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No, that is my whole point. Knowing the multiplication table up to 9*9 is a prerequisite for the following:
Anyone who is arguing that kids do not need to learn what 7*8 is denying the need for learning arithmetic at all. (There are probably people arguing that you do not need to know arithmetic because computers exist. I disagree. Knowing how to arrive at a result without a computer in principle, even if a million times slower and error prone, is useful in itself. Besides, with LLMs, this argument generalizes against anything you learn in high school.)
I am fine with busywork which is an instrumental goal for something worthier. I learned my multiplication table because I knew that it would enable me to do long multiplication in base 10. Then in fifth grade, they tried to make me learn 2..9 * 11..19, and I was not having it. If you are in base 10, the utility of knowing non-trivial products of integers takes a sharp dive after 99. I felt that if I gave in and learned what 137 was, they would come around in sixth grade and demand I learned by rot what 1315 or 237 was. (For factorization, the useful thing would be to quickly identify a prime factor in any integer in a given range, but even this obviously does not scale.)
A lot of the stuff kids learn in math class, painful as it may be, is actually useful in some fields. I don't think that youths should be forced to learn what a logarithm is, but they should be clearly told that any subject up to the softest social sciences will expect them to know, so that they can make an informed decision.
Other stuff has value because it teaches core concepts of math, such as proof by induction or the definition of real numbers. Sadly, people only learn what real numbers are when they go to university, and proof by induction in high school is turned into a mockery by turning it to just another pointless algorithm which can be used to prove increasingly pointless sum formulas by rote.
Wait - you value this, but not polynomial division? They're both things you can just ask the computer to do for you instead, but at least polynomial division requires you to hunt down a computer algebra system; long division capability come pre-installed on every phone.
Polynomial division is IMHO a curiosity.
Long division crops up every time you need want to split the bill.
Polynomial division might crop up in the wild if you want partial fraction decomposition, which I guess you might want if you are dealing with rational functions and want to numerically evaluate them or calculate their anti-derivatives. While I am sure that rational functions have their uses, my gut feeling is that they are both too narrow to pop up in physics a lot (where you will frequently have square roots placed so that your functions can not easily be transformed into rational functions) and too inconvenient to be preferred for empirical models.
Factorization of an integer is a hard but finite problem. Factorization of a polynomial is in general just not possible exactly. You can test if 7, 13, 17, 19, 23, ... etc happen to divide 3071 to factorize it. You can not test if x-1, x-1.1, x-sqrt(42+sqrt(42)) etc are factors of 5x5+4x4+7x**2-2x-2, because there are countably many algebraic numbers which could be a root.
I think that we learned both polynomial division and solving quadratic equations around grade eight. Solving quadratics in something which I would call bloody useful. Quadratic functions are the first non-trivial functions students can tackle, and quadratic equations pop up all the time in high school physics.
I strongly disagree with the sentiment that math skills which are less readily automated are more valuable. To grok (I'm reclaiming that word) how multiplication and division work doing long multiplication and division is definitely more useful than just using a calculator. Nobody needs the numeracy to be actually excellent at these operations any more. Anyone whose job actually requires them to multiply five-digit numbers will hopefully have the good sense not to try that by hand.
My more general point might be that I do not want students to be excellent at applying any algorithm. They will always suck very hard compared to the simplest of computers. Still, it is useful to demonstrate that you can apply an algorithm, even if it is just at toy-sized problems.
Also, applying a pre-learned algorithm is not math. Some algorithms (e.g. solving an equation for a variable) are genuinely useful in proofs, and thus are valid technical skills to learn to be able to engage in math, same as being able to write symbols with a pencil. And of course, 'can you apply an algorithm halfway reliably?' is also a good way to check if someone has a basic understanding of the algorithm in question (even if it does not probe if they understand why it works), which is why rotating trees by pencil is a staple in CS exams.
Still, for school math, I feel that 'can apply pre-learned algorithms' should earn a passing grade, not an actually good grade, which should require thinking.
This was an excellent explanation, thank you.
One side note:
Piecewise-rational functions are very popular for two big categories of empirical model: anything where the true behavior can have asymptotically-polynomial singularities, and CAD models. Being able to do sharp corners and spectral approximation refinement and exact conic sections all with the same backend is a very useful trick.
This is irrelevant to your points, though; even when working with NURBS, polynomial long division doesn't really come up.
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Where do you have proof by induction in high school?
IB does proof by induction in high school.
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In Finland in the mid 90s.
Of course that was by far the most useless thing they ever taught in high school math here (and in fact one of the few useless things).
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Any East Asian education system would touch upon proof by induction, and most certainly of any Math-specialized classes. Seems like AP Calculus briefly touches upon it. IB Math Higher Level and Further Level certainly does. The British A-Level Further Maths also does. India probably does.
Obviously teaching material quality varies and I'm not a math expert but proof by induction might not be accessible but definitely within reach for most high schoolers out there.
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