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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?"
There are plenty of self study guides that one can purchase.
Another option, if it fits into a persons schedule, would be to take classes through a community college. Costs are lower, classes are generally smaller than a university and schedules are often flexible for working adults.
Another thought I had is home schooling materials. I've never personally been involved in homeschooling, but as I understand it these kids can earn a highschool diploma at home. So why couldn't someone put themselves through such a program just to learn the information? I'm sure there are lots of resources out there for this, a quick google turned up this one.
It's hard to believe that's how Micronians are made. Why don't we see it right now by having you both kiss one another?
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.
http://ocw.mit.edu/
Does it go on forever?
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.
And the whole 3 = 1 thing...
If I have nothing to hide, don't search me
There's probably a community college in your area that teaches courses in all of the above and beyond. The fees are low (my local community college charges $20 per class credit) and there's usually no requirement that you formally enroll, declare a major, etc. The advantage is that you have an instructor who can answer your questions, plus who assigns you homework. In my experience, the only reliable way to learn math is to do it, and it's too easy to get lazy with self-directed study.
Breakfast served all day!
Any sort of advanced math is very easy in which to develop bad habits. Advanced math "build", unlike other subjects in those same grades. If you didn't "get" Death of a Salesman, you still have a shot at understanding Moby Dick. However, if you did not "get" fractions or percentages, then you really can't go a lot further.
If your math skills (or, rather, lack thereof) are holding you back, think of the tutor as an investment.
On a side note, you would be surprised at the proof of "bad math skills" that can be seen in the corporate world. People rarely / never stop to do a reality check. "Can it be that 105% of the people required to take the training have taken it?" Ugh.
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
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
And our good friends at MIT - http://ocw.mit.edu/OcwWeb/web/courses/courses/index.htm#Mathematics
There is a quandary here (in your reference to getting a book) that i've been confused about for a long time. Since every game console out there is essentially a mathematics imaging system, and given that they are pretty common and rugged, how come there isn't a sweeping line up of interactive educational math titles that let you play with the problems in realtime parameter tweaking, or in context, or visually, or what-have-you..
Seems like every math class in america should have a playstation 2 with "Calculus: The Beginning" stuck in it. Cheaper then the calculators and computers per student and the student can play it at home if they want. What's not to like?
In the larger case though, i would just like to have such a thing as an entertainment option to, like the submitter said, keep a sharp edge on the skills.
CS majors know the time/space tradeoff, but they never get taught the 3rd, crucial, tradeoff of the set: comprehension!
http://www.saxonpub.com/ .
they've changed their URL, but it redirects pronto, and the new one isn't rememberable. .
Diff between these and the normal ones?
One concept, one lesson.
Big concept? broken into several components, and distributed over several lessons.
Syncopated plan: one gets the chance to get a knowing into long-term-memory/function before one hits the next lesson that relies on it.
having tried many, and lost my math in some brain-damage I got in my teens, this is THE required one.
Find the book you need,
by doing a placement-test,
then get the ISB# for that recommended book,
then find a second-hand copy on http://www.abebooks.com/ for cheap.
Try also my gallery: http://photo.net/photos/AntrygRevo
The way I always heard that explained was Pi = 3, for sufficiently large values of 3.
-jcr
The only title of honor that a tyrant can grant is "Enemy of the State."
I just finished taking the GMAT test. the quantitative (math) section covers almost all of the math you are looking to learn. A good book (like the official guide to the gmat) has problems arranged in order of difficulty and explains all of the answers in a step by step process.
GMAT math covers basic athrimetic, geometry, algebra, combinatorics, probability, word problems and data sufficiency. I haven't done long division
by hand in probably 15 years so I found the steps to be quite helpful.
One plus of using the gmat math as a stepping stone is that if you ever want to take the test yourself then you will be pretty well prepared for it.
Another plus is that there is a ton of free material out there for gmat math preparation - study guides, practice tests, quizzes, etc. that can all be downloaded for free.
If you are going to teach yourself, I highly recommend firstly finding out how you learn. Knowing that you learn better by reading, or by hearing, or by drawing, modelling or however can save you a lot of time later on. A quick google search shows a few sites. As with all internet quizzes, never rely on one, but do a few. My girlfriend recently went back to Uni and after determining her learning sytle is doing much better now.
That said, I do maths at Uni and still occasionally forget some of the specifics about the basics. For that reason, I still have all of my high-school text books and even a few second-third-forth hand. One of them is particually good at one thing, another is concise at another. So, my suggestion is to go to second hand book stores and garage sales and pick up a couple of these. Few people want these after school and if the textbook was fazed out, they wouldn't of been able to sell it. As a result, you can often pick these up for $5-$10, especially if you aren't worried about it being brand new.
I went to graduate school at Cal in Math, and I couldn't agree more with the previous poster. I was the head TA for Calculus and a regular TA for discrete math. I think discrete math should be taught in high school along with probability and statistics. It's more fun and more useful to most people.
The materials mentioned are quite good, but never forget that math is learned by working problems. My advice: go to your nearest college bookstore and buy the text book for whatever course is appropriate for your level. Read it, in order and work the problems. I also recommend creating your own "lecture notes", with the book closed, for what you just learned. Do not ever skip move to the next section until you you absolutely understand it cold. Memorize nothing (other than defintions and terminology). Math is very natural to do self paced like this, and there's a good chance you'll enjoy it more this way. Just don't get impatient.
I saw several people here recommending tutoring, college courses, and college text books. I don't recommend any of these to begin, although they are good if you want to continue.
What I recommend here is the "low-brow" way. The easy, the "dirty" way that purists and snobs will turn up their nose at. This is equivalent to the advice of those people who give children comic books to encourage them to read. The method works, right? This will work for you too, and you'll enjoy it as much as comic books.
The key, essential text, is a book written a long time ago, called "Mathematics for the Million". It is still in print, and is excellent. It takes you from early chapters on counting from one to five, and works up through simple geometry through to algebra, logarithms, trigonometry, spherical trigonometry, calculus, and ends off with combinators and linear algebra. It is written in a great style, easy to read, but packed with information. It has lots of interesting stories and applications of the math, but not any fluff. This is the key text. It is 800 pages long, and worth every page. The price is astoundingly cheap. A chap on a desert island could rebuild much of civilization if he had this book with him. If I was on a desert island, this book would come second on my list, right after the Bible. With each chapter, it puts the mathematical developement in historical context, showing how real people developed the math out of the math that went before it, which will be fresh in your mind from the chapters you already read.
After that, you may want to work through these books: "Algebra The Easy Way", "Trigonometry The Easy Way", and "Calculus The Easy Way". In the "Easy Way" series of books, each concept is introduced in the context of a story and a practical application, as a group of people "discover" these fields of mathematics for themselves, to solve their problems. It is set in a fantasy setting with kings, queens, dragons, etc.
Finally, for inspiration, and "fun", I recommend all of the mathematics books by Martin Gardner, Ian Stewart, and A.K. Dewdney. All three of these men ran a very successful mathematical amusements and puzzles column in Scientific American. Their books are compilations of their columns. They make math interesting, showing interesting relationships between the different bits of math that we are told are "important". And they show interesting applications, puzzles, and pictures resulting from the mathematics. One Martin Gardner column that really stuck with me was the one on the "super ellipse". It has the interesting property that it looks like it should tip over, but it actually keeps itself balanced, and resists being tipped over.
As an earlier commenter said, you can't just read about math. You have to do it. You have to practice. If you are willing to practice though, the books I listed above will get you where you want to be, with a minimum of head-scratching.
Good Luck!
It isn't true unless it makes you laugh, but you don't understand it until it makes you weep.
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.
I know this post will be lost in the shuffle, but I want to thank you all for taking the time to reply to this topic. You have given me a ton of practical advice, and more importantly, hope.
Perhaps I should have replied earlier to this topic to give a little more background on my situation, some details were omitted by myself or Slashdot editors. But I'm actually glad I didn't get too specific because of the breadth of answers I have received. Many others will benefit from them, so I thank you for your indulgence.
Some of you wanted to know more background, well here it is for the interested.
I moved around a lot as a child, five different school systems up to junior high. Mismatched curriculum was always a problem, each school I'd start at was more advanced than the last, but my real problems didn't begin until I stopped moving. I went to a very reputable New York high school in the mid-80s. In my latter years there I was diagnosed (perhaps incorrectly) with some vague, undefined "learning disability". They'd no doubt label it ADHD today. I do seem to have dyslexia, but personality conflicts with my teachers had a bigger impact on my learning. Their anger and frustration with my obvious ability versus my lack of performance had a very negative effect on me. It didn't matter that I had an IQ of 136 or that I scored 1390 on my SATs, my grades were always terrible because I resented having to do what I thought was pointless busy work (something I regret today). By my twelfth year I was cutting classes everyday to spend my time in the library, learning what I wanted to learn about science, mathematics, and computers. If I am interested in a subject, and have the proper material, I usually have little difficulty learning and excelling in it.
My specific problems in mathematics classes were varied. Part of it was not being presented with practical applications. Most of it was not doing the home work, which severely penalized my grades and crippled my overall retention. Although I did well on tests, I wasn't learning. Having a literal nervous breakdown during my analytic geometry finals didn't help anything. All that said, I LOVE mathematics. I love its purity, its elegance, its logic, and its lack of ambiguity.
Fast forward to today, I'm a clever and skilled programmer, graphics designer, and game developer, 26 years as a hobbyist, 10 as a professional, with no formal education in those fields. As I expand my skill set in game programming, I'm finding more and more that I don't possess enough basic mathematics ability to truly understand topics like kinematics, physics, artificial intelligence, and statistics, even if I almost blindly employ them everyday. The practical applications I craved as a child are squarely in my lap, and I'm so rusty now that I couldn't tell you the difference between a derivative and a determinant. I may know more about fractals and ray tracers than any of my friends, but I couldn't possibly explain them or think about them critically because I don't speak the language. I liken it to being able to play jazz, but not being able to read music or talk about music theory in a meaningful way. This needs to change, my lack of mathematics skills are holding me back.
So there you have it, in too many words or less. Thanks again to all the respondents, and to Slashdot for posting this topic.
+0 Meh