Why Math is considered more difficult that it is.

One of the largest books in my library was the Encyclopedia of Mathematics, and I'm not sure that was the correct title. It was 4 inches thick made of telephone book paper, and it was oversize. Its not a simple subject. I had a difficult time with Mathematics, because I couldn't grasp some mathematical concepts. I do much better if someone teaching fluid dynamics uses a wing rather than a graph. I get the concepts better and faster. Stress, injury, and disease also have a lot to do with comprehension. If you lead a somewhat sheltered life you probably are able to learn, comprehend, and retain easier than if your life is turbulent, or the disease is debilitating.

There are a lot of factors that go into understanding - anything. You have to be hitting on all cylinders to get it right: the information you are ready for, taught by a prepared teacher using the right materials, while the student is awake and attentive, not sick, not hungry, and not concerned with other dire circumstances. In my life that was almost a rare instance. I considered myself blessed to have lived 21 years. Mathematics was then somewhere down the road in order of importance. I had to get back to it, and mostly it was difficult for a good long time.

 

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I recall the answer had no multiplication in it, just addition.

I think you should start with a general approach: assume \( a,b \in \mathbb {Z} \), then assume \( a + b = a - b \).

Then prove that \( a = b = 0 \) is the only solution . . . ?
As I recall though, that wasn't the "right" answer, the answer depended on some ring axioms, one of which was the existence of the additive identity.

 

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I broke my math bone in the fifth grade. It never healed properly.

 

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Neither did my question.

There are a couple of different approaches, but they depend on what you are allowed to write. So I asked.

The obvious one would be to write 0 + 0 as (a - a) + (a - a) and 0 - 0 as (a - a) - (a - a), and do some algebra. But I can't tell from your description whether such parentheses can be introduced and removed. Is writing, say, 0 - (a - a) as (0 - a) + a allowed, for example.

Then multiplication was involved, and the necessary associative laws.

 

Rereading above, that was unclear. Simpler:

Is this sequence allowed:
0 + o
(a - a) + (a - a)
a - a + a - a
a - a - a + a *
a - a - (a - a) *
(a - a) - (a - a)
0 - 0

Note the role of an intuitive grasp of negative numbers - without one, the asterisk-marked steps are hard to follow, let alone invent.

 

I tutored a kid a couple of years ago who was really struggling with his high school math and physics courses, and it was almost halfway through the year when I started, so he had to overcome not only a learning deficit but also a grades deficit from his existing assignments and exams. He'd also had tutoring from other students and instructors who themselves had no trouble with the material but couldn't successfully transfer their understanding to him. He ended up passing both classes, I was so proud, and his dad was beyond thrilled.

Right off the bat I identified fundamental problems in the way he was learning the material; I blamed most of those problems on the school system and on teachers who themselves lacked a proper understanding of the material they were trying to explain. Like most of the other students in his class, who were also struggling, he was learning by rote, whereby they'd be given a certain kind of problem and shown a series of steps in order to solve the problem, with next to no explanation of the reasoning involved. I dug deep down into the fundamental logic and started imparting an understanding of why the teacher was doing those steps, teaching him to reason the problem out and apply formulas from scratch instead of just reproducing the teacher's argument.

I saw tremendous improvement from this student over the months that I taught him, to the point where problems that used to completely stump him had become routine, "yeah yeah no problem, I got this" kind of stuff. He ended up doing well enough on his remaining assignments, quizzes and exams to pass both classes, and I hope one day he'll choose to continue developing those faculties and applying them to whatever he does later in life.

How you learn is every bit as important as what you learn. Most of my knowledge even at the university level came from independent reading rather than what was discussed in class, and I always sought logical proofs and derivations for whatever claims were being made in those classes rather than simply accepting them at face value, blindly memorizing formulas and applying them.

 

But there are an infinite solutions to a + b = a - b.
b = 0 but a can then be anything, as you're left with a +0 = a - 0.

 

Excellent opening post!

 

The teachers are often themselves victims of a system built on the assumption that narrative or informal comprehension, the thinking behind stuff, the "proper understanding", requires unusual intelligence or ability. And they took only "ordinary people" math classes, themselves. (If one demanded a background of math classes that inculcated an informal comprehension of "high level" basic theory, the kinds of classes that might routinely lead to a "-1 is a rotation, like a flip" informality in the non-specialist student, it would cost 3/4 of the elementary school teachers in the US.)

They are also in a situation in which it is impossible, in time and motion, to provide very many multiple viewpoints to the class or different viewpoints to different individuals.

And they are under considerable pressure to produce test results, at least in the US. "Proper understanding" takes more time to translate into reliably accurate test responses - and the test is coming up soon. The test is always coming up soon.

So they postpone. The "proper understanding" is of course important, but instruction in it will happen next year, at the higher levels, whenever.

And it never does. Fast forward ten years: you have students looking at graphs of first derivatives going up and down through "stability points" or functions with negative and imaginary exponents, while thinking with minds prepared by "minus times minus equals plus" and images of vertical thermometers interchanging 0 and 32 paired with a horizontal "number line". Math is going to be hard, for them.

 

Excellent

 

I think Mathematics' level of difficulty doesn't really change - it has always been tough ever since. What makes students think it's considerably hard is the anxiety associated with it. According to studies, "Math anxiety can cause one to forget and lose one's self-confidence." I know people who fail Math subjects because of too much stress topped with anxiety.

 

There's a bit of blaming the victims in the subject line. I love history, it bores the paint off my wife.

 

And yet one frequently encounters examples such as CptBork's in post 46, of long-struggling students making sudden and even dramatic progress after some changes in the manner of their instruction and/or study.

One also frequently encounters the circumstance of students who do first struggle and then master commenting that it suddenly seemed easy.

There seems to be a chicken and egg problem here. Which comes first - the difficulty (and associated failure), or the anxiety (and associated failure)?

Thing is, although math is demonstrably intrinsically difficult for humans the performance standards are concomitantly low (compare the relative complexities of successful performance in mathematics and successful performance in dinner table conversation), and though it is undeniably difficult for many people, the struggles of most of these people begin very early in their education and often seem to pivot on relatively simple and basic matters - much simpler than the issues involved in learning to tie one's shoes, or cook a hamburger, or read a newspaper. So whatever the intrinsic difficulty of the subject, that does not seem to be the central issue in most people's difficulties with it.