A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made—all without the advice or participation of a single working musician or composer.

Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.

As for the primary and secondary schools, their mission is to train students to use this language—to jiggle symbols around according to a fixed set of rules: “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”

In their wisdom, educators soon realize that even very young children can be given this kind of musical instruction. In fact it is considered quite shameful if one’s third-grader hasn’t completely memorized his circle of fifths. “I’ll have to get my son a music tutor. He simply won’t apply himself to his music home-work. He says it’s boring. He just sits there staring out the window, humming tunes to himself and making up silly songs.”

In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in scales and modes, meter, harmony, and counterpoint. “It’s a lot for them to learn, but later in college when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high school.” Of course, not many students actually go on to concentrate in music, so only a few will ever get to hear the sounds that the black dots represent. Nevertheless, it is important that every member of society be able to recognize a modulation or a fugal passage, regardless of the fact that they will never hear one. “To tell you the truth, most students just aren’t very good at music. They are bored in class, their skills are terrible, and their homework is barely legible. Most of them couldn’t care less about how important music is in today’s world; they just want to take the minimum number of music courses and be done with it. I guess there are just music people and non-music people. I had this one kid, though, man was she sensational! Her sheets were impeccable—every note in the right place, perfect calligraphy, sharps, flats, just beautiful. She’s going to make one hell of a musician someday.”

Waking up in a cold sweat, the musician realizes, gratefully, that it was all just a crazy dream. “Of course,” he reassures himself, “no society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial; no culture could be so cruel to its children as to deprive them of such a natural, satisfying means of human expression. How absurd!”

…

Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done—I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.

Everyone knows that something is wrong. The politicians say, “We need higher standards.” The schools say, “We need more money and equipment.” Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, “Math class is stupid and boring,” and they are right.

Well I generally don't drink caffeine because it tends to contribute to serious anxiety episodes, but if tea is required maybe I will look into it.

Later courses build aggressively upon those course objectives, and you are expected to have a basic understanding of terms and concepts.

Eh, yeah, but it's far more important to have a grasp of the fundamentals than it is to memorize some specific application. If you have the fundamentals, it's not that hard to figure out the specific application. Even if you memorize your way through the objectives, you aren't going to remember them that well in the next class, and you won't have the tools to relearn them.

I'm actually not sure if we're talking about the same thing.

I generally think university math courses are taught alright. High school math courses are so bad that mathematical literacy would probably improve by their abolishment.

This is correct, and my earlier comment wasn't meant to imply that memorization should be avoided in math.

Every mathematical book that is worth reading must be read "backwards and forwards," if I may use the expression. I would modify Lagrange's advice a little and say, "Go on, but often return to strengthen your faith." When you come to a hard or dreary passage, pass it over; and come back to it after you have seen its importance or found the need for it further on.

So much of being in school is about being told to keep your chin up, despite all the circumstances.

This predisposes people to think that they are bad at a subject without considering the possibility that it was their teacher who was bad at the subject.

Because, I mean, what's more depressing--that you tried and failed, but made a noble attempt and can respect others who succeeded where you didn't, or that you were never given the proper chance to succeed in something because of circumstances beyond your control? That's a dark thing to accept about the reasons for your relative strengths and weaknesses, and one that would be hard to live with, since it makes you wonder if your entire career path is simply one big rationalization of mistakes of others.

I'm fairly confident that I will do fine in University math courses if I revisit the stuff from high school in a perspective that doesn't make me want to die. But I'm a millennial, so instead of putting in that work, does anyone have a infographic that has all of the formulas and properties on it? That way I can just repost it on social media and be all proficient in math.

Also I will never forgive Descartes for starting that convention of using letters for variables instead of something more whimsical like drawings of elephants and dicks, maybe some swastikas?

George Chystal put it best in his 19th century classic book (on what today would be called a mix of "precalculus" and real analysis), called Algebra:

George Chystal put it best in his 19th century classic book (on what today would be called a mix of "precalculus" and real analysis), called Algebra:

THE SCHOOL PROBLEM

Most people consider school to be a grim necessity to be accepted, endured and survived. School, as nearly everyone freely admits, is dull, unpleasant, and designed to build mediocrity. It is a mapping of the world of ideas into a sequential bureaucratic presentation system, with generally awful results.

1. The Curriculum

The very system of curriculum, where the world's subjects are hacked to fit a schedule of time-slots, at once transforms the world of ideas into a schedule. ("Curriculum" means "little racetrack" in Latin.)

A curriculum promotes a false simplification of any subject, cutting the subject's many interconnections and leaving a skeleton of sequence which is only a caricature of its richness and intrinsic fascination.

2. Teacher as Feudal Lord

The world of ideas is carved into territories, and assigned as fiefdoms to individuals who represent these territories (called Subjects); these lords and ladies in turn impose their own style and personality on them. The pupil must pay homage to the Duchess of History, the Count of Mathematics; and if you and these individuals do not like each other, you will almost surely dislike the subjects they control, which take on their stamp and personality. Each feudal lord has absolute power to bore, offend, and sever access.

The teacher controls access to the subject under his or her own viewpoint. If you find this viewpoint unfriendly, unpleasant or confusing, that subject becomes closed to you forever.

These two principles-- the crushing of living subjects into curricular caricatures, and their bestowal to feudal overlords--effectively guarantee that whatever is taken in school becomes and remains uninteresting. Everything is intrinsically interesting, but is drained of its interest by these processes.

Thus follow both the dreariness of education and the crippling of the mind as we see it everywhere today. Education is typically the process of successively ruining subjects for you, and the last subject to be ruined determines your profession. An educated person is someone who says, "I don't know anything about that, I never took it." Whereas a free-minded person can become excited about a new idea, in any subject, whether or not he or she ever heard about the idea or the subject before.

What is perhaps even worse, this system imbues in everyone the attitude that the world is divided into "subjects;" that these subjects are well-defined and well-understood; and that there are "basics," that is, a hierarchy of understandings which must necessarily underpin a further hierarchy of "advanced ideas," which are to be learned afterward.

This outlook could not have been better designed to crush people's mental spirits, to keep them from becoming involved with ideas, from thinking, exploring, conjecturing, taking interest.

I also think there's some pedagogical quirks of math education that are just impossible to avoid. I have a friend who's in community college and wants to study mathematics. In one of their algebra courses, one immediately before calculus, they were taught http://mathb.in/149164 this formula for the line between two points on a function. Which is confusing and worthless unless if you have the concept of limits and can build that definition into the definition of derivative. So the problem wasn't the content, it was that it was unmotivated.

But, I don't see a good way to fix it. There isn't a way for the college to figure out exactly who's going to take calculus and motivate things better, or explain why certain things will be important. And I can see this plainly in math when looking back at old material, many things seem pointless and unmotivated until you take some abstract algebra and real analysis and realize their importance. Another example: manipulating inequalities and absolute values. Completely unmotivated until real analysis.

Remark. I believe that the above argument [a concrete demonstration of the Euclidean algorithm] is more convincing than an abstract proof, and just as precise. In order to be sure that the abstract version is sound, I needed to see how it works in a typical case. But having seen a typical case such as the over above, I then tend to find the abstract discussion--which simply substitutes Greek letters for the numbers 123, 54, etc.-- unnecessary. In addition, abstraction does not add anything significant, and definitely interferes with the learning process.

Marc Kasowitz, President Trump’s personal attorney on the Russia case, threatened a stranger in a string of profanity-laden emails Wednesday night.

The man, a retired public relations professional in the western United States who asked not to be identified, read ProPublica’s story this week on Kasowitz and sent the lawyer an email with the subject line: “Resign Now.’’

Kasowitz replied with series of angry messages sent between 9:30 p.m. and 10 p.m. Eastern time. One read: “I’m on you now. You are ****ing with me now Let’s see who you are Watch your back , *****.”

In another email, Kasowitz wrote: “Call me. Don’t be afraid, you piece of ****. Stand up. If you don’t call, you’re just afraid.” And later: “I already know where you live, I’m on you. You might as well call me. You will see me. I promise. Bro.”

Kasowitz’s spokesman, Michael Sitrick, said Thursday he couldn’t immediately reach Kasowitz for comment.

ProPublica confirmed the man’s phone number matched his stated identity. Technical details in the emails, such as IP addresses and names of intermediate mail servers, also show the emails came from Kasowitz’s firm. In one email, Kasowitz gave the man a cell phone number that is not widely available. We confirmed Kasowitz uses that number.