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."
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.
Test 1 2 3 4
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.
>>TV had taught me that math was difficult.
Go watch PBS you victim of TV
http://en.wikipedia.org/wiki/Jury_nullification
Google: "Feynman mathematics"
A summary of Richard Phillips Feynman
Amazon search for Richard Feynman
--
Mod +1 informative -5 Karma Slut
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
learning a more efficient way to think is better, as any other learning, at a young age. my expience tells that not efficient, but rather practical way of thinking is more widespread in adults. I mean, if I need some specific things done, I know how, that's not always efficient though.
root of all...
"One, Two, Many" works for me. Or is it "one too many"...or "one to many"???
rewriting history since 2109
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.
no more brain power than, say, learning to follow a recipe which is why a burger flippers should have a PhD!
Engineering is the art of compromise.
Including Mathnet!
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?
There. I said it. I really, really, truly suck at math. I mean, i can add/subtract and multiply/divide. other than that, I'm horrible at it. I too would be interested if there was another method of learning math so that I could be better with it. Any "out there" suggestions would be welcome. Oh, I'm good with literature, art, history, etc, just so you know what my strengths are (which are commonly opposite that of those who are good with math.) Any help would be truly appreciated.
Never monkey with another monkey's monkey.
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
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.
Try to find a textbook with a wide range of difficult in the problems. Start with problems that you think you should be able to do. If you have difficulties, don't hesitate to try easier ones first. If you feel confident with some problems, move up to more difficult ones (but be honest about that, try to get them 100% right, not just the "yeah I got the concept"-feeling).
If you can't figure some problems out, go ask s.o. Don't hesitate to go to your teacher's office hours to ask him/her about problems; teachers are usually happy to help students like you who do work on their own. Consider tutoring. Good tutors can help a lot, and your university may well offer some inexpensive tutoring.
I don't think you can learn math just by reading a great explanatory book. (Such a book might give great motivation for math, though.) You won't be able to tell what you understand already and what you didn't get yet before you tried it out in problems. And even if you understand it immediately, you will feel much more confident about it after you have successfully applied it yourself in problems.
I have done quite a lot of high-school tutoring, and it was always amazing to see how quickly students could move on to more difficult stuff after they had found some solid grounds of problems they could do.
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
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 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".
... yeah I was suppose to be working but it's one in the morning, so what they hey ...
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
>>> "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
""I've always been an avid reader but my math skills were poor[1], 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'm solving that particular problem through self-education via what would be considered children's level educational software. The plus is that good software is adaptive to your level of skill and rate of learning. Palatable.* Non-judgemental, and more importantly. You can go at your pace. Once I'm through with that then I can move to the books, of which I have plenty.
*Usually graphical, but also not intimidating.
[1] I've bolded the above because I've always wondered if there's a link between the two. Do math people have problem with words?
Here's another thought. Fingermath is an innovative way to bring the tactile to math (to a point naturally).
Oh well since I'm doing ads, here's another. Invention Highway Creative Thinking Family Edition
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.
I have a degree in math. I was also an engineering major.
Math has many aspects to it. There is the mechanical aspect, like adding numbers, or long division. To learn the mechanics well, you need to simply solve a large number of problems. One aspect of calculus is mechanical. You apply mechanical rules to find derivatives, etc.
A second aspect is the application of the mechanical rules to solve more generally stated problems. Traditionally these problems are called "word" problems. These require some lateral thinking and practice.
A third aspect is pure math, were various "truths" are proven by starting with fundamental axioms and logically deriving the required result. This can help for a deeper understanding of a particular truth, such as the Pythagorean theorem.
The engineer will be most interested in the second aspect. The first aspect isn't used in practice due to computers. And frankly, engineers do not care about the correctness of the math in the same way that mathematicians care. So the third aspect won't help an engineer very much.
My advice is to solve a large and varied set of word problems, that range from basic algebra, to calculus, and then vectors. This is really how you develop a strong feeling for math and how to apply it to engineering. If you also solve a large number of mechanical problems that may have eventual application in solving the word problems, that would help as well.
Just realize that confusion is usually caused by a lack of familiarity. Solving a large number of problems is the first step towards clearing up the confusion. Books with a lot of examples are the best. Go straight to the examples, see how a particular problem is solved, and then apply the technique to problems listed in the excersises.
Look for anything by William Poundstone - he goes into ideas about paradoxes, probability, game theory, AI, all sorts of things that should probably appeal and it'll encourage you to read more maths related works.
Very True. Especially connecting back to arithmetic. That doesn't happen until you study group theory, usually after calculus and diff eqs. While studying calculus, the best way is to think in terms of physics and the real-world applications.
Math is just a series of simple rules. When put together in the correct order something larger is created, just like a computer program. I have a natural aptitude for math and always found the little rules to "help" more confusing than beneficial (x100 to get a percentage confuses me, x100% makes perfect sense, maybe I'm crazy.)
None of it is hard at the level of doing it, just forming the picture/design/structure of the solution might not be immediately straightforward.
I think the best way to approach learning math is to find a problem you have trouble with, and learn the ideas behind solving it. I know people who started doing better in Calc when they threw away their silly picture filled BA Calculus book and used the "harder" one from the Math/Eng courses. (Guess the 2 colour graphs are intimidating)
> 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.
there is no need to sign your posts. this isn't usenet. your username is right there above your post. stop it.
Statistics I've found is more difficult to grasp. As I've come to understand the key terms and concepts and worked out some problems on paper sure, its become easier. I've picked up Statistics Demystified (Highly recommended) and its helped. I need to brush up before my next Stats class though (bear in mind doing more 'applied' and 'actual' research with it more in the Liberal Arts type and not in an actual Stats dept level) .... any suggestions (books, exercises, websites)?
IMHO, there are a number of 'open' textbooks on the web now by (mostly) US college/university professors. These are often better than the store-bought ones. Anyone wanting to brush up on some math or learn it, highly recommended.
The amount of math that you learn and the level you attain are proportional to the work you do. There is no royal road to learning. If you're not learning math, it's not because you lack talent. What you lack is probably confidence. It has been noted that most people discover that they have no talent for math the same year they have a bad math teacher. If you doubt the relationship between expertise and work, check out http://www.freakonomics.com/pdf/DeliberatePractice (PsychologicalReview).pdf.
The most amazing math teacher I ever saw was Charles Ledger. The back wall of his classroom was lined with trophies. He taught in a pretty regular public school (ie. mixed population). He didn't have elite students and yet his students seemed to own the national math competitions they entered. I've seen those competitions. They require quite high levels of problem solving. Most university freshmen wouldn't do very well on even the grade 7 and 8 competitions.
Charles started every class with ten minutes of drill. Those kids really knew their number facts. At the end of the class, he posed a problem with an obvious, but wrong, answer. The idea was to make the students quit jumping to conclusions and start thinking analytically. Charles made two main points: 1. Students who are expert at arithmetic have an easy time with algebra. Students who are expert at algebra have an easy time with calculus. Forget the calculator. Fluency in the basics does matter. 2. Confidence is really important. If students don't believe that their hard work will result in success, then they won't work hard. It is really important to demonstrate to the students that they can succeed. http://www.spiritofmath.com/t_train.html Click on the 'Summary of Drill System' link. It really is worth you while to get good at the basics.
Another success is the Jump Math program started by John Mighton. http://www.jumpmath.org/ John has also written a book: "The Myth of Ability" http://www.anansi.ca/titles.cfm?pub_id=206 Where Charles Ledger works hard on the basics, John Mighton's technique is to take beaten-up students and convince them they can succeed. He teaches them something slightly above their grade level. When they succeed at that they are willing to continue working. If they don't know the seven times table, he gives them problems that don't need the seven times table. He breaks each technique into such simple steps that the students can't fail to do them right. Usually within a year or so, his students are working at grade level or above. Based on this success, Mighton can honestly assert that anyone can learn math.
I once watched one of my buddies learn some math that he wasn't supposed to be able to learn. He was a mere engineer working on his master's. He found himself in a math class full of math students (not at all like engineers). The prof made a gentle joke of his lack of background. He should have been out of his depth. On the other hand, engineers know how to work if nothing else. His technique was to read an example, close the book and attempt to solve it, open the book and see if he was right. He would repeat this until he got it. He did this for every example in the book and for every problem for which the answer was given. He made a point of skipping nothing. He totally nailed the class.
The bottom line is that you can learn math. All it takes is work. Depending on your background, it could take a thousand hours of serious effort. That's three hours a day for a year. You'll be ready for engineering math and you'll fly throgh it with ease.
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.
The number you have dialed is imaginary, please rotate your phone 90 degrees and try again.
I thought that was Barbie's job.
I see even classic Slashdot is now pretty much unusable on dial up anymore.
When arrived at my school a lot of (CS,PMATH) professors told me that they didn't believe that Calculus should be anyone's introduction to mathematics. Unfortunately, a lot of Calculus tends to be taught with "tricks" (this includes a fair number of proofs) that while are neat to look at as a math major provide very little benefit to most people.
One of the best things that happened to me was our first year classical algebra. It starts from the very basics of logic, surveys elementary number theory, modular arithmatic, complex numbers and puts a very large emphasis on constructive proofs. It was a course where you actually began to understand mathematics rather than just use it. I think I realized I was studying the right subject when I spent about six hours in our tutorial centre proving some arbitrary property about GCDs with about ten other people. Anyway, after I had finally solved it, I rode my bike home, collapsed in my bed out of mental exhaustion. Of course, when I awoke I realized how much I had learned in the process of the proof.
After learning logic and the basis of proofs, Calculus begins to make *a little* more sense. A lot of people get scared the first time they see something like the precise definition of a limit; but a strong logical background makes it quite seemless.
Anyway - it might be worthwhile to see if the school has a first year "proofy" class that you can get yourself into. Otherwise, it might be worthwhile to read a book on basic mathematical logic or number theory. Even if you can't recreate proofs, there is a lot of benefit in reading them. Of course, if it is just manipulating equations that you need help with I'm sure it will come in time.
(That course is Math 135/145 for anyone at Waterloo)
I too had troubles with math. In fact, I failed my Algebra 1 class the first time I took it. But, the second time I took the class, I had a different teacher. Thanks to her, I gained a fondness for math (of which I had outright despised for years. Her main reason for getting me interested in math was for very similar reasons to yours. She let the students teach the other students whenever she was having trouble getting a concept across. Plus, she was only pro-active in our studies. She wouldn't just let us goof off in class, but, she did that in a good way. She made us want to keep looking and "getting" what we were learning. And to this day, I thank her for being the person to finally get through my math deprived mind. And also thanks to her, I am considering pursuits in college (Being a senior currently in highschool) more oriented towards more complex maths. So... it's all in how you are taught, and honestly, you could teach yourself, I find some of the best ways to learn a concept in math, is to see different variations on a problem, but backwards. More or less, "reverse engineer" the problem to come to where you started. GR
"Paranoia is the flaw and gift of man. Heed its advice, but do not live by its will."
"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.)"
So why didn't you come up with a physical "jigsaw" puzzle?
Math (at least high school and undergraduate in college) is not about radical concepts and philosophy of numbers, it is about solving the problems. Basically if you work enough problems you will understand more than enough to do the test. In most college courses (except for Linear Algebra) the final number did not even need to be calculated. You could integrate/differentiate/etc and just slap the proper numbers together with a bunch of parenthesis and leave it. But anything can be mastered with enough practice. In college doing the homework twice (once throughout the class and a second time the week prior to the exam) led to me getting mostly 95 and above on the exams. There are tons of tutorials on Trig/Algebra all over the internet with practice problems. If you made it to pre calc through the college placement exam you should have the fundamentals. Basically with Trig you just need to memorize a few pieces of the sine wave along with the tangent function.
There are several elements to this
1) Education is inherently a problem in communication
2) Some Data is NOT intuitive, and needs to be broken down into appropriate byte size pieces. Different humans have different byte sizes.
Add to this the concept of missing fundamentals. Sometimes the data you need is included in earlier study/work that was not completely of fully digested. You need to make sure you cover this.
Sometimes the data is new, and you will need to work it out with sufficient practice so that you can develp observation on how it actually works. Then you can put it together. Application to real world situations is useful as well.
"It is a greater offense to steal men's labor, than their clothes"
First and foremost, you do NOT think logically. This you must accept as if you did, math wouldn't be so difficult for you.
... and hours, and then some more time, studying math. Also, please note that studying math does NOT just include reading, and memorizing the definitions. It _mostly_ includes actually doing problems.
;)
Secondly, you must understand that no matter how good you are at math, you must spend hours and hours and
Also, reading a math text is different from reading any other non-science book i.e. It is NOT a dime store novel. What you must do is absolutely scrutinize each and every sentence in the book and make sure you understand them. When I was first studying (and this still holds for more advanced books today) it might take me over an hour to go through a page or two.
What I'd recommend, is get a textbook and take the time to read through a section. Then do every problem. When doing a proof, at each step write down possible paths that you can go down to find a solution (or if it's simple enough, just remember them). Then one by one, pick the path that seems most resonable (at first this won't be the right path). Then one by one, pick off the wrong ones until you reach the solution.
This may seem like a massive waste of time, but it's really the only way to learn. Just know that as you get better (i.e. have done lots of problems), you won't have to keep that list, you'll just know what path to take, you'll just see it.
This is somewhat what I did when I went back to university. I had to re-learn highschool math plus try to keep up with the intro calc class and the calc based physics class and... Needless to say, for the first couple months, it was get to the U at 8:00-8:30am and leave at 10:00-11:30pm.
After that I was golden. But, I (and everyone else as well) had to go through pain to get there. The only difference between the people who take math at a U, no matter how good they are, is how much pain do they have to go through. Because _everyone_ has to go through some.
Good luck and have fun... eventually
There is no silver bullet. The most sensible thing is to practice all the problems in all the books. Learn to use whatever calculator or computer they will let you use to solve the problems. At least learn the algorithm, even it you do not understand the principle. For instance, every DC circuit problem can be solved by drawing the picture, plugging in the right values into the calculator, and having it in reduced row echelon form . Even if you do not understand Gauss-Jordan Elimination, the problem can still be solved. Again, try to understand the algorithms even if the math is a bit fuzzy.
The mention of Feynman is interesting. The man was a brilliant physicist and mathematician. The proof of the equivalence of competing quantum theories. Equally groundbreaking was the restatement of the process of solving QM problems, an oversimplified description would be graphical solutions. The idea is solid though, as stylized graphical representation can be the basis of solving the problem. If you can draw the stylized inclined plane properly, the solution is obvious. If you can draw the proper stylized circuit, the solution is obvious.
So it goes back to practice. Practice drawing. Practice the math. If you don't know algebra, practice that. If you don't know negative numbers or fractions, practice that. Read textbooks and look for clues that will help you understand. But engineering is not a field of reading. It is a field of practicing.
"She's a scientist and a lesbian. She's not going to let it slide." Orphan Black
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.
Tom Swiss | the infamous tms | my blog
You cannot wash away blood with blood
First observation/piece of advice: Rote and repetition are not the same thing. Rote implies that you're not trying to understand what you're doing. When I tutored, I had an almost foolproof method of developing skills, which went like this:
1. Have the student work the easiest problem in the set. Help the student as needed until the correct answer/method is completed.
2. Hide the results from step #1 and have the student work the problem again.
3. Repeat step #2 until the student gets through the problem effortlessly. This may take 5-10 attempts.
4. Go to next problem in the set and do the same thing. You'll find that as you progress through the problem set the number of reps will go down to about 2-3, and the student will pick up speed.
Why this works: weaker students have a tenuous hold on each advance they make on a problem. Repetition solidifies what was done in each previous problem. Since math problem sets are designed so that later problems build on the skills developed in earlier problems, this makes later problems easier. I routinely had D students advance to B+/A- in a week or two. You have to be committed (i.e. trust the method) for it to work, otherwise you'll lose concentration. If you have to take a long break in the middle, start over at the beginning to "warm up" your brain, though you don't have to repeat every problem.
Second: When you think you understand a concept/technique, find someone to teach it to. There's nothing that forces you to *understand* what you're doing like having to explain it.
I'm almost in the same boat here. The only difference is I'm up to Calc II. There's alot of great advice here, but I have to add a little of my own:
Whatever you do; don't wait to take the next math class in the summer if you have any, ANY amount of time inbetween it and your current level class (semester, quarter, whatever.) In fact, never "take a break" or anything from math classes. Don't make the same mistake I did. Consider bridging the years with a summer course, too.
That's what I learned in geometry.
Math is
- a set of propositions
- a set of ways to manipulate propositions
The game is to use the manipulations on the propositions to reach the answer.
Learn truth tables.
Learn proofs by induction.
Learn axiomatic linear algebra without matrices.
Learn metric topology.
Learn real analysis.
Learn complex analysis.
Use Wikipedia.
I used to tutor math at the University, and I have to say that your approach is among the best I have heard of or tried.
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
Some math books do a poor job at explaining what they are trying to teach. Not showing the appropriate examples on what you're trying to do is one problem. Not following the proper format students are expected to follow when showing their work is another problem the books have.
They need to strengthen math skills while in k-12 schooling. If they can extend math class by 40% per day, that gives more time in which students can receive help and be given more clear instructions on how to solve the problems.
I believe if our schools relied less on homework and put more emphasis on using quiz and test scores to determine one's math grade, it would relieve a lot of stress students may go through just to finish the homework when they could take a casual approach to it. Doing it at their own pace, tests would be done on certain days and they'd know way in advance to prepare for it.
There was a Professor at the University of Minnesota in the '70's named Pedoe, who taught a class in Non-Euclidian Geometry over 3 quarters. The first quarter, the descriptions were oriented around numbers (for those who understood and liked numbers, arithmetic), the second quarter the descriptions were oriented around Algebra (for those people who liked general recipes and principles), and the third quarter the descriptions were oriented around actual visualizations as in graphs and geometric diagrams(for those whose primary understanding was visual). He said there were always three approaches to understanding Mathematics, and applying your preferred descriptive method and translating into the two others would make you both a better mathematician and a better communicator.
A very good example of this is the way people learn Ohm's law: Usually, this is taught by visualizing the equation E=IR in a triangle or circle with E over a line, I and R located below the horizontal line and separated by a vertical line. But if you think about it, this is also the description of the "basic equation" x=yz. Almost every equation is solved by simplifying it to the basic equation, so this is a good entry into Algebra through visualization.
Another possibility is that your preferred learning strategy is kinesthetic rather than visual or auditory. Do you learn best when you do things "hands on"? Does something have to "feel right" before you know you understand it? If so, make models you can manipulate out of clay, wood or plastic shapes, or even objects on your desk, and use an abacus or slide rule (remember those?). You will master your subject in a VERY effective way. The best Engineers and Tecnicians I know have a "feel" or "gestalt" for the "whole thing" in a perfect description, and can "feel" or "sense" the abberations between what the perfect model is and what is out of place in the current reality.
"The mind works quicker than you think!"
Now, in maths you could compare learning the notes to understanding the notation and some basic concepts. But the point about passion was a good one: you have to work by yourself. You can compare the lectures to teaching you how to read sheet music, but to really learn how to play, you need to do real practice. Concerning that, there were some good points above. Also, what I like to do with problems that can be treated analytically, is to start looking at what the most simple solution to some equation looks like. Insert zeros, insert one and a zero..., look just at the first terms, maybe doodle some graphs... see how those behave to build up intuition on the problem, then go after the full scale beast of greek alphabetical soup looking at you on the page.
Think that you have to exercise your brain, as a muscle. It's a good analogue, anyway. Compare reading math books to reading an exercise manual. You can't get fit by just looking at those instructional images; you have to get to the floor, grind your teeth and start doing push ups / what ever. Reading about the beauty math in some nice books is probably a waste of time, at first. You'll see some of niceties yourself if you work hard. Compare this to reading a travel book about the Grand Canyon, and going rafting down it yourself. Not to say, that you shouldn't read those books. They just probably wont increase your math skills, not in the initial stages, anyway. You'll also enjoy the good books better, if you really understand beforehand what the writer is REALLY talking about.
It's going to be painful, but the upshot is that you can and probably will learn, if you have the will and discipline. You have to accept that the elementary practice stage is going to take some time, like a few years or something, before you can really play something neat. Then, perhaps at some point you might find some practical or abstract math related problem that you really enjoy thinking about, and at that point, it turns to intellectual joy. Don't read maths next to your computer. Find some desolate chair and a desk where you won't be interrupted. Collect enough non-digital reference material so you don't need to check some concept at wikipedia. If you suddenly notice that you don't understand something, go back to where you lost it, and start working again. Always have a pen and pencil, and do notes, while reading. Also, coming up with your own memorization rules will probably help a lot. Spatial, graphic, really silly poems... what ever seems to work best for you, you'll have to discover yourself. Some stuff you come across, cannot really be deduced except through some really loong proof that you are going to forget anyway. Do the proof once, so you see what's happening, and then accept the truth of it and use the memorized end result.
There's probably some point beyond which it's really difficult to increase your math skills but it's likely waaay ahead.
I don't think people who are good at math have some radically different way of thinking that ordinary people are incapable of. They're just using the same kind of common sense logic people use every day, for example when solving puzzles. However, what makes them good is that they are very determined to understand *why* something is true, instead of just accepting it on the authority of some textbook or some teacher. If a bad math student is solving x+2=9, he'll say, "the equals sign is a bridge, and when the 2 walks across the bridge it has to change its sign." wtf?? Some people actually teach it this way. If a good math student sees this equation, he'll think, "okay, x+2 is literally the same number as 9, so if I subtract 2 from x+2 then I'll literally get the same number as if I subtract 2 from 9, so therefore x is 7." Or maybe he'll just notice that 7 satisfies the equation, therefore the answer is seven. (But wait---is that the only number that satisfies the equation?? Yes, if x is any bigger than 7, then x+2 is too big, and if x is any smaller than 7, then x+2 is too small.) Someone who's good at math will often ask: how could I have figured this out? How do I know it's really true? Could it be some mistake that has never been caught? Also, people who are good at math are always checking things with specific examples. Suppose you forget whether or not sqrt(a+b)=sqrt(a) + sqrt(b). (This is a common mistake.) If a good student is unsure if this is true or not, he'll just check it with some numbers. Is the square root of 4+4 equal to 2+2? No. Yet somehow students make this mistake again and again on exams. (I was a TA, I saw it a lot.) Everyone has this kind of common sense ability. But most people aren't willing to spend the very large amounts of time that it takes to understand math well. It takes a lot of time. I can understand why they don't do it, because it's hard work and they have other things to do. But if you want to learn math, chances are very good you can do a good job of it if you just invest a lot of time. Start hanging out at www.artofproblemsolving.com. That place will make you good at math.
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.
Seriously, the main problem is the notion that maths is so out of your league. The reality is that it IS scary, however, nobody is really good at it! It is painful how we humans try to grasp the concepts of maths, or rather, usually we don't. By various twists and turns we sometimes manage to proove some simple conjectures - but even those usually don't give us any idea of what they really mean, no real understanding. I mean in the sense of those autists who can tell you in an instant wether 1285982340578978943768747897897488899827 is a prime number or not - must of us have sit down and do the calculations. Then we have proven it is prime or not, but it doesn't really mean anything to us.
So quit your unrealistic expectations, that is all there is to it. Try to just enjoy the process of problem solving. By and by you will aquire a toolset which you can apply to all kinds of problems (ie natural induction, equation solving, etc.).
In my own experience (MSc in Mathematics), learning higher mathematics was not so much understanding it step by step. Rather, it is a gradual process of getting used to it. Looking back, those first year problems suddenly seem simple, yet at the time, you were struggling.
>I believe that I have a flaw in the basic way I think about numbers. This is where the real problem lies, in your belief. You can't expect to learn math when you *know* you cannot learn it. You are who you think (truly believe) you are. You can't change who you are, unless you think differently about yourself. When you do, you change. And of course, you aren't who you think (believe) you aren't, so you can't become who you want to become, unless you stop thinking that you are flawed/wrong, wired differently or have a different concentration of neurotransmitters inside your brain. People have a flaw in their basic physiology - they don't have wings, yet everyone can fly for a few hundred bucks. Some people even walked on the moon! So its not who you *are* that matters, its who you think you can (or cannot) become (do). Math isn't a wild beast that cannot be tamed. Math is just a collection of abstract conventions and proofs, invented by men for men. People tend to think that math exists in the universe and it's important and it governs the laws of nature, but in reality it's just the way people have come to relate to the world in abstract terms. Nature doesn't have laws, people do, and we use the laws of numbers to approximate the reality that surrounds us. Which makes math exceptionally interesting. Fellow Mathematicians will agree that math delivers great amounts of intellectual pleasure, some of which are profound and revelatory. So go ahead, dive into the numbers with curiosity and wonder and you will discover an entirely new realm of thought in math. Don't be afraid of math and it will open itself up to you. It takes the first 'click' to understand its beauty. Just stop thinking that you are flawed !
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
Meta will eat itself
Learning the practical skills of Calculus, particularly integration, is like learning a foreign language. The methods of integration are like the syntactic structures of language: they're ways of getting certain things done. Not the only way, just ways that work for some people.
Facility in a language comes when you can use its structures to think with, without thinking about them. When they become automatic, you've mastered that part of the language. That means while you can learn about a language by studying its vocabulary and grammar, you can only learn to use a language by drill.
And like most drills, doing a huge amount at one time is not as good as doing a little bit on a regular basis. Three problems a night is better than twenty problems on the weekend.
Personally, I was very good when mathematical proof was demanded of me, very bad at actually using calculus. So I decided that I would practice by redoing all my problem sets, even the questions I got right. When I told my teacher I was going to do this, he was dubious I could learn something by going over something I'd done already. But it turned out to be a very time effective way to improve my performance. Rather than struggling with new problems, I reinforced the patterns that worked, and fixed up the ones that almost worked but didn't.
If you think of math as a language, this makes perfect sense.
Post may contain irony: discontinue use if experiencing mood swings, nausea or elevated blood pressure.
As a mathematics undergraduate at a British university, perhaps I'm in a position to make some recommendations. At school I read a couple of books which really spurred my interest in mathematics and motivated me to take it as a major at university.
My first recommendation, "How to Solve It, Polya", is a cult classic. It introduces new methods of thinking about mathematical problems and is accessible for someone with just basic high school mathematics. A strikingly powerful piece of literature.
"The Art and Craft of Problem Solving, Zeitz" is a more modern book. Zeitz has trained several American IMO (International Mathematical Olympiad) teams. It introduces both approaches to solving difficult mathematical problems and the machinery required to do so. Aside from that, it lists several techniques of improving your mathematical ability from a variety of angles (word games and so forth...)
Hope you enjoy. Best of luck!
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.
"Some days you just can't get rid of a bomb."
No, integrals are incredibly simple to show to a student. You walk up to the board and draw a function. You then ask the class what an integral is. When they don't know (assuming noone has had AP or failed, and remembers) start shading the area under the curve.
Remember kiddies - math has meaning. It isn't just an abstraction. Make your teachers show you the meaning and relationships.
The Feynman lecture on algebra you probably heard of is in chapter 22 of volume 1 of the Feynman Lectures on Physics.
You got numerous responses, and many good ones. As many have said, the idea that math is inherently harder to learn then other things is a myth. But, it may well be that it's like learning French as an adult. What comes naturally to French children can be very hard for english speaking adults. It can take a huge amount of effort to become fluent. One method that seems to help many people is to learn computer programming. Program requires you to think in terms of algorithms and abstract structures. And you can go a long way in math with those approaches. Someone mentioned Maple, and Maple would be good if you already know how to program. But the Maple programming environment is kind of clunky and not at all ideal for learning to program. The problem is you need to know a lot of math to use Maple effectively. If you don't know any programming take an introductory course. Then take a data structures course. That's where the bang for the buck will be in terms of your "math brain."
I am working to be analytical about animations using quaternions. The brain SUCKS at remembering visual stuff. Instead, the brain is great at shop & compare. That is why artists use easels by the way - it's not just to hold up the canvas, but because visual memory is so bad, but comparison is so good, so the artist's work can quickly be compared to "the real thing".
a tion/Dynamic_graphs/1276.html
an example animation - http://www.theworld.com/~sweetser/quaternions/qem
the project - http://quaternions.sourceforge.net/
Working on new views of old physics at http://VisualPhysics.org
First, forget the notion that some people are unable to learn math. Some may be better than others, but like language, the human brain is wired to understand mathematical notions. The timing required to track the trajectory of a ball and catch it is a very sophisticated math problem that even very young children can master. Second, remember that math is a tool we've constructed to solve problems. For most people, having a detachment between learning something tangible and learning a tool to solve it, short circuits the natural process of learning. Learning, or at least learning well, virtually always requires an interest or need in the subject matter. So treating math as just formulas and rules and a vocabulary of terms to remember is a huge problem. To understand math, you should understand the underlying concepts and how they apply to the real world, and being able to conceptualize those ideas when you're looking at formulas, etc. A good teacher, math or otherwise, will implicitly to this for you. However, there really are very few good teachers around, either at primary or higher education levels. And colleges are increasingly money machines designed to crank students through the system, and even good professors succumb to the pressures of getting as many students through courses as possible, instead of taking the time to help people understand. After all, by college, shouldn't you be prepared to do all that work yourself? Or so goes the thinking. Start with understanding basic ideas of linear rates, and find a parallel in real life, e.g., like driving on the freeway. See how the position and speed of your car on the freeway corresponds to a linear function that intersects the x and y axis of a graph. Then move to understand acceleration, and what that looks like on a graph. Then think of two cars in different places moving toward each other at different speeds, and how lines on a graph that intersect can help you understand when they will crash. Now what if one is accelerating? If you're not interested in cars and driving, pick something else of interest to you. At each step of the way in learning math, if you make sure you understand the underlying concept and how to visualize in action in day to day life, you will make learning much much easier.
Do you think verbally or with imagery, or both? Do find yourself needing to move when learning something? How good are your reading skills?
If your reading skills suck, work on that first since they are so fundmental to everything.
If you use a lot of imagery, learn to draw pictures. Reading equations doesn't work so well for me until I can draw a model, even a simplified model, of the situation (what do n orthoganol vectors in n dimensions look like? Like a sea urchin).
Logic and verbal descriptions help me too. But simply reading an equation without understanding what is going on behind the scenes doesn't work for me. I can't just take stuff at face value. Recast what is happening in your own words, write an essay or come up with rhymes.
If you are more physical, set out coffee cups, bottle caps or other physical objects on a table top and use them to represent things in the eqautions. Shuffle them around and group them as needed.
Find or form a study group. It really really helps.
And above all, never give up. The nice thing about college is you can retake classes. If you get a D or worse, don't drop out. Sign up for the class next semester. You will, in fact, have a head start when you retake it.
putting the 'B' in LGBTQ+
I've gone back and forth on being (relative to my peers) very strong or very weak at math. Probably pre-calc was my weakest time. Pre-calc for me meant a hodge-podge of trig and bits of algebra and stuff. It was a mess, and I barely passed. I barely passed the first two semesters of calc also. However, by the time I started grad school, I was significantly ahead of the group in my math ability.
There was no trick to it for me, but a variety of things. First, just stick with it. Keep doing it and do the best you can in each class. Keep progressing, but balance your progression. You want to move on to classes with more advanced math, but not that require skills you failed to pick up earlier. Even if you suck, a lot of it will come back to you later.
Try to mix up pure math and applied classes. It's an enormous help to see the weird stuff you did as just "math" used for something. But once you start to get the feel for it, the pure math classes can give you cool, fun insights.
Draw lots of pictures. I actually recommend graph paper, rather than software tools. If you're doing probability, draw Venn diagrams. If you're doing calculus, plot the functions. If you're doing something with a geometric component, draw the shapes.
Work with others. If you find a good partner/group, this is fantastic. If you find a mediocre partner/group it's still helpful. It gives you the opportunity to explain your methods and solutions, and explaining is a fantastic way of learning. It also gives you an immediate source to turn to when otherwise you'd spend 9 hours pulling your hair out about a problem. You can either learn from someone else's insight, or at least feel better because it's hard for everyone.
I'm a gamer (board & role-playing games) and a programmer, and so the first college math classes which I really enjoyed were probability theory and discrete math. They gave me the math for dealing with things I already had a good intuition for. If you can find similar courses which apply to some knowledge you have, that can help.
Anyway, for me there was no "trick" or applying a different way of thinking. It was just plugging away at it. Good luck.
Seems to me the real problem is that you cruised for six years instead of engaging the material. Those years were there for you to develop a way of approaching math.
Yes, you need to go to your professor's office hours (and your TA's, if you have one.) But you also really, really need to find your university's academic services office (or whatever they call it). Get a tutor. Explain the situation, in complete honesty. You need your own problem-solving method; you can't just lift it from Feynmann (who made it up on his own, remember) or Polya. Only someone working one-on-one with you while you solve problems can point the way out.
> I am struggling. I believe that I have a flaw in the basic way I think about
> numbers. [...] 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?
The difference is that an average person thinks math is all about performing various calculations with numbers, and a mathematician thinks math is about understanding the world.
I was lucky. I don't know if the teacher just explained it right, or if it just hit me, or what, but I got it in seventh grade: math isn't about numbers, at least, not mostly. Numbers are just a handy source of concrete examples. Most folks don't realize this until they get further along, at least to multi-variable algebra and more often after getting through Calculus. But it's an important realization, and it makes math much easier once you've grasped it. (In less than one year math went from being my worst subject, which I hated, to being my best subject, which I loved.) Math is about seeing the similarities between one situation and another -- isomorphisms -- and then determining whether what you know about one particular case is in fact a generality that applies in other cases as well. Math is about developing useful ways of understanding the world in which we live. Math is very cool, once you understand it.
Math is abstract. Concentrate on understanding the underlying theoretical concepts. Once you see the conceptual patterns and understand what is going on, the numbers will fall into place. Don't just memorize the rote method (at least, not most of the time): concentrate on figuring out *why* it is that way, and then you won't have any trouble remembering it (or improvising slight variations for special situations, or combining it with the stuff from other chapters).
Cut that out, or I will ship you to Norilsk in a box.
That's the recipe!
Good judgement comes from experience, and experience comes from bad judgement.
- W. Wriston, former Citibank CEO
Open Source Mathematica/Maple: http://maxima.sourceforge.net/
Open Source MATLAB: http://www.gnu.org/software/octave/
They work great and are free.
Learning math is just like learning a spoken lanuage. Math has it's own vocabulary, and it's own set of rules. And (at least at the pre-calc / calc level) these rules are totaly consistent.
So, How do you learn a language? By exposure, if you want to speak spanish well, live in Mexico for a while. Sadly, it's much harder to imerse yourself in math. In fact, one of the only avenues availible is... Going to class, and doing your homework. (sorry) I just came off a stint of teaching precalc and trig and I can tell you as a straight fact, none of my students who consistantly came to class and did thier homework got less than a B.
However if you hate rote you probably also hate page after page of math problems, so the homework part may be a stretch for you. I was the same way, and I still managed to get a bacelors in math. You simply need to find some other way to imerse yourself in math. In my case I conned my way into tutoring math (even though I knew nothing about it) teaching others is a great way to learn, since it forces you to really get your thoughts together. You might also try reading books on math, I highly recomend Kline's "mathematics for the non-mathemtician".
But honestly, the homework/class thing is the easiest way to go.
Ptolemy once asked Euclid if there was not a shorter road to geometry than through the Elements, and Euclid replied that there was no royal road to geometry.
That was true then, and it is true now.
You will find a mountain of contradictory advice. I can only offer what has worked for me (I have a BSc in CompSci/Mathematics, and an BE in Elec Eng).
1. Learn to abstract - from concrete objects, to numbers in a number system, to algebraic symbols in an algebraic structure. Almost all of mathematics is abstraction from simpler concepts.
2. Learn to reason - learn how to prove things, directly and via reductio ad absurdum.
3. Do the exercises. Some people have a knack for understanding things straight away - the rest of us need to work hard.
There's no royal road - even those who have a knack have developed this knack through practice. Most people have a psychological block regarding mathematics, because it is probably the most poorly taught subject, starting from kindergarten. Practice and develop confidence - it then becomes a "virtuous cycle" where confidence encourages you to practice more, the extra practice turns into ability, and the ability into even more confidence.
There are people like Feynman that are like magicians - they produce brilliant results without giving any indication of how they came to produce such brilliance. But even they have to work at it. Feynman was a master showman, and loved to confound people with seeming flashes of brilliance that really stemmed from very simple ideas (eg, see "Surely you must be joking", the anecdote where he competes against the abacus operator by mental calculations - using very simple tricks, he must have appeared to the abacus operator to be a walking computer). The point here is that Feynman was a brilliant man, but not above a little showmanship. So don't be intimidated by showmanship in mathematics - sometimes things are simpler than they appear, and not necessarily beyond your abilities - it's just a question of working out the trick these magicians use...
Oh, and finally: real mathematicians can't do arithmetic. They may be able to do tensor calculus on multivariate curved manifolds, but they will struggle with basic numeric arithmetic like most people. It's a badge of honour for mathematicians to get arithmetic wrong occasionally, because arithmetic is "mindless"
Life is chock-full of lies, but the biggest is math. That's particularly clear in the discipline of probability, a field of study that's completely and wholly fake. When push comes to shove-when you truly get down to the core essence of existence-there is only one mathematical possibility: Everything is 50-50. Either something will happen or it will not. When you flip a coin, what are the odds of it coming up heads? 50-50. Either it will be heads, or it will not. When you roll a six-sided die, what are the odds you'll roll a three? 50-50. You'll either get a three or you won't. That's reality. Don't fall into the childish "it's one-in-six" logic trap. That is precisely what all your adolescent authority figures want you to believe, that's how they enslave you. That's how they stole your conviction, and that's why you will never be happy. Either you will roll a three or you will not; there are no other alternatives. The future has no memory. Certain things can be impossible, and certain things can be guranteed-but there is no sliding scale for maybe. Maybe something will happen, or maybe it won't. That's all there is. What are the chances your sister will dies from ovarian cancer next summer? 50-50. (either she'll die from ovarian cancer or she won't). What are the chances your sister will become America's most respected underwater welding specialist? 50-50. It will happen, or it won't. There are two possibilities, and both are plausable and unknown. The odds are 2:1. The facts are irrefutable. Quasi-intellectuals like to claim that math is spiritual. They are lying.
I took precalc in high school and battered my way to a Bachelors in ME.
I've since gotten a MSME at one of the nations best ME schools and have gathered an appreciation for higher level math.
I'm now working on my PhD and have passed the qualifiers which includes a math part created by the Math department and it typically has a high failure rate - I passed with one of the highest scores.
Considering my development of math skills, I'm convinced that my brain was still developing in my early twenties and I just didn't get it. I now get IT at a much higher level.
Here's what I'd recommend:
1) If you don't get the concept, STOP, and ask the teacher. If the teacher can't explain it to your satisfaction (i.e. you get a light-bulb moment) then go somewhere else. In my experience, engineering professors explain math better than math professors.
2) Only work on your assignments when you GET the concept. Once you understand the concepts the work actually becomes enjoyable but it's EXTREMELY important to get the concept BEFORE. You don't have to but and you can work out the problems until you see the pattern but it is MUCH harder that way. Trust me, that's how I learned.
3) The best way to learn this is to write down a study-sheet after each homework assignment highlighting what you just learned. Presenting that to the teacher or engineering prof might be a good idea also.
I learned Math at a totally different level studying for my qualifiers as it forced me to go back and study about 10 semesters of higher level math on my own and make study sheets. My brain races all over the place when I just think but when I'm forced to write it down, my brain slows down and I'm able to think things through properly. Try that and see if it works.
MOST IMPORTANTLY AND UNIVERSALLY TRUE!
If someone can't explain it in a way that you understand it or they can't break it down into something easier then THEY DON'T GET IT ENTIRELY THEMSELVES or they don't care enough to explain it to you so don't take it as a sign of your weakness but theirs. It took me a LONG time to figure this out - just find someone that has the time and understanding. For me, these people are usually in engineering because they've seen Math applied in so many different ways that they have tons of examples.
ULTIMATELY - nobody can do the thinking for you so you'll HAVE to struggle with the concepts and ideas and do the homework but once you get it you'll realize that Math is nothing more than a very concise (and cool) language to explain things that verbally takes much more effort.
You know what the real problem with math is. The teachers! Yes, they suck. The books normally expects to have facts, then the teacher MUST explain the facts, but what I found is that the teachers do not known about math so facts are studied as math and math became facts. Therefore, after summing all the facts you get bad teachers that show only the facts in the books that hold only the facts so the teacher can teach you the facts (such an infinite series). So about your question, the problem is that you cannot see what is math all about (yes for real) they said math is everywhere (and really is everywhere, yes). So how you can get your math groove? Well I do not know
There as many ways to understand math, it all boils down to how YOU understand concepts and exploiting your own neurophysiology as your would a wireless router with a default login/pass. There is, for example, synesthesia, which is the crossing of senses. I am a synesthetic and I have very strong color/shape associations with letters, numbers, symbols, words, sounds, etc. E is orange, 8 is indigo, 9 is red, etc. If you have perceptions like this (and many people do without ever realizing they percieve the world differently to most people) you can train youself to be more "intuitive" with math based on color concepts. Feynman himself is a synesthetic and percieves equations in different colors depending on their structures. Even if you don't naturally have this ability, it can be trained to a degree, though how far I don't know. I have heard of people cultivating the ability through concentrating on existing associations, regardless of how weak they are, and eventually they intensify. Hallucineogens can also induce temporary and, with enough use, permanent synesthetic perceptions. We all have unique neural structures cultivated from our life experiences, its just a matter of how much you "know thyself" so that you can exploit your own understandings to better compute math. Try to find likenesses between the math you are working on and your stronger cognitive abilities. If you are spacially strong, thing about math in 3D terms if you can, etc. Hack thyself!
If you are struggling now, you wont make it, let go.
My story: I was always good at math, and so as a senior in high school I ended up in the highest math course offered. The teacher was great, I mean really great. Like state teacher of the year great. This guy Dr Corbin Smith, taught only the highest math and remedial math....nothing in between. Any way, we used to sit and discuss the mathematics of odd solids (without the calc) it was very fun. But I was failing his class, badly. I worked my butt off and I still couldn't get the pre-calc he was teaching. It was somewhere in the middle of the year he said something I will never forget: "Sera, you have a great love of mathematics, but you suck at it."
My college pre-calc I got a C, Calc I was differential calc, the only reason I passed the class was that the final was multiple guess. I was the last one to leave the exam hall because I integrated every answer and picked the closest one. Calc II, well, let us say after taking the class 4 (yes four) times I still dont really understand it. After the second time through Dr Smith's words came back to me. Needless to say I tried everything and eventually I realized - I am just not wired for higher math. It happens. I am a great programmer, and a huge geek able to grasp concepts quickly that others just cant get at all. You have strengths, math is not one of them. Find out what yours are and run with them, you will be happier in the long run.
Sera
Slashdot, where armchair scientists get shouted down and armchair theologians get modded up.
There are, as you probably realize, many ways to approach learning. Sometimes it helps to get an explanation of what the concepts are, that is to say 'what is trying to be accomplished,' before delving in to find out the nitty-gritty How parts.
- Berlinski/dp/0679747885/sr=8-1/qid=1160231231/ref= pd_bbs_1/002-9559630-6664847?ie=UTF8&s=books
q =richard+feynman+synesthesia&sourceid=opera&ie=utf -8&oe=utf-8
As one previous poster mentioned above (in probably different terms,) understanding a base point is usually sufficient for building up to the bigger thing and what you'll really remember is the path of logic that gets you there. What is to be developed in this case is an understanding of how to build up from the base, to learn enough of the entirety of the system to grasp the larger concepts. This takes time, and some people don't 'get it' until some trigger of knowledge is learned that sets all the other previously learned bits into alignment. This can happen at any time, early on or much later. Don't give up just because it's not instantly understood!
If you want to learn the Calculus, maybe you should consider learning -about- the Calculus. It has been a few years since I last read it, but David Berlinski wrote a fantastic book called "A Tour of the Calculus" - I recommend it. Amazon link below:
http://www.amazon.com/Tour-Calculus-Vintage-David
You mentioned Richard Feynman, a fantastic visionary, who attributed much of his success in understanding the concepts and abilities to perform works in the Calculus at a very young age (12 or so, if memory serves) to the education his father instilled upon him. The focus was patterns and behaviors. For example, many of Richard's peers growing up might have learned the names of various species of bird found in the nearby woods, being able to identify them on sight... Richard's father instead would have taught him to study the bird and piece it into it's environment, what is the importance of the call it makes, how does it 'fit in' to it's surroundings to keep itself and it's family safe and well fed, etc. I'm probably doing a horrible injustice to this as I read his book some time ago, but the main point is Feynman had a deep sense of logic and strongly developed problem solving abilities, which served him far better than methodical approaches to learning. He is also likely to have had synesthesia, which is a tremendous leg up on the competition:
http://www.google.com/search?client=opera&rls=en&
Remember also that things you lack a natural talent for can be supplemented by lots of hard work. I believe the Scientific American had an article not so long ago that explained some research that showed 10 years was the approxomate amount of time for nearly anyone to become an expert in almost any field - musicians practice feverishly for a long time before they become talented enough to be recognized, chess grandmasters play the game with intent passion for about as long, so it is with pretty much any field. Yes there are some with natural advantages, but so it is with anything - keep at it and you'll find success!
"We are not always what we seem, and hardly ever what we dream."
Schmendrick the Magician
In my experiences with math, from high school to college, the instructor is the determining factor.
Some people are visually oriented; pictures such as charts and diagrams help engender understanding. Other people are better at abstract symbols; pictures aren't much good to them. Some people learn best by observation of how something is done; some people have to roll up their sleves and get their hands into the learning experience. But, always, the teacher is the key.
I've always had trouble with algebra (because of a succession of algebra teachers), but little trouble with geometry, trigonometry, or calculus (because of a success of teachers).
Registered Linux user # 170078
I liked Maple when I could use it on the University computers. It is not free, however (though can be hacked).
Maxima is an Open Source computer algebra system that can do most of what students need and is free.
My theory on this is that there are many subtle neurotoxins in our environment that we are in denial about. They disproportionately impact the lives of poor people, but they also are increasingly present in the bodies of all who live in modern societies unless we take steps to ban them.
Perhaps this is why people from certain nations seem to be excelling at math now while others from richer societies are falling behind.
I read recently that there are entire regions of the US that are contaminated with lead from old lead mine tailings. These tailings reduce IQs by an average of seven to ten points and there is nothing that can be done about it besides moving away.. (even aggressive cleaning, HEPA vaccumming, wiping, etc, do not result in the reduction of blood lead levels they would need to to avoid this, its probably because the particles in the air and water, etc. are too small to be stopped by filters)
We need to eliminate lead from the environment because it also effects neuron growth in older people (adults)
Mold is inside of many buildings and several mold toxins effect brain development and the aquisition of new knowledge dramatically.
See this month's National Geographic for a good article on the effects of environmental toxins..
When I look at nearly all the stumbling blocks I've conquered - algebra, writing skills, engineering school topics - I got over each and every one only after some serious one on one time with a real person.
In a similar sense, read multiple books on tough topics. Don't just grunt through the explanation from the one 'official' course book or lecture. Find 10! Eventually, one of the sources will have an explanation that fits you...someone who got stuck on the same mental snag as you. THEN go back and read the rest...it will fall into place.
Lastly, the best way to learn is to teach. Work in a study group - not to do less work - but to exercise your grasp of the subject. If you can teach it, you know it, and vice versa.
Does it hurt to hear them lying? Was this the only world you had?
I think you will benefit from reading Beyond Numeracy by John Allen Paulos. His earlier book, Innumeracy, is more of a social commentary, but Beyond Numeracy explains a lot of the reasoning behind the different areas of math, such as algebra, geometry, and chaos theory.
I think a lot of textbooks teach math like a cookbook. Just mix all the ingredients and put it in the oven. Don't ask what the baking powder does. In Beyond Numeracy, Paulos doesn't just say that the area of a circle is pi times the radius squared. He explains how that formula was found without calculus.
Beyond Numeracy does a good job at communicating mathematical concepts, but I don't know how easy it will be for you to apply what you learn to your math classes. I have noticed that people who claim to be bad at math get frustrated when they work on a problem for five minutes, but a person who thinks they are good at math feels challenged by a problem that they can't solve in ten minutes. Then they become almost obsessed with finding the solution themselves. They really don't want anyone to tell them the answer or explain it to them.