Over in this thread
a lot of people seemed to gravitate toward math they were forced to learn.
I loved math and learned things that seemed pretty useful—the Pythagorean theorem, for instance. When people say, “Why should I learn math?” I sometimes reply, “Remember all those word problems?” IMO that’s is how you could use it. But not all things are obvious or immediately useful to the average person.
Got any gripes?
Kramer’s Law. Sure, it can be used to solve systems of equations, but it’s horribly inefficient, even on a computer, and when done by hand, tedious and mistake-prone to boot. The fact that it works at all is an interesting curiosity, but practical it’s not.
For that matter, matrices in general, at least the way they’re usually taught. Matrices are, in fact, incredibly useful. But between my own schooling, and years of substitute teaching, I’ve seen many, many presentations of matrices to high schoolers, and every single one has stopped about two pages short of the part where they get to their usefulness.
Similar to Kramer’s Law is Hero’s Formula. Again, it’s interesting that it works, and there are some interesting results that you can derive from the fact that such a formula exists (but without actually needing to know the formula itself). But there are almost always easier and more practical ways to find the area of a triangle. I remember one time in high school, we had a homework assignment to find the area of a triangle given coordinates of three vertices: The teacher expected us to use the distance formula and Hero’s formula. I instead did it in my head, using three trapezoids, before the end of the period… and thus convinced the teacher to cancel the homework assignment.
I assume you mean Cramer’s Rule, not Kramer’s Law.
(There! I just used my knowledge of Cramer’s Rule!)
Isn’t all that a little bit of a mischaracterization? E.g., you are not going to use Cramer’s Rule to solve a system of equations by hand. Why would you? You are actually going to use Cramer’s Rule to, let’s say, prove that some algorithm has a certain asymptotic running time, or that some Grassmannian has the structure of an algebraic variety, etc.
Similarly, I think there can be no doubt about the utility of matrices, and if that is not clear in the high-school or college textbook then it must be a very poor one, indeed.
I’m sure one could start compiling a list of ostensibly impractical or esoteric concepts in mathematics, like, I don’t know, when was the last time you needed to use the classification of finite simple groups? But I’m sure that (a) it gets its rare uses and (b) many of the techniques involved, to say nothing of the very concept of symmetry groups and their representations, are so ubiquitous in mathematics, the physical sciences, and even beyond that it goes without saying. Analogous things may be said for Fermat’s Last Theorem.
That’s the point. I assumed matrix math was important for solving complex problems, but that wasn’t part of the course. An entry level class in applied matrices would have been useful.
Cramer’s Rule is extremely useful in a wide variety of proofs in discrete math, differential geometry, and most famously to derive the variation of parameters method of solving inhomogeneous ordinary linear differential equations; however, in any first level science and engineering course using matrix algebra (linear circuits, dynamic systems, mechanism kinematics, et cetera) it is introduced as a way to solve a system of 3x3 linear equations even though it is prohibitive for anything larger and not otherwise useful for numerical calculation. I probably had to spend a lecture in four or five classes “learning” Cramer’s rule (not including straight math classes like differential equations and linear algebra) before going onto the more useful method of Gaussian elimination via matrix inversion. I suppose you could argue today that you wouldn’t do either of these in practice with a numerical system by hand; you’d plug the matrix into Matlab, Mathematica, or Python/NumPy and have the computer grind through the operations because it is infinitely less likely to make an error, but if you are handling closed form relations in, say, mechanism design you might need to work the transform manually (although still probably more reliable with Mathematica or SymPy).
i think the problem is that matrix algebra is really only useful for solving systems of equations, and very few introductory science classes actually have students solving more than two simultaneous equations, so there is no apparent application and matrix algebra just seems like an abstraction. I was through introductory differential equations, basic mechanics, and intro modern physics before I really understood the very broad application of matrix methods toward solving nearly any real world problem involving linear systems of equations (and Fourier methods for approximation, but that is another topic), and it was a revelation how easy it made the calculations.
However, most people are never going to use matricies, or integral calculus, or differential equations, or indeed basic algebra in any explicit way, and so these areas of knowledge and their methods seem “useless” to the general public even though they are crucial in understanding how the world works and modeling it in useful ways.
The most useless thing I could remember in math was the “FOIL” method of solving second order polynomials. It probably didn’t help that by time I was in algebra I’d already taught myself that topic and applied trigonometry as part of programming a celestial calculator (in Microsoft BASIC on TRS-80 CoCo after running out of space and patience on a little Sharp hand ‘computer’!) and could calculate the parameters for the quadratic formula in my head much faster than factoring the limited number of quadratics with integer roots. I recall the teacher standing over my desk in the middle of class screeching at me “FOIL it! FOIL it! FOIL THE PROBLEM!!!” and then very performatively writing a big “F” on my assignment because I shrugged at her. I’m still not sure what the point is other than a simplistic way of teaching algebra by people who don’t really understand math.
Stranger
I probably goofed - I was referring to determinants
The “FOIL method” is just an acronym that helps you keep track of how to multiply two binomials together (“First, Outer, Inner, Last”). It’s not really necessary to know the mnemonic (except for communicating with other people who use it), but being able to multiply binomials together is a pretty basic skill that comes up a lot in any math that involves algebraic manipulations.
Looking at a presumably well-known book about matrices, Horn & Johnson’s Matrix Analysis, determinants are introduced on page 8 out of nearly 600 in Volume 1, immediately following the definition of matrix multiplication, so that should give some idea of how important they think it is.
So why did I spend endless hours multiplying all those numbers? I could never figure out when to stop.
I see it as, if someone tells you to solve a quadratic equation or factor some polynomial by completing the square, maybe it’s not really necessary, and a so-so teacher may not be able to explain why you would do it that way, but if you really really understand Galois’s method you would find you not only understand what it means and where the quadratic formula comes from, but that you can just as well apply it to cubic, quartic, etc. equations, for example.
Wouldn’t that require a different approach to math instruction?
If you are talking about systems of linear equations, I agree that a more practical way is Gaussian elimination, but let’s look at a problem like finding the eigenvalues of a matrix. Don’t tell me the first thing you think of is not to compute the characteristic polynomial \det(x-A).
Some elementary class is not going to get into abstract algebra, but in any case there is definitely no reason to intimidate someone or give out F’s because the calculation didn’t ‘look right’ on the paper; that does sound like a hallmark of someone who does not really understand the subject.
I just remember that she had a very specific way that solving quadratics had to be done using this ‘method’ and if you didn’t use the exact steps in precise order (which started with a list mapped to each letter in the acronym) she would mark the solution wrong. I’m not sure why you would need a mnemonic to multiply every term in one binomial by every other term in the other binomial; that seems confusing and error prone versus just separating the equation into two separate binomials and factoring, and of course it doesn’t work at all for higher order polynomials. I found the entire issue, her dogmatic view that this was the only way to solve the problem, and her effort to humiliate me into doing the work her way so offputting that I basically blocked her out and nearly failed the course even though I had beeb using algebra and trig to solve actual problems for years. I think now that she probably had only a tenuous grasp of mathematics and rote obedience was her only way of ‘teaching’ a subject that she didn’t really understand.
Stranger
I’m of the mind that pretty much everything I learned in school, including very many years at universities, is useful. Maybe some of the things I learned are not the best, or most practical methods. However, knowing this low-quality stuff at least gives me an understanding of how techniques have evolved over time*.
*For instance, genetic algorithms (GA). For a while, and to some extent today, they were the go-to method for solving certain types of problems, only…so inefficiently. I’m re-reading a novel from the late 1990s that mentions using genetic algorithms and “supercomputers” to solve the knapsack problem (fill a “knapsack” with as many beneficial objects as possible, while staying within weight or size constraints). Today I’ve got numerous other, better ways to solve this problem and try to avoid G. A. whenever possible.
Sorry, It’s not.
You expect me to remember the correct spelling of a numerical method I’d never use? 
OK, I’ll take your word on it… Even with an advanced degree in a STEM field, none of that was ever relevant to me, so I don’t know of it… But I have no problem accepting that, to some in other fields, it’s quite useful. But surely, the proper time to introduce it is when one’s going to use it for one of those things, not at the high school level when all it can do is things that can be done better using other methods.
And I kind of agree about FOIL… It’s really just a simple special case of the procedure for multiplying two arbitrary polynomials, and the general case follows straightforwardly from the distributive property. If you’re ever going to find yourself multiplying polynomials larger than 2, it makes sense to just learn the general method, and not even worry about calling it FOIL.
But on the other hand, most students won’t ever multiply polynomials larger than two terms, and a lot of those students have difficulty with organizing things systematically. So in a high school classroom, it makes sense anyway.
A question about the Pythagorean Theorem: in hindsight it can be proven literally dozens of ways; but what led Pythagoras to suppose that there was an easily expressible relationship between the sides of a right triangle? Why right, of all possible angles? Or perhaps it should be asked in the converse- what led anyone to suppose that if the length of a triangle’s sides are squares, this means the triangle must be right?
OK, for me it was Differential Equations.
As an engineer I needed four semesters of “serious” math. I don’t mind - teaches you how to learn, how things work, discipline, everything college should be about. Calculus applies to engineering.
But DiffyQ? In 40 years I’ve never had call to solve sin or cosine based integrals or differential equations. So for me, that was most useless - a lot of stress, a lot of studying, and I missed an A by two points (strict grade curve in this class), and for what?
My understanding is that the Egyptians (who needed to accurately survey land every year after the Nile floods) had various “rule of thumb” methods for identifying right triangles, but no general idea of the relationship between the lengths of the sides (so they knew of the 3-4-5 triangle and that it was right). Pythagoras was therefore inspired to go further to understand the right triangle more abstractly.