NT math is like trying to see in the dark
I think there is truth in this, and it is why I did a lot better in maths once I took it to a higher level. I remember when doing GCSE maths being told that GCSE maths involves learning what to do, while A level maths involves learning why you do it. So I went on to do A level maths and then it all made more sense. It seems an odd way around to do it though - surely it would make more sense to explain the why's first. But I think a lot of people are quite happy just to learn the methods and don't care about why, whereas I need to understand why first. And people like myself, who need to understand the 'why', might get discouraged at school, as it is all about the 'what' for so long, so they might give up and not take it a higher level, because they don't realise how much easier it becomes. I wouldn't have taken it for A level if my teacher hadn't told us that A level is about the 'why'.
I'm about 5 months or so older than 'new' (post-Sputnik) math. I view math as a tool; give me one way to solve the problem, I'll solve it, and be done with it.
But the whole emphasis (at least when I was in school) was how many different ways are there to solve one bloody equation!...
I was left with the feeling that math is arbitrary and capricious, instead of the exact and methodical science I feel it should be.
Geometry was a problem for me; I saw the solution. Don't ask me how to explain it, but the gears whirred in the noggin, and I'd see the answer. Explaining step by step was beyond me.
Never got past college-level statistics, which was a lot easier (mainly looking up things in tables).
I still do multiplication from left to right, when I'm doing it in my head....![]()
This is one of the reasons that trigonometry annoyed me at first. My thinking is so concrete that I honestly couldn't relate it to anything useful until I had absorbed enough of it to put it together for myself. I have essentially dug information out of the subject that I'm not supposed to know until a BS and several years of dedicated study later. Now I just do trig problems for personal enjoyment, but it would have been much easier for me if the concrete premises of it had been explained to me first. The instructors have always told me that it would just bore the other students and waste their time, though, so I try to be forgiving.
In 7th grade a pre-algebra teacher told the class "We solve by steps(and display them) because in harder maths that is the only way to do it."
I've had almost as many mathematics courses as her and I can assure you there are many ways of getting the answer. For me, I fill my short term memory with the problem and solve at many times the speed of a normal student and that's calculus I, II, for long division I can solve to an arbitrarily high level of accuracy as fast as I can write the numbers. The thing that trips me up the most is having to write down the steps, because I feel like it's taking an eternity to solve.
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When I was a kid I always liked seeing what was actually happening with math, sort of like a raw process of values. For instance 40/2 was basically forty (dots? blocks? notches?) divided into two groups, twenty each, or two goes into forty, twenty times. Things like that made sense to me, because I could see what exactly was happening to the numbers and visually keeping track of what was happening to them, like where and how they were flowing in a sort of way.
But then there was long devision. When the teacher started talking about long devision, I did not see the purpose. I did not see the relationship between these new steps and what was actually going on behind them. She would start saying things like "you carry this number over" and things like that. I did not see how all these steps (that seemed random to me at the time) would come up with the answer at the end. I could not associate the random middle steps, with the beginning and the output. So instead, I would just do reverse multiplication in my head because I could see what was happening.
I think this is a general problem for teaching of mathematics: A majority of pupil are only interested (if at all) how to do it, whilst a minority (perhaps the vast majority of Aspies) do better understand something if they understand how and why it works.
To stay with the example of a long division: The method is easy to understand if are aware that you divide that e.g 54031 / 34 can be written as (5 * 10^4 + 4 10^3 + 0 * 10^2 + 3 ^ 10^1 + 1 * 10^0) / (3 * 10^1 + 4 10^0). Suddenly the number-juggling does makes sense and appears as a practical way to get a result quickly.
I had the same problem in primary school and the first years of secondary education. Later when the courses for mathematics were split into a course for basic maths and a specialist course, in which the theoretical basis was much better laid down, the problems disappeared.
I've had almost as many mathematics courses as her and I can assure you there are many ways of getting the answer.
I had the same problem in primary school and the first years of secondary education. Later when the courses for mathematics were split into a course for basic maths and a specialist course, in which the theoretical basis was much better laid down, the problems disappeared.
You guys need to start a project. I wanted to say "write a text book for Aspies," but the powers-that-be would never pay for it. Only the powerless student would appreciate the need. So it would have to be a free website. A public service.
I'm serious. The hard part would be to figure out a format. You'd need to make it accessible.
Just picture it: A bunch of people with a passion/obsession, cooperating on a series of explanations, starting at jamesp420's level, and just keep going until you run out of steam.
I had a really hard time with math all thru school until 10th grade. That's when we studied geometry and I just had some sort of natural affinity for it, I found it very easy and it made total sense to me. The weird thing was that here I was, someone who had previously had struggled with algebra (getting C's mostly) now helping out the kids who the year before had been getting A's in algebra but were now struggling with geometry.
Conversely though, with algebra and other forms of math, if the rest of the class got something right away, some new concept the teacher was introducing, I'd really have to struggle to learn it. BUT, if the rest of the class was having a hard time with it, I'd get it right away and then whatever came before it that I was having trouble with would suddenly snap into focus for me and I'd "get it".
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Give me ambiguity or give me something else.
I'm always doing maths differently than everyone else. My teacher says that in order to expand algebra like say (x+11)(x-9) you have to multiply the first parts of both brackets together which would x*x which equals x^2, then multiply the first part of the first bracket with the second part of the second which would be x*-9 which equals -9x, then multiply the second part of the first bracket with the first part of the second which would be 11*x which equals 11x, and then multiply the second parts of both brackets together which would be 11*-9 which equals -99. Then I'm supposed to simplify it by grouping together all of the like turms, so the first bit would be x^2 since there is only one x squared part left, the second bit would be -9x+11x which is 2x, and the final bit would be -99 since there is only one non algebraic term. Then I have to finally group together all of the answers which would yield the answer x^2+2x-99.
But why go through all that when you can just add the 2 numbers together and then multiply them to get the final answer of x^2+2x-99 in seconds. Every time I see a question about expansion, or any maths related question, I know the answer instantly while all the other NT's (there’s another aspie in my class that works the same way and at the same speed as me) have to take about 5 minuets just on one question.
Last edited by robo37 on 29 Mar 2009, 2:21 pm, edited 2 times in total.
@ robo37
I think that is an excellent way of doing it and thanks for sharing with us. People who love playing around with numbers invariably develop shortcuts that work for them. Extrapolating new ways of achieving the correct answers, especially if they are more efficient---requires turning the problem over in your head and looking at it from a different perspective.
It seems to me that people with math ability tend to have a file of skeletal results for each type of exercise or problem. Then all they do is plug in values to those skeletal formulas they carry around with them in their brain. But for most NTs this isn't an option. It is quite enough for them to remember one set of rules they can use to solve the majority of exercises or equations. The situation is that when you throw one at them in a slightly different form, they don't know what to do with it. Whereas you and your friend would simply extrapolate a few general principles from your mental file of possible results...And you would have a pretty good idea how to solve the new problem in a different form.
My father was nearly a math genius and he tried to show me how he solved math problems. He was a very good teacher but I think you have to find your own ways of doing it, and not rely on how other people do it. I think it requires a kind of intuitive leap to figure out your own more efficient way of getting the correct answer.
But why go through all that when you can just add the 2 numbers together and then multiply them to get the final answer of x^2+2x-99 in seconds. Every time I see a question about expansion, or any maths related question, I know the answer instantly while all the other NT's (there’s another aspie in my class that works the same way and at the same speed as me) have to take about 5 minuets just on one question.
There is an obvious flaw with your method, it does not scale on as well to later years. So the NT's in your class might be behind now, but later on they will benefit. The quickest (and most reliable) method is probably the table method, which serves you well all the way into degree level maths.
@ Kangoogle
Not necessarily. robo37's explanation of how to solve the problem conventionally proves he knows how to do it that way, should it be required. If you can properly teach a thing, surely you know how to do it. And he can teach it. At more advanced levels, he may either revert to solving the conventional way or develop a brand new short-cut for dealing with the next level.
@ robo37
But don't listen to me robo37. Kangoogle knows a lot more about these things than I do.
At some point I realized that I do mental math backwards. I add from left to right and do some other things that are harder to explain. But the point is it works, and I can do it fairly quickly, so I don't see what the problem is. In school I was more or less forbidden to use this kind of creative math - I had to do it the textbook way and show my work and blah-blah-blah. And the problem sets! *headdesk* I guess other people learn things by doing them many many times? Not useful to me at all. Just very, very tedious.
I took one math class in college. The professor thought I was a genius because of my spatial intuition. But ultimately I only got a B. Oh well.
Not necessarily. robo37's explanation of how to solve the problem conventionally proves he knows how to do it that way, should it be required.
He knows an algorithm that can solve some problems, yes. But it will hold him back in the long run, for example what happens when he sits an exam and uses his methods. Inevitably the examiner will not be familiar with them and he will almost certainly drop marks.
And next week or two when he has to go and expand say (2x + 2)(3x +1) he is back at square one and has to learn the conventional method anyway. Why learn loads of different methods when you could become really proficient at one in half the time?
Not in the higher levels of maths. You could teach someone how to prove something without being able to solve the problem yourself.
Still does not necessarily mean he understands it.
Its a really bad strategy when you get to the higher levels. You see its a lot harder to question spot the exams, so you need a more general approach.
Its fine at a lower level. But wait until things are not commutative, i.e. where we cannot say that a + b = b + a in general.
In school I was more or less forbidden to use this kind of creative math - I had to do it the textbook way and show my work and blah-blah-blah. And the problem sets! *headdesk* I guess other people learn things by doing them many many times? Not useful to me at all. Just very, very tedious.
It helps in the long run and trust me I got equally irate at the time...
I have a few more entertaining ways of convincing my professors that I am a genius. There was the time when I showed up pissed and late then just solved all the problems calmly...
I am absolutely horrible at math. It really started hitting me in seventh grade, and I failed it for the first time in eighth grade. That was pre-algebra. I re-took pre-algebra in my first semester in college and failed it once again. However, I ironically got a B+ in Algebra I/II in ninth grade, so I don't know what was going on there. And I got a decent grade in Geometry in tenth grade, but that was only because I was cheating and never got caught once the entire time...
