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Different Ways to Conceptualize Math?
rook a asks: "I've always been an avid reader but my math skills were poor, and TV had taught me that math was difficult. I knew only the concepts of the basic operations. From seventh grade through high school, I did only what was needed to get by and so my math skills remained below par. Now, as a freshman pre-cal student, I am struggling. I believe that I have a flaw in the basic way I think about numbers. I can think logically, but it does not carry over to math. I read somewhere that Feynman gave a lecture on arithmetic but I could not find it. I believe that different people have different thought structures for the same ideas. Has there been any research or books on the difference between how a mathematician, or a Richard Feynman, thinks about math and the way that the average person thinks about math? Or, did any of you initially find math difficult in college but go on to higher maths? If so what changed for you?"
"I wanted to be an EE and want very much to be good at math but if my ability does not increase I will not be able to. I am willing to do anything to increase my skill. I hate rote and do not want to be merely 'good' at math, I want to speak it. If math is a mindset then it's one I want to be part of.
This is similar to another question, however I found several interesting books but no comments toward learning a more efficient way to think."
This is similar to another question, however I found several interesting books but no comments toward learning a more efficient way to think."
However, the idea that one needs some special cognitive ability or conceptual skills is a complete myth. Once you have absorbed the concepts and vocabulary from one level, moving to the next level should require no more brain power than, say, learning to follow a recipe in a cook book or installing a plumbing fixture.
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Look up "Mathematics: From the birth of numbers" by Jan Gullberg. It should do the trick. Incredible book.
Many people don't "get it" with math because they are not cognitively wired to absorb stuff the way it is presented. Yet, if something is presented a bit differently they might then "get it" and be able to move on to the next step.
I was very fortunate to have an excellent math teacher. This teacher was able to teach kids who had previously not done well at math and get them scoring As. I think his secret was this: He used many different wasy to explain things to the kids. Some would get it immediately. Some would only get it when he explained things differently. Quite often he'd explain things in thee or four different ways. Now sometimes he'd be stumped and could not get an idea across.... So here's where he was different from other math teachers..... He'd get one of the kids that "got it" to sit and explain to the kid that didn't "get it", and he'd watch and take notes. Eventually someone would manage to get through. Better still, the teacher would then have a few more ways of explaining things to future classes.
Engineering is the art of compromise.
Keith Devlin addresses your concerns. His recent book "Math Instinct" looks at the conundrum of mathematics being easier in practice than in theory.
I haven't read it but I have read his "Math Gene" book looking at innate abilities for mathematics.
Slashdot: Where nerds gather to pool their ignorance
Different people have different brains. Some people just can't do math after a certain level. A lot of stuck-up geeks will tell you it's just that you haven't learned the lower level math well enough -- that may not be true. They probably have a brain that is well-suited for doing math, and they think that everyone must be just like them, that math is easy, and anyone who says otherwise is lazy or doesn't care.
:( My point in saying this is that my 'math' mind works visually. I had no problem doing geometric proofs as long as we were looking at figures and drawing. However, when it comes to reading 'number sentences' with abstract symbols, and solving equations, I'm sunk.
;) I have a BA in Anthropology and Religious Studies.
;) You might see it worthwhile to try to get good at those maths, or, you might just find something that is more suitable to your natural abilities.
I consider myself to be a geek. I have always had a nerdy, intellectual personality. However, I had math difficulties since day one, starting with addition.
In high school, we had a geometry class. There were hardly any numbers in it, just images, compasses, and protractors. A lot of our assignments were proofs. I got an 'A' in the class. I remember one assignment in particular at the beginning of the class. There was a figure that was a bunch of triangles, and we just had to count how many different triangles we could find. Most kids got 12-15, but me and a few other kids who were good at art counted into the late 20s. There were actually 32 in the figure. The next year was Algebra II, and I got a C.
Another area of geekiness is reading and language. I taught myself to read before school started. I never had a problem with reading or writing assignments -- I typically did them the night before, skimming. That got me a magna cum laude degree in the honors program at Ohio state (in the honors program, you could only take classes that were designated 'honors' -- less than 30 students, taught by a professor, or a graduate level class. ) I took my math at a local community college and transferred in so as not to ruin my GPA
I'm pretty good with computers, but companies aren't very interested in a computer guy without a BS. I am doing alright with my LAMP job, but I will probably go back to school and get a masters in linguistics. I took a few classes and found it fascinating. I did really well with the grammar parts, such as diagramming sentences. From linguistics, I can use this as a launching pad into other areas that I am interested in, such as artificial intelligence or speech recognition. I couldn't get into those areas through CIS.
I guess my long winded point of all of this is just because you might not be good at certain types of math, doesn't mean that you aren't smart or aren't a true geek
Computers are useless. They can only give you answers.
-- Pablo Picasso
Wow, there are just so many ill-informed and probably very unwise pieces of advice here.
Let me make this clear to start with - I've worked teaching Maths to people struggling with it for many many years, starting with private tuition, through the Education Department and as a University Tutor/Lecturer. I've seen teaching of maths at all levels above primary. And I'm damn good at it, as I've only had a handfull out of hundreds of students over 20+ years who didnt show marked improvement, in their grades and in general life as they've realised that they're not stupid after all.
The things that are most important to get a "unified" and "intuitive" understanding of maths are the simple rules you met at primary school. You didn't meet their names back in Primary School (commutivity, reflexivity, transitivity, associativity) but these properies (and their abscence) are what make all the different algebras different (or the same). If we wish to be very technical we could say these properties allow us to identify isomorphisms and homeomorphisms between spaces, but that's not what matters. What matters is that by the end of primary school you have learn't all the basic principles that "higher maths" is made up of, it's all just a matter of putting it into perspective.
As another example, consider garden variety subtraction over the positive integers. This gives us a lovely view of the ideas of openness, closedness, a non-commutative operation. We have to "borrow" to perform some subtractions in primary school - and we use exactly the same technique when solving a quadratic equation by "completeing the square". Later you can use exactly the same idea to solve hairy beasties using tensors and Kroneker's Delta.
Multiplication, division, addition and subtraction of the integers is isomorphic to the algebra of the Polynomials over the Integers - anything you want to do to polynomials is EXACTLY the same as you did to integers in grade 4.
Understanding your Calculus I & II courses is easy if you forget the "formulae" and look at the geometry and the quanitities. Most people get terrified by hairy looking "formulae", to the extent that they develop "formulitis" where maths has degenerated into a mass of formulae to be applied (or mis-applied) on demand. A better approach is to learn what is happening to the quantities being discussed, and then learning how the formulae are just generalisations and shorthand for exactly the same things. For example, look at using increasing numbers of rectangular strips to find the area of a squiggly closed curve, using paper and scissors (try this!!). Then you can see there's a way of writing this down in terms of the 1st rectangle, the last rectabgle, and a huge number of "arbitrary" rectangles that go between them. At that point you've got yourself about to take a Reimann Sum, and you've almost understood the definate Integral - but you haven't got any scary expressions with capital sigmas, lower case epsilons and trying to count an infinitely large number of infinitely small things and get your head around how it's not the same as 0+0+0+...+0+0 being something that isn't zero.
I've heard the "don't try to understand this, it won't make sense to 99.9% of people, and even if it will make sense don't expect it to do so for a few years yet" excuse for mediocrity many many times. My experience is that it's not true. In particular there's nothing you'll meet in maths in an engineering degree that can't be reduced to a series of operations you met at primary school. The wonderful thing is that as you overcome the Fear and Loathing, you will start to see the lovely patterns, and you'll start to see that the different methods and techniques are all just different approaches to essentially the same problem - describing the realtionships between observable quantities.
The most productive acts you can do to improve your maths skills are:
1) Always do a back-of-envelope estimate before you punch the (hopefully correct) butt
The important thing to understand about maths is that it isn't an intrinsic ability - our brains are not designed to deal with even counting, and certainly not with abstract mathematical concepts. We adapt various neural modules such as language, spatial perception etc by constantly using them in unique ways to consider mathematical concepts.
As an example, the notion of a "number-line" as something on which all natural numbers have a place is introduced at an early stage in teaching. This is later developed to deal with non-integers, and then extended backwards to develop an understanding of negative numbers (and how they're not "different" numbers, but a continuation of the line). Then at a higher level this is further developed to include imaginary numbers as a perpendicular axis to real numbers, and the notion of complex numbers is introduced. Through all of this, it is the spatial-perception module that is being used and thus adapted to deal with abstract space and its relationship to number.
One of the most important mathematical concepts to develop (though few high-school children do) is to stop thinking of numbers as abstract things in themselves, and see them more as names of matched sets of objects - four elephants can be "matched" to four marmosets on a one-to-one basis (unlikely and unproductive though that might be), so those menageries are in the set of all things that can be matched in this way, but they cannot be matched to any abstract "thing" called Four. Four is just the name of the set. This is a simple way of approaching the basis of Set Theory, which is irrelevant at high-school but vital at Uni. Admittedly, it might not be so useful for EE, but IANAEE.
One of the key areas you will need to master for EE, I suspect, is algebra. This is closely linked to the language centre of the brain, so you will find it easier to learn if you consider it as a language. Start with simple expressions and learn how to translate them either way, gain a familiarity with the most basic ones so they become second nature, and progressively move on by expanding your vocabulary and the complexity of expressions. When you face a challenge, slow down, break it down and try to translate it. Eventually you will become fluent and - more importantly - it will be like a second language in which you can converse without difficulty or any real conscious thought.
Interestingly, a lot of our perceptions and methods of thinking about mathematical issues are conceptually conflicting, and that is a barrier that is difficult to overcome. As an early example, moving from algebra to graphs to vectors & matrices is a serious stumbling block for many children - they can handle any concept individually and with practice they can translate one to the other, but until the mental connections are made they will find it difficult and obscure. Once those connections are made it is a rapid revelation, and they find their understanding and enjoyment of both topics is enhanced (as you might have guessed, this is precisely what my missus seeks as a reward for her hard work).
I mentioned algebra as a key player in EE. There are obviously other areas you will need to grapple with - trigonometry and graphs being obvious ones - and they will require different approaches, but if you find you have trouble with any of them then I strongly recommend you call in the professionals. Uni-level course books and materials tend to present the facts and concepts in a very clear way, but they do not tend to be very forgiving or understanding of those who have difficulties - if you don't get a concept, you will fall down later when you need to build upon it. The best thing you can do is enrol o
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