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The kid I am teaching math (subtraction for large numbers right now) just said this is all too easily done by a calculator, why don't we use it?

Well, I did tell him that you can only learn more stuff by learning this, but this seems like an extremely wrong answer. I am lucky that he is interested in learning, otherwise for a kid who does not have a keen interest, this may have been a devastating answer, because the kid may think learning this only brings more learning.

So, what should be the correct way to teach someone that it's better not to use the calculator?

My question was just for small children, but I believe the concept can be carried forward to higher standards to the limit when computational mathematics is a necessity.

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Ask the child why anyone bothers walking when we have cars that can take us places. – Dave L Renfro Jun 2 '14 at 14:17
Why do we learn the meanings of words like "dog" and "cat," when we could always use a dictionary to look them up? – Ben Crowell Jun 2 '14 at 14:30
To avoid near-death experiences with tutors who upon seeing you add zero or multiply by one with a calculator wish to cause you physical harm ? (not me, but I had a friend) – James S. Cook Jun 2 '14 at 23:52
@BenCrowell show me one dictionary which can explain the meaning of those words to someone who hasn't seen either. .. – vonbrand Jun 3 '14 at 1:01
I forgot to take a calculator to a chemistry test once, and I still got 97/100 because I know how to do long multiplication/division. They aren't useless. – Miles Rout Jun 3 '14 at 11:32

14 Answers 14

up vote 42 down vote accepted

To the student who wants to know why he or she should learn addition/multiplication/other mathy thing when they can use a calculator or computer:

It is a matter of independence. You will not always have access to your preferred tool. Being able to compare prices in a store without shamefully pulling out a calculator will boost your confidence that you can make do in an increasingly numerical world. The older you are, the more you will appreciate this.

You will be able to ballpark-verify outcomes. If you want 49 times 51, and you mistype and your calculator says 100 instead of something closer to 2500, you will know this is wrong. A less numerically literate person is resigned to accept the calculator's word as law, which fill help a fat lot when it says ERR.

You can prevent being cheated. (*) People will try to take advantage of you in many ways, not the least being sleight of hand with numbers. Being skilled enough to quickly do these calculations in your head will help you verify or counter numbers fronted by others, and can prevent nasty surprises.

(*) This is close in meaning to independence, but I find it helpful to stress both self-sufficiency in everyday situations and being able to recognize fraudulence.

You are not asked to disavow your calculator. It is not evil or corrupting. Do fall back on it when a calculation takes too much time or you just want a second opinion, as we do too. But you will be a more rounded, self-sufficient individual when you are able to do head calculations or even paper calculations.

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"You will not always have access to your preferred tool." That depends a lot on what calculations we are trying to do. If it can be done only within short-term memory, that's a valid point. But if it requires paper and pen (if "large" in the question means 20+ digits, for example)... How often do you find yourself with a piece of paper, but without a calculator of any kind (computer, cell phone...)? The only example that comes to my mind is no-calc-allowed exams, but it is hardly convincing in this very debate. – T. Verron Jun 4 '14 at 9:07
"How often do you find yourself with a piece of paper, but without a calculator of any kind?" Fairly often, actually. I like having my phone off. And I'm leery of turning on a computer: once it goes on it never goes off again, and I get nothing done for the rest of the day. – TRiG Jun 4 '14 at 9:28
@TRiG, That's funny, I never get anything done at work until my computer is on. :P – Brian S Jun 4 '14 at 14:29
I'd add two things: First, by understanding how the operations work manually, you get a better understanding about how numbers "work". Second, you also practice calculating small numbers in your head, this will be relevant because you will never want to use a calc for every small addition/subtraction. – Cephalopod Jun 4 '14 at 18:00
I would like to add a point about learning concept. You can add and subtract many more things than numbers. Think of sets, polynomials, vectors, matrices etc. Numbers are just a convenient starting point into the math. – Thinkeye Jun 5 '14 at 13:53

I think the issue fundamentally is about a student's comfort with symbols and notation. A student who relies on a calculator thinks of it as a magical oracle that returns an answer to the question being asked. So, when that same student is asked to solve even the simplest of algebra problems, since the calculator can't answer that problem, the student is lost. And since they're not comfortable with symbols, they don't know how to start.

Also, when someone learns subtraction by hand they should simultaneously learn that to check the answer they do the addition by hand from the bottom up. That way, the idea of addition and subtraction being opposites is cemented from the beginning. That way when a student is asked to solve $x+3=5$ they know the should subtract to get rid of the three. And by know I mean at some deeper level than "my teacher told me so".

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"since the calculator can't answer that problem, the student is lost" so long as the student doesn't think to turn to a "smarter calculator", like Mathematica... Neither case (being lost or finding a better calculator) is good for the student in the long run, though. – Brian S Jun 2 '14 at 16:41
@BrianS my math test, at until 16years old (then is you should understand when/how/why use calculator) was never in te form "solve Y() for x" but was a little situation, sometimes even with "not needed data", so you have to think about what variable you need; many that was learning what formula you need to use just looking at the dimension of the given variable and know formula (and not thinking) was tricked by that – lesto Jun 4 '14 at 14:25
"A student who relies on a calculator thinks of it as a magical oracle that returns an answer to the question being asked." Your quote could equally apply to calculators or some traditional methods for accomplishing multi-digit subtraction. Research has shown time and again that students often treat algorithms as "a magical oracle." All the way back to Erlwanger's influential work. The problem identified is not in the technology, it is in the instruction. The calculator's presence is therefore technically irrelevant to this particular problem. – JPBurke Jun 8 '14 at 0:34
+1 for "That way when a student is asked to solve x+3=5 they know the should subtract to get rid of the three. And by know I mean at some deeper level than "my teacher told me so"." I agree. Strongly. – Tutor Jun 12 '14 at 23:02

People who ask why calculators don't resolve the (very low-level) issues well enough are quite accurate in their skepticism, I think, which makes it hard to present an "absolutist" defense of alternatives. For that matter, seriously (!), why is it ok to use Hindu-Arabic numerals and the weird associated algorithms, rather than "honest" manipulations of hash-marks or pebbles? (As a kid, I was required to do a bit of arithmetic in Roman numerals... ah, a lost art!?!) As a small kid, I did do quite a lot of seat-of-the-pants arithmetic by visualizing dots or marks, e.g., multiplication of small integers, while resisting memorizing multiplication tables, and even resisting, for a time, the algorithms. :)

Indeed, schools don't explain why these algorithms work, and, thus, could be claimed to exactly fail to teach "understanding". What they teach is execution of algorithms with a certain interface.

My point is that the defense of learning how to manipulate the standard algorithms with Hindu-Arabic numerals cannot be mathematical, cannot be absolutist, but can only refer to particular contexts, _if_any_, in which there is some immediate, practical advantage. Not fake in-principle advantages, not marooned-on-desert-island advantages.

In particular, it is hard to defend a claim that the standard algorithms with Hindu-Arabic numerals give "understanding" more operational than facility with a calculator. Although I once would have claimed so, I capitulated at a point when each month's bank statement had 50 checks, and my (on the whole very quick and accurate) mental or on-paper standard calculations simply had too many chances to fail... while my wife's (for professional reasons) extreme facility doing arithmetic on a calculator did truly afford a much lower error-rate. For 20 years, I have not checkbook-balanced by hand.

And, since I am essentially always near a computer, if I need to do any computation (numerical or otherwise) beyond a certain very, very low complexity, I fire up Python or Sage or...

The immediate defenses of learning the standard things are just two, and somewhat flimsy: first, many older people will judge one on this, and this (dubious) expertise will be used as a filter/weeder (as in the "do you want to be a manual laborer?" defense), and, second, for a person interested in science/technology, much older literature (and other context) assumes familiarity with such stuff, as implied context. The latter reason to pay attention is less invidious than the former.

In reality, I (as a professional mathematician) think there's scant reason to pretend, especially to small kids, that there are absolutist reasons to acquire facility in execution of visibly peculiar algorithms in manipulation of Hindu-Arabic numerals... even though "all the old people" know these things. Any sensible kid will believe anything else you say less after any kind of rant that attempts to claim that they shouldn't depend on computers, much less calculators.

A similar issue arises in having the first-year calculus be so caricatured that many different software packages, many "free/open", can do it all far better. That is, what the heck should we convert these courses to... rather than be foot-draggers and insist on a highly antique, stylized, unconnected-to-reality drill that few of the participants take seriously?

Indeed, the real problem, to my perception, is that as machines can do more and more algorithmic things, the "easy" (algorithmic) stuff that once was a fine human province is no longer available. Only more difficult things are left... and it becomes ever more difficult to pretend that we'll insist that the whole population learn lots of things that machines cannot do. An awkward situation.

Summary: I'd recommend not trying to defend any prohibition of calculators. Rather, grant calculators, and then you can ask the next level of questions. When the kid gets access to and learns how to use symbol-manipulation software, then those issues, too, are taken for granted, etc.

Yes, I know, your local curricular constraints... Still, I do often find that clearing one's head by thinking what would make sense if there were not random exogenous constraints is, first, soothing, and, second, does put the necessary compromises in perspective. :)

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In principle, I see your point here, but, I wonder, in practice, after decades of use of TI-83 etc in our highschools, are the students really going on to the next level of graphing questions? Is there calculator-based improvement masked by other educational degradations? – James S. Cook Jun 2 '14 at 23:47
@JamesS.Cook, my anecdotal evidence indicates that both personnel and textbook limitations prevent conversion of "mathematical education" to an arguably higher level. As we know, anything for high schools has to be mass-produce-able, which requires near-universal teacher-training curricular reform, in the first place. Second, the "parental objection" obstacle creates a lot of noise and consumes much time+energy, as is widely documented. Perhaps the latter will change in another generation, when/if hand-execution of algorithms loses the sanctity of the past, and machines are more taken-for-... – paul garrett Jun 2 '14 at 23:51
... granted. But, yes, in fact, facilitated by even the simplest graphing calculators, (engaged) kids can experience far more, with less effort, than I ever could, 50+ years ago (computing values by hand, sometimes using trig tables, plotting points...) Most of what I learned from plotting by hand, or multiplying or dividing big numbers by hand, is that it was easy to screw up, and that it was not uplifting or edifying. But there was no choice. Kids these days know much more how graphs look than aeons ago. A good thing. But how to test...? :) – paul garrett Jun 2 '14 at 23:53
I find this answer perfect to defend my own use of calculators. But for a person who constantly confuses $12$ and $21$ and forgets $16-9=7$, there can be not a single more horrible answer because if we promote calculators rather than demote them I do not believe that they would ever actually learn maths and then science and may not be able to really do something useful in life. – Rijul Gupta Jun 3 '14 at 6:15
"Indeed, schools don't explain why these algorithms work" Frankly, when I was a child, I rarely surprised why things work. I was always interested in how they work and how to use them. And only when I got older, had loads of knowledge of "how's", I began to wonder why they appear to work. I think this would be the same for most children, so the schools/teachers don't have to teach a general kid why the addition algorithm works — but should do this only if asked. – Ruslan Jun 3 '14 at 11:54

The reason we teach strategies for multi-digit operations is (in a large part) because students are learning to manipulate the symbol system we use to represent numbers, and they need to see that there is meaning behind these strings of digits. That understanding carries through not just for subtraction, but for anywhere they'll use multi-digit numbers.

To put it another way: practicing subtraction is an opportunity to learn about how these number representations are made up (their structure). They're learning how that structure yields to their desires as mathematicians.

This is precisely why teaching subtraction as a specific algorithm fails. Subtraction as a "standard algorithm" actually isn't any better than using a calculator. Both substitute a mechanism (rote steps or calculator clicking) for understanding something about the actual mathematics behind multi-digit numerical representations in a base 10 number system and its implications for performing an operation. Standard multi-digit subtraction simply makes the student into a kind of computer that follows subtraction steps.

However, if you're teaching the student ways of breaking down multi-digit numbers to make subtraction easier (understanding the base 10 number system, decomposing, manipulating, and composing a large number, and using it to your advantage) then there is a lot of actual mathematics going on, and decisions based on mathematical knowledge.

In short (and this might be obvious), the answer depends on what you're teaching.

Let's say that in your classroom a student sees this:

\begin{align*} 2367 \\ -1989\\ \hline \end{align*}

... and it would not be unusual for them to respond with:

"I can make this problem simple. 1989 is really close to 2000. If I add one, I get 1990. I can add ten to that and get 2000. Ten plus one is eleven. I think of 2367 as my starting place. If I'm trying to subtract from it, I can add any number to it, as long as I remember to subtract that out again, too. So I can write the problem like this:

\begin{align*} 2367 + 11\\ -1989 + 11\\ \hline \end{align*}

"... and that's really this:

\begin{align*} 2378\\ -2000\\ \hline \end{align*}

"...which is simple. The answer is 378."

It's hard to argue against the view that:

  • This student is thinking mathematically, not just following an algorithm.
  • This student did some problem solving, involving decisions based on number sense and understanding of the place value system.
  • This student got some practice with arithmetic.
  • This is very different from what a calculator does, and difficult to say you could replace this reasoning with a calculator.

Of course, maybe your student doesn't just say this right away. Maybe there is some work going on, different representations like this along the way:

Getting from 1989 to 2000 in two decimal hops

... it doesn't really matter as long as it's good mathematical reasoning.

The point is: if the students are reasoning mathematically, that's something a calculator cannot do for them. In that case, it is a lot easier to both see and argue that the calculator is not appropriate for the activity. Other times and in other cases in math class, there are a lot of ways it is appropriate to use or explore with a calculator. But if the only thing they're doing is something that a calculator can do for them, and that's why it's hard to see why they couldn't just substitute the calculator in, that's not a problem of explaining to the student why not to use a calculator. That's a problem with what the student is being asked to do.

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I don't think there is one "right" answer to this. But one possible answer (assuming that we agree you are right!) is that you want to be able to know whether you made a mistake, or at least to recognize when things are suspicious. There's no auto-correct for subtraction on a calculator, or computer.

I usually use this argument when it comes to differentiation and integration, but the same principle applies. Naturally, this reason won't obtain in every situation, and of course you don't want to do 100-digit subtraction by hand - the calculator will be more accurate than you are, assuming you type it in right (also a necessary condition for doing it correctly by hand). Yet, the better you understand what the tool is doing, the better you will use the tool.

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100-digit calculations are not very likely to work on a handheld calculator. A computer would require using the right libraries to get an accurate answer. – jpmc26 Jun 2 '14 at 21:49
100-digit calculations where one need an accurate (not approximate) answer only occurs in pure mathematics. – kjetil b halvorsen Jun 3 '14 at 8:30
@jpmc26 The sentence about computers is not true anymore, "big ints" are nowadays a part of standard libraries for many programming languages. – dtldarek Jun 3 '14 at 12:34
Yes, Sage certainly supports arbitrary stuff out of the box and I assume that is true for the proprietary folks as well. Oh, and I don't think that 100-digit precision is just used in pure math, otherwise people wouldn't ask for that precision with special functions. Error propagates... – kcrisman Jun 3 '14 at 14:48
@jpmc26 Your comment is only technically correct. A similar claim would be: you need right libraries to print anything to standard output (e.g. libc is a library). You make it seem hard, while it is an easy thing to do. – dtldarek Jun 3 '14 at 15:30

All of math is a logical progression. One should master a type of calculation before turning it over to a calculator or computer. Can I multiply 4386x934754 by hand? Of course I can. And it was important to learn to do this in 4th (?) grade, but soon after, no need for long multiication or division.

Every day, I'm intrigued at the disparity between the students who rely on the calculator for the simplest of problems vs those who take pride in doing it in their heads. I can't quite claim cause and effect, but in my opinion, it's rare to find a heavy reliance on the calculator and a high mastery of the subject going hand in hand.

What's missing from the conversation so far is exactly when it becomes appropriate to use the calculator. What scares me is when a student's answer is grossly wrong and he doesn't have the skill to see that for example, a rock wont take 1000 seconds to fall from the tallest building on earth. Or that when you get a result of .84 for sin(1), you really meant to hit 2nd sin asking for inv sine to give you 90 degrees. And the calculator should be on degrees, not radians. A certain level of mastery should be required on a process before letting the calculator take over.

The above translates to "tell the student it's important to understand how to do math at each level before trusting a calculator's answer."

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"in my opinion, it's rare to find a heavy reliance on the calculator and a high mastery of the subject going hand in hand" In my experience, the opposite usually holds true, namely that high reliance on a calculator usually goes hand in hand with high mastery of the subject, but that may just be because the people with high mastery of analysis tend to be very skilled with computers, and thus tend to outsource their useless, unenlightening grunt work away from paper and onto silicon. There may also be a generational gap at play. – DumpsterDoofus Jun 3 '14 at 22:00

For young students especially, there is intrinsic value in all 'physical' work done with numbers, and that intrinsic value is the steady accumulation of number sense. I don't mean that kids should be endlessly crunching numbers, but a balance should be met.

Students who are moved towards calculators at the first instant that they're more convenient often end up on a slippery slope. I've seen high school students punch expressions like $\frac{4*3}{2}$ through a calculator, which betrays severe innumeracy, and is an enormous waste of time besides. If this step occurred in a more serious 'problem', then the time of punching these calculations into their calculator is also likely to have been enough to interrupt their working memory and distract their train of thought from the more abstract (and interesting) thing that they were supposed to be working on. This is an enormous problem.

It's possible to explain the difference between 'problems' and 'exercises' even to young children, and if there is good rapport, it's possible that they can be convinced to trust your judgement that prescribed exercises are for their benefit.

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When doing a problem like 8x6, how many students do you witness adding eight sixes together?

It's been my experience that most students will instead memorize a series of reference points and skip the repetitive addition. The question then is, what is gained by spending months memorizing the results of an equation (for instance, times tables) as apposed to spending a few minutes learning how to work with a calculator?

I empathize with the argument against calculators. It's much harder to prove somebody knows what multiplication means when they can blindly punch a few numbers and get the same numeric results, but maybe that just means we have to update our tests to match current technology.

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To be clear, do you believe the 10x10 multiplication table isn't worth memorizing? – JoeTaxpayer Jun 2 '14 at 23:16
As a child I loved math. I loved doing it, and I loved learning it. I was fortunate enough to see the fun and application of math in areas beyond schoolwork. So I must have gained a lot from my mental math lessons, right? No, in truth I found I could speed up my results and increase my accuracy with a watch calculator which I kept with me at all times. I've been constantly calculating my whole life, and the only real use I've ever found for multiplication tables was passing elementary level math classes. – DiscOH Jun 3 '14 at 17:09
When I do 8*6, I double 24 in my head. Strictly speaking, I've never memorized 8*6. And it's never been a problem. – JPBurke Jun 3 '14 at 21:03
@JPBurke but if you always used a calculator you likely wouldn't have the acumen to convert 8*6 to (8*3)*(6/3) in your head first. – Dean MacGregor Jun 5 '14 at 12:07
@DeanMacGregor To be fair, yes, my understanding is more sophisticated than that now, but around 3rd grade it was simply the application of a rule. – JPBurke Jun 5 '14 at 12:17

When you solve problems without a calculator, you are training your mind to solve that class of problems. It warps the way you think about numbers, and forces you to abstract problems to solve them more easily and quickly, until one day they become trivial to solve in your head. Rinse and repeat for every new concept you learn in math. A calculator robs you of this experience. Learning about math isn't about solving a singular trivial problem, it's about being able to solve the next problem faster.

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I myself am not asking why we should not encourage usage of calculator! The question is how to teach children to stay away from these calculating beasts. – Rijul Gupta Jun 2 '14 at 16:51
I disagree with "it's about being able to solve the next problem faster." – Mark Fantini Jun 2 '14 at 17:13
@Fantini you just take longer to get to consider the next problem... – vonbrand Jun 3 '14 at 0:55

I think you have to separate the issue of whether the student understands numbers at all from the use of calculators. I once had the rather thankless job of trying to teach continuing studies physics to someone that was convinced that

0.5 + 0.5 = 0.10

Their answer when I pointed out to them this wasn't right - get out their calculator !! Needless to say, trying to educate this student didn't go so well. The person had never understood numbers at all, and still in their 30's didn't. It's not so much about can we use a calculator - but ensuring that the student doesn't get through by mindlessly pressing buttons and copying answers off the screen whilst having no clue whatsoever what they are doing. Unfortunately, over reliance on the use of calculators can lead to this sort of total failure of mathematical education. Once the student has masters numbers decimals and fractions, then in my opinion if they want to use calculators - let them.

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It seems like a calculator would have been the best way of showing this student that they were wrong... – jwg Jun 4 '14 at 13:29

Given that the time alloted to math in school is finite, I'd be glad if the children learned more meaning and how and why to sum/multiply/integrate than mindless application of rules by rote. It is certainly much harder to teach/test...

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I would argue the correct way to answer this is is the same answer that one might use in teaching the foundations of any discipline. Everything starts with the basics - eventually yes, it will be easier to use the calculator, but if you never understand the basic way of thinking that all math relies on - then you will never be able to do anything with math.

Present the student with the concept that in any job, any personal endeavor they ever undertake, they may likely use some kind of math, be it algebra or basic addition/subtraction/etc. In some cases, a calculator may or may not be at hand (these days, some form of calculation device or app will likely be easily available in the developed world). Someone who has some basic grounding in mathematical concepts - a comfort with numbers - is less likely to get cheated when buying a car, a house, a microwave, a video game, and is going to be better able to cope with computers, which operate entirely on a mathematical basis.

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Computing $3^{2^{32}}$ by hand, using algorithms learnt by rote having not the slightest clue why the results computed are right is a fine way of "learning the basics"... – vonbrand Jun 3 '14 at 0:53
I agree with this. Sure, there are computer theorem provers, so why try to prove anything by hand in first year undergraduate mathematics? Because you have to start with the basics. – Miles Rout Jun 3 '14 at 11:35
@MilesRout perhaps one day the "automated provers" will get to point of doing the routine proofs for us (given reasonably natural-language description of the stuff to prove, or of a proof to check, or of one to fill in details) – vonbrand Jun 4 '14 at 1:41
@vonbrand And whether or not they do get to that point, students will still have to prove things, as they should. – Miles Rout Jun 4 '14 at 5:52

One point that seems worth emphasizing is that learning to perform an operation (even a mechanical and uninspiring one, like long addition/division) helps you to understand better how the the objects you operate on really work. Being able to perform addition is about a lot more than being able to magic up the sum of two numbers. It also involves knowing what would happen if you were to perform addition, without actually doing it.

Suppose you want to do something a little more exciting than add integers base 10. Suppose, for example, that you are learning about a different number system, and you want to add numbers base 3. The standard techniques for adding two numbers carry through without much modification. The approach "use a calculator" fails miserably. Suppose you want to write a computer program adding two very long integers (i.e. program the very calculator!). You have to know how you would do things, in the first place. This may fall under the heading of "learning stuff to learn more stuff", but may be useful for people who like computers, or are mature enough to learn about different bases. (I would be so bold as to suggest that mentioning different bases would be very helpful to some students in understanding the "standard" base.)

As another example, consider a situation when your input is not just a bunch of random digits, but is something more structured. For a trivial example, say you are adding $1230000000000000$ to $4560000000000000$. It may be obvious that the answer is $5790000000000000$ (i.e. the zeros stay where they are), but if it's not, then imagining you are doing long addition is a sufficient argument. A less trivial example is adding $1$ to $999999999$, or multiplying $111111111$ by $11111111$. Or you can derive the criterion for divisibility by $9$ by imagining you are performing long division, without ever using the word "modulo". Or adding two very "sparse" numbers. Or... etc, etc. To rephrase this: if you are operating on numbers that have some regularity, this can help you if you are doing things by hand. It won't if you are using a machine.

Finally, there are also a lot of problems where you are supposed to find long numbers where some digits are unknown and a single equation is given (One example I just found is to find a solution to: SEND+MORE=MONEY, where each letter is a digit; there are also simpler ones.) The only way to approach these is to know about long addition.

By the way, if you are adding two numbers, but are only interested in the digit at a given position, for whatever reason, then long addition allows you to find this digit with minimal effort. Again, calculators do not provide this.

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Teaching math isn't all about teaching numbers, it's about teaching a step-by-step approach to solving larger problems. 8x6 = 48 is one thing (is this math or just retrieving memorized information?), but think about bigger problems later on with "solve for x" problems in algebra. PEMDAS is a good example: step 1, step 2, step 3, [...], you're done. Problem solving is what you're really teaching. We apply this same line of thinking to other subjects, like mechanics. If you open the hood of a car, you see lots of complex machinery. However, if you can follow directions, you can fix all sorts of things on your own. Math teaches us problem solving skills early on, unlike most other subjects.

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Re-read the question please. – Rijul Gupta Jun 2 '14 at 16:56
What you describe is "following an algorithm", I consider "problem solving" more in the line of coming up with a correct, reasonably efficient algorithm. – vonbrand Jun 3 '14 at 0:57

protected by quid Jun 2 '14 at 23:15


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