[SebNag_]

Trying to figure out the proof of a² + b² = c² by myself, without looking up the solution, was somehow exiting. Being exposed to a riddle and trying to find the solution is kinda cool.

However, not solving it after 10 minutes left me feeling a bit dumb... :)

[jacobolus]

This is a general problem with mathematics education. See http://toomandre.com/travel/sweden05/WP-SWEDEN-NEW.pdf

Real (non-trivial, non-obvious) problems that someone hasn’t seen before can take hours, days, weeks, years, whole careers, or sometimes centuries to solve. Some of them later turn out to be impossible (and for many we still just don’t know).

Real math education would have students grappling with relatively open-ended problems that take significant amounts of rumination and some cleverness to solve. It would explicitly encourage/reward close critical reading, creative brainstorming, planning, strategic thinking, generalization and specialization, executive control (e.g. time management), error checking, and clarity of exposition (including when asking for help after being stuck). There would be no shame in throwing out incorrect hypotheses, asking for clarification, getting stuck on a problem, making subtle mistakes which could serve as good examples for future improvement, etc. But skill and stamina at such work must be trained slowly, starting from an early age.

The problem is that current (US) math education instead pre-chews everything, assigns students lists of exercises almost identical to what they saw someone solve before, and mostly tests memorization/recall and willingness to do the same trivial task over and over for hours despite being terribly bored, under purely extrinsic motivation.

For people used to such math homework, the standard response to a single problem which takes >5 minutes to work through is to give up.

[Retric]

Depends on what the goals are. I don't think most counties want a large fraction of their population becoming mathematicians. They basically want most of their population able to use math not discover new areas.

[jacobolus]

What do you think it means to “use math”? If you want to train human spreadsheets you’re wasting everyone’s time: the spreadsheet already exists as a product for electronic computers, and is much faster and more reliable than a human will be at doing huge piles of arithmetic. (But anyway, even if you wanted to train expert performance at arithmetic, the way schools currently go about this is horribly inefficient and doesn’t take into consideration research on neuroscience / psychology.)

If you just want to train people to be unthinking drones who can follow narrowly specified procedural rules without understanding their context or meaning, then I guess the current system is a relative success.

In general, the point of mathematics education in primary/secondary school is not to train future mathematicians, but to teach people important problem-solving skills. The same skills are (to some extent anyway) useful in essentially any field you might name, from childcare to plumbing to legal analysis to fine art.

In particular: self-confidence that hard problems can be tackled and that anything one person can do another typically can also with training and effort, time management, lateral thinking, learning when to keep trying a strategy vs. when to switch and try something else, salvaging useful partial results from failed efforts, drawing diagrams, careful record-keeping of works in progress, more generally externalizing problem state so that it can be worked with outside your head, converting fuzzy problems into precise formal terms to they are amenable to careful logical analysis (including making explicit all of the assumptions involved in the model chosen), exploring the relationships between different problems, investigating specific concrete examples of general rules and generalizing from particular cases to abstract theorems, searching/skimming published literature for solutions to problems that are too much to handle or finding relative experts to ask for help and knowing how to do so productively, clearly explaining an original problem and its context and any simplifying assumptions and then clearly explaining a solution step by step, checking solutions by solving a problem multiple ways or doing quick sanity checks, .....

I could probably keep listing more here, but you get the idea. Anyone who plans to do any kind of real-world technical work will be at a huge advantage if they have significant amounts of problem-solving practice going back to childhood.

You might similarly protest that we should not bother reading and analyzing novels in school because few careers explicitly require reading/writing fiction, or that we should not bother with physical education courses because few careers require playing dodgeball, or that we should not bother with music courses because few careers require skill at playing the recorder, etc. etc.

[Retric]

There is a wold of difference between solving for X and proving P =/!= NP. Collage level math like DiffEq is still the kiddy pool with well known approaches that work.

Writing a new graphics engine, or baking a cake takes applying existing techniques, but not the fluid exploration of unsolved frontiers. So, when I say use math I mean take advantage of what exists not nessisarily add anything new.

So, yea we want problem solvers, but not thinkers.

[jacobolus]

Why do you think (paraphrased) “I think secondary school mathematics should focus on solving non-obvious problems” has anything to do with “proving P ≠ NP” per se? I’m obviously not suggesting that we should assign famous unsolved research problems to secondary students. I would instead hope we could assign students a variety of problems taking them between 10 minutes and a few weeks to solve (at their current level of skill), with emphasis placed on smart problem-solving efforts rather than on sorting students based on who gets the most right answers.

Most American undergraduate differential equations courses are taught as a list of recipes with little room for thought. Rather comparable to elementary school arithmetic drills frankly, though obviously involving more built up preparation. https://web.williams.edu/Mathematics/lg5/Rota.pdf

However, it is possible to assign difficult problems to students at any level from age 3 onward (see http://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf for an example of real mathematics instruction for preschool students; for primary students look up the work of Dienes; at the middle school level I think some Russian programs are pretty good https://bookstore.ams.org/MAWRLD-7/ etc.). It just takes more work for teachers to provide feedback about student solutions to such problems, it’s less amenable to grading by rote (and therefore not easy to check via standardized tests), and it takes more significant focus/attention/decisionmaking by teachers from moment to moment (and ideally more teacher background preparation). The students learn the subject more deeply, enjoy the process more, and learn significantly more transferrable skills.

Porting a graphics engine from one platform to another very similar platform (after having ported many other software projects between the same pair of platforms) or baking a cake just like all the others you have baked before might take nothing more than skillful application of well-established procedures, but making a new graphics engine in the first place (assuming it does something novel) or inventing a recipe for a new type of cake definitely takes problem solving skills.

[Retric]

I am saying there are different classes of what you are lumping under problem solving skills. Coming up with a new cake recipe does not involve building a new oven or measuring system. It's a well constrained problem. Use known techniques and apply some time and money gets a new recipe.

P vs NP on the other hand might not be possible to solve.

[mgoetzke]

You might like http://worrydream.com/KillMath/

[dagw]

The problem is that current (US) math education...

Worth noting that the US tried the kind of math education you are suggesting (called the New Math initiative) and it failed miserably. The math education we are seeing today is largely born out of a counter reaction to that failure.

[jacobolus]

The New Math was something fairly different than what I am suggesting. It was an attempt at an alternate curriculum for primary/secondary school based on higher-level / more abstract mathematical topics, partially displacing study of arithmetic.

The New Math curriculum per se wasn’t so terrible, though it certainly had flaws (like anything invented from scratch out of context and not slowly developed and tweaked over time in response to feedback in a real-world setting). The bigger problem was that the proponents of the New Math didn’t have much buy-in from students, parents, teachers, school administrators, or the broader society, didn’t really do any outreach or teacher training, didn’t really produce enough supporting materials, and just dumped the curriculum on schools without support.

Parents and teachers didn’t know what to make of the curriculum (were unqualified to teach with or assess it), and didn’t feel involved in the process, and as a result there was a lot of opposition.

But what I’m talking about is not teaching different subjects per se, but teaching whatever subject in a different way, focused more on solving problems and thinking than on precisely mimicking teacher’s demonstrations or memorizing formulas. The current typical math pedagogy is patronizing, emphasizes memorization/recall and very careful attention to details (sometimes irrelevant details about formatting), teaches students that they shouldn’t try to think for themselves and teaches them to conflate getting the right answer with being “smart” or “good at math” and that anyone who makes a mistake or doesn’t know how to get the answer is “stupid” or inherently incapable.

[n4r9]

Are you referring to these?

https://en.wikipedia.org/wiki/New_Math

https://en.wikipedia.org/wiki/Secondary_School_Mathematics_C...

I'm not in the US, but both of those seem to be greatly focused on the curriculum. OP is instead talking about the style of teaching and learning, which could be applied to practically any curriculum.

[srtjstjsj]

That's incorrect. New Math was about teaching math from axioms, not the experimental mathematics parent poster was promoting.

[dannylandau]

One of the few times that I've read a comment on education that I totally concur with.

[escherplex]

The intuitive geometric description I sometimes use with people is to have them imagine two duplicate rectangles and cut a diagonal in each to form four equivalent triangles. Mentally label the hypotenuse c, the long side b, and the short side a. Then arrange the four triangles into a square with the perimeter consisting of sides of length a+b. Now from the outside area is simply (a+b)^2. But computed from the inside area will consist of 4 * (1/2 a*b) + the area of the central tilted square with sides c or c^2. Simple algebra yields a^2 + 2ab + b^2 = 2ab + c^2 or a^2 + b^2 = c^2. A more fun problem is imagining why there are 180 degrees in any triangle. And for that matter where did that 360 degrees in a circle come from (last one hint: imagine 6 duplicate equilateral triangles and arrange them into a hexagon, assign the Babylonian magic number 60 to each vertex and sum the interior central angles, after using one side to draw a circle around the hexagon).