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Best Way To Teach Oneself Math?
An anonymous reader writes "In high school I failed two out of three years of math classes and eventually dropped out of school completely. I earned my general equivalency diploma as soon as was legally possible and from there went on to college and beyond. That was many years ago and my most basic algebra, trigonometry, and geometry skills are slipping away at an alarming rate. I'm looking for a self-guided course covering the equivalent of 4 years of high school mathematics including calculus. My math skills are holding me back. How can I turn this around?"
The way I kept my math skills fresh was to invent new problems to solve. Also I would derive every new formula instead of just memorizing it. Some random examples off the top of my head:
Derive newton's method.
Find the formula for the circle that passes through any three arbitrary points
Derive all the trigonometric identity functions
I don't have a great answer for your question. However, for me the key to learning math was to stop being intimidated by it. I don't think they do a great job of teaching it in school where they take a very linear approach. They tell you about a concept (e.g. integration) and show you how to do it in certain situations, etc. If someone from the beginning had told me how to visualize what integration was, I think I would have gotten it immediately. Instead I was worried about writing down every little thing the teacher said. Having now gone through six years or so of advanced math, it's somewhat difficult for me to completely empathize, but I guess I would start with the basics. Wolfram, wikipedia, whatever are all fine resources for math. Start reading the simple stuff and if it's confusing, don't be afraid to move backwards and get even simpler. We all forget that stuff now and then.
Get a math textbook. [Hungerford's 'Contemporary Pre-Calculus' worked for me. For Calculus, Larson's 'Calculus' is keen.]
:My $0.02.:
Set aside 30 minutes a night.
Work the problems out with pen and paper.
Where necessary, remember formulas however best suits you.
Avoid technological fixes.
Perhaps life really is full of possibilities.
I think that taking courses at a community college is the best idea. In fact, take it for a letter grade. Although the grade doesn't really matter, this will give you an incentive to do the work and stay with the class.
I think it's only too easy to just pick up a math book and tell yourself you're going to do the work, only to get frustrated and abandon it a few weeks later. By having an actual class that you have to make time to attend, you're making more of a commitment and are more likely to stay with it.
I concur, Good study guides and good courses will put you on the right track.
No matter what you do, realise the Mathematics is not a spectator sport. I continuously fall into the trap of reading about Mathematics than doing Mathematics. Do the exercises and do some more. One thing I did do which was invaluable was a bridging course that reviewed much of final year high school Mathematics with plenty of exercises and a great teacher. Recognise your wakness and go back and make sure you understand whatever is being assumed at the level you are having diffculty with and again, do those exercises. For example, if you are having trouble with trigonometry, review the ways of deducing angles for triangles and bisected parallel lines. Review Pythagoras's Theorem, fundamental algebra, etc.
Slashdot: Where nerds gather to pool their ignorance
I was consistently at the bottom of my class in high school math. I had to lie cheat and steal to get into community college. I eventually made it through a BA, and now a few years later I find myself in a full-time MBA program where math proficiency is a foregone assumption. I told myself before I started my MBA that a couple of "...for Dummies" books and some online courses would get me caught up with the pack. I was wrong. It has taken a herculean effort through private sessions with professors and other students to keep me from failing out of Accounting and Statistics. As great as online courses and the like are, there is no substitute for a good teacher. You will be amazed by how much more effective a tutor is than taking a self-directed online tutorial. If you are the kind of person who is bad at math, you'll probably always be bad at math, but you do have to learn how to get by when necessary. Get yourself a private tutor, suck up the cost, and see the results for yourself.
12 years ago I got an 800 on my math SATS and got A's in every math class I took in high school and college. These days, I struggle with the simplest day to day mathematical problems. I imagine it's just a matter of practice, but it's alarming nevertheless.
As a math teacher, I'd say you're better off getting help from someone competent than going it alone.
That being said, and the understanding that you don't want to pour in the money required to get a good teacher (craigslist looking for a math tutor is a place to start. If you start off with one and it doesn't feel like a good emotional fit, then get a different one. A good tutor will try to get a solid grasp of where you are now, and then start taking steps to get you moving forward from where you are. A great tutor will help you when you're stuck, but also give you specific resources that you can use to work on exactly what you need to be working on right now in your time away from the tutor), here's my advice.
First off, understand what exactly it is you are trying to do. You are trying to build abstract thought paths in your brain. This is hard to do. Many of the math problems you were presented with in high school were an attempt to get you to make the leap from specific application of concepts in lots of different ways to the abstract concept itself. In algebra, you do tons of factoring and other ways of solving the quadratic equation. The point of all those problems was that you would, through many problems approaching the concepts from different angles, fundamentally understand what parabolas are all about. Accurate quadratic thinking is much much harder than linear thinking. When you see a line, you know it's a line, but when you see a curve, it might be quadratic, cubic, exponential, logarithmic, or any of a host of variations.
So, do a bunch of problems to build your skills and gain fluency with the concepts. Then try to figure out exactly what it is that's really going on. There's often some really obvious reason that something works the way it does, if you can find it. For instance, the whole FOIL method for multiplying binomials like this: (x+3)(x+2). If you draw a rectangle, and put the x+2 on top and the x+3 going down the side, and break the rectangle into an x part and a 2 part vertically, and an x part and a 3 part going horizontally, then you'll get 4 rectangles that all add up to make the original rectangle. Their areas are x^2, 2x for the first row and 3x, 6 for the second row. Those are, respectively, the First, Outer, Inner, and Last products of the FOIL method. If you draw the picture, it's really obvious, and you'll wonder why you struggled with it for so long (if you did). A good tutor can help make it all easy for you by showing you the really obvious reasons why things work the way they do.
Good luck
That's the key question. What tasks are you doing regularly that your past failures to learn high school math are stopping your from?
I use some form or another of "math" regularly, but I'll tell you one thing: most of high-school math isn't very useful for me. I've never needed calculus, and barely ever needed geometry. Algebra is ocassionally useful, but the very basic bits of it are good enough (I remember that there is such a thing as the quadratic equation and factorization of polynomials, but I've never really needed to use them).
On the other hand, graph theory, mathematical logic, lambda calculus, probability and statistics have been very useful, and I suspect abstract algebra would also be so if I understood it. But guess what? None of those are regularly taught in high school. (Hell, mathematical logic isn't even regularly taught in university math departments.)
Don't just assume you need high school math. Make some effort to figure out what kind of math would be useful, and go with that. If you're into programming, you may want to try a discrete mathematics textbook.
Are you adequate?
Attending a class also allows you to ask questions for topics that you may not understand completely, even with studying the book. I know that most math books are written by math PhDs, and although the topic is covered, it may not make sense. That's why it's so important to have an interactive learning environment. Like the parent says, you are less likely to get frustrated and give up.
Change is inevitable, except from a vending machine -- Robert C. Gallagher
Pi DOES equal 3---to one significant digit. You compsci people are always forgetting about significant digits. The fact that better approximations were available at the time is irrelevant. Better approximations than 3.14 exist today. The most accurate of which has orders of magnitude more digits than would be polite to include in a slashdot post.
Can you be Even More Awesome?!
Go to law school!
;-)
That way, you can afford to hire an accountant...
In all seriousness, I was a geek in high school and did well in every subject except math. I aced AP Computer Science and, yes, received full credit. I aced Geometry without any real effort - it made sense to me, and I could apply it to a real object. But when it came to algebra or any form of math I could not immediately apply to something that mattered to me I simply could not get my head around it. I just didn't care unless I could actually use it.
I realized this was a weakness of mine, and shifted away from computer work to other areas. If math is your weakness, but you have strengths in other areas, you may want to consider doing the same. I'm sure I could be good at math if I really put my mind to it, but I just don't find it enjoyable - why kill myself when I can make a living at something I enjoy more?
College books are not cheap, however. [/payed $450 this semester]
Whatever, a college textbook is probably the cheapest thing you can use. Buy the 3-year-old previous edition off half.com or something for like 8 bucks.
How much has math changed in three years? It's not like the problems matter since no one is grading them. I mean, 10th edition, 11th edition, they're practically the same damn thing but one costs $139 and one costs $9.
One other way to save money is to buy an older edition of a math book. It's not as if there's been great advances in trig or algebra over the last ten years, and you can sometimes get a used math book that might be an edition or two behind for only a few books. Use that extra money to get the solutions manual. Someone else said it here, but it bears repeating: Do as many problems as you can stomach. For the type of math you're trying to relearn, there's no better learning method than trying as many problems as you can. And stretch yourself by doing the harder problems that usually come at the end of the problems list. I wish the original poster much luck in reaching his goal.
Check out the web sites at MIT and UC-Berkeley, which are the #1 private institution and the #1 public institution, respectively, in the USA. There is a good chance that they offer on-line videos of the lectures.
Buy the same textbooks that the students at those universities use. For the pre-calculus mathematics, UC-Berkeley would be your best bet. MIT caters to only students who have already taken calculus in high school.
Why would that make them good resources for someone who wants a remedial education? If you want to catch up on barely-remembered stuff from high school in your spare time you don't go for a course that expects the best and brightest and will try to weed a quarter of them out early on. I'd be wary of the textbook choices, too. Professors don't always pick the textbooks that are easiest to learn from. This goes double if the professor writes their own textbook -- I have a signal analysis book by an MIT prof that's written in a deliberately dense and formal style. Amazon.com reviews are much more helpful for textbook selection, IMHO. Going to a local library and checking out a couple is also a good idea.
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And remember that being good at maths is part aptitude, part attitude, and part doing it. Just like you won't become a good musician without having a minimum of talent, liking music and lots and lots of voluntary exercise, you won't master math as long as you dislike it and don't do more than you have been asked to do.
If there's something in math you don't understand, take one step back and play with what precedes it, over and over again, until you truly master it, and it leads you into what you don't understand. Then you'll get the "a-ha!" experience, and everything will become much easier. In math, you must understand all the foundations before you can proceed to the next level. You can't pick that up later, or you'll end up just going through the motions with no understanding, and you will become lost and unable to apply your skills if a similar but not identical problem comes along.
I know that calculators killed me. I used to be able to do all kinds of math in my head but found I was losing it. Now, I only use a calculator if I really need to. I try and do more in my head or on paper. Seeing it is different than punching buttons on a calc. Now, I'm finally able to again add up the entire shopping cart of goods so I know what to pay at check out. Division and multiplication are again a snap. Trig still requires the rule or calc but I try and use my ole slide rule again because it forces you to do more in your head. I also find that when somebody else uses a calculator and makes a mistake that I see it almost immediately while they trust the number on the display. Calculators ruined math for me but, by not using them much, it does come back.
Banjo - The more I know about Windoze, the more I love *nix
Like the author, I dropped out of HS at age 15 and got my GED right when I turned 16. I eventually went to university and earned a BS in Computer Science, and now have a job as a Software Engineer in the Video Game Industry. The time frame from GED to University was about a decade and when I started classes my math skills where dull to say the least.
The best advice I think has already been given. Go to a community college and retake College Algebra, Trig, and how ever many calculus courses they offer. A probabilities course wouldn't hurt either. If you are getting into Software I would strongly recommend a Linear Algebra course as well.
In the end it will cost about a grand or so and take about a year, but at the end you'll have most of the math knowledge you need in non-academic settings. If you are a self disciplined kinda person then just buy the text books and go through them completely. But the structure of a class will help.
Now I've seen Everything
Above all, practice (Math is a muscle). Find some problems that you think are interesting, and solve them. If something doesn't work out the right way, try it again. Try and explain the problem to someone else -- that almost always helps.
I highly recommend this book: The Square Root of Two by David Flannery. It's an excellent book which gives some real good insight into how to think about math problems, and is a pretty fun read.
http://mathforum.org/dr.math/ is a great web site for helping with homework.
Also, don't get discouraged, Math Is Hard.
Know yourself, and how you learn. People either tend to be visual, auditory or kinesthetic learners. Figure out which you are and make sure that you are getting that kind of information. All people benefit from all styles, but you will have one that you learn better form than the others, and you should make sure to make use of that. So if you are an auditory learner, don't just read a book. You need to go to a lecture as the hearing is an important part for you to learn.
Don't shy away from calculators, embrace them. I know too many people who try and learn higher level math (and too many teachers) who don't want to use calculators because they don't want to rely on them. Ok, there's something to that, but because of the immense amount of calculation involved, you will really cripple your learning without one. You need a calculator to quickly take care of the simple stuff so you can use that to solve more advanced problems. Also, programming a calculator to do something is a good way to learn it. In general, if you understand a concept well enough to write a program for it, you've got a fairly solid understanding of it. Don't just put everything in the calculator to get the final answer, but do use it to simplify things you already understand. For example if you can do division, there's no reason to do long division every time you need an answer, just let the calculator handle it and work on the problem.
Make sure to get applications for the math explained to you. At the level you are talking about, I think essentially everything has a real world application. Make sure this is taught to you. It can really help your understanding to get some real world examples. I always had a really hard time with imaginary numbers in high school because I couldn't understand them (or why you'd need them if they were imaginary). Wasn't till many years later I learned what they actually are, and that they aren't imaginary at all.
Now, all that said, you need to ask yourself why it is you think math is holding you back. What is it that a higher level of math understanding is preventing? I ask this for two reasons:
1) You need to focus on what to learn. Many people think there's a certain, immutable, order you need to learn math in, or that you must know certain fields for no good reason. That's not the case. While math builds on more basic concepts, you do reach a point where you can learn only certain parts. If you are talking about math related to programming, then calc really isn't so useful, that's more linear algebra. Figure out what you need to focus your studies on. Not saying you can't learn more for fun, however if the point is to improve in something you need, make sure you learn the right things.
2) In most fields you need way less math than you think. I took through calc 2 in university and I use basically nothing past what I learned in 6th grade (algebra) in my life. There just isn't a lot in the world that requires more than basic math. If you aren't in a field that does, or don't want to move in to one of those fields, I don't know you'll find it that useful. My math skills have dropped way off through disuse. To the extent I use higher math at all it is usually solving a problem just for fun, one I could easily look up a solution to.
Please don't misunderstand, I'm not trying to discourage you from learning, I just want you to consider why so it is as successful as possible. I'd hate for you to struggle through learning new math, only to find that it does you no good at all.
Because one thing to remember is that it really isn't going to be any easier. If you take the advice of others and get a good teacher, that'll help a lot, there are plenty of lousy highschool math teachers, however you probably just don't have much of an affinity for math. Like most things, there are just some people that get it, some that don't, and a whole range in between. Unless your failure the first time was related to drugs, teenage rebellion, inattention, or something like that you'll probably still find it hard. Nothing wrong with that, I just don't want to see you getting frustrated for no reason.
My answer: math is boring, make it fun ! I was uninterested in mathematics until I discovered Martin Gardner's articles. At this moment, I became the major in my class. I recommend you any of his books. Once you'll understand that you can have fun, you'll concentrate on the domain your are more interested, since math is a very large domain.
I have to disagree with you about just reading through the parts which seem easy enough for you. Like the GP said, you HAVE to work on the problems to get a good understanding of the concepts. Unless you are talking about the most basic of stuff, you are very likely to miss something important when you skip the problems and just read through the theory part. Most of the time the theory will seem very simple once you start following the logic, but once you start doing problems you will encounter all sorts of problem in putting that theoretical knowledge to practice.
.." you are in for a surprise when you get to work on the problems in tests/quizzes/exams. I really can not stress the point enough, do the problems yourself. You don't have to do all of them but make sure to do as many different types as possible. The most obvious advantage of working on problems is that you are less likely to forget the theory and its applications in this case as opposed to if you merely read through the chapter and moved ahead.
Usually any problem set in a decent book has more than a handful of "problem types" where there is a specific trick required to pply the theory to get the solution. If you just look at the problem and think that "Oh I will apply the theorem and it will turn into some-format and then
Politicians and Pedophiles: Two groups of exploitive bastards who are most dangerous when they're thinking of children.
Step 1: Figure out what you want to know and why you want to know it.
You are probably living a rich, full life without knowing advanced group theory. So you are probably thinking about learning math for a specific reason, either for professional advancement or curiosity. If you are going to be successful, figure out what it is you really want to know or what it is that piques your curiosity. Are you frustrated because you want to save for retirement but don't know how to handle investment returns? Do you just want to not be embarrassed when you have to do simple addition and subtraction in front of your peers? Are there specific problems that crop up at work?
Once you've identified these issues, then refer to the advice from the other posts and put together a game plan.
The key is to pursue the things you're interested in. The approach is the same as, for example, you want to know more about cars. Finding out about auto mechanics is much easier and more interesting when your car is broken and you've got a specific problem to solve. Or if you have friends who are grease monkeys and you want to be able to talk to them on their own level.
Pick some problems in the books or classwork, but also just pick little problems that crop up in your life and try to work them out while you're on the bus, waiting in line, at the gym, whatever. And be sure to talk to other people who know more. Don't be embarrassed. If you don't meet someone in your class, join in online forums. Trust me, people who enjoy math really enjoy talking to other people about math. Like learning a foreign language, you can't learn it by reading a book. You have to do it and you are most efficient when you engage other people in your learning process.
I base this advice on experience: I stopped taking mathematics courses in my sophomore year in high school because I found it boring. (Unfortunately, the way high school math is typically taught, it usually is boring). Later, because there were things I was interested in, I took it up again in college and went on to earn a BA in mathematics, probably one of the best choices (both for my intellectual enrichment and my professional life) I've ever made in my life. I kept my focus by finding things that made me curious and following up on them and have never looked back.
-- My choice of computing platform is a symbol of my individuality and belief in personal freedom.
-1: flamebait should really be -1: inciteful
I managed to get my bachelors in math, but I was a struggler, not a natural. At first, I did absolutely awful in college - although I did very well in high school. I managed to do better in college, by improving my study skills.
1) Make use of other people. Unlike many other subjects, with math it can really help to have something explained by a live person. Make use of teachers, tutors, and fellow students.
2) Don't fall behind. Unlike many other subjects, cramming seldomly works with math. You can get hung-up on some concept and not be able to go any further. In math, you are always building on what you have already learned.
3) If one source doesn't work, use another, and another. If you read on books explaination, and it doesn't make sense for you, get another book and read that explaination. Read a few explainations.
4) Of course, do as many problems as you can.
5) If you having trouble, do your best to isolate exactly where the problem. That way you can explain to somebody else much better. Also, the process of isolating the difficulty will lead to the solution.
6) Sometimes it helps to know the history of certain areas of math.
...and I hardly remember how to do any of it now that I've been out of college and in the workforce for 10 years. As an IT manager, I need to use nothing more than an Excel spreadsheet for all the math I deal with on a daily basis. Once upon a time, I used to be able to solve complex systems of third-order differential equations and could show you all the math necessary to describe a multistage rocket taking off from Earth all the way to placing its payload satellite into a particular orbit around Mars. But I barely remember any of that stuff since it's completely irrelevant to my livelihood and career. If you don't use it, you'll lose it for certain.
Ah, a topic of discontent.
You know I can remember thinking about mathematics and the legends behind the basic foundations in analysis, calculas and the like. (i.e. Euler and Newton and Kepler et al.)
I thought WOW I must be stupid, these guys just picked up Mathematics no problemo......
Well....not quite. I mean, make no doubt, Newton, Kepler and Euler all where very adept at Mathematics.
But, they also worked VERY....VERY very VERY hard at it.
Can you imagine the PAIN and SUFFERING, Kepler had to go through in building even the most basic elementals of planetary motion by doing the same calculation sometimes 100 times to prevent error?
Even then, he got the calculations wrong for the orbit of Mars and missed the eccentricity factor that would have been a shoe in while he was testing different shapes of orbits for Mars: namely an ellipse.
It would take Kepler WEEKS to perform these calculations, which now I can do in a fraction of a second on my laptop.
The labor required in those days to do mathematics was intense, and highly error prone.
Newton would lock himself away for DAYS barely eating anything performing every possible experiment, and when not satisfied with just experimentation, he wanted quantitative results from the experiment as well.
Has anyone, I mean anyone here gone for days barely eating anything working non stop on a mathematics problem for 18 hours at a time?
You know the "greats" in Mathematics worked at it with super human resolve and zeal, only if you would care to read about this HISTORY of mathematics you would find it as so.
Expect to put in at LEAST as much effort if you want to really join their ranks.
I would like to point out that with tools like: http://www.gnu.org/software/octave/ you can bypass the pain and labor of mathematics and get to the core of the matter MUCH faster than Kepler or Newton ever could. So you could literally "cheat" out of the labor these guys had to put in, and put the machine to work doing the calculations to develop methods of computation much quicker to solve problems.
So, although no doubt, these men became literal geniuses, if you look at their lives and what governed their passions with regards to numerical studies, they put in huge amounts of time to the problems they wanted answers to. They earned the right to be called geniuses, it certainly wasn't given to them at birth.
Keep this in mind the next time you are stumped on any sort of mathematics problem. Also keep in mind that like the "greats" you have to be stick with it, and never give up!
-Hack
Got Geometrodynamics? Awe, too hard to figure out? Too bad.
No, golf is boring crap. It resembles geometry a little, but the courses are too irregular, and the balls too highly influenced by the wind, to be all that mathematical.
A much more interesting, and more fun to play, game which involves lots of geometry and physics is Pool. Best of all, all it requires is a room and a table, with some balls and a stick, whereas golf requires an overpriced membership at some stupid club where hundreds of acres of prime real estate have been wasted on growing grass.