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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?"
Why not just stop by your local college bookstore? Just pick up a math text book, go through it, do the problems, check your answers, etc etc. Millions of students have used them. Probably will work out for you.
When growing up, I was forced to do pages of simple math problems - just simple addition, subtraction, multiplication, and division. Imagine sheets of paper with 20 rows and 3 columns filled with questions. I would then get timed to see how quickly I could complete these questions. This was done time and time again until I didn't have to think in order to solve such problems. I benefit from this even today..
The thing is, when you're learning math you want to focus your efforts on the subject at hand - not the other simple math that accompanies it. For example, when a prof is going over a question on the board you don't want to waste time with the simple stuff. It takes away from what you should really be learning.
So I guess my suggestion is this - make sure you know the basic stuff really well. You will always have to use it and without it you will always be at a disadvantage.
Willy
In order to learn it on your own, you want to enhance your curiousity at any chance you get. If you get the feeling that you're forcing yourself through it, you might not continue. To maximize curiousity, i suggest you find several math books. Each day, you set aside some time to do something, anything, without a preconception of what it will be (unless there's something you're really keen on doing). When you sit down, you bring out your 3 or 4 books and you flip through until you see something interesting and work on that.
Sometimes you'll find something that requires previous concepts that you don't yet have. This is fine, because now you can go look up those concepts with a sense of purpose. This will help you to your larger goal of the more interesting thing that you flipped to in the book. I did this when i picked up a book on fractals...lots of bright pictures, it seemed interesting. In there, they talked about integrals, which i hadn't learned yet, so i set out to find out what those were.
As for practical tips when you're learning one particular concept, reading textbooks is sorta like reading manpages in unix. it takes a certain mindset, and you usually want to pick out the relevant pieces from the page the first time around and then go back for specifics later. Textbooks are usually written very precisely and they sometimes have a lot of formal jargon or formulae that aren't useful the first time you read it, but can be helpful when you go back to get more details. So read it with that in mind. The first time through, don't expect to understand everything there. Just skip past the parts that are too hard and continue on, trying to get the general idea.
Next, do some of the easiest questions at the end of that section or chapter. Sometimes those questions may seem too easy, like you can just look at them and you think you know how to do it already. I suggest doing some anyway rather than skipping them. There's a difference between knowing the concept enough to recognize it in the questions, and actually knowing it well enough to do the questions quickly and correctly. Doing more questions is always good practice even when they seem easy at first glance.
When you've done several of the easy questions, you start to get more of an intuitive feel for the concept. Go on to the medium questions, and now you'll probably better understand the parts of the text that were difficult to understand on the first time you read the section. I suggest that you try hard to really understand the concepts in one chapter before you go onto the next one. If you have a solid grounding in the beginning, then the later stuff will be much easier and it'll be easier to get that intuitive understanding that lets you see the direction to the answer right from the start.
If you have several textbooks to choose from each time, then as you work your way through bits of each of them, you'll start to see the connections between different areas of math. This is something that most people don't get in their normal classes because they tend to focus too closely on one topic. If you wander through several topics following your curiousity, i think you'll get a better broad understanding of the connections, and it'll help you personally keep your motivation up so that you can continue to do it. remember to have fun with it. if it turns into a chore, then you'll stop doing it before you reach your goals.
have fun!
"The value of a man resides in what he gives,
and not in what he is capable of receiving."
--Albert Einstein
I don't think they do a great job of teaching it in school where they take a very linear approach.
I'm not currently a professional teacher, but I have been one, at a Big Technical University that you have heard of, for four years. My skin crawls when I hear people demeaning a linear pedagogic approach because, frankly, and you can take this as an expert opinion by someone who has won awards for teaching, there is no better way. Period. People learn depth-first by cycling down from coarser details to finer ones. They learn in steps. To quote Prof. Patrick Winston of AI fame, you only learn that which you almost already know. Trying to teach in fuzzy alternate ways, teaching by trickery, emphasizing word problems or case study, teaching two or three paths at the same time, all of that stuff does not work for technical and mathematical subjects, pure and simple.
For the basic mathematics that the original post is inquiring about, the concepts are reasonably simple and straightforward. What they require, however, is what often appears to be mind-numbing repetition. It's work. While I applaud this fellow's current initiative, the effort should have been put in when he was a teenager because it's a lot easier then. It sounds like he's understood the mistake and is currently, as an adult, trying to correct that, which is definitely commendable. Unless he's the sort of person who developed phenomenal self-discipline later in life, however, the best bet is to get to a classroom. There are any of a large number of adult education services in every city I've been to. Often local high schools will have evening adult-ed classes as well. Or, as another poster suggested, the local community college can be a good resource. But basic mathematics requires a lot of rote work. It can be a joy to know that you've learned everything that was used to get mankind to the moon, a tremendous joy in fact, but it takes work.
Put my fist through my alarm clock with its ding-dong death inside my ear. - The Blackjacks.
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.
My best advice is to try a two-track approach: non-discrete mathematics and discrete mathematics. Traditionally high schools teach only non-discrete mathematics: e.g., trigonometry and calculus. Since you are studying the material on your own, you could improve upon the standard curriculum. Read a good book on discrete mathematics first. It will build your intuition of mathematics. Then, study the standard topics in non-discrete mathematics.
Discrete mathematics and non-discrete mathematics are quite different, but the reasoning in discrete mathematics will hone your skill in handling mathematical proofs, which are central to both branches of mathematics.
For a real challenge, after you finish your studies, try to determine whether P = NP.
I was someone who was once considered to be exceptional in math. Unfortunately, I made the mistake of stopping at calculus.
To regain my mastery of mathematics, I decided to take a single math problem very seriously. I figured that I would try to
understand the solution by grounding all ideas down to postulates.
I figured that this was a great way to learn mathematics anew and really get advanced. I soon learned that there were wonderful
math resources on the web. Wikipedia is really great. There's also MathWorld.com.,
PlanetMath, MathForum.org, and
Cut-The-Knot.org.
Being pretty ambitious, I chose Fermat's Last Theorem and Andrew Wiles's solution as my jump off point. I started this adventure
in 2004. Since then, because the problem is so tough, I started blogging through the different threads of the problem and I find
myself recreating the history of mathematics from the perspective of number theory.
I am not sure that this approach would work for everyone but if you are a solid problem solver, it can really make advanced
mathematics more fun. If you are interested to see what I came up with, you can check out my blog a My math blog.
I also started a general math blog.
Best of luck in learning mathematics.
-Larry
You know, I wince when people say that. Yes, math is hard, but then, music is hard. Creative writing is hard. Any subject is hard if you don't get it, and even if you do get it, any subject needs hard work to get good at it. Yes, math needs abstract thinking, and some folks are better at that than others, but then, some people are better at pitch and rhythm than others. Picking on math in this way is sowing the seeds of defeat.
One of the math books I have (I can't remember which one) starts with a riff about how most folks want to drop math as soon as they can, but then it lists a whole list of subjects (things like "how to avoid getting ripped off", "how to play the stock market", "how to save time and effort by taking shortcuts on common problems", "having fun with games and puzzles" and so on) and speculates that pretty much everyone would want to take a few of those options. The trick is, of course, that they're all math. I'm convinced that the reason most people hate math is because it's taught in an almost completely abstract way (because the teachers have to get through the syllabus in a limited number of class hours). Teach it the other way -- take real problems and show how math can solve them or generalise them, and I reckon a lot more of the students would go along for the ride.
A friend of mine used to teach remedial physics to a college class. He wasn't much older than the students, so he started the first class by pretending to be another student and mixing with the others as they came in. In the process he discovered that most of them were bikers who had to get the physics qualification to support a motor mechanics apprenticeship they were doing. After some consternation when they discovered he was really the teacher, he started by asking them how they would tune a 2-stroke engine; what effect the things they were doing would have on the engine, and how they would measure the effects. This led them through all sorts of physics, from friction and levers to gas laws and fluid flow. He got every student through the exam, because he made it relevant. The same can be done with math, and it makes it a whole lot easier.
Quidnam Latine loqui modo coepi?