A phillipic against equations in math teaching

It isn't true that most people hate math. Most people have no idea what math even is. In most math classes, no math whatsoever is taught. So people end up hating something artificial, something they have all the right to hate. The area taught in "math" classes that's probably the most removed from real math are equations.

How equations are important for ordinary people

"Math" textbooks try to convince you that what you're learning is useful. They do this by introducing word problems. Word problems let you judge how useful what you're learning is. The more convulted the word problems, the less useful the topic. For example, calculus textbooks have no problem with word problems. They ask about velocities deduced from graphs of position, they ask about sound loudness based on how far you are from the source... All those things are problems you may run into in physics and calculus is the best way to deal with them.

But what are the word problems for equations? Train problems? Workers working together (Bob builds a house every 4 hours, Tom builds a house every 3 hours, how long will it take them to build 147 houses if Tom arrives 2 hours late?)? You never come across such questions.

Which equations do non-mathematicians need? Pretty much none whatsoever. If you do simple physics, you will want simple manipulation of equations, but most of the work here is just letter/number juggling. Quadratic equations may appear in acceleration situations, but nothing worse than that. If you do serious physics, see the mathematician section, the part about calculus.

How equations are important for mathematicians

Not as much as laymen would think. There are polynomial equations, equations of the form:

axⁿ + bx^n-1 + cx^n-2 ... mx+n = 0

where a,b,c... m are any numbers (a shouldn't be zero) and n is any positive integer.

The interesting part is that the way to find the solutions changes according to the value of n. If n is 1, 2, 3 or 4, there are formulae for determining the solutions. The formulae are:

For n = 1, this isn't even a formula.

ax + b = 0

ax = -b

x = -b/a

Simple enough.

For n = 2, we have the quadratic formula, often memorized and tested without any explanation:

There are two solutions, one with plus and one with minus. The reasoning for this involves a little strange manipulation called "completing the square", but it was still known even before the number zero.

For n = 3, things get somewhat scary. This time, there are three solutions. In fact, this is a general rule: there are (at most) n solutions for a polynomial equation with the largest exponent n.

These formulae were invented in the 16th century, which means they were invented before the plus and minus signs. Seriously. Apparently, they had P and M, instead. It was common to just write expressions out, so instead of the quadratic formula, you'd have:

"minus b plus or minus square root of b squared minus 4 times a times c, all that divided by 2 times a". Admit it: the formula is better. Imagine writing out the cubics. The students today have it so easy... Anyway. I have to stress this. This is NOT what mathematicians do most of the time, since

a) It can be simplified, somewhat.

b) You almost never actually come across these equations, so once the general solution is found, mathematicians no longer care.

Oddly enough, few teachers find the bravery to teach this.

For n = 4, things get ridiculous. The four solutions' formulae are these:

These formulae were found together with the cubics, but presumably not written out fully until the invention of modern math symbols - I suppose the amount of ink needed for that would buy you a nice little house in the 16th century.

Let me quote Wikipedia here:

The Soviet historian I. Y. Depman claimed that even earlier, in 1486, Spanish mathematician Valmes was burned at the stake for claiming to have solved the quartic equation.^[3] Inquisitor General Tomás de Torquemada allegedly told Valmes that it was the will of God that such a solution be inaccessible to human understanding.^[4] However Beckmann, who popularized this story of Depman in the West, said that it was unreliable and hinted that it may have been invented as Soviet antireligious propaganda.^[5] Beckmann's version of this story has been widely copied in several books and internet sites, usually without his reservations and sometimes with fanciful embellishments. Several attempts to find corroborating evidence for this story, or even for the existence of Valmes, have failed.^[6]

As you can see, quartic equations are a big deal.

I don't think anybody has ever used these formulae in this form (you can consider this a challenge). But they exist, and we have to deal with that.

Now you might expect some even huger formulae for n = 5, but they don't exist. It's not that they haven't been found yet; they don't exist. There's no combination of roots, +, -, * and / that yields the solutions. The reasons are really complicated and involve some quite advanced math (Galois theory, to be precise), but there are no formulae for n>4. See? This is the kind of stuff that might be interesting.

Mathematicians care about equations on a more general level, so that instead of solving a million equations, you get a few types generally solved, at which point all has been done and there's no more need to care.

As for other equation types, there aren't many. Much of what is taught as equation solving is just a lot of different ways to move things around in equations to make them polynomic. It is only a glorified movement made to ensure students can do "math" without ever thinking at all. There are differential equations in calculus, but they're a completely different deal than what you're taught in algebra. Quadratic equations pop up occasionally through their solving - and they are responsible for some duality in nature, such as two possible values of spin for electrons. Trigonometric equations are important in geometry and waves, including sound waves. What's characteristic of math education is that this is covered very little. So what is covered?

What is taught

There are SO MANY types of useless equations. Complicated fraction equations, where you have to do a lot of simplifying only to get some polynomial equation. Odd square root combinations, where the way to solve the equation is "Move all the roots on one side, raise both sides to some power, hope things cancel out, if not, keep repeating until they do, eventually get 20 possible solutions and check painfully whether each of them is a real solution or a fake one that appeared as a false positive because of the method used". Sometimes, the equation only works for a specific situation, because something "happens to" cancel out. This defies an important principle in math - be general. Oftentimes, changing a tiny thing, like adding x to one side, renders the equation almost unsolvable. Sometimes, equations are portrayed as needing a smart idea, when all they need is muscle power, patience, formula knowledge (Who would think that (x³ + y³) = (x + y)(x²-xy+y²)? Nobody, the teacher just tells you that and you can check) and memorizing a thousand tricks.

Why that's terrible

One of my saddest memories is asking my math teacher what was the point of learning something and hearing "you never know what might come in handy". Discovery for its own sake is fine, but not uninteresting, specific, memory-heavy discovery. This creates a distorted idea of what math is about and causes people to dislike it. The time spent teaching equation solving (that computers can do for you, unlike some other math) could be spent productively. You could teach children actual interesting stuff. Or you could give them a break. You could do pretty much anyting but boring stuff. Math can actually be interesting! M