450% copper without any hint of how ridiculous that answer is.

It's what the calculator says, so it must be right.

Having just defended a dissertation on calculator usage and gender, I have to totally disagree with on many facets. You are not considering the learning styles of students when you "outlaw" the use of calculators, because they provide visual images that allow students to learn through discovery. I've conducted three years of qualitative and quantitative data, and there are also gender related differences regarding the use of calculator. When students are not taught how to use a calculator, they will not function well or perform well with a calculator. I don't test calculator skills, but neither do I test multiplication and division at the high school level. Many of my tests require a calculator section and a noncalcutor section. Most of the colleges and universities that I am associated with require a graphing calculator, so I don't know where you are that is operating in a manner that doesn't allow students to use calculators. As one of my students indicated, "You must be smarter than your calculator."

The only people who think that graphing calculators should be banned are educational instructors.

Here in the real world, the math problems are much more involved. If you had to do something by wrote, it would take you too long. A university that bans the use of graphing calculators are really denying a skill set to their students that will be needed when they need to apply their math in the real world.

I agree with not using them in algebra, but anything beyond that, it should be integral to the teaching.

Think about it this way, in the real world, how many times do you think you are going to use 1 and 2's for calculations? How many times do you think you will be using complex numbers and punching them into digital devices?