Why Math is considered more difficult that it is.

IMHO e^(πi) = -1 is beautiful , also Pythag's theorum. Up to those points in algebra(?) and geometry it was just a tedious slog for no apparent reason or reward.

 

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The damage comes from failure, which is baffling and frustrating and threatening to the student - you risk persuading the kid that they're stupid and can't do real math, when all they are is too young in some way (age varying by individual). You also risk fixing a misunderstanding, implanting some schoolkid strategy for getting by which will never be corrected.

Most math notions are pretty simple, after all - after you "get" them. Unfortunately, at the bureaucratic high school level comprehension can be faked - and we see that many successful graduates of US high school calculus programs do not actually comprehend limits, which more than a couple math teachers and tutors of my experience have attributed (in my hearing) to them having been "taught" the concept before they were capable of understanding it, and getting by ever since on whatever flawed conception or collection of arbitrary rules was good enough for passing tests in that never-repeated class.

It's not just fancy stuff like limits, either - for example, negative numbers are often taught to the very young and then never taught again, so the kids who weren't ready then never do actually get a handle on them. Imagine complex numbers faced by a student whose understanding of negative numbers comes from a mental image of a thermometer and some arbitrary memorized rules for multiplication, division, etc. ("minus times minus equals plus, the reason for this we do not discuss"). If you are ever dragooned into teaching an introductory calculus class at a large State university, you can give yourself a bit of a jolt by rigorously evaluating your students's ability to handle negative numbers in arithmetic and algebra - at least in the US.

 

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Depends on the school system. Let's say 16/17 in general, ignoring "foreshadowing", but there's a push to " teach calculus" at younger ages in many places.

This has mostly come up for me in the cases of students who have tripped in that particular hole in their background while facing college level math requirements or the need to do well on some standardized test, and care about it. Volunteers who want to learn, in other words. Not tried with many young children.

For such students, one approach that seems to work is building a new picture of the number line based on describing multiplication by negative one as a rotation - a degenerate one, that amounts to a flip over the origin or an inversion through it (a counterclockwise rotation of pi radians, with nothing in between because we are working in one dimension). All operations involving negative numbers are then - for instructional purposes - split into the operations with the positive numbers and multiplications by negative (or positive) 1.

Two flips gets you back where you started, then. The null rotation: Plus.

There can be lot more narrative of course, introductory and otherwise, all informal and heuristic rather than rigorous theory, but this approach can handle associative and distributive laws, expand easily to multiplication by i (rotation in two dimensions), bring trig functions and vectors/complex numbers into intuitive relation, and help with a variety of other sometimes troublesome matters.

Simply having a picture that makes sense, where before there was the insecurity and stress of not having a coherent intuitive idea, can benefit.

 

That's actually a very nice trick I haven't heard about before. Cool!

My personal solution is to introduce a new mathematical object, called the "whatever you need"-object. Its value is defined as -1 where needed, and +1 otherwise. Just stick that in front of everything, and job done!

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The - 1 x -1 thing is not an issue for my son. The logical idea of a double negative is pretty easy, I think. (Reminds me of the story of a class in which the lecturer was explaining that the negative of a negative is a positive, and then he went on to say that, of course the positive of a positive does not make a negative, hahaha. And then a voice from the back of the hall said, "Yeah. Right.")

I learned differential calculus for O level, which I sat at 15. I think by next year my son should be able to grasp it. But not yet: the basic rules of algebra are still too shaky.

 

Mathematics while important , does not seem , when taught , to have a practical application , when your 17-19yrs. old .

 

Its ubiquitous confusion in vernacular speech indicates there may be more trouble buried in that "logical idea" than the well educated can remember from their childhoods.

And it seems difficult to employ that metaphorical understanding in dealing with complex arithmetical expressions.

Native speakers of English who are careful can have an advantage, there, over languages in which double negation is standard grammatical negation itself - and has to be unlearned in order to intuitively apply such "logical ideas". But most native speakers of English are not careful.

Compare: All are careful/all are not careful // most are careful / most are not careful // many are careful/many are not careful // some are careful/ some are not careful // few are careful / few are not careful // least are careful / least are not careful // none are careful/none are not careful.

Another casualty of dropping "vocational education" from the American high school curriculum.

 

When I was young (and even now) the only people who didn't desperately need maths were the arty farty types.

 

Agreed. I suspect it's partly due to idioms. For example, I have no difficulty in understanding "I ain't got no money", but would with "I haven't got no money".

Anyway here is a mathematical proof I came up with for another forum I sometimes inhabit.............

Let's assume we are talking the integers. This means that

1. There exists an operation conventionally called "addition" and written \(+\)

2. There exists and operation conventionally called "multiplication" and wrtten \(\times\)

3, There exists integer \(0\) and \(1\)

Note that "subtraction" is NOT defined for the integers. So when one writes \(a-b\), say, one really means \(a+(-b)\)

So........

Consider the sum of products \(ab+a(-b)+(-a)(-b)\).

By the associative law write this as \(ab+[a(-b)+(-a)(-b)]\) and by the right distributive law write this as

\(ab+[a+(-a)](-b) = ab+0(-b) = ab\) from the multiplicative property of zero

On the other hand, using the exact same rules

\(ab+a(-b)+(-a)(-b) = [ab+a(-b)]+(-a)(-b)]\)

\(=a[b+(-b)]+(-a)(-b) = a0+(-a)(-b)=(-a)(-b)\) (although you use the left distributive law here)

Which implies that \(ab = (-a)(-b)\) as desired.

So returning to the OP, write \(a-(-b) \equiv a+(-(-b))\) or better still \(1a+(-1)(-b)= a+b\)

 

You haven't provided a clear and intuitively useful notion of what "- b" is, compared with just "b".
That makes all use of associative laws, distributive laws, and even multiplication itself, intuitively meaningless - gibberish, of a kind - to many students.

How, for example, did you come by this step:

 

I paid no attention in school, in any subject, yet naturally found math easy. Some teenagers, who strived to be a+ students, really struggled. They couldn't develop an understanding of numbers no matter how hard they'd focused. Numbers are an art, as I see them.

 

I really struggled with math in grades 1 -12. The explanations the teacher gave seemed like gibberish to me. Some people seemed to just 'get it' and others like me did not. The teachers tended to just shrug and give the impression that some people can get math and some can't. I found that very sad since I liked science and realized that science was not in my future since I didn't have a 'math mind'.

I went into the Navy in nuclear power and things changed. I went to Navy nuke school for a few months, which was math, physics, electronics, mechanics and steam systems. The method of teaching the concepts was very 'nuts and bolts'.

Math definition of rational numbers:
If r and t are rational numbers such that r < t, then there exists a rational number s such that r < s < t. This is true no matter how small the difference between r and t, as long as the two are not equal . In this sense, the set Q is "dense." Nevertheless, Q is a denumerable set. Denumerability refers to the fact that, even though a set might contain an infinite number of elements, and even though those elements might be "densely packed," the elements can be defined by a list that assigns them each a unique number in a sequence corresponding to the set of natural numbers N = {1, 2, 3, ...}..

Navy definition of rational numbers:
If you can write it as a fraction it's a goddamn rational number.

The light went off in my head, "math is simply another language and I don't understand the language". To be truthful the math definition above is still gibberish to me, but that strict definition is not really important unless you are a mathematician. I went on to college after the Navy with my new found insight and aced 3 semesters of calculus and 1 semester of differential equations, not to mention using those skills in my engineering and physics courses. I went from getting below average grades in math in high school to being a paid math tutor in college. It is not that I am particularly smart, I'm not, it is just that I learned how to translate the language of math to something I could understand.

If you have the right teacher and apply yourself - anyone with average intelligence can learn math.

 

It is a well-known property of the integers that for every \(x\) there exists a \(y\) such that \(x+y=0\). It is customary to say \(y=(-x)\). I am sorry if you are unfamiliar with integral domains - in fact the integers are a Ring (multiplication without an inverse is a defined closed operation), and I was making use of this fact

 

Mathematics was my college major & favorite subject. I still remember much of what I studied many decades ago.

I believe there should be math courses for those less adept and/or less interested in the subject, especially now that computers are available to almost everyone.

Computers allow display of fractal graphics, construction of magic squares, & other mathematical issues which are easy to understand & likely to be interesting to folks not adept at the discipline.
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Martin Gardner was the author of Mathematical Games in Scientific American. There is at least one book which contains articles from that column, most of which are understandable & interesting to folks who are not adept in mathematics.

 

I remember an algebra assignment question which was about proving that 0 + 0 = 0 - 0, I got it wrong (zero marks) and it took a while for me to understand the right way to answer such a question. So what is the right way, given you aren't allowed to assume 0 = -0?

 

Are you given the associative laws ? That is, can you write x - a - a = x - (a + a) ?

 

Just noticed I forgot the key parentheses: intended (x - a) - a = x - (a + a). Careless.