For any layperson who wants to understand what mathematicians think about math, I cannot recommend _The Mathematical Experience_ highly enough.
It is accessible to a high school student. Yet has insights for a PhD in mathematics. And more than any other book I know, it captures the experience of mathematics. Both good and bad.
I like to say that I went into mathematics for reasons given in that book. And I left it for reasons given in that book.
The comparison of early maths (algebra in particular) to spelling is spot on. It also reminds me of a somewhat wry observation from one of my mathematics professors: that if we taught spelling and grammar exclusively through crossword puzzles and competitive Scrabble, many fewer people would learn to spell and write grammatically.
The analogy is to teaching algebra through endless, dry repetition of problem solving. Some of you won't get what the big deal is; you're the ones who are good at 'crossword puzzles'.
I usually compare it to being forced to play through the first three levels of Super Mario Bros over and over again. Then they put up posters saying "Video games are fun!"
I had a fantastic calculus teacher in high school (Freddie Irani, many thanks if this somehow finds you). He's probably the only math teacher or professor I ever had who made math both fun and intuitive. We had no homework, and regular quizzes.
He taught math by talking about the history of the problems we were trying to solve, and how people went about solving them. It was the concepts that mattered; the equations were secondary. He would put a problem on the board, and ask the class to solve it. He would then give us pieces of the puzzle, until we had enough that we could figure it out ourselves. I found the class participation to be key. With most math classes in college, the instructor works through 3 or 4 problems for a given topic on the board, and the students spend the entire class furiously copying down what he writes. It's pointless. I get that it's driven by the need to cover a lot of material, but I'd rather the class work through two examples then the instructor work through 4. Also, regular quizzes were a key part of Irani's class that made retention much, much easier. Instead of the typical weekly homework, two midterms and a final, I would much prefer weekly quizzes, 3 (or 4) midterms, and a final worth a much smaller chunk of the grade.
Last semester, I took linear algebra. We had weekly assignments that I did with a tutor (bad decision), two midterms, and a final worth 45%. I went into the final with ~40% in the class, crammed for it, and ended up with a mid-60 in the course. It was pointless, and I don't feel like I learned much. I would much prefer the system Irani set up, but he's not a university professor, and he doesn't have a PhD. He taught himself calculus with an inch-thick pocket book he picked up from a second hand store.
Any specialized field becomes obscure to the uninitiated. While it may be possible to use metaphors to give them a glimpse into the field, it's ultimately a false picture. And so, analogies should be used to whet appetites, not attempt to explain expert knowledge.
Feynman on magnetism: "I can't explain that attraction in any of the terms that's familiar to you. For example, if we said that magnets attract as if they were connected by rubber bands, I would be cheating you[...] and if you were curious enough, you would ask me why rubber bands tend to pull back together again and I would end up having to explain that in terms of electrical forces."
Conrad Wolfram gave a great talk at Ted about how to reform math. He is spot on. I think this is one way Mathematica could really jump to the next level... develop a full on home-school curriculum from age 3 through 12th grade.
I've used Teaching Textbooks (http://teachingtextbooks.com/) for math up to Precalculus. It's a self-paced, rather engaging math program with good word problems.
For elementary school, their curriculum has a more interactive program with word problems, quizzes, etc. and in middle school and high school it becomes self-paced lectures with problem set / workbooks. Another good, similar program for high school is k12 which I've heard good things about too. There are some good / new curriculums out there for homeschoolers.
Wow. This post is a revelation for me. I have frequently explained to people that have bothered to ask (and probably a few who didn't) that the reason I love programming has a lot to do with my love of language - the elegant syntax, the rules and their exceptions. Despite rigorous math training (I'm an electrical engineering student), until now I had never been able to see it as more than a means to an end. I hated math classes so much, but they allowed me to do interesting things with electricity so I put up with them. On occasion, I have maybe been struck by a part of what this person is referring to, but this puts it together in a way that I had never really understood before.
I'm the opposite, I love programming, but dislike languages. I can recall in junior high wondering why these certain rules existed. "This is simply more verbose and doesn't disambiguate anything as far as I can tell." The problem I encountered with natural languages is I rarely could get an answer as to why we have a form for this word. Whereas with programming, I can usually get an answer for almost any question I have about a language.
Yes. The analogy only goes so far - natural languages are inherently more complex than any programming language I know of for a number of reasons. Nevertheless, it's just as easy to find examples of code which are "simply more verbose" and that do not "disambiguate anything" as it is to find examples of the same in natural language - conciseness and clarity are valuable everywhere.
Natural language is evolved (and therefore haphazard and merely "good enough"), whereas programming languages are designed. Natural languages are to programming what biology is to robotics.
Agreed. In fact its funny you make that analogy... in that whenever I think about doing some biology I want to use nanotechnology. My brain just has more trouble settling into emergent systems.
That, like mastery of music, language and art, takes persistence, time, practice, learning, error and experience - make the most of your time to devote yourself to furthering what you know. The expertise you seek will come to you.
As I read this, I noticed the similarity between mathematics and programing. In fact, if you substitute the word "mathematics" with "programming," the post sounds like my experience with my computer science degree. Algorithms and programming languages were unnatural at first. I spent most of my formative years fighting with programming languages instead of using them. But after a few years, I moved past the point of fighting algorithms and languages and started using them to actually solve problems.
Writing software doesn't help you discover the laws that govern everything, but it affects people on a much more pragmatic level - software satisfies our curiosity, tells us how to go where we need to go, and affects every aspect of my day-to-day life. And that's not half bad.
In the film Matrix you discover an alternative reality and choose to take the red pill. When you are able to think mathematically you discover a new universe and you can choose to take the red pill to evolve, but that universe is only accessible to those that can understand maths.
Great explanation. The only shortcoming, or maybe not, is that he could have pointed out that the parallel he drew is true for anyone that specializes and transcends towards ideas rather than frets mechanics of a field. I'm much this way about computer science, engineering, economics, finance, etc. The things they teach you in an undergraduate education are just tools in your toolbox, the bread and butter so to speak. It's the ideas are what's important.
Honestly, it seems like people on HN so often assess things based on their raw usefulness. It makes sense, given that HN is read by many people who think like engineers, but in another recent thread on /r/math, people are talking about how the reason mathematicians study mathematics is not because it is useful, but because it possesses an inherent beauty/logic/whatever that makes the people who study it want to keep studying it. I'm doubling in math and CS and I have to say that the reason I am studying math is not because of any potential applicability to work I might do in software later in life, but because it has an allure for me that is unmatched by almost anything else.
hehe I unoffically doubled in math/chemistry, and I gave up on math because precisely due to its inapplicability, I was competing with people who were learning calculus at age nine. Chemistry was a better choice, because most sane parent do not allow their sixteen year old children to tinker around with carcinogens and explosives.
In retrospect, I should have been a coder. Although all my coding adventures post high-school have basically been autodidactic, I jumped in the deep end and took a high-level undergrad/grad CS class and coded my own web server... I was the only person whose server didn't allow the user to drill through the directory structure and hack the server because at the time I didn't know the posix file functions and actually wrote my own string parser/handler.
I wonder if anyone here has studied the Trachtenberg Method of Speed Mathematics, or Vedic Mathematics. If anyone here has, I'd like to find out what they thought of the way those methods are taught.
I have taught a series of lectures on the Trachtenberg method in high school. It has some cute tricks, but really this is not mathematics, it's arithmetic. Don't waste your time with it. If you're interested in maths, study maths not mental arithmetic. I promise it is much more interesting.
A friend's father was educated in Vedic Maths, and she and her sisters grew up having 'Beat the Calculator' competitions with him. Eg, they'd take turns giving him a series of random numbers to add/subtract/multiply/divide, and he'd do it in his head while they'd do it on the calculator. She said he almost always won.
She attributed that not only to the fact that he was very smart, but also that he had mastered the Vedic techniques that let him quickly organize, simplify, and compute long strings of numbers and operations in his head.
For years I've heard the argument that Mathematics should be Maths (being Australian). That's the first time any one has made a sensible argument (to me) for the alternative.
I've never heard anyone call Gymnastics "Gyms". :P
Interestingly, in Spanish and French the unambiguously plural terms 'las matemáticas' and 'les mathématiques' are commonly used, so the 's' in 'mathematics' may be of a different kind than that in 'gymnastics' or 'economics'.
It is accessible to a high school student. Yet has insights for a PhD in mathematics. And more than any other book I know, it captures the experience of mathematics. Both good and bad.
I like to say that I went into mathematics for reasons given in that book. And I left it for reasons given in that book.