Math for Programmers: 3D Graphics, Machine Learning, and Simulations with Python

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I sometimes remark that mathematicians become programmers (mathematica, matlab) while programmers, for the most part, forget math. 12:27 AM, March 18, 2006 Anonymous said...

To put this in perspective, think about long division. Raise your hand if you can do long division on paper, right now. Hands? Anyone? I didn't think so. It may not be a Computer Science degree(Actually its Bach of Information Technology -- but no-one comment on that) But still requires abit of knowledge.. Interesting stuff. I fully agree with self directed learning of any subject. But I think your distance from your youth has blinded you to the real complexity of math, particularly from the perspective of a student encountering the theory for the first time. Yes, gifted students can and will advance faster than the teacher be teaching. However, you must remember that they are teaching to ALL the students in the class. Also, the point of the depth first teaching is to get students to learn one thing well before encountering more difficult and abstract mathematical concepts. It's easy enough to state that integration is summation of a continuous function. It's another to teach kids to recognize that velocity is the integral of acceleration and that all these functions can be related graphically and are fundamentally connected mathematically. Mathematical Logic. I have a really cool totally unreadable book on the subject by Stephen Kleene, the inventor of the Kleene closure and, as far as I know, Kleenex. Don't read that one. I swear I've tried 20 times, and never made it past chapter 2. If anyone has a recommendation for a better introduction to this field, please post a comment. It's obviously important stuff, though.Here is some of the information that I discuss and demonstrate in the book. I don't claim these to be original discoveries So they're right: you don't need to know math, and you can get by for your entire life just fine without it. think in a much more logical way when I am doing math on a regular basis. And yes Steve, I use ONLY And you know what? They're absolutely right. You can be a good, solid, professional programmer without knowing much math.

I initially hit a wall in re-learning all this stuff, and now I see that it's because I'm trying to learn it the way it was originally taught to me in college. I'm still getting a feel for how many exercises I want to work through by hand. I'm finding that I like to be able to follow explanations (proofs) using a kind of "plausibility test" — for instance, if I see someone dividing two polynomials, I kinda know what form the result should take, and if their result looks more or less right, then I'll take their word for it. But if I see the explanation doing something that I've never heard of, or that seems wrong or impossible, then I'll dig in some more. I Disagree with you saying maths teaching is wrong. Maths needs to be taught basics first, can you remember what the first Maths you were taught in school? To Count next to add/subtract then that multiplication is adding several times and division is how many times you can subtract one number from another. By the time you get to colledge level the idea is that you start to specialise, here in england for a standard maths A-level you can specialis in stats, discrete, mechanics. The problem is most children don't see how certain basic pieces of maths fit in later on and get board learning them before they get on to the interesting/fun Maths (I'm 16 and I would be there if it wasn't for som of my maths teachers).

The point is to realize that the prime numbers are the atoms of integers. With this fundamental idea many areas of whole-number mathematics become transparent. 1:18 PM, March 17, 2006 Anonymous said... How about showcasing books as you read them and posting your thoughts on them as you go through? Keep the comments open. I have daydreamed about a solution. I think what should be done is to restructure math so that a student would take a one semester course in polynomial algebra, where they mastered thos basic skills. No roots, no rational expressions (algebraic expressions which look like a fraction), just polynomials. Many more students could handle that. Then they would take "Polynomial Calculus". Polynomials handle a majority of the real life situations, and you could demonstrate the power of Calculus. Math and science majors would then need to take more clases which would teach them to deal with more complicated algebraic functions, then apply the Calculus concepts to those functions. 11:22 AM, March 19, 2006 Anonymous said...

Basically this guy is telling us to learn on our own, like we should be doing anyway. The reason math is not interesting to people is not any failure of the teachers, it is a failure of people to open their eyes and think of learning as something they do outside the classroom. You don't have to tell me to use Wikipedia to find interesting things. What I personally like about funky math is it seems to stretch my brain in whole new ways, and I end up coming out with a better understanding of how the whole universe fits together. And when you see concepts from different areas coming together (linear algebra for multi-variable calculus, tensors of inertia looking like covariance matrices from statistics) it's pretty cool. I appreciate what you are saying. I'm coming at this from the angle of, "how am I going to teach my kids math?". My wife and I homeschool our kids, so we have the luxury of teaching math in whatever manner we prefer. I've read some compelling arguments for starting math relatively late, say 10 years old. Prior to 10, teach the "grammer" of math. The vocabulary of it, if you will. This would be the "breadth" approach that you describe so well. Then, when they are well versed in the "language" of math, teach the application of a particular discipline. At my University we get this course in our first semester of our first year, "Applicable Mathematics for Computer Scientists", pretty much comprising of what you have suggested. 12:20 PM, March 17, 2006 Anonymous said...A great book on mathematical logic, or modern symbolic logic as I have known it, is "Logic : Techniques of Formal Reasoning" by Kalish, Montague, and Mar. I enjoyed your article, thanks! So now come university, And i'm doing discrete mathematics as a first unit, rather easy to understand straight off the bat unlike many others who thought they were exelent at maths the last 2 years..

I think it is safe to say that your ability to move easily from one subject to another owes much to the teachers that spent countless hours drilling into our heads the fundamentals of mathematics. You can learn algebra without calculus, but not vice versa. 7:42 PM, March 17, 2006 Anonymous said... And has anyone pointed out that Computer science IS mathematics? 9:54 PM, March 17, 2006 Anonymous said... My way to math was thru number theory (long story). Not necessary any one subject that you learn in school, but it's mostly an exercise in looking for patterns in number.. Math to me is to recognize pattern in abstract space and it's a matter of strategies which help in solving problem and writing programs 11:09 AM, March 17, 2006 Anonymous said...By the way, my hand is up. I can do long division on paper, right now 2:56 PM, March 17, 2006 Anonymous said... Keep working. No matter your route to mathematics, it will pay handsomely in the end if you keep it up. If you're suddenly feeling out of your depth, and everyone appears to be running circles around you, what are your options? Well, you might discover you're good at project management, or people management, or UI design, or technical writing, or system administration, any number of other important things that "programmers" aren't necessarily any good at. You'll start filling those niches (because there's always more work to do), and as soon as you find something you're good at, you'll probably migrate towards doing it full-time. Take a look at Michael Mitzenmacher's "Probability and Computing". It covers randomized algorithms, Chernoff bounds, Shannon's Law, stationary ergodic processes, and all of the really interesting stuff from a computational perspective. It's challenging stuff, but it's a tremendously well written book.