Upper level Mathematics as a core subject is one of the most perplexing parts of the American curriculum. The level niche, obscure, and abstract thought found in classes like "Algebra 2" and "Trigonometry" is taught as if it were as fundamental as language, the basics of history, and the entire concept of logic itself. The concepts learned past the basics associations with solving for X, measuring simple shapes, and basic plotting serve very little purpose to the average person in their personal and professional lives. The amount of time sunk into these subjects also is absurd, with most of the upper level classes serving no additive purpose in themselves. The amount of time used on these subjects could go to any other form of good knowledge such as philosophy or the students own interests.
But the point of math isn't the concepts themselves, it's to get you to solve problems!
This is a common cope yelled out by mathophiles who insist their knowledge is of practical ability due to its encouragement of a "problem solving" thinking methodology. This is akin to the arguments that those who teach language and writing might say about making their students write papers. "We're not teaching them to write this specific paper, but rather how to make good papers and good research categorically". So, this argument, if the predicate was true, isn't inherently flawed. However, Mathematics by no means a "problem solving" discipline when at the lower level. Yes, there are those trying to use math to solve architectural problems and real cryptography problems, but those aren't in your average high school classrooms and thus aren't even the concepts being talked about by the general public. No, instead students are memorizing algorithms made by smarter people and applying them given niche cases. Students are acting as computers, not the designer of the software that is running on it. When a student is good at math, he is merely a good calculator, not necessarily someone who is good at problem solving. Your average student, when faced with a quadratic equation, won't think up the quadratic formula or some form of factoring all on his own as a means of solving said equation. Instead the student will recall upon what they were told by a teacher in how to do it. They will be repeating and memorizing certain moves and actions rather than synthesizing any moves or actions of their own. This is fine when you're tasking students with learning addition and subtraction, tools used quite often in daily life, however this is utterly pointless when that student, past the class and other frivolous classes said student will take in the future, will not use the tools memorized. If one wishes for problem solving using math, give the student a problem and make them synthesize an algorithm to solve it. That is problem solving and that is how students will learn to problem solve, not by doing the grunt work calculations that we as a species have long since thrown onto computers. Better yet, show a student arguments about epistemology or the existence of God that they may have to think through and understand completely as to come to logical conclusions. Show something where they aren't merely applying formulas and algorithms, but instead critiquing and thinking about formulas and algorithms. That's surely a way to teach someone to solve a problem.
Math's Syntax Problem
Even if we grant that math exercises problem solving skills, we still
find problems. Math has to be one of the most ugly forms of logic,
contributing to its ineffectiveness at being a useful logic in terms of
digestibility. If math were a programming language, it would have been
abandoned, ridiculed, and abstracted off of for its sheer amount of
fragility. Math, by structure, is rigid, hard to work with, and
punishing of even minor mistakes that, in other methods of logic, either
wouldn't result in catastrophically incorrect results (programming
languages usually provide fail-safes and have easy tools to debug with)
or would have the true conclusion not entirely lost as there are other
fail-safe statements to make a conclusion true (language). If you take
an equation like
, it is a
syntactic disaster in both solving and
solution.
One where everything is shoved together in such an ugly way as to invite
you to lose a step in the mess. Going through the process of solving an
equation like this is hell because of this error-prone fragility.
Mathematical problems (assuming they are problems and don't have some
heuristically, unoriginal algorithm to perfectly solve it) ought to
spacious and inviting to look at so you don't lose yourself in the
condensed mess of modifications upon modifications. Philosophers and
Programmers1 both figured this problem out by writing their
logical statements in sane ways. Programmers use words as symbols for
meaning (instead of mere numbers), break problems down into individual
statements and steps as they're being solved so they're easy to
analyze and understand, and break problems down from macro-level
monstrosities into smaller problems that are easier to work with. Some
of these concepts may be integrated into how we teach math currently,
however, that simply isn't happening. Philosophers, at least
philosophers who write, have a tendency towards enumerating concepts in
a very verbose way, allowing much space in the content and ideas
themselves so they may be properly processed with meaningful statements
rather than obtuse symbols. One can have a math equation described
linguistically to them in how it functions and it will make complete
sense, however, the second that you put those symbols on a paper it all
falls apart. That failing is due to the sheer failure that mathematical
syntax is. Some may say this is a "stylistic critique" and "a matter
of opinion" that should be discarded. There is some truth to this, yes
I would like the logical notations I use to not be an obstacle in the
way of doing good logic. I rather not be stuck with absurd symbols
strewn about in needlessly complex ways when just a bit of space and
fluff would make everyone's life easier. There is a certain level of
"yes, this is my opinion, and yes it is the correct one" that comes
with this as the nature of how math is presented effects its ability to
be comprehended. Nothing in math is too complex until you're forced to
memorize insane symbols and formulas that can't be associated with
anything real.
The Test Problem
Going by the ACT scores in 2017 (though generally all scored for quite a few years have been similar), 20.7 was the average score. A 20.7 (rounded up) is only 30 questions out of 60 correct.. This is a 50% pass rate for the ACT's Math Section. An absolutely horrible score by all measurements. If a student received a 50% on a math test, that student might as well give up on passing that class. With scores this low, and the fact that this is average, implying that there are people who make even LOWER scores, it starts to beg a question about why this sort of thinking is even tested. Why bother holding someone's financial future in terms of scholarship behind their ability to do pointless computations and memorization. It's absurd. A hypothetical critic may say that it's to make sure that only the brightest get into college, this would be true if 40% of Americans didn't go to college, effectively guaranteeing that the population of colleges already aren't the brightest and best even though we give such tests. This idea that college is an "intellectuals only" zone is also insane. America isn't really a nation where 40% of its population is intellectuals because of the fact that 40% of people go for a bachelor's degree. Instead, it says that 40% of people need a bachelor's degree to live. There's a staggering pay bump for merely being college educated. Judging by the average expenses people go through, this pay bump may be an absolute requirement to do something as simple as starting a family. Adding any more obstacles to this system as it stands already (that includes absurd math requirements) only makes life harder for your average person given our modern world.