Wikipedia:Reference desk/Archives/Mathematics/2017 January 16

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January 16

Maths in school, maths in the real world and maths do by mathematicans.

What are the important similars and differents with maths in school, maths in the real world and maths do by mathematicans? --Curious Cat On Her Last Life (talk) 14:43, 16 January 2017 (UTC)

That's a fantastic question and I don't believe I can do it justice, but offhand I'd say that:

• Math in school is the worst because its defining feature is that it is easy to test. School is a grades factory, and if you can't test it and put a number on it, it's out, no matter how beautiful or useful it is. That's why it focuses on practicing techniques rather than true understanding - testing mastery of a technique is easier than testing understanding. But the techniques are not so important, the ideas are.
• Math in the real world is practical. Its only requirement is that it gets the job done. It can be very interesting and varied, because different jobs requires different kinds of math. There is always a breadth of low-hanging fruit that are fun to explore, because there are always new ideas for how to apply known mathematical concepts to solving the plethora of challenges that life has in store for us. Understanding is important because if you don't understand it, you won't be effective in getting the job done using it. But there comes a point where something seems to work and you use it without really understanding how it works, or even being absolutely sure that it's fundamentally true. Results are everything. And it's somewhat limited in depth, because the vast majority of mathematics has no known applicability.
• Math done by mathematicians is limitless in profoundness and beauty. Everything goes - an idea does not need to sully itself with the real world, as long as it is elegant, interesting and ties in to the great tapestry that is math. But it can be hard to find the motivation to explore abstract notions that are many inferential steps detached from what we are familiar with. Trying to actually contribute to the body of knowledge can be an arduous task, because so much has already been discovered, and the rabbit hole leading to the frontier of mathematical knowledge goes so deep. Mathematical research is the ultimate pursuit of truth, because a result must be proven rigorously to be accepted. This is a blessing to those who value correctness above all else, but to others it can seem like needless technical pedantry that detracts from focusing on the actual concepts.

-- Meni Rosenfeld (talk) 18:02, 16 January 2017 (UTC)

I think this you give a very nice answer to this question actually. Math in school is often considered boring and/or difficult, with lots of rules and hard work, but once you see the elegance of some of the derivations, you'll remember those. The problem is that most people don't, especially not in school where the grading is important, not the comprehension. Rmvandijk (talk) 12:43, 19 January 2017 (UTC)

You might want to break up "math used in the real world" to "math used by the average person", and "math used for jobs that require additional math skills". In the first case, I wouldn't expect much beyond compounded interest, logic, and basic geometry (say to calculate square footage of a house) is needed by the average person. But for those whose jobs require additional math, it can get quite involved, especially in the sciences, except that math proofs aren't often required outside of mathematics itself, but rather they would use existing, proven math techniques. StuRat (talk) 19:29, 16 January 2017 (UTC)

• In the general public (i.e. people who don't use advanced mathematics in their jobs), there are two "types" of mathematical ability that are considered important: numeracy and statistical literacy. Numeracy is the ability to perform mathematical calculations, and statistical literacy is the ability to understand and evaluate the results of calculations. Schools are generally good at teaching the former, but very bad at teaching the latter. This is why, for instance, politicians and marketers can often get away with making dodgy claims (eg these graphs), and why gamblers often incorrectly estimate their odds of winning. Here is the mathematical content of the English national curriculum, roughly in the order they're learned:

Basic arithmetic (counting, adding, subtracting, multiplying and dividing small numbers) - used in basically everything. If you can't do these, that's a serious problem (numeracy)

Number sense (knowing whether one number is bigger or smaller, and by how much). Again, a crucial life skill. (numeracy)

Measurement (reading scales, converting between units) - telling time, cooking, DIY. (numeracy)

Geometry (calculating areas, volumes and angles) - DIY, map reading. (numeracy)

Graphs (drawing and reading) - understanding data presented in articles and adverts (statistical literacy)

Advanced arithmetic (long multiplication, long division, working with complicated fractions) - this used to be important in a lot of everyday life, but calculators have mostly displaced it as a vital skill. (numeracy)

Ratio (proportion, interest) - cooking, DIY, finances (numeracy, but key to statistical literacy)

Probability - gambling, buying products like insurance (statistical literacy)

Statistics (different types of average, interpreting graphs) - understanding data presented in articles and adverts (statistical literacy)

Trigonometry - DIY (numeracy)

Algebra (rearranging formulas, solving for unknowns in simple equations) - in theory, useful for solving problems like "If my car does X miles to the gallon, do I have enough fuel for an hour on the highway", but most people generally don't use it - at least in a systematic way (numeracy)

There are also a few useful life skills that schools don't tend to teach - most don't teach much in the way of accounting, even though that's essential to planning a household budget, quick approximation techniques (for instance, if some politician says, "We spend $X billion on welfare, and that's [too much/too little]", it's very useful to have a rough idea of whether that's big or small compared to the government's budget and how much that is per household, without having to research the exact number), or how to look for signs of misleading data. Smurrayinchester 16:15, 18 January 2017 (UTC)

As an educator myself, I need to make a minor quibble with your description of numeracy. Numeracy is the mathematical analogue of literacy (for reading) and fluency (for speaking). Just as true literacy is more than being able to mechanically pronounce individual words as written on the page, numeracy is more than the mechanical act of being able to perform calculations. Literacy requires reading comprehension above all, the ability not just to understand the correspondence of written words to sounds, or even of being able to understand the definition of a word in isolation, but rather the ability to extract meaning from a passage of writing, and be able to understand the abstract connections between the written word and ideas. True numeracy must incorporate some form of number sense, not merely the ability to remember and perform a rote algorithm, but to understand the meaning of a number, and the meaning of the relationships between numbers and operations. Schools are good at teaching and testing for performing algorithms, but not at developing number sense; as noted, that's because it is complex to test for. Not that it cannot be tested for; just that it cannot be tested for on a scantron, and schools are not willing to put in the resources to assure it is assessed correctly at a systemic level. --Jayron32 13:09, 19 January 2017 (UTC)

That's a fair point. I was trying to make the difference between numeracy and statistical literacy clear, but I did oversimplify. Smurrayinchester 13:26, 19 January 2017 (UTC)

Comment. I know of one difference between math in school and math in the real world. In the real world, it's good to be accurate. In school, directions that say "don't be too smart" can occur; the most common is "use 3.14 for pi", meaning that you must answer problems as if pi were by definition the decimal 3.1400000, not 3.1415926.... Any other example of such a direction common in school math?? Georgia guy (talk) 17:51, 19 January 2017 (UTC)

On the contrary, in the real world we use answers to the correct level of precision, and do not introduce false precision. See significant figures. In school math, we assume every digit matters. In the real world, we know when they don't. --Jayron32 18:00, 19 January 2017 (UTC)

And the key fact is that schoolwork assigns the value 3.14 for pi with the assumption that it has an infinite number of significant figures. I want to know of another example of school math that has the direction "don't be too smart". Georgia guy (talk) 20:05, 19 January 2017 (UTC)

That probably goes back to Meni's "easy to test". The grader just has to look for one answer, with scribbles that look like they're probably right, and then can mark it correct.

On the other hand, it could be a valuable practical lesson — if you just need three sig figs in the answer, don't (usually) bother using seven sig figs in any of the inputs. It will just slow you down, to no benefit. --Trovatore (talk) 20:19, 19 January 2017 (UTC)

But the direction is still popular even though it's a remnant from before scientific calculators became common. Georgia guy (talk) 20:21, 19 January 2017 (UTC)

In the days before calculators, the usual instruction was: "take $\pi =$ 22/7" and the numbers were designed to cancel. Dbfirs 20:41, 19 January 2017 (UTC)

I remember it as 22/7 too! You could of course just leave pi in your answers until the very last step, but in practice I found that the cancellations were very useful: if they didn't happen, I had probably made a careless mistake somewhere. I must disagree with Georgia guy here: when areas of regions involving circles are involved, the problem is of a sort where in the real world, you would probably need that much precision: 22/7 is only about 0.04% off the right value, after all. Of course, in more purely mathematical problems, this would be nonsense. No one would ask for angles in radians where pi is approximated as 22/7 – or at least no one should. Double sharp (talk) 15:16, 20 January 2017 (UTC)

P.S. In the real world, it does not really matter that the expansion of pi never terminates. I know the first 50 decimal digits of pi (and I know some people who know even more), and I cannot think of a single real-world application when that accuracy would be insufficient. 51 orders of magnitude from 10⁰ to 10⁻⁵⁰ takes us from the estimated diameter of the observable universe way past the diameter of a proton, although it doesn't reach the Planck length yet. (This naturally excludes "unnatural" questions like tan(10¹⁰⁰) or tan(10¹⁰¹⁰⁰). Double sharp (talk) 15:22, 20 January 2017 (UTC)

StuRat say correct that maths in the real world can be maths use for every day life and maths use for job. Good answers about maths in school and maths use for every day life. Meni Rosenfeld say maths in school must easy to test but I think also must easy for young people? Example before learn complex numbers must pretend cannot have minus square root. Also I think in school maths question give the numbers and you find the answer, example solve quadratics equation, but in real world you must ownself find the numbers to put in the equation? Now hope for more info about maths use for job and maths do by mathematicans. --Curious Cat On Her Last Life (talk) 13:26, 21 January 2017 (UTC)

Small part of an answer, I'd say read Simpson's Paradox and regression to the mean (and then the Sports Illustrated cover jinx) to get an idea of how statistics may allow people to understand things that aren't obvious from basic numeracy. Although many "statistics" concepts are less about mathematics than about human behaviour. Goodhart's law is more about how people behave when faced with a target than about maths, really. Blythwood (talk) 16:36, 21 January 2017 (UTC)

Reading that article, it sounds like the "jinx" has more to do with people crippled and killed soon afterward in accidents. The regression to the mean thing may appeal to mathematicians, but it seems relatively irrelevant to the alleged phenomenon. Wnt (talk) 18:18, 22 January 2017 (UTC)