As most of you know, I'm a high school student (freshman).
This year I complete my 10th year of being a full-time student (including kindergarten), and, I must say, I am sick and tired of the way people are "educated".
Take your typical mathematics class (my favorite subject inside and out of school) in the United States.
The teacher starts by checking the night before's homework (most of the time checking if people have done it), asks people if they're having any difficulties with it. If so, then he/she does that problem on the board and explains it. Then, introduces a topic like "rational functions", "imaginary numbers" or "the Pythagorean theorem", from the "textbook" which has a few examples in it.
The lesson usually consists of beginning by showing the formula for that chapter or section or unit or whatever the heck its called, and then having the students see a simple exercise in which they plugin a few numbers into this magical formula which then spits out the answers.
Then, the teacher then picks a more "difficult" problem, in which the same formula is used, but, its a "real world application", in which students find things like the distance between two campsites or figure out how long it will take Bob to run around the track if he's going at so and so miles per hour.
And, the homework is handed out, which are exactly the same problems, just redressed with different numbers, and the cycle begins once again the next day.
So, what do people learn at the end of the chapter?
Here's the problem.
First of all, the students are never given the idea about what how the formula actually came into existence.
Basically, the current system dictates that it doesn't really matter where it came from, for all your concerns, someone just thought about it one fine day and wrote it down, you have a test coming up next week, so you'd better stop getting distracted and get the formula memorized or you'll fail.
So, this causes students to go about the "textbook" skipping everything but the formulas, and then memorizing those. Then, when the test comes along, those who had time to memorize their formulas do excellent, and those that had something going on get low grades.
Then, as soon as they're done with the test, they put in all of their efforts into memorizing the next set of formulas and have nothing left from the last set that they memorized except "I love *whatever the topic is*, I got a 96% on that test".
And, the "textbooks" have "real world applications", which involve doing things that people would never even DREAM about doing (there are an incredible amount of problems about outdoor activities), because they're just plain stupid (do you measure the distance between the department stores in your town with the Law of Cosines?).
So, this causes students to think "when am I going to use any of this stuff anyway?", which is something I would think about very often if I hadn't taken the time to read a bit more about math.
These facts need to change.
First of all, you might have noticed that I keep saying textbooks in quotes, and there's a reason for that.
The textbooks suck.
Pick up any standard, run-of-the-mill textbook about "Algebra II" or "Precalculus", and you'll see a very similar picture.
The chapter begins, promising incredibly lackluster "real world applications". Then, goes into the first section, in which it tells you the formula in a conveniently highlighted box, which basically tells students not to read the rest of the passage, since its only there to make the book fatter anyway.
Then, there are pictures of people doing sports and things considered "fun", and are related to the subject of matter extremely lightly, but, the publishers and authors assume that if you put in some pictures about sports, the students will "learn better" (a.k.a memorize formulas in a shorter amount of time).
Then, there are a few examples in which the formula is used (again, no explanation of where it came from, or why it is needed), and ends with a horribly stale application, and then commences onto the exercises.
There are 110+ exercises in each section, about five of which are actually worth doing, since all the others are just slight deviations from the examples.
At the end of it all, you haven't solved a single problem by thinking it out yourself, all you've done is plug in some numbers.
The biggest thing that these textbooks and curriculum lack are real problems.
Right now, almost every high paying job (with a good success rate; e.g. you cannot count football players, since less than 1% of them make a high pay) in the world deals with solving problems.
We solve general problems like "An easy way to share pictures is needed; we have to build it", but, we also solve problems like "The search results are taking way to long, is there some kind of way we can speed them up?"
Obviously, these are difficult problems, and there's no general "formula" to plug things into that'll spit out the right solution.
We have to invent things as we go along.
When a patient comes to you, you don't have a set method of identifying "he has so-so, and he needs this to fix him up".
First of all, he gives you about 98 different things that are happening to him, and from there, you need to understand which ones are important and then find out what's wrong with him (assuming that the symptoms you said are important really are important)
You have to take a small amount of information given to you, and synthesize a solution.
You have a set number of rules you have to follow, and some conditions given with some more information, you need to derive a conclusion.
All of these things are what the current curricula lack.
There are no difficult problems that one must think about in order to find the solution, they are just things that as long as you don't make any mistakes, you'll be perfectly fine.
This makes students entirely neglect the point of mathematics!
The point of math isn't to find out the probability of Jimmy wearing red socks on Tuesday, or to memorize the Rational Root Theorem word for word, because you probably will never use those things in your entire life, but, the point of math is to show how clear headed thinking and problem-solving are executed.
Given a difficult problem, one must invent, synthesize and derive in order to find a solution, which may involve failed approaches, but, the result is much more lasting.
And, this cut-and-dried approach to math is useless, because as soon as you take advanced math courses in college, you'll realize that the "proof" sections in the book actually mattered!
Okay, so, I've stated all the problems, now, how about a solution?
First and foremost, get rid of the "Algebra II" and "Geometry" subdivision junk that's going on, its entirely useless.
Math wasn't invented like that, and therefore should not be taught that way.
Mathematics isn't a bunch of blocks of knowledge that you use parts of to do exercises, its this whole continuum where a problem, if not solvable by the methods in one field, can be moved to another.
Second, the textbooks are horrible and need to be rewritten.
To do that, we begin by placing a limit on the size of the textbook to 300 pages or so (sounds a bit Legalist, no?)
Then, we take out the horribly expensive and useless color pages (except for 4th grade and below; I really liked the colors then, because the textbook didn't really mean much), and replace them with black and white.
Since its all black and white, we'd might as well throw out the pictures of people surfing as well, since people draw mustaches on those anyway.
Then, we rid ourselves of the idea of sections, since they get in the way anyway and constrict our overall "view" of mathematics too much.
Now, each chapter deals with each topic in a very conversational way, where the motivation of each step (all proofs included) is shown, and how one step leads to the next is very clearly shown.
There is one textbook that is currently in use that I would like to point out that does this incredibly well, which is Linear Algebra and Its Applications by Gilbert Strang.
Matrix multiplication isn't just thrown out as a formula, but, its developed conversationally from the idea of linear systems (certain higher level textbooks do much better in this regard).
And, the topics within a chapter are connected, one cannot just jump, everything is continuous.
However, chapters can jump around math as much one would like, from Number Theory to Geometry and then to Combinatorics.
All the exercises need to replaced with a few, well-selected sets of problems which not only extend the material, but, provide new insight as to how to take a simple idea and use it on a complex problem (Pigeonhole principle-esque stuff).
Then, the section of textbooks would be quite solid.
Teachers need to be retrained, and paid a heck of a lot more than right now (think six figures), to make it a much more competitive field. Teaching needs to based around problems, starting lessons with problems, and then allowing the students to have a go at the problem, giving hints in between to lead them closer and closer to the solution.
If we are able to make a large portion of these changes, we would be far better off than right now in terms of what students can take away from mathematics classes, regardless of what they study in the future.
That's my idea of how this would work, and I would love to hear yours, so, please drop it in the comment section below.
If you liked this post, please tell me about it, so I'm encouraged to write more, and please follow (there's a button at the top right).
Questions, comments, etc. all welcome!