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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 Bible. It contains more math than you can shake a stick at and it's pretty entertaining too!
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.
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.
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!
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
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.
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?
http://jumpmath.org/about/myth-of-ability
John Mighton, a math PhD and award winning playwright, founded a math tutoring program called Jump Math. It has been very successful with all kinds of student. In particular, it has worked for adult learners in jail. "The Myth of Ability" gives the basic philosophy of the program. Once you have read it, you will have the clues you need to direct your own math learning program.
Almost all the things we think about as intelligence are a result of pattern recognition. We really don't work by logic. Master level chess players, for instance, don't work out positions by logic. They can't work out moves much farther ahead than non-experts. What makes them experts is that they have studied thousands of games and they recognize situations when they see them. The way they got to be experts was by 'deliberate practice'. That's how you are going to learn math. http://www.nytimes.com/2006/05/07/magazine/07wwln_freak.html?_r=1&n=Top%2FFeatures%2FMagazine%2FColumns%2FFreakonomics&oref=slogin
Once you understand the underlying principles of how we learn and once you understand that the effort required will almost certainly lead to success, you will be much more likely to put forth the effort required.
As someone who has had to ramp up his math skills recently, I admire what you're doing and wanted to share my experience. The main thing that struck me is that you're looking to do an entire high-school equivalent math program, which to me seems like a daunting and boring approach.
Instead of looking for a curriculum, it sounds easier to find some relevant problems and work backwards. You mentioned that your lack of math is holding you back. Why not identify some specific cases of this, and learn enough math to overcome whatever issue made you feel this way? Doing this enough times will give you a solid background in math, I think.
In my own case, the reason I had to ramp up on math is that I was taking a pretty hardcore machine learning class during my masters. The course assumed a much deeper knowledge of linear algebra than I had. I literally had to do hours of research to understand many slides from the lectures which were really intended to be background and proofs, not the meat of the course. You can imagine that by the time the course ended (I got an A- which was a big deal for this class) I had a much stronger foundation in linear algebra and other math concepts than I did initially - even though I didn't set out to learn that stuff. Call it just-in-time learning. Now I am studying for the CFA (Level 1) and it also has some math, although nothing too hardcore. Still, the first volume contains a quantitative methods section which talks about statistics and the like. So again, even though my goal is to learn Finance, not math, I ended up refreshing a bit of math in the process.
Maybe this "just in time" learning isn't for everyone but it seems good to me in that it forces you to learn math that's the most relevant to your life, and it in a sense forces you to make sure you've learned it well, given that you'll be applying it immediately.
Also, MIT has some online courses that you should check out. I know you talked about highschool level stuff but why not be even more ambitious? For example, there's a series of video lectures with dr. Gilbert Strange about Linear Algebra. I don't think the course requires too much other background (and again, if he talks about a concept that you don't know, this is a great opportunity for additional just-in-time learning).
The main thing I am trying to say is that you should set a goal for yourself that's narrower than "learning everything". Define a concrete problem and solve it. For example, your problem could be as simple as watching all of the lectures mentioned above, or reading some calculus text. Instead of spending years learning everything everyone tells you that you need to know before you can do calc, just do the reading and then branch out into understanding pre-requisites as you encounter them in the text. I think this is a much more structured and motivating way to do it.
Good luck!
http://ed.markovich.googlepages.com
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
I've found a number of helpful math lessons on youtube recently. Some are actually pretty good. Just search for algebra or whatever you're looking to learn. Last week I got refreshed on statistics.
Obviously there's a signal-to-noise ratio problem, just skip over the noise.
O lord, bless this thy holy hand grenade, that with it thou mayest blow thine enemies to tiny bits, in thy mercy.
For Self Teaching- don't do it. Your main problem is finding out what learning mechanism works best for you and then finding a compatible mentor. Don't go to a local college and merely buy the textbooks there, you will get through the first chapter then realize you wasted $100 on a book you have no idea how to read.
Also, you need to decide how far in math you need to go. For calculus not all books are created equal. Find a simple book that has easy to understand examples but does not go too far. Make sure it has a few chapters on limits only- you need to know these to know calculus. On the other hand, you likely do not need to know how to check if an integral is converging or diverging, knowing how to do Taylor series, Laplace Transform, Invariant coordinate systems, etc. The book you select should have basic differential and integral calculus but nothing too advanced. Take baby steps. If you can work your way (with someone) through these things you will have a better chance to succeed and know what types of math you need to specialize in and how much.
Also, tell us what types of problems you are running in to or else we can't pin down a specific way to help you. What types of applications are you doing and what do you need to find out? You may only need differential and some basic integral calculus do to the work you need.
That's my advice for self-teaching, but I would suggest going to a community college or finding a mentor who will (maybe for a small fee) teach you the math.
Finally, if you do not understand the math you will not be able to use it in your job. Make sure you don't waste your time going down the wrong path. It's essential to have someone to ask and review your work so that you find out you are not doing things backwards and upside-down.
Learning math is similar to learning a language, although the constructs are vastly different between the two. It doesn't happen through osmosis and it's hard to get a good understanding of the "pronounciation" unless you have someone you can go to. Again, seriously consider taking some precalculus classes at a Community College then going on to calc. Without the foundation for the more advanced stuff you will get nowhere. De toute façon, on chance!
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
I've taught a number of community college classes out of Michael Sullivan's "Algebra & Trigonometry" and overall I'm pretty pleased with it. Currently on edition 7+ (so well edited & typo-free), contains all the basic stuff you mention (algebra, trigonometry, analytic geometry), pretty comprehensive.
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
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.
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?
I am from the opposite end of things, someone who did math competitions from elementary through undergrad and who misses having them in graduate school. That said, The Art of Problem Solving books might work for you. They are intended to help students prepare for middle and high school math competitions, have solution manuals, and are $73 for both books and their solution manuals. There is also a new strictly algebra book available. My main reason for recommending this is that the whole point of most math competitions and these books is to teach you problem solving techniques, you will learn algebra, geometry, trig, etc, but also learn more of how to apply them to more interesting/applicable problems. http://www.artofproblemsolving.com/
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 used to tutor Juniors, Seniors and Grad Students in Math and Physics. ALL learning is self-learning. No one "teaches" you; YOU do the learning. Remember to practice, and I suggest spending a lot of time doing word problems, since they are the reality of math.
OK, Arithmetic: "The Trachtenberg Speed System of Basic Mathematics" by Ann Cutler and Rudolph McShane. This will teach you to do Addition, Subtraction, Multiplication, Division and Square Roots, much of it in your head. Learn to use an Abacus/Soroban. It helps to bring arithmetic into focus. there are a couple of computer-based practice utilities on the net to help you memorize the rules and gain quickness in TSS.
Algebra: "Programmed Reviews of Mathematics" by Flexer and Flexer. Six small books with a good introduction to the basics of many Math concepts.
"Algebra", "Functions and Relations", and "Trigonometry and Analytic Geometry": "Pre-calculus Mathematics" Vols I, II, III by Vernon Howe.
Calculus: "Quick Calculus" (Wiley Self-Study Guide) by Kleppner and Ramsey, and also "Calculator Calculus" by McCarty.
Most of these books are older and you will need to look for them. Most of them are "programmed instruction books", which is not a popular Thing to publish these days. Programmed Instruction was developed by B. F. Skinner and Norman Crowder and has been used to teach almost any subject imaginable. The information is presented in "frames" with questions and answers, on the principle that people learn faster in short, successful segments than they do with larger difficult presentations. Programmed Instruction seems to have fallen out of favor about the time that B. F. Skinner was castigated and demonized for his rigid behavioral views. I have never known anyone to NOT learn from good programmed instruction, if they could read the material and understand it. You might want to check with your physician to make sure you don't have an issue like dyscalcula (similar to dyslexia) or some other learning disorder that needs to be overcome first. If so, that could explain much of your frustration and can be handled.
Good programmed instruction takes a long time to develop and test. Each frame should lead to 96%+ success for people taking the course. Many older books simply broke up their information in short segments and asked a question without actually testing the goal and result. I am least satisfied with the Wiley Self-Study guides, but they are usually adequate for learning.
Good luck!
"The mind works quicker than you think!"
Get some high school or college textbooks. (Algebra/Precalculus, Calculus, Geometry, whatever you want to learn)
Get the solutions manual for each book.
Work through the textbook. I really mean work, so write down and think through all of the examples in each chapter. Then, do 'enough' problems at the end of the chapter. Check your answers with the answers in the solutions manual. If you didn't get it right, do it again. If you still didn't get it right, then read through the solution provided. If you STILL can't get it, ask someone, possibly on a forum online or in person.
It worked for me--6 years ago I was a B- high school math student and now I'm taking graduate level math courses.
These are the books that Richard P Feynman used to teach himself math.
Algebra for The Practical Man.
Geometry for The Practical Man.
Trigonometry for The Practical Man.
and Calculus for The Practical Man.
They're old self study guides and they're the best I've seen. I've seed the Idiots and Dummys guides, they're horrible. The Practical Man series really explain how it all works, not just memorizing formulas. I found them on Amazon.
I don't think you'll find all of the high school math in one book, but I do know of a great book that I taught myself Trig with because my school won't let me take it. It's "Trigonomerty: An Analytic Approach. By Drooyan/Hadel. Amazingly awesome book that not only teaches you the circular functions and other stuff people associate with Trig, but it does a damn good job of showing you how everything works, and why. The first chapter is about the unit circle in case that helps clear up what I'm trying to get across. :)
:)
I'll admit it isn't the most visually appealing book or the easiest thing to read, but if you spend a few hours really working to understand the contents of a chapter, it's totally worth it, because in the end you'll have a very very deep understanding.
Also, the Blitzer: Precalculus book is very good. Great if you need to refresh your algebra skills. It also has a great Trig section.
For calculus, I would suggest, "The Complete Idiots Guide to: Calculus" (to get started), it was surprisingly good. And most of all, Calculus: An Intuitive and Physical Approach. That last book is practically my bible.
Anyways, good luck, you learn your math, and I'll struggle to get my stupid high school to let me take an interesting math class I won't be bored in.
If con is the opposite of pro. Then isn't congress the opposite of progress?
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
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
When you get to the calculus level, check out "How To Ace Calculus". It has a lame sounding name, but is a fantastic book that keeps everything in the real world vocabulary. Now, I did use this book alongside a course in real life, but I am a very independent learner and would have gotten at best a C otherwise.
:-)
P.S. One of my favorite parts is how the authors will say stuff like "your teacher really means this, but the other way makes them sound more important"
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.
Bottom of the math class? Could barely get into a community college? Failed by a book written "...for Dummies"?
Thank you, you've gone a long way towards explaining what kind of people get MBA's.
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
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.
There are some very interesting replies here. Two I'd like to repeat because I find them particulary true:
1.) Don't be intimidated.
2.) Stay curious. Find ways to get curious about certain fields of math.
These are from different posts, but I think they go good together.
The truth is, math is a mess. It's a historically grown mumbo-jumbo of countless variations in notation. The problem is that with programming languages - no matter how crazy they may be - they allways come with a reference manual to explain their syntax. In fact, that is the main element by which we judge the viability of a PL. With math on the other hand academia kind of expects us to understand what the Professor is writing on the blackboard without even addressing the issue of a solid reference in which I can look up the meaning of the sum-symbol or what a limes means and how it looks like. It's like music-notation. Somewhere back in the day - often a few hundred years ago - someone came up with a certain notation and since then that's the rule of thumb by which everybody sticks to sorta-kinda 50% of the time. If he feels like it. These notations are mostly literally bolted on to terms and expressions in the most chaotic and hideous way one can imagine. It's like trying to understand a Perl obfuscation contest without the manual.
This is IMHO the single biggest problem in grasping math. Especially for Computer Geeks who are used to strict syntax constraints.
I' currently studying the first semester of BS-CompSci and am glad for having finished my German GED just this summer, with all the accelerated math (barely made it with a D+ due to the time-constrained tests) still in my head. I can just about keep up with the lectures. We allready have quite a few students bickering about the lack of a symbol and notation reference.
Bottom line:
Math is a mess. It is a non-trivial science and takes work to understand, but it's a mess none-the-less. If one keeps that in mind without using it as a cheap excuse not to fully work out and understand the details then learning math is much less frustrating. That's how I feel about it anyway.
We suffer more in our imagination than in reality. - Seneca
For anyone seeking to master mathematics this is the one way I have found to start.
Google: On the study and difficulties of mathematics by Augustus De Morgan... you can download it for free from google!
after reading that title I suggest reading De Morgan's Trigonometry and double algebra title also available for free from google
Followed with elements of algebra also by Augustus De Morgan
followed with elements of trigonometry by De Morgan
I would also supplement this study with project MATHEMATICS! by Tom apostol
Then work through Tom Apostol's Calculus ( M.I.T. uses this text for their theory calculus courses) you can find this ebook
floating around on most bit torrent sites.
I would also suggest you have a look at Dover Publications, they have great reprints of math classics including some by De Morgon.
Truly, once you get the basics firmly in your head, the more advanced topics come much more readily.
I hope this short list can help others as much as it has help me in the self-study of mathematics.
I wouldn't recommend self study through books, as you have nothing pushing you to do the work, such as assignements. The Open University does a very good maths course (MU120 I think). Your only problem will be doing the exams if you're not in the UK, but the course teaches you up to University level.
Course details are: http://www3.open.ac.uk/courses/bin/p12.dll?C01MU120
It will cost you around $600 if you can afford that, but is far more effective in my view. You get a tutor and set texts all online, plus messageboards for the other students and tutorials if you are in the right country.
This is what city colleges are for, just refresh yourself on the math you did take. Like pick up a trig textbook (either at the city college library, since they also do those low level courses, or buy a cheap used one) and do a handful of problems for each chapter. Once you've done this part, sign up for calc I and beyond. Picking times that work for you, some classes can be held online at the swankier city colleges. But night school works just as well for must of us working stiffs.
“Common sense is not so common.” — Voltaire
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.
SAT prep books are great for reviewing basic math concepts, especially ratio and percentage problems, which come up in everyday life. (The kind of algebra that can help you spend money wisely.) The SAT will also give you an idea of what is considered fluency with math at the high school level.
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.
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.
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