Solve this algebra problem to graduate. What's the answer?

[Shagnasty]

@DSYoungEsq

Your lack of understanding of modern cognitive thought theories, not to mention modern pedagogy theories is considerable, to judge from this post. I won’t bother to categorize them. But you are making a simple basic error when you say that the only reason someone might have problems learning Algebra is a lack of desire. That statement is laughably wrong, and unfortunately buys into a stereotype of what holds some people back in a given subject area.

I agree. I think a good analogy is that many extremely talented musicians can’t read sheet music. They just know how to invent and play music intuitively and that is the real goal.

I get really turned off by the X factor in algebra. Yes, it is a variable but it is a horrible one as any software engineer will tell you. That type of notation isn’t allowed in any type of modern software design because it is too abstract and unreadable to most people. I stop reading any posts on this board and others when someone starts naming people A,B and C. My brain simply doesn’t work that way. Call them Mary, Jane and Billy for all I care but people don’t have single letter names.

I don’t think algebra is that hard either in real-world applications. However, I truly believe that the current teaching methods for it are horrible and worse than useless. That applies to a lot of STEM fields as well. Teach trigonometry by measuring lines from a flag pole or something. Teach chemistry by doing experiments (chemistry does better at this than most already) and teach physics by doing cool things that the students have to figure out. Most students don’t respond well to having to memorize obscure formulas without any context. The people that write math textbooks don’t seem to understand that most people don’t think like they do. You can’t just vomit up 200 pages of obscure formulas and expect most people to appreciate what they mean in a deep sense.

Speak English for God’s sake! I work in an analytical field and have always been great at practical and word problems but I couldn’t give two shits about factoring polynomials. That is what MathLab is for.

Science and math don’t have to suck. Real scientists, doctors and engineers hardly ever have to use Algebra II or above and the few that do are more than capable of learning it through elective classes. It is telling that the quantitative section of the GRE (Graduate Record Exam) for college graduates looking to go to graduate school is much easier than the SAT that high school students need to take to get into college in the first place. Most people in college take only a couple or few math courses and quickly forget everything they were force-fed in high-school because higher math has little value for most of them.

That is completely different from writing proficiency that is an essential skill for almost everyone.

[watchwolf49]

@DSYoungEsq

Your lack of understanding of modern cognitive thought theories, not to mention modern pedagogy theories is considerable, to judge from this post … [snip]

I’ve made no claim otherwise …

@DSYoungEsq

[snap] … But you are making a simple basic error when you say that the only reason someone might have problems learning Algebra is a lack of desire. That statement is laughably wrong, and unfortunately buys into a stereotype of what holds some people back in a given subject area.

Yet, you make no statement as to what is true … what other reasons would an able student have troubles passing Algebra such that they couldn’t simplify the problem in the OP … as a sophomore in college …

Pedagogy doesn’t answer the question of why a class should be taught … does it?

[Derleth]

@Shagnasty

That is completely different from writing proficiency that is an essential skill for almost everyone.

I rearranged the post to put this on top. I think it’s that important.

I get really turned off by the X factor in algebra. Yes, it is a variable but it is a horrible one as any software engineer will tell you. That type of notation isn’t allowed in any type of modern software design because it is too abstract and unreadable to most people. I stop reading any posts on this board and others when someone starts naming people A,B and C. My brain simply doesn’t work that way. Call them Mary, Jane and Billy for all I care but people don’t have single letter names.

That is indeed a problem with how grade school and high school math is taught now, and I think it is because the current curriculum is based on a goal that’s too far away: Calculus, as in by-hand integration and differentiation, in all its awful glory, can only be approached through years of algebra, and, as you say, the algebra we teach no longer has much practical application. In school, once you have the equation, the work is only begun, because the scutwork of symbol-shuffling is still ahead of you. In the real world, getting the equation means your work is over, because Maxima has existed for decades and can solve any realistic equation in an instant. We give them years of algebra so they can do the algebraic parts of calculus faster and more efficiently. We build to calculus because it’s calculus, the great and powerful, vitally important to beating the Soviets to the Moon.

Absolutely none of the math you learn up through second semester calculus (basic integrals and infinite series) is best done by a human. People try to save arithmetic by saying it gives people a sense of numbers, but that isn’t what we teach. We teach “calculator emulation”, as in having humans emulate calculators, and maybe the brighter students get a feel for numbers, in an intuitive sense, instead of teaching estimation as a respectable and useful mental tool in and of itself.

Algebra can be saved, but we must apply it to a closer goal. Statistics is that goal. You can do useful statistics, with real-world examples, without needing a ton of preparation, and it teaches actual logical thinking much better. It teaches adversarial logical thinking: The teacher said this. I have an argument which shows something else. I should prove the teacher wrong. That’s how the class should work, because that’s how real life works: People are actively trying to bullshit you, and you have to defend yourself. In writing. Coherently.

(Yes, this will weed out the stupid and the asshole teachers. I hope to see them driven before me and hear the lamentations of their unions. Bye-bye, petty martinets. We never needed you.)

And statistics can lead into calculus. It can lead into linear algebra, if you want it to. Math is connected like that. However, most students, including a great whack of STEM students, can give calculus a miss entirely and never miss it. Statistics is both more important and easier to jump into early.

[Chronos]

I get really turned off by the X factor in algebra. Yes, it is a variable but it is a horrible one as any software engineer will tell you. That type of notation isn’t allowed in any type of modern software design because it is too abstract and unreadable to most people. I stop reading any posts on this board and others when someone starts naming people A,B and C. My brain simply doesn’t work that way. Call them Mary, Jane and Billy for all I care but people don’t have single letter names.

Thank you, Shagnasty, this is actually a really useful insight. It’s something that’s extremely easy to fix, and which could potentially save many students struggling with algebra, if your difficulty is a common one.

It calls to mind Misner, Thorne, and Wheeler’s Gravitation, the definitive graduate-level text on general relativity. They don’t call the Riemann tensor R, as is the custom. They call it Riemann. Likewise for all of the other significant mathematical objects. It means that their formulas take up a bit more space on the page, but it makes it abundantly clear what each thing is.

And if it’s good enough for a graduate-level course on general relativity, then it’s good enough for high-school algebra.

[DPRK]

True, and, if you recall, Knuth published a lovely little book on mathematical writing, illustrating how writing and notation can make the difference between incomprehensible and crystal-clear for the same material.

But what is the “extremely easy fix” in general? Say in case of the following (arbitrary) exercise: “find all solutions of x² + 2y² = x² + xy + y² = 2.” It’s not out of a physics textbook or word problem, so is it insightful to rename the variables?

[Chronos]

…True. I didn’t think it through all the way; it is useful to learn the general (and hence abstract) case.

[Thudlow_Boink]

@DPRK

But what is the “extremely easy fix” in general? Say in case of the following (arbitrary) exercise: “find all solutions of x² + 2y² = x² + xy + y² = 2.” It’s not out of a physics textbook or word problem, so is it insightful to rename the variables?

I was wondering how it would help in a case like the problem this thread started out being about. Would it make things any easier if the x and y in that expression were renamed Xavier and Yolanda?

[Leo_Bloom]

@Thudlow_Boink

I was wondering how it would help in a case like the problem this thread started out being about. Would it make things any easier if the x and y in that expression were renamed Xavier and Yolanda?

Ask Bertie Wooster, who lucubrates thusly with Jeeves:

“A problem has arisen in the life of a friend of mine who shall be nameless, and I want your advice. I must begin by saying that it’s one of those delicate problems where not only my friend must be nameless but all the other members of the personnel. In other words, I can’t mention names. You see what I mean?”

“I understand you perfectly, sir. You would prefer to term the protagonists A and B.”

“Or North and South.”

“A and B is more customary, sir.”

“Just as you say. Well, A is male, B female. You follow me so far?”

“You have been lucidity itself, sir.”

“And owing to … what’s that something of circumstances you hear people talking about? Cats enter into it, if I remember rightly.”

“Would concatenation be the word for which you are groping?”

“That’s it. Owing to a concatenation of circumstances B has got it into her head that A’s in love with her. But he isn’t. Still following?”

“Yes, sir.”

I had to pause here for a moment to marshal my thoughts. Having done so, I proceeded.

“Now until quite recently B was engaged to --”

“Shall we call him C, sir?”

“Caesar’s as good a name as any, I suppose. Well, as I was saying, until quite recently B was engaged to Caesar and A hadn’t a worry in the world. […]”