Slashdot Mirror
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."
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.
SJW: a person who perceives an injustice, and while correcting it, commits a greater injustice.
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.
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*.
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.
Microsoft cheerleader, blue flag waving, you got a problem with that?
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
watch this
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