# Is there a virtue to learning how to compute by hand?

I have been professionally tutoring a wide range of students (from elementary school through graduate school) for many years. Most of them are from the United States. I generally focus on helping my students develop procedural fluency built on a conceptual foundation (i.e. "now I can do the thing and I won't forget how to do the thing because I understand what it is and why it's useful"). I think that a conceptual understanding is very important but only because it serves a utilitarian purpose of helping students remember methods and feel empowered as they work. With this is mind, I happen to think that knowing how to add, subtract, multiply, and divide by hand is important. However, I am beginning to think that perhaps I am simply biased because of my upbringing, and that such things really aren't valuable anymore.

The vast majority of my students are younger than 20 years old, and the way they've been taught math is almost unrecognizable to me. Most of them have (for as long as I've known them) never taken a math or science test during which a calculator wasn't explicitly allowed. As a predictable result, they rely on their calculators to do all calculations. I suspect that most of them could rattle off some of the times tables if their life actually depended on it, but I've seen all of them type single digit multiplication and addition problems into their calculators. I know for a fact that many of them never learned the old trick for multiplying by 10 (you know, putting a zero at the end), and almost none of them know how to check for divisibility above 2. On top of that, my grade school and middle school students generally don't remember any method for conveniently adding, subtracting, or multiplying by hand. Some of them can multiply by drawing boxes or number lines, but the ones who can don't understand what they are doing.

For older students, we generally have bigger problems and more abstract topics to tackle, but for my youngest students (4th - 8th grade), I try to encourage them to develop a familiarity with the simplest forms of by-hand addition, subtraction, and multiplication (you know: one number on top of the other, carry the 2, etc.). While the methods are usually familiar to them, this sort of encouragement generally goes nowhere because their teachers don't care and because they always have either a dedicated calculator or their phone handy.

I tell students that calculators aren't always allowed (maybe that's not true anymore). I tell them that knowing is faster than typing it in (the difference is admittedly minimal). I remind them that different calculators have different syntaxes, so it's important to have a way to do it by hand if only to check the calculator (true, but not persuasive). I even joke that the world might end, and they'll need math when the electrical grid goes down (they remind me that their calculator has a little solar panel on it).

Truthfully, it's depressing, and what's worse, I'm not sure I'm convinced anymore either. I mean, I don't know any convenient algorithms for computing square roots or logarithms by hand but that never held me back.

So, to restate the question, is there actually any demonstrable, practical value in learning to add, subtract, multiply, and divide by hand (even if the reason might not be persuasive to a 10-year-old)?

• One point in favor of basic calculation skills is that small calculations will unavoidably be a part of later math (ex: developing guesses when factoring) and familiarity with the mundane aspects will help them concentrate on the real issues at the higher level. As far as the computational tour de force of calculating something like (14.321 * 15.686) by hand, I don’t think there is value to that. The reasons why the usual “line up the numbers” algorithm worked were not apparent to me as a 7-year old and the later explanation does not superficially resemble the “line em up” method either. Mar 1 at 19:53
• As far as basic calculations go, there are still plenty of everyday examples where being able to calculate by hand (or in your head) is useful. E.g. I have $10, how many \$1.80 chocolate bars can I add to the basket? How much change am I expecting if I buy three? etc. Surely (I hope) your students would not have a habit of pulling out their calculators at a cashier. Mar 2 at 4:15
• @Steve "As far as the computational tour de force of calculating something like (14.321 * 15.686) by hand, I don’t think there is value to that." — I'd say it is about 225. Sure, there is value in it, when you are buying a Christmas card with 15% off, and the register does not have info about the discount, and the clerk cannot figure out how much to charge (true story). Anyway, one must understand who was interested in arming students with calculators: HP and TI, first of all. With graphing calculators selling for \$100 even today when a smartphone costs \$40, this is just business. Mar 2 at 5:21
• @Steve I know I'm odd but I do a lot of mental arithmetic when cycling - estimating arrival time given average/current speed, distance so far, and remembering total distance. This needs approximation, subtraction and division/multiplication in bases 10 and 60, and I typically calculate to a precision of about 2 minutes. As I'm riding, it's completely hands-free, mental exercise to go with the physical Mar 2 at 11:25
• I have never encountered a student who both has a dependency on calculators to do basic arithmetic and understands the basic arithmetic operations well enough to know when to use which. I'm sure you're all too familiar with a student reading a word problem with two numbers and asking "Should I add these or multiply them?" Those operations are so dissimilar it's hard for me to imagine a scenario where it isn't absolutely clear which to do. But if you only know how to push buttons on a calculator, multiplication and addition don't seem so different. Mar 2 at 18:19

I couldn't agree more with @Steve's comment. The following response is written with elementary-to-high-school mathematics in mind.

• A lack of a decent number sense really does encumber making sense of and parsing word problems, as well as the process of exploring solution strategies. It is akin to interpreting a passage written in a not-so-familiar dialect: attention to the actual substantive problem at hand is being competed for.
• Performing a sequence of algebraic manipulation involves micro-decisions along the way that don't require a calculator, but definitely do require facility/flexibility with arithmetic and fractions, to be achieved efficiently and effectively.

Using the calculator as a crutch significantly atrophies the student's numerical fluency and impairs the development of their intuition and mental dexterity. With a weak number sense, even basic procedures consume valuable working-memory space (sometimes causing a bottleneck), interfering with higher levels of competence (interpreting problems, drawing connections, devising strategies, reasoning correctly, etc.).

The student who is comfortable computing by hand avoids these pitfalls.

On a related note, a side effect of being over-reliant on the calculator is the subconscious tendency to reach for and jab at it as a distraction from actually trying to make sense of problems.

So, while I agree that long or non-simple computations should be outsourced to or aided by a computer/calculator, I believe that, especially in elementary school, students ought to be given ample opportunities to cultivate numerical fluency and number sense (and the attendant self-confidence). Calculator-prohibited tests aren't a bad idea to prevent calculator “addiction” down the road.

Summing it up: while mathematics is certainly not just computation/arithmetic, being adept at the latter enables us to focus on—and informs—the more creative aspects of the former (e.g., recognising patterns, discerning structure, comprehending & modelling & solving problems, communicating with accuracy & unambiguity & conciseness & persuasiveness & wit). Students who learn to depend on a calculator for trivial computations will also have a curtailed risk comprehension and statistical and scientific literacy.

Echoing @DaveLRenfro's comment and the other excellent replies, another invaluable virtue of learning to be non-reliant on the calculator (developing numeracy and competence with ratios & percentages) is the facilitation of (independent) sanity checking, estimation and error-checking—not just at the end of a problem solution, but throughout the process—not just on paper, but also in routine day-to-day situations.

• I strongly second this, and am distinctly reminded of a high school math class where my peers were dumbfounded that I could do 2x2 matrix multiplication with small integers faster in my head than they could on a calculator (and for that matter that I could do trig problems on a slide rule faster than they could on a calculator). Mar 2 at 12:34
• This is a great answer. Though I think you hinted at it, I’d like to make it explicit that, in my mind, there’s an element of psychological stance around how you engage with any problem mathematical or otherwise. Do you try and solve it yourself, or do you immediately reach for the person or tool closest that you think can solve it for you? When you’re young and learning, an emphasis on the former is crucial; placed on the exploration versus exploitation axis (often discussed in the context of learning systems), a student should seek to explore what they can do, not exploit the nearest tool. Mar 2 at 22:29
• I wish that when I was in grade school, I'd been made to practice much more mental math and much less how to solve things with a pencil and paper. As an adult, it is much easier for me to grab a calculator than to sheet of paper, but doing a calculation in my head (even if it's an approximation) is easier than both. I also think it would have helped my "number sense" way more; I'm not really convinced e.g. long division helps with that at all. Mar 3 at 20:50
• This is very much what I thought. I was appalled when I heard that they were phasing out hand arithmetic in favor of calculators because "you need to emphasize conceptual reasoning". Yes, you do, "but dammit, you don't get it - it's both/AND! not either/XOR!" The most headpoppish thing and I'm not at all surprised at what the results of this have been. Seriously, if you can't even multiply one-digit numbers ... sheesh. Mar 5 at 1:45

I find the ability to estimate calculations quite useful and I think you need to be able do do calculations to estimate them. If you are keeping a grocery budget, I would suggest you should know what your groceries will cost within $$10\%$$ before they are rung up. To know that, you need to be able to add and multiply in your head. You don't need many digits of precision, but you need to be able to do it. I was in engineering during my career and it helps a lot to know what is important and what is not.

• That's the most important point IMHO. People relying 100% on calculators won't be surprised when the results are off by many orders of magnitude, e.g. because of . vs ,, or 1e4 vs 1*e*4 or Windows Standard calculator still being too stupid to calculate 1 + 2 * 3. Mar 2 at 7:41
• @EricDuminil to be fair, Windows Calculator displays "3" as soon as you hit "x" and displays "3x3 = 9" as end result, so at least you know what its doing... Mar 2 at 8:05
• @Polygnome: Good to know, thanks. It's a bit less wrong now. Mar 2 at 8:27
• This. I tutored in college and the number of 3rd or 4th year students who needed help with basic algebra problems blew my mind. Not every job requires linalg or differentials (even in stem I don't use a lot of what I learned from upper division courses) but a minimum of algebra & geometry are necessary for daily tasks. Relying on a calculator for everything is akin to only learning the bare minimum vocabulary, then carrying around a dictionary to look up unfamiliar words. Sure, it's technically possible - but you're going to miss 90% of what people are saying. Mar 2 at 22:33
• @ChrisH: Yes. Last year, a soon-to-be-engineer from our university was pleased to announce that the photovoltaic modules on his small roof would produce more than 15 000 000€ worth of electricity each year. :-/ Mar 4 at 15:43

Yes! But the virtue doesn't lie in being able to do the calculation but in gaining a feel for numbers as well as algorithmic thinking.

I teach Computer Science freshmen and one of the first things we need to do is introducing base 2 as well as number systems working with modulo (two's complement). We also introduce basic circuitry to do addition/subtraction and multiplication/division.

• In order to understand why "shifting" left/right in binary is multiplication/division by 2, it's helpful to have seen the same concept with 10

• understanding the, very frequent, check for even/odd in binary is much easier when you have a basic grasp of how number systems work

• binary subtraction is a hard one for many, I frequently need to re-teach them decimal subtraction first. If I give them a subtraction $$a-b$$ with $$a and ask them to do it by hand, they don't manage it (since we create infinitely leading 9s/1s). This is an important insight when implementing algorithmic subtraction.

• Understanding algebraic groups like the one set up with two's complement (equivalence classes using modulo) is much easier.

• Long division and written multiplication gives easy insights on why, e.g., multiplication of two $$p$$ digit numbers requires at most $$2p$$ digits. Similarly, why addition can only increase the number of digits by 1.

• Finally, half/full/carry-ripple adders are intuitive if you have worked with written addition (carry digits, etc.)

So, rest assured, all of this is useful. However, it's a whole different ordeal to convince your students. Depending on their level you can try to ask them to "algorithmicize" addition (and even program it!) or ask questions like "whats the maximum number I can have if I multiply two $$n$$ digit numbers?", with emphasis on $$n$$, i.e., not a fixed number. They also need to explain why.

If you can find exercises that actually make them need to use written calculations, they will be motivated to learn these.

I taught at the elementary and high school levels. At times we used calculators and at times we didn't. Students benefit from experience both ways. Students need to learn that calculators are only a tool and they still have to think. Students also need to learn that having a calculator doesn't guarantee that their computation will be correct. Finally those that can't compute have trouble with algebra.

I usually allowed my 5th grade students to use a calculator for our chapter on mean, median, and mode. I taught them that if you are adding 10 numbers to find the mean, you should add more than once because there was no guarantee that you would get an accurate answer the first time. After a test using calculators the students were always surprised that even with the calculator the test was still hard.

In 9th grade we introduced graphing calculators. Students who didn't understand the order of operations had trouble figuring out when to put in parentheses, resulting in lots of wrong answers. Students who didn't understand that 0.33333333 is an estimate of $$\frac{1}{3}$$ couldn't understand why their answer wasn't exactly right. Similar mistakes were made with pi.

When I taught algebra, I discovered how important it is understand computation with fractions. A student won't be able to add $$\frac{x}{3x+2}$$ and $$\frac{x+1}{3x-2}$$ unless he already knows how to add $$\frac{1}{3}$$ and $$\frac{2}{5}$$. As mentioned above a student who doesn't understand order of operations will have trouble simplifying expressions. A student who doesn't understand that $$\frac{x}{3}$$ is the same as $$\frac{1}{3}{x}$$ will have trouble solving equations.

There is a place for calculators, but it is only a tool and students need to be able to calculate, estimate, and solve problems. Students need to learn to use the calculator efficiently and effectively.

Brian D. Rude, "The Case For Long Division." 2004. HTML link.

This is a somewhat long (unpublished) article (which I haven't studied carefully), but maybe the excerpt below suffices to give the gist of it. Before this excerpt, among his closing sentences are: "But a calculator should be more than a paperweight. Let’s teach for understanding."

"My argument is that long division is just one procedure, among many, that promotes the usefulness of arithmetic, that helps to make arithmetic a life skill rather than just one more subject forgotten, one more door opened but then allowed to close again by lack of relevance to everyday life. I will make my argument for long division purely on the utilitarian aim of education. My argument is that even the person who has no intention of learning anything more than perhaps ninth grade algebra should learn long division. [...] But my argument has very little to do with being able to divide one number by another when an actual division problem arises in everyday life. We’ll use a calculator for that. My argument is that we should not expect anyone to really make sense out of arithmetic if they can’t do long division. If you can’t really make sense out of arithmetic a calculator is nothing more than a paperweight."

I'm not evaluating his argument; just posting as someone's strongly held opinion. Although I certainly agree that we should "teach for understanding"!

I attended the Computer based math educational summit back in 2016 and found their ideas interesting. I agree with some of their points and disagree with others, but it is certainly interesting to look at the following diagram from their website.

They argue, that if we let computers (proper ones, not handheld calculators) do the repetitive calculations, then we can focus more on setting up problems and interpreting results. Moreover, we can tackle more interesting problems. This is a different type of thinking, but logic, reasoning, algorithmic thinking is certainly part of it. Their website is https://www.computerbasedmath.org/

• Why not learn both? Once you start dealing with algebra, it's essential to have a good sense of the simple calculations and fractions in order to deal with coefficients and non-polynomial formulas. Sure, we might not really need things like long division anymore, but 1-digit multiplication and 2-digit addition are incredibly helpful. Math isn't just about getting the right answer, but understanding how to get there. Mar 3 at 16:37
• @Beefster I also think that a right balance is useful. However, I would make a distinction between numeracy and mathematics. Most of the discussion in this thread is focusing on the reliance on handheld calculators. This is in the category of numeracy for me. A strong numeracy background is indeed important as a foundation. However, what we do once we have these foundations can be different. In my opinion an early introduction of some kind of coding in a mathematics curriculum would be useful. It would enhance critical thinking. Mar 4 at 6:22

Most of the real work in doing math is understanding and conceptualizing the problem rather than in computing an answer. This will be apparent to anyone who has read a mathematical journal article which included no numerals anywhere in its text. This will also be apparent to a high school student in algebra or calculus who spends most of their time "setting up" their problem rather than "solving it." The value of a calculator is to make the final step of a problem (i.e. solving it) easier; the value of pen and paper is to make the conceptualizing part (i.e setting it up) easier. To forgo learning pencil-and-paper computations early on is to build habits that make conceptualization more difficult later.

In learning arithmetic, conceptualizing a problem is often as simple as determining whether to multiply or divide. Writing this down seems like a trivial formality even to (perhaps, especially to) children. As a result, it can appear to a student that mathematics is merely the process of figuring out what operation to perform and then doing it. To an extent this may be true, but figuring out what to do is generally very hard. In fact, it is so hard that we emphasize word problems at all stages of math education precisely to train this skill. In later courses, it becomes so challenging that pages of hand-written work are often necessary if for no other reason than to keep a record of what you were thinking in case your answer is later determined to be incorrect. However, because of the seeming triviality of arithmetic, it can seem that the conceptualization is trivial too and that we may as well skip the formality of writing it down.

This is exactly where the rubber meets the road. Figuring out whether to multiply or divide two numbers is anything but trivial - especially for children just beginning their mathematical education. And the skills and habits they learn at this stage will inform their habits later on. Do we really belive that a child who never learns to bear the yoke of long division will have the patience to sort through integration by parts? Or worse yet, can we really believe that children who are not instilled with the virtue of showing their work will suddenly learn to keep meticulous notes when it really matters to themselves or their colleagues?

By-hand computations are useful for many reasons: building familiarity with useful ideas, improving estimation skills, and creating a sense of independence and self-reliance. As others have pointed out, all of these things are important and valuable in daily adult life. However, one virtue stands out to me as most valuable: learning to show your work. What will a student who has never learned to show their work do when they encounter a problem (mathematical or otherwise) which can't be solved in one step? I think many will throw up their hands in frustration. I think many will believe that if they can't do it in their heads or on their phone then it is too challenging to be done at all.

There will come a day in a student's life when writing out the details of multiplication will cease to be important, but careful and well-documented thought will never lose its value. And if we're not teaching children to be careful, accurate, and useful thinkers, then what are we doing?

• "What will a student who has never learned to show their work do when they encounter a problem which can't be solved in one step? I think many will believe that if they can't do it in their heads or on their phone then it is too challenging to be done at all." <--- THIS. Fantastic answer. Mar 3 at 6:41
• Not (properly) showing their work (documenting their mathematical thinking for review/communication) is analogous to bad programming style (writing unreadable code that is hard to debug/understand). Such habits/practices are preferably cultivated as early on as possible. Mar 7 at 2:46
• What will a student who has never learned to show their work do when they encounter a [real] problem which can't be solved in one step? I think many will throw up their hands in frustration [and decide] it is too challenging to be done at all. I run into this ALL the time. Perseverance and determination are things that aren't readily taught since that comes from within. This: youtube.com/watch?v=pmteqkbBfAY would not have been possible without long, hard hours of extensive research and development with two spectacular failures along the way. May 18 at 11:49

Beyond having worked as a programming teacher I have no experience with math education, but this is a topic I have been fascinated with for years.

Arguments in favor of mental/manual arithmetic can be typically categorized as:

• ### It's important in daily life

The argument typically goes that you need to be able to do arithmetic a lot in daily life with the most common example being buying items in the shop. It's good to be aware that physical shopping has been on the decline for decades and even in physical shops handheld scanners that tell you the sum on the go are on the rise.

As someone who has relatively bad arithmetic skills and seeing how most people function with even worse arithmetic skills in daily life I dare say this is just not true nowadays.

• ### It's important to be able to do arithmetic for maths

Let me very clearly state that this might be true for students who will be studying pure maths at university level. For everybody else the most important life skill is typically being able to reduce a problem into steps that allow coming to a conclusion. The typical math education example of this is word problems. The ability to express a real problem as a maths problem is typically the hardest part, but any time spend teaching students arithmetic takes away from the time spend on teaching "problem solving". Understanding for example what integration means and when it's useful is far more valuable than the months we spend on actually doing integrations by hand.

To this day I value the ability to express something in mathematical terms, as the next step is going to a calculator or wolfram alpha to actually get to a conclusion/the next step. In my experience most mathematical educations are incredibly focused on the mathematical computation and the 'solve actual problems' is barely touched upon.

• ### It's useful to catch mistakes

The argument goes that being able to do mental arithmetic makes it easier to notice mistakes. This is without a doubt completely true. At the same time I have noticed that nearly the same benefit is gained when you learn to just do 'order of magnitude estimates' which I do myself.

At the end of the day it boils down to teaching an intuitive understanding of the mathematics itself, rather than teaching a student to become a human calculator. Being able to look at "find ∫[x sin(x)] dx" and being able to imagine how x sin(x) would look, and thus how the integral of it would kind of look is incredibly valuable. That means that regardless of how you calculate and use that integral, you will be able to tell whether the result is complete bullshit or not.

The current highest voted answer calls using a calculator a crutch, but I would really want to ask everybody to consider that using a CAS calculator or WA is just offloading the 'stupid, but error prone' part of mathematics and leaving the user with the truly hard parts. Computers and calculators aren't a crutch, they are - if properly taught - a stronger foundation. In my own high school I did the highest level of maths and every single student in my class was better than me in actually "doing" traditional mathematics, but 10% of our final grade came from two essays where we had to mathematically model, analyze and interpret simplified real problems and I comparatively aced those essays as my programming interests had always created a greater focus on the bigger picture (where and how can I use what I am taught). That disparity where I was helping students who were far better at mathematics than me fascinated me for years which led me 7 years ago to computerbasedmath.org which posits that far too little time is spend on definition, abstraction and interpretation, because all time in maths education is spend on computation.

• There's one more argument in favor of mental arithmetic: mental exercises are useful in combating Alzheimer's. Puzzles, crosswords, and Sudoku are all examples of mind exercises. Mar 5 at 2:23
• @AlanCampbell There are also studies suggesting that adoitness at mental arithmetic merely masks early stages of Alzheimer's. But for sure, one must use their brain to grow (and maintain) their brain, especially during the developmental stages. Mar 7 at 3:49

Calculus student had a final result of

$$\frac{1}{2\pi}$$

Which she told me was 1.57. I immediately realized that she had calculated

$$\frac12\pi$$

from keying in 1/2$$\pi$$.

There’s no going back on calculator use, I realize. What I strive for is to have the student who performed a series of calculations (for, in this case a related rates problem in calculus) to manage to recognize that final calculation should result in something less than .166 or really at least less than 1. The 1.57 should have struck her as an incorrect answer.

The issue is also one of calculator skills. I try to maintain an approach that in life, any tool needs to be used carefully. A knife is important in the kitchen, but sloppy usage will cut your hand. Calculator use requires mindfulness as well. Using parenthesis properly is part of the process as is an understanding of the expected result as in my example.

Similar to this simple parentheses issue, I’ve seen trig exams showing multiple steps indicating an understanding of all the rules the student was expected to know. The final answers? All wrong. The student’s calculator was in radian mode while the questions were all in degrees. I think that the calculate by hand skill would build the math sense that would lead to the recognition that a series of answers make little sense.

By hand =/= in your head. Having some faculty for mental arithmetic is good. Having a fluency for deconstructing a 'problem' into basic mathematical operations is pretty essential. However many people learn, and continue to benefit, by writing out the 'sums'.

If Fred drives at 45 miles per hour, how far will he go in 45 minutes?

I expect most people here have done 3/4 of 45 in their heads before reaching the end of the question. (And never had to worry if the units were correct.) But if you can't jump to the easy/obvious method then what do you do? One way is "Obviously not as far... Aha! 45 60ths... Now that's 45 over 60

 45
----
60


and times the speed

 45
----  X  45  =
60


So now I have a calculation plan." What happens next is anybody's guess: Personally I would use a calculator to check my guess of 34-ish with 4 5 ÷ 6 0 × 4 5 =. I'm getting old and the 45/60 = 3/4 = .25 mental bit (see the error!) is not worth the strain. Some people retain information by reading it. (Others by doing or hearing.) Now that 'template' is available for the next time Fred drives for 1.3 hours at 47 mph.

This approach means that students need a pencil and paper. Eeek! A good foundation for pencil and paper is sums in an exercise book. Fractions and calculators don't mix. Fractions are a key part of our mental grasp of worldly things.

• That looked great at first, and don't you think the strategy; the "calculation plan" is beside the point unbless at then end of the day you're calculating basics long-hand? Isn't the strategy the same whether 45/60 x 45 is solved on paper, in your head or with a calculator? Mar 4 at 20:15
• (a) People use the weirdest methods to fumble towards an answer. (b) Having a map is valuable. For some people that path is best visualised. Such skills come with practice. Starting all in your head+calculator is a recipe for randomness. Mar 6 at 7:51

You really need to be able to do sums in your head when debating or negotiating. To prove this, show your students these famous car-crash interviews: (1) Diane Abbott (Labour) floundering horrifically over numbers she could neither remember nor even estimate in her head. The coolness and speed with which the interviewer questioned her numbers must itself have put her in a mental tailspin, leaving her completely unable to think straight. (2) Natalie Bennett, Green Party leader in 2015, doing similarly appallingly over the cost of building half a million homes. Both these ladies probably still find the memory of these moments keeps them awake at night.

I think if we can introduce a variety of algorithm's for teaching addition and multiplication, it'd be beneficial for the student. No matter what mathematics level you are, algorithmic thinking is always important. When I say algorithmic thinking, I mean to say that we have a set of rules to tackle a problem such that applying the rules in the correct order always solves the problem.

But, other than accuracy, I don't think we should try to 'test' the computational limit of students like asking them to multiply / add large digit numbers, we should stick to working with small numbers fast and quick because that is the skill which is important in real life for example for buying something.

At a higher level, I think for applied science subjects, we should just stop putting it as an exercise for students to calculate as it simply check if student understands the topic or not. If a question in a test comes in such a way, then it tests the students ability to execute a standard algorithm rather than their grasp of the topic (the second being the important thing when dealing with an applied sciences).

Truthfully, it's depressing, and what's worse, I'm not sure I'm convinced anymore either. I mean, I don't know any convenient algorithms for computing square roots or logarithms by hand but that never held me back.

As for this, I think introducing some higher level mathematics help a lot. For example, introduce differential calculus and Taylor series as a method to overcome this. I am a believer that we can introduce the concept to a person who simply has done basic differentiation and grasped polynomials (I've written such an introduction here) though I'm not sure how easy it is to explain differentiation to a ten year old. There was some discussion on a similar problem in this MSE post.

Other ways include introducing how that topic could be applied into other fields, for example consider this theorem:

For any compose number $$x$$, one of it's prime factor must be less than$$\sqrt{x}$$

And other number theory / modular arthimetic results

To be honest, I wouldn't say this answer is completely original, these ideas had been implemented by professor Greist in his online MOOC course on calculus where he assumed knowledge knowledge of basic differentiation and then began with Taylor's theorem. If you see through the playlist, he brings in the applications of calculus in various fields such as economics, thermodynamics , life sciences and more

• Also as a side note, while all of this sounds great on paper. Most teachers just don't care enough to teach children more than a set of rules, so the whole answer is 'idealistic' but I think it is doable for a teacher who is motivated. Mar 2 at 9:10

You're entirely correct and there is huge value to being able to compute by hand.

For a most basic instance from the real world, here in the UK I once interviewed a candidate for office manager who had spent about 10 years working in a tax office, and been promoted three times. To me, that should have meant he ate, breathed and slept numbers.

I asked him what 7x8 was, why it was special and what other number grouping had similar characteristics. To me, anyone happy with what US Americans would call junior high school arithmetic should have known all three answers.

My candidate, however, spent several seconds looking for a calculator and failing to find one, resorted to pen and paper - remember, this was for 7x8.

As it happened, this was in an open-plan office and everyone else was both ear-wigging and either more my age, or from South Africa; theoretically a generation or so "behind" the UK.

Even with pencil and paper, my candidate was so clearly uncomfortable with 7x8 that I asked him - slightly less bluntly - "Do you realise, everyone else in the room had all the answers before you even realised you might need a calculator?"

He replied: "Oh, my God! Now I see why older people are always complaining about calculators. I just wish someone had explained that before."

• What's special about "7x8"? It's the most difficult entry in the standard 10 by 10 times table for people to memorize. The other grouping with a similar characteristic is "8x7".
– Mark
Mar 2 at 23:39
• @Mark Sorry, I've no idea why 7x8 should be "most difficult… to memorize". It should be one of the easiest because it allows the silly trick answer, 56=7x8. Of course that's childish, and the point is any (ordinary) child should be able both remember and to understand it. The other number group with the same characteristic is 12=3x4 and in both cases, it's the very triviality that matters. Mar 2 at 23:50
• I've got no idea why it's so hard either. But people who aren't taught that one specific trick answer have a harder time memorizing it than any of the 99 other entries in the times table.
– Mark
Mar 2 at 23:55
• To be honest, I do not see why it is a shortcoming if someone had not been told in his life: "look, how interesting, 56=7*8". For me this is checking "did you hear this before", not "how logically can you think". Maybe I am missing something. Mar 3 at 0:11
• I’d have blurted out 56 instantly, but probably thought nothing was ‘special’ about that product. I continue to be a bit taken aback at how students can be in HS and not have memorized the multiplication table. Mar 3 at 0:34

You can't be overly reliant on technology.

I once bought an item from a shop during a power outage. Cash register was non-functional. Not enough light to power a solar calculator. The cashier had to compute a 10% discount by hand.

She did it.... the long way.

• by doing it the long way she avoided a possible argument that could come if there were any disagreement in the method. Mar 3 at 3:39

I remember getting my first 18 inch slide ruler. I had worked hard, poison oak filled, hours making fire breaks in the Oakland hills to make enough money to get it. It was aluminum, had a spring loaded cursor that slid like it was greased, and the C and D scales (I think) lined up perfectly. It was huge and had scales I was never going to use. I loved it.

So what do I do first? I was in the kitchen an looked down at the kitchen table. I would measure the width of the table, the angle to the opposite corner, and calculate the length of the table. I was trying to figure out how to set up my protractor to measure the angle when my dad walked in. My pop was a very successful travel agent but he didn't know or care what 8 times 8 was. That's what adding machines are for. He stood there watching me. I'm sure he thought I was insane. "Why don't you just measure it"? "Well", i said. "Suppose this is a plot of land and there is a river going through here." "OK". He replied, and continued standing there. I got the angle, to what I hoped was the nearest half a degree, picked up the slide rule, moved the scale, moved the cursor and read off a number. Knowing I couldn't be that accurate, I picked up the slide rule again and converted the fractional part of the answer into sixteenths of an inch. "51 and three quarter inches", I proclaimed. I held the end of the measuring tape on one end while my dad pulled it over to the other end. "54 inches" he said. "Well," I said. "With the tools I was using, "That's not too bad." To my surprise, my dad was amazed. How did you do that? I knew better than to go into more detail than saying this slide rule has scales on it that convert angles into numbers that you can use it to calculate distances and multiply and divide numbers.

I almost never used that slide rule to calculate an answer. First, I calculated the answer myself, then I used the slide rule to check my answer. Yet I treated it with the same respect that a marine treats his rifle. It was always clean, properly lubricated, and the cursor was always lint free.

Eventually my slide ruler went the way of land lines, record players, and tv sets that you used pliers to change the channel with. It's a shame. I learned a lot about mathematics by figuring out how slide rules worked. Maybe now we need to show them how spreadsheets work. And hand-held plotting calculators.