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."

Comments

Math is not difficult

[Catamaran]

Math is unique in that there are many levels of abstraction, and you can't understand the higher levels without first acquiring a pretty good understanding of the lower levels. At each level, a certain amount of study and memorization is required, just as in any academic discipline.

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.

Re:Math is not difficult

[Quaoar]

True to an extent, but organizing your brain so that you can call up the knowledge necessary to solve a particular problem is something that is very difficult for some people. This is mostly a problem on math tests, where not only do you need to know what to do, you need to be able to follow the steps quickly enough to complete the test on time. It's just something that some people are not naturally very good at.

Re:Math is not difficult

[Lazerf4rt]

It doesn't have to be difficult. I think the reason it is or isn't for most people is emotional, or psychological. I for one loved math as a student. It was the only subject where you were either right, or wrong. I could walk into an exam, write it, verify my answers, and be sure of how I did. The teacher couldn't slant, because if there was a mistake in the marking, it could be proven a mistake.

On the other hand, there's a friend of mine who hates math. He's no good at it, and he can't learn it because when he tries, he spends too much time worrying about the fact that he's not good at it. He calls it a mental block. It's probably the same reason why a lot of nerds are no good at sports.

I'd suggest to the submitter to stop looking for "different ways" to conceptualize math, and actually just follow through with one way.

Some answers that worked for me

[Marxist Hacker 42]

Numerical Methods. It's usually taught as an advanced, post calculus course for computer science majors. But it gives alternative methods for all sorts of things from trigonometry to calculus, and it does so in methods that can be programmed in Basic or even Assembly (you do know, don't you, that at a very basic level the most complex math any computer can do is 1 And 1 is 1, right? And that all the other math computers do is built up from simple AND gates?)

In addition to this, I also recommend Godel, Escher, and Bach: The Eternal Golden Braid for a totally different way to think about mathematics, philosophy, and religion.

Re:Some answers that worked for me

[MrSvenSven]

Sorry to troll, but it's a NAND gate, not an AND

right...

[Nyall]

>>TV had taught me that math was difficult.

Go watch PBS you victim of TV

Info on Richard Phillips Feynman

[davidwr]

Google: "Feynman mathematics"

A summary of Richard Phillips Feynman

Amazon search for Richard Feynman

--
Mod +1 informative -5 Karma Slut

Many

[JustOK]

"One, Two, Many" works for me. Or is it "one too many"...or "one to many"???

This book will change your life

[LoonieMiami]

Look up "Mathematics: From the birth of numbers" by Jan Gullberg. It should do the trick. Incredible book.

Different people learn differently.

[EmbeddedJanitor]

Sure there are levels of abstraction etc, but I think you got lost on the "cognitive ability" bit.

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.

Re:Different people learn differently.

[computational super]

I'm with you up to "know your professor's office hours and ask questions there". College professors "encourage" that sort of thing (by saying you should do it), but I gave up on it after several years of trying. First of all, the questions I want to ask (and I doubt I'm alone here) are usually along the lines of, "Everything you said in the lecture made sense. The textbook made sense. All the exercises and homework problems made sense. I still tanked on the last exam. What's up with that?" If I had a question like, say, "how does the chain rule for derivatives work again?", I'm sure he'd have a great answer, but I can look that up in the textbook. Instead, I end up trying to formulate a semi-intelligent sounding "I don't get this whole 'analytic geometry' thing" type of question. He goes off on a tangent, reciting what he already presented in class, and sums up with "did that answer your question?" I can either say no and go back to step 1 or nod enthusiastically and go back home and try to divine from the textbook what he might end up asking me on an exam.

Some thoughts

[Anonymous Coward]

First of all, do you know your learning style? Auditory, visual, kinesthetic? Your writing suggests visual. Did you find geometry to be easy, or difficult? If the answer is easy, there's part of your answer - relate calculus and linear algebra to geometric problems. Hint: most EE math can be reasoned about algebraically (equations) or geometrically (pictures).

See if you can get your hands on a demo of Maple. There's a student version available, I don't know if it's crippled, but I know that it's a disgustingly great deal. It got me through EE school. Mathematica has better marketing, but I always found it to be a horrible program (at least, its syntax requires you not know anything about programming languages). Maple has some great somewhat-interactive graphing modes too. You can't / shouldn't use it for the math courses, but for EE courses, you'll need a really good math program to help you out.

Also see if you can get your hands on a HP48GX calculator. Real engineers use old-school HP calculators. Posers use TI. You'll thank me come EE exam time. I'm not convinced that the currently selling HP calculators fall in the "real calculator" camp, but they might be okay. You want RPN. Trust me, if you're an engineer, RPN is your friend. It also tends to keep people from swiping your precious calculator ;)

See if any of the professors in the EE department teach math classes; usually there'll be a few people who have a foot in each department. Make friends and see if they'll help you out during their office hours. In general, I have found that math professors can't teach math worth anything. Or at least not to engineers. It's just a different mindset / world view. And the result is that they're teaching math the way they think of math, and you're just going W-T-F?! The EE professors can teach it with an engineering spin, and they have the very distinct advantage of being able to map math problems to the real world EE problems you need that math to solve. The worst math professor phrase is "suppose you want to..." - well, suppose that I don't, ya damn hippie! EE profs can put the horse back in front of the cart and tell you WHY you NEED to do this or that math, and that insight alone makes it much easier to learn.

In general, I must emphasize that EE is a math intensive major, and it gets very very much uglier than basic calculus. If you truly aren't good at math and you aren't willing to put yourself through dramatic pain and sufferring to learn it anyway, change majors now. Really, seriously. If you're going to hit your limit and change majors, you're better off doing it while you're not as far along and don't have as much work to throw away. If you decide to stick with it, good for you, just understand that it's going to get *a*lot*worse*.

Re:Some thoughts

[RzUpAnmsCwrds]

See if you can get your hands on a demo of Maple. There's a student version available, I don't know if it's crippled, but I know that it's a disgustingly great deal.

Absolutely. Maple is your friend. The student version is every bit as good as the full version (it's the same program), and it's $100. Not bad for a CAS that does just about everything.

Mathematica has better marketing, but I always found it to be a horrible program (at least, its syntax requires you not know anything about programming languages).

Mathematica is not bad if you live Mathematica. For the rest of us, Maple is easier to use, has a better interface (tabs, advanced yet easy to use formatting, etc.) and is much more like the programming languages you're likely to know anyway.

You can't / shouldn't use it for the math courses

That almost made me spit out my Diet Coke. Here at CU, we aren't allowed to use CAS programs on exams (or any calculator at all, for that matter), but on homework assignments they are absolutely essential for checking your work. In fact, we have three labs per course that absolutely require the use of a CAS system - the Applied Math department pushes Mathematica, but I use Maple.

Also see if you can get your hands on a HP48GX calculator. Real engineers use old-school HP calculators. Posers use TI.

Most engineers I know use neither. Numerical computation can be better accomplished using purpose-driven software. Many EEs would be absolutely lost without Matlab, a SPICE simulator, and countless other software packages.

There's nothing wrong with a 12MHz M68000-based portable computer with a math-optimized keyboard, 240x160px screen, and 256k of memory. It's a fine device that works very, very well. You can compile for the calculator using GCC, there's a big standard library, and the built-in software is generally very, very good.

Trust me, if you're an engineer, RPN is your friend.

There's nothing wrong with RPN, but assuming that it is "natural" or "superior" is like saying that we should all use DVORAK keyboards. Use both, choose what you prefer. I'm an algebraic entry person myself, because I like the input to match the problems - it helps me visualize what I'm actually doing and helps eliminate error. Pushing numbers (or symbols) on to the stack is more abstract and, at least in my case, more prone to error. A few saved keystrokes don't mean that much to me.

See if any of the professors in the EE department teach math classes; usually there'll be a few people who have a foot in each department.

No, no, no. I have had mathematics professors who were also EE professors - they tend to spend too much time focusing on specific applications and gloss over the fundamental mathematics. Real-world examples are great, but you need to understand the concepts first - and EE profs, in my experience, frequently do not understand that their students do not.

At CU, we have an entirely separate engineering math department (Applied Mathematics), with different courses and different textbooks. Our text is filled with engineering sample problems and our professors use them in class. But our Applied Mathematics professors do nothing but teach mathematics - they know their material (and how to teach it) very, very well because that's all they do.

In general, I must emphasize that EE is a math intensive major, and it gets very very much uglier than basic calculus. If you truly aren't good at math and you aren't willing to put yourself through dramatic pain and sufferring to learn it anyway, change majors now. Really, seriously. If you're going to hit your limit and change majors, you're better off doing it while you're not as far along and don't have as much work to throw away.

At CU, we call engineering "pre-business". It's not for everyone. If you don't like it now, you won't like it as a career.

It's not math anymore.

[CDarklock]

It took me a long time to figure this out.

The math you learned in primary and secondary school, where it's numbers that have distinct values, is no longer really applicable. Don't try to "grasp" the concepts. It's not a small step like algebra was, it's a quantum leap. You are working with a fundamentally different question, which is the question of infinity. You need to learn new rules. Don't try to use the rules you learned with numbers; they don't apply. Your way of thinking needs to be fundamentally altered.

Where I always screwed up in learning higher mathematics was in trying to somehow relate it back to arithmetic. That doesn't work. If you keep trying to connect those two dots, you will be perpetually frustrated. Just learn it for what it is. It doesn't matter if you understand it any more than it mattered if you knew why 2 + 3 was 5 in elementary school. Trust me: you will be able to understand it later, once you know a certain critical mass of concepts, but you need to have enough dots before you can connect them into anything remotely like a picture.

This will take roughly your entire pre-calculus class and probably half of your first actual calculus class. You will be confused. It will not make sense. You will feel like you are learning nothing. The answers you give on exams will feel memorised and formulaic, almost like you are cheating.

But eventually, you will have that "Aha!" moment where you really do finally understand what a definite integral is. You just have to trust that the material you're learning is going to get you there, even if you don't know how.

Likewise, it's not really true that higher mathematics doesn't connect back to arithmetic. It just won't connect back for a really long time, and it's not productive to look that far ahead right now.

Re:It's not math anymore.

[exp(pi*sqrt(163))]

Don't try to use the rules you learned with numbers

This is the worst advice ever.

Most of the time when you're doing EE you'll be working with equations in which the variables represent numbers. It's important to bear in mind, that every stage, that these aren't just meaningless symbols. An unknown variable, x, satisfies all the properties that all numbers do. For example xy=yx because 2*3=3*2 and 5*7=7*5, and (-1)*22=22*(-1) and so on. Sure, you can forget about this, and just use the rules of algebra to manipulate these symbols. But as long as you do this you'll have no insight and you'll be like a brute force chess playing machine that has to search out all possible sequences of moves. Keep in mind that these symbols are actually numbers, and all that's happening is that you're doing arithmetic, then you can let your intuition about numbers guide you, even if your equation doesn't even contain any numbers.

eventually, you will have that "Aha!" moment where you really do finally understand what a definite integral is

It's much easier to understand the concept of a definite integral than to memorize and use the rules for manipulating them. Properly explained, the idea is incredibly simple. And once you get the idea, many of the properties will be plainly obvious.

It just won't connect back for a really long time...

Your teachers must have been awful. And despite the fact that I have a PhD in math, you must have had way more stamina than me to learn all of this stuff without connecting it back to arithmetic until much later.

Re:It's not math anymore.

[CrankyOldBastard]

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

You're psyching yourself out

[GuyMannDude]

The fact that you state "TV has taught me math is hard" and that you have a problem with "numbers" yet are good at logic leads me to believe the problem is in your mind. Mathematics really has very little to do with numbers. It's symbolic logic. Equations are just concise, precise statements. If you can do logic, then you can do math. The only time numbers comes in is at the very end (for engineering and science) when you plug numbers into the final result.

I'm not good with numbers and I have a poor memory but I have a Ph.D. in applied mathematics from one of the top institutes of science in the entire world. There's no magic to it and don't let popular culture tell you that mathematicans are somehow different from everyone else. Just take a deep breath and relax a bit.

I'm not going to recommend any books or tell you to meditate or anything else like that. You just need to have some faith in yourself or dig deeper to find out what the real problem is. When you say you're good at logic, what are you basing this on? Are you a whiz at logic puzzles or something? Most of math is logic, a little creativity, and a lot of hard work.

By the way, if you're struggling in a class, here's an idea to try. Go to some of the already-solved example problems in your textbook. Write down the problem on a piece of paper and close the book. Try to solve the problem. Write out all your thoughts, crazy ideas, questions, etc. Struggle with it for a good half an hour at least. Then open the book (assuming you didn't solve it) and look at how they solved it and see if your scribblings were even close. The act of trying to work through the problem will make your subsequent reading of the solution that much more meaningful.

GMD

Think of it as a language...

[Aelcyx]

I like to think of math as a language for anything quantifiable. When people "talk math" they use these math terms because these terms precisely project their thoughts into words. I think the best way to understand math is to really contemplate everyday physical phenomena. Think about vector fields in your car when the A/C is blowing and trying to reach everyone in the car. Think about parabolas when something is thrown into the air. Hell, try to do your own experiment and figure out the parameters for it. You'll soon find that you'll be looking into a lot of things that change with time and hence, require derivatives. This should segue into your pre-calc learning.

For starters, I'd say look at the basic definition of a derivative: lim[h->0] (f(x+h)-f(x))/((x+h)-x) and compare it to finding the slope of a line: (y1-y2)/(x1-x2)=rise/run. A derivative is nothing more than finding the slope of two points on a curve as the two points get closer and closer together until they lie directly on top of each other (this gives you the slope of a line tangent to a point on the curve which is equivalent to the rate of change at that point on the curve). This is the only hard conceptual part about pre-calculus, really.

And a couple other notes on learning. Intelligence, imho, is just the ability to break things down into smaller and smaller parts or to divide concepts into many little parts. Any field you learn has two parts to it: concepts and vocabulary. When you come across something "hard," figure out what is stopping you: the concepts, or the vocab. If it's the concepts, have someone explain it to you in laymen's terms. If it's the vocab, look it up at mathworld.wolfram.com or of course, www.wikipedia.com.

Keith Devlin has looked at this issue.

[Anthony]

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.

TRUE FACTS FROM THE MATH INSTINCT When a dog runs along a beach and then jumps into the water to retrieve a ball thrown diagonally into a lake, it instinctively solves a problem that humans need calculus to solve. Lobsters have a built-in positioning system that is the equal of the hugely expensive and mathematically rich high-tech Global Positioning System (GPS) human travelers use today. Within a couple of days of being born, human babies know the numbers 1, 2, 3, and can distinguish between a correct addition or subtraction such as 1 + 2 = 3 and an incorrect one such as 3 - 1 = 1.

Brains are different

[lawpoop]

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.

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. :( 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.

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 have a BA in Anthropology and Religious Studies.

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 ;) 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.

Re:Math is not difficult ... err, yes it is

[pbhj]

Right, background: excelled at mathematics in primary school (up to age 11) but got bored as I'd finished (the concepts of) all the course texts and didn't like doing actual work. Was top set in secondary (up to 16) but never really shone until that final year. Did double maths A-level (maths and further maths) and went on to do Theoretical Physics and maths degree. Some of it came easily to me - complex numbers, fractal geometry, differential equations; some not so easy - quantum field theory, fluid dynamics.

I've never really considered that I could have a different approach to numbers that would make maths easier. Maths and Physics I loved at school as I have a very poor memory and could always go back to basic assumptions and build from there. Later on (eg fluid dynamics) I had to try and really on some rote learning as the stuff was too abstract for me grasp.

I don't really have a visual grasp of concepts - I've often tried to envision a four dimensional hypersphere or a fractional dimension without much success. When I turn my mind to dimensions folding in on themselves the images are often just (barely) 3D. But somehow I grasp many of these concepts ... I guess it's that step of going from "this is an electron, a solid minute particle orbitting an atomic centre" to "this is an electron a four dimensional probability wave".

>>> "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"

Hmm, that's a very _now_ statement. I'm sure that if all you're trying to do is pass an exam that's true. If you're trying to understand and develop, indeed push the boundaries of, a concept then I don't think that's true. Have you ever just picked up a recipe book (for soufflé say) and just tried to follow the recipe. Sure you know what the words are and carry out the action, but you can just lack the knack to perform it well. It's a terrible analogy but I think as with musicality, a sportsmans eye for the ball, an artists abstraction of images to capture their essence, there's a mathematicians feel for the equations and their beauty or otherwise.

What was the question again ... yeah I was suppose to be working but it's one in the morning, so what they hey ...

If you want to improve...

[nbritton]

Watch 'Algebra: In Simplest Terms' hosted by Sol Garfunkel, PhD.:
http://www.learner.org/resources/series66.html

26 half-hour videos covering all topics of Intermediate and College Algebra. The webcast videos are free (registration required), just click on the VoD symbol to watch them. If you use SDP ( http://sdp.ppona.com/ ) you can download the ASF steams for repeat viewing. BTW... I got an A+ in my College Algebra class... It's absolutely critical that you fully understand advanced topics of Algebra before starting a Calculus class.

Well, here's one hurdle you need to overcome

[bunions]

> I hate rote

This insane allergy people have to simply memorizing some things gets in the way all the time. Just get over it. Despite what new-age bullshit you might be used to about how rote learning is 'just' memorizing lists of facts, it remains important to memorize those facts. Some things you just have to memorize, and math is full of them. What edges of the triangle a cosine relates to is an example. Once you start committing this stuff to memory things will start to fall into place. Worked for me. Got a degree in math and everything.

Puzzle pieces versus steps

[Daniel_]

I've been tutoring math from calculus to basic arithmetic for a number of years now. I also am drawing on my own experience when I first took an honors math analysis course. There is a radically different approach between how math (really arithmatic) is taught between high school and college.

High school typically chooses a rote approach - learn the steps required to complete the problem and regurgitate on request. Even some college courses are taught this way. You are given a collection of steps and are expected to remember the steps that are applicable for each problems. I have found, tutoring, that the best approach by far is to teach a collection of 'pieces' - a particular approach to a particular sub-problem - where my students also have to learn why it works. I then encourage each of my students to visualize any problem as a jigsaw puzzle where existing pieces are combined to find a solution for the problem at hand. (i.e. There exists a sequences of steps using known 'pieces' to solve the problem and the student is expected to eventually pick up an intuitive understanding of what kind of techniques to apply when facing a new kind of problem.)

I've experienced a great deal of success teaching with this technique and recommend it whole heartedly. Create a notebook listing every technique for solving a sub-problem you have been shown to date. Each technique should have a name, a set of conditions when it applies, and how to implement the technique. If you plan to remember the techniques for an exam, also include a description of why it works - preferrably worked out / thouroughly understood by you.

Obviously, this is what I have found to work - YMMV. But I have found that, as long as an individual is capable of viewing problems abstractly enough to grasp the approach, it has been an effective problem solving technique.

Why is TV always to blame?

[unitron]

"...and TV had taught me that math was difficult."

I thought that was Barbie's job.

Miller's "Popular Mathematics"

[Mr. Slippery]

See if you can find a copy of the 1942 book "Popular Mathematics" by Denning Miller. It goes from arithmetic to calculus, taking generally a more geometrical, physical, and historical approach than most math classes do these days.

I was pretty good in math, up until I hit differential equations; I bought this book just for curiosity, so I can't really say if it will help you. But it looks like copies can be found on eBay for just a few bucks, so I'd say it's worth the gamble.

It has to do with passion.

[Starker_Kull]

All the people who have said that there is no difference in ability, and any arbitrary person can advance to any arbitrary level of mathematical ability are pretty unrealistic. I base this statement on my own abiding comfort and love of the subject, as well as five years tutoring and teaching it at levels varying from elementary school level to graduate school. That said, here are my own personal observations as to which people succeed in their math goals and which people fail.

First, what people said about practice is partway true. But HOW you practice is as important as how much. Many people think that if they do the same problem over and over and over, perhaps with minor variations, this will somehow improve their mathematics ability. Except for at a very base rote level, this is untrue. A far better challenge would be to INVENT problems like the ones you have been solving, and see if you can solve those. Frequently, the 'canned' problems you are given for most mathematics instruction below second year university level are designed to have 'neat' answers. This very quickly becomes a crutch for students, because they are so used to looking for the 'neat' answer that they are unable, or don't trust their ability (almost the same thing, in practice) to work a problem when it is unclean. In addition, when you start designing problems, you start to focus on the crucial idea of whether you are right or not. Having an answer handed to you is almost useless, because it short-circuits the other half of problem solving - how do you know whether you have a right answer? If you don't understand how to check your answers, you aren't qualified to be doing the problem! Right there, that suggests a different method of problem solving - trial and error. This is not to be scorned, but encouraged, because it means your brain is engaged again, and you are not just regurigitating the motions.

Second, most people who are good at math like it. What this means is that they are practicing far more often than people who don't like it, because they have some part of their mind on math problems throughout a day, or they find problems that have mathematical solutions. How do you get to like math if you don't? Tough question - I found that good teachers who enjoyed explaining how they got to an answer, what makes it fun or interesting, how it applies, or just how neat it is are better than the rote type. But at some level, you have to start figuring what you want to DO with your math - frequently, practicality and application focus the mind and make it easier to learn and enjoy it.

Third, don't let people who are better at it than you get you down. REAL math is messy. When solving a problem that has not been solved before, mathematicians go through all sorts of detours, false starts, unnecessary constructions... messy, messy, messy! But after thier adventures through the mathematical jungles, after they get the prize, they clean up the mess. They don't mention the false starts, the extra logic that really isn't needed, the play with ideas that turned out to be useless. They just show the clean, sparse, neat path. This is a modern fashion, and I think a bad one, because it removes the human element of play, adventure, and imperfect effort. Learning math is messy - you need to experiment, make mistakes, try to fix them, try different ideas, and PLAY with the stuff. They don't tell you this in the textbooks, at least not the modern ones (of course, there were flowery extremes on the other side - read Cardano for an illustration of 99% prose and 1% math! But he does tell you of his false starts, his dispair, his mistakes, and the joy of his ultimate triumph). AFTER you have made mistakes, tried alternatives, and played with other ways of solving a problem, then the 'standard' way of doing it makes much more sense, and you appreciate the WHY vs. the HOW. This is why, if you don't know how to check your answer for sure, you are not at the level where you should be attempting such a problem.

Feynman

Memory, time and visualization

[waterbear]

Have you already tried to check out which thinking methods fit with you?

For example, do you already have the habit of trying to find rational patterns, and enjoy visualizing them? If not, you could try that out and see if it fits with you. Visualization may be a two-edged weapon when it comes to math. Some people (including me) do it a lot and find it helpful. (But others handle math topics that may defy visualization, and claim that the visual-modeling habit ends up a hindrance.) Maybe you could find it stimulating to visualize. To find out, you could try reading about classical geometry and working through the essentially visual proofs there, and then go on with coordinate geometry. Visual modeling based on geometry helped me through calculus.

Then, do you have very good numerical memory? Would it stimulate you to try extending it anyway? Can you do simple mental arithmetic really easily, like adding up your purchases without needing the machine? You might try it regularly and enjoy making it come more easily.

But most of all you probably need to spend a lot of time with a chosen subject, and try to think about it and analogize it in lots of different ways and see which ways stimulate you and work for you.

Good luck.

Ask a Secondary (High) School Maths Teacher

[tygerstripes]

...like my missus. She's actually not that good at maths, but she understands how people think and learn about maths pretty well, as will any good maths teacher. There are hundreds of books on the subject, so find a Maths PGCE/Teaching course syllabus and look for the Recommended Reading section - that should give you some good grounding.

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

If Pre-Cal is bad...

[thebdj]

then you should quit now.

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.

You will never make an EE with bad pre-cal skills. You have yet to hit Calculus and are struggling already. Most every EE I know, and that was my degree so I know quite the few, were taking Calculus in high school. It will only get worse until Differential Equations, and if someone told you EE was not a lot of math, they lied to you.

Have you considered the option that maybe EE is not for you? I whole-heartedly suggest that you go and find a counselor or advisor and get their opinions, but I am pretty sure any one from your College of Engineering, will tell you that it probably is not a good idea to pursue EE (or any other engineering) if you are struggling with Pre-Calculus. I know I have completely skirted your question, but this is something you should really consider. If you are not good with Math, engineering is not for you and trying to learn math now is a bit late in the game.