One comment: there's a difference between mental arithmetic, which is as useful as ever to be able to perform quickly/effectively, and the pen-and-paper variant (which I assume the cited assignment required), which is not. Everyone should, of course, know how to do the latter, but doing long pen-and-paper operations usually doesn't save time over reaching for a calculator, and is error-prone. Unfortunately, schools (at least here) seem to endlessly practice pen-and-paper while giving essentially zero attention to mental arithmetic.
I have to disagree with this a little bit. The skills you use for mental arithmetic are very similar to the skills for pen and paper. I would view pen and paper as training wheels.
It might be argued that mental arithmetic should also at least be taught, but the only way I know of to do that which will tend to generalize to a majority of students is to teach them to use an abacus (seasoned abacus users eventually stop needing to actually have the physical device). And yes, we should almost definitely be teaching children how to use an abacus.
It's my impression that doing basic mental arithmetic effectively tends to depend on understanding what happens to the magnitudes of numbers after various operations and making good approximations, which has very little resemblance to the brute force long multiplication/addition/division that is taught, but is teachable; yet approximations are almost never acceptable in school, despite being very useful in real life. Meanwhile, more advanced stuff requires random numerical tricks like these:
The first thing you described is what is called "Fermi estimation". Basic mental arithmetic is being able to do addition, subtraction, multiplication and division in your head. That kind of arithmetic is strictly a prerequisite for Fermi estimation. The main purpose of Fermi estimation is not to make the calculation easier (although it can be used for that too), but to get answers while not having very good data.
For example, the other day I was trying to figure out, supposing I built a small 4-foot deep swimming pool in an apartment (not on the base floor), if I could expect it to weigh less than what the builders would assume the floor needed to be able to tolerate over that area. I didn't know the mass of a cubic foot of water off the top of my head, so I went off of guessing at how many 2 liter bottles would fit in the volume, and how much a crowd of people would weigh if they all clumped together tightly. Based on these estimations, I knew my answers would be within an order of magnitude of reality. Since the first answer was several orders of magnitude larger than the second answer, I knew that the final answer to my question was "no".
I generally do this kind of thing several times per day, because my brain is filled with crazy ideas like building swimming pools in apartments.
The answers Feynman came to could be done with ordinary mental arithmetic, assuming you have good tools (better than I have) for moving numbers between working memory and short term memory, and the tricks he used to get there faster are shortcuts you simply learn with time and practice.
But basic mental arithmetic is very similar to pen-and-paper arithmetic. The main differences for me are small mechanical adjustments to make use of results that I have cached. For example, if I'm doing division, it might be easier to multiply the number by four first, and then divide by four at the end. And of course if I'm getting a rough estimate, I'll do some rounding here and there to make things more convenient.
The other major difference is that I usually reduce a fraction as far as possible before doing long division, since that makes the multiplication steps (necessary for getting the remainder) simple table look-ups instead of long-multiplication.
Metric would have made your indoor pool example way easier.
4 foot is around 1.2m,so you have around 1.2 metric tons of water per square meter. Most buildings are designed for at most a few 100kg per square meter.
> The skills you use for mental arithmetic are very similar to the skills for pen and paper.
Speak for yourself. Pen-and-paper arithmetic, to me, is a completely mechanical process, pure symbol manipulation, whereas mental arithmetic is very, very heuristic-heavy and involves a much richer mental model that often borrows from geometry.
Do calculators even routinely work with infinite precision (like (sqrt(2) x sqrt(2) = 2)? I guess you could achieve some results with Mathematia or something like that, but I have my doubts about the average calculator.
Sometimes it is still useful to do calculations by hand, also to understand what is going on.
(Edit: just tested the OS X calculator, sqrt(2) x sqrt(2) works out by chance, but sqrt(200) x sqrt(200) fails).
Well, I was talking about basic arithmetic, addition/subtraction/multiplication/division of integers. Doing algebra (which is usually where you'd find square roots) on paper is certainly still useful, but usually the numbers aren't very large, so it's essentially orthogonal to the type of work mentioned here. Algebra and more advanced math has multiple ways to go about dealing with an expression and requires understanding to proceed, but long multiplication or addition is just rote application of an algorithm.
(I suppose the fraction in the problem in the article makes it a bit trickier, but I suggest that the dilemma of ordinary four-function calculators not being able to do exact fractions is not a common real world problem, or they would be able to do them.)
Edit: Sorry, I just noticed that actually I was missing the point - I thought your comment was in response to vacri. Leaving my comment here, just in case.
I think your comment misses the point. It's important to learn when to use a calculator and when to simplify the equation instead. Your example obviously should be simplified first on paper, resulting in 2. This can easily be enforced by requiring the intermediate steps, such that a calculator won't work. The point of parent is, that learning pen-and-paper calculation is of basically no value (except if you go into computer science, where remembering those algorithms can come in handy). Your example is not calculation, it's arithmetics, a necessary step towards algebra.
I agree that probably training too much manual addition and multiplication is a waste of time these days. However, a bit of it still seems to be useful. For example I am often able to calculate the bill in a restaurant (not always, though). And sometimes it is just faster than going to the calculator, depending on the situation.
Also a lot of the process of addition and multiplication might also be good practice for arithmetics?
A current-generation Casio scientific handles this well - the display shows exact numbers such as 1+√(2) which suggest it's doing the precise calculation where reasonable.
The problem is that something like sqrt(2) can not be represented as a number in a calculator. So as long as the calculator only operates with numbers, instead of doing algebra, it will do it wrong. In the case of sqrt(2) x sqrt(2), most calculators will convert sqrt(2) to an approximation of sqrt(2), then multiply those approximations. If you are lucky the result will be rounded to 2, but you will not always be lucky.
No, I don't believe you are correct: a calculator can represent sqrt(2) as sqrt(2), just the same as we can on paper (perhaps you are calling this an algebraic representation).
In fact, I'm confident that Casio - probably the market leader in scientific calculators - is doing exactly this, storing the exact number.
Casio does not appear to approximate simple surds as floats until it becomes necessary, and stores the exact number sqrt(2), just as it stores exact representations of rational numbers wherever reasonable (and, it correctly distinguishes whether it knows a number exactly or only as a float - it refuses to convert inexact floats into fractions, but will convert exact numbers).
Calculator phone apps don't compete at this level - which is surprising, I would have thought smartphones would have put calculators against the wall.
Well that is also wrong, or at least the .0 shows that the calculator is not doing the equations, it is just calculating. It doesn't know know that sqrt(2) x sqrt(2) = sqrt(2 x 2) = sqrt(2^2) = 2
And that is the kind of thing you get to understand if you do the calculations by hand.
