It used to be that to be a composer, first you learned the piano. The piano was thought to give one an appreciation of the nature and nuance of music. From there you could much more easily learn other instruments and then learn how to put them together into something beautiful.
Math is the new piano. Learn symbolic manipulation -- and I'd say that means first-year calculus -- and you can work with the kinds of symbolic systems you'll find everywhere else in the modern world.
I used to do some tutoring when I was a kid, and I think this level of understanding is achievable by 90% of the population given the right environment. The real question is "why are the structures and environments for education that we create so unable to accomplish this?"
Dumbing down the system isn't the answer. It's not like you can dumb down the world to make up for your inability to prepare students for it.
Let's be honest: calculus is taught for its application in engineering, not to teach deduction and symbol manipulation. If that is your goal, there are more suitable maths to teach.
I think proofs are compressed because of lack of margins. Don't laugh! Authors seem to think they need to cover lots of essential results, and if you do motivations, intuitions, insights for all of them, already long math books will be impossible to hold in your hands. In my opinion the obvious solution is to omit most proofs and do far more non-formal discussion around important proofs. Which won't happen because their goal is to teach particular math results, not mathematical thinking, contrary to what they say.
Everybody does not need to understand this and I dare (with fingers crossed) say a good percentage of those on HN don't either. There needs to be rationalization within the educational system for what is taught, with relevance to education for the sake of knowledge and for how material can be used for future employment prospects & general life skills. That may play to the lowest common denominator or the student mean, but that's largely and unfortunately what public education has become over the past fifty years.
But is it the best way educators know of to do this? Because at the end of the day, whether math is a good basic skill that pays dividends in logical thinking and symbolic manipulation (dividends beyond what other methods could deliver) is an empirical question. What you say sounds plausible, but theories can be plausible and dead wrong.
Let's rather hope they're a passing phase, please.
Meanwhile, being good at (actual) math is rapidly become a prerequisite for having a bright economic future. School math is worth increasingly little (computers do it better than you). Whereas true math is deeply creative, and computers suck at it.
This separation is also why so many programmers say "programming doesn't take math". It doesn't take school math. It rests deeply on true mathematical reasoning.
This continues to spill over until it becomes a negative selector, driving a preference for liberal arts type subjects over science and engineering, not by the positive virtues of the former, but by the perceived (and, probably at this point, real) difficulty of the latter - and further augmented by the fallacy that any college education is better than none.
We're happy to drill reading in schools, but loath to drill even the simplest maths. Only the most abject failures are allowed to go through school without being able to read a newspaper article at a reasonable speed and give a summary, but if someone breaks down having to do 12 * 9 in his head, it's fine, he's just not very "sciency".
Do they really teach that somewhere?
I don't think FOIL should be given much weight as some sort of eternal mathematical truth, but it's a useful stop on the journey.
(a plus b) times 'whatever'
a times 'whatever' plus b times 'whatever'
Perhaps it would be best to refer to a completely non-technical education as trivial since it ignores more of the liberal arts than it incorporates.
Then again, other troublesome behavior is quite accepted as well, like, for example drinking alcohol. How often haven't I heard a parent proudly tell that his or her child just had her/his first beer.
Some bad behavior seems just socially and culturally accepted. And changing culture is hard, so, even though there have been programmes stimulating STEM for decades now, there seem to almost no real improvement.
It goes without saying that almost every non-trivial project requires serious thinking about time and memory complexity costs, and I've been surprised to find theory of computation useful when working even on simple parsers (nothing fancy from compilers classes, just straightforward reading of proprietary formats).
Despite that, I'm going to college currently for my degree. I just got an "A" in intermediate algebra, though it was an up hill battle the whole way. The struggle, aside from it being a 6 week summer class, was trying to convince myself that I wasn't wasting my time; that I would learn something applicable to life outside that or the next three requisite classes. Something, that is, besides helping my kids with their math homework.
I've come to the firm conclusion that college math exists purely to perpetuate itself, like a virus in our education system; The only job it prepares someone for is to teach the damn class!
Please try to share your methods via a blog post/elaborate comment if you can spare some time.
But I'll be sure to put together a follow-up instructional blog that skips the vitriol, as nobody should actually have to just believe me. I do think that the very general attitude toward each problem is the only constant feature, and each topic has to be treated as a brand-new exploration, so I'll need to coagulate a topic to start off with. I'm leaning towards degrees of freedom, coordinate systems and the magic of Euler's formula. Sound good?
And if not, I'm serious about the last sentence. What topic gave you the most trouble when you tried to learn it?
I had multiple false starts on Linear Algebra, Probability,Graph theory,Set theory, and Logic. Somehow books on these topics tends to run into hundreds of pages with multiple exercises of similar type which kinda encourages me to drop it typically when it starts becoming repetitive.I am not sure how to cross this plateau.
Of course, that's just an artificial way of making the problem harder, basically, so that it's more interesting than safer but repetitive sets of exercises. The best way to be interested, by far, is to actually want to solve some problem. For now, anyway, I heartily recommend looking for problems that these disciplines can solve, and choosing one to solve yourself.
But obviously, Calculus was not developed the way it is taught. I found historial approaches more motivating. That means either using infinitesimals (17th century approach), or starting from construction of the real number, say Dedekind cuts or Cauchy sequences (19th century approach).
I suspect that part of explanation has to do with computers. The word "computer" was originally a job title (see, for example the Harvard Computers: http://en.wikipedia.org/wiki/Harvard_Computers). It changed meaning in the middle of the twentieth century when mechanical (and later electronic) computers displaced human computers.
It seems like much of "school mathematics" is vocational training to work as a computer. Unfortunately, we don't employ people to do computation anymore because the cheapest and worst computer you can but will be thousands of times more productive at computation than a human being.
If our mathematics curriculum was designed to train people to be computers, and that profession no longer exists, it's no wonder that school mathematics seems pointless and disjointed.
I once solved these problems for a small group of these children to whom I taught algebra from first principles: all grammar was explicit (no elision of parentheses or operators) and we used no numerals. The reduction rules then were able to fit down one side of a whiteboard, and after five one-hour sessions, self-professed math phobes were solving nontrivial equations.
Checking it out now.
I suspect it's the lack of arithmetic skill more than the conventions that are the heart of the problem. I think kids should be required to learn to do mental arithmetic before they're allowed to use calculators, but this may be impractical.
Clearly you were never a kid.
I also worry that smart kids will work out that calculators are ubiquitous before they force themselves to become proficient at mental arithmetic (e.g. Google and Spotlight are both far more powerful calculators than anything I had access to as a student).
Math and Applied Math is more than numbers and math for the sake of math. It is a way to bring your level of abstract thinking higher - much the same way learning languages does.
Math is so much more than algebra and calculus. It will teach you pattern recognition and deduction - two highly sought after skills in engineering - but can be applied to any profession.
If anything I believe we should bump up the level of math that high school children learn.
Frankly, I agree with Hacker: quadratic equations are boring. Teaching number theory will do better job of teaching pattern recognition and deduction. More Fermat's little theorem, more Chinese remainder theorem, less quadratic equations. I found undergraduate number theory to be far easier and more fun than calculus, and you need no calculus to start on number theory.
No. The point was that it is much more than what you call the content.
This is not just math, it's virtually every subject you learned in school. Imagine the following at a cocktail party:
"I had to read Joyce, Whitman, and Hawthorne, but I never got the point of any of them."
"I took a lot of history, it was a bunch of dates which I could never remember."
"I took three years of Spanish, but I don't remember a word."
A lot of people just don't click with formalized education at all in their early years. They just couldn't throw themselves behind all those authors and facts they're "supposed to" be up on. Some of them are too busy learning informally by dealing with the world around them -- i.e. with things like nature, work, and you know, other people.
Then slowly grow to form their own view of the world, and seek their own path to obtaining a more structured comprehension of it -- which might well mean starting (on their own volition) formalized education sometime in their 20s or 30s -- or perhaps not.
A great many who choose this route, or have it forced upon them may fail or flounder -- but a great many also don't, and end up with accomplishments for others to read about or hear about in those books, classes and received intuition that some people think are the only valid starting points to figuring out how to make a dent in the universe.
An example of learning without boundaries:
One day, I decide I want to play the guitar. I get my hands on one and fiddle around. Perhaps a brief history on the instrument will give me some better context as to why the instrument is shaped as it is, and what all the features are. An exposure to famous guitarists and their music improves my appreciation of music and gives me context to work with. As I continue to struggle with notes and basic music theory, it may be a good time to take a brief detour into the realm of physics - a lesson in simple harmonics. If interested, I can dive further into the required algebra relevant to harmonic equations. Hmm... maybe learning to use a computer program can help me figure out how messing with these waves changes the sound...
And so it goes. A familiar story to most hackers, but sadly an approach that most educational systems do not encourage.
Is this exploratory form of learning superior? My theory is that the associations formed between nodes of knowledge are necessary for any sort of actual deep learning to occur. By sandboxing subjects, the standard school curriculum limits the possible associations that students think are allowed, thus limiting the probability of successful links between nodes. Even an above average student can graduate from college, and be left with nothing but a bunch of barely reachable random islands of knowledge, floating further out to sea every day.
In other words: school sucks, learning rules.
Based on your example they could then go on to construct some instrument (say, a flute) by writing a little program and print the instrument with a 3d printer or have a rough version cut by a laser cutter. The question is: how to integrate ideas like that in the current school system? ...
We already have such a world - it is the Web. And we know it is a mess, a time sink, and a minefield. However, with focused effort, it does function as an amazing learning environment.
Unfortunately, it is not a suitable alternative to school yet. Telling kids to drop out of school and hit the 'Net instead is most likely a bad idea.
Perhaps the future role of Teachers, will be to function as Guides in this virtual world of information. Most awesome field trip ever!
I see in his description more of a tech tree. It'd be more akin to what you'd see in Diablo 2, but expanded by a 1000. The tech tree method would also explain prerequisites. Ideally, if you can prove you have the knowledge of a node, you could pass through it and continue learning.
No clue how we would go from factory farmed schools to that though. It's another educational pie-in-the-sky thought.
I think refactoring current educational systems - let's call them assembly lines - into this model is very difficult, and perhaps not worth the energy. A traditional master/apprentice system might be a better base to build on top of. A master can provide the ad hoc guidance that is helpful during free exploration. With the Internet, a distributed master/apprentice system can become a reality.
Don't get me wrong - math is awesome, and everyone, especially young people, should have the opportunity to explore it. Perhaps it's helpful to think of math as a cultural activity, which is mostly removed from practical concerns.
Calculus is probably more relevant than symbolic logic for many kinds of engineering, for example, so the course is already being offered. It's also seen by (most) mathematicians as a more central subject, so it's what goes in the core if the mathematics department is given responsibility for designing the core. Additional complications arise over what to do with things like statistics (in some places it's in the math department, while in others it's a separate department).
For example, a friend of mine was a forestry major, and the only reason she needed calculus was for population modeling. She knows exponentials and logistics better than I do! But she had a hell of a time passing calculus because she had trouble with the other 90% of the course. Keep in mind that this is a discipline that until maybe a decade ago required zero math.
Maybe at least at the lower-division level, applied math needs to be a bit more targeted...? But that would require us letting go of the notion that math is intrinsically edifying, and that people "ought to", "must", or "should" know it.
I always recommend http://steve-yegge.blogspot.com/2006/03/math-for-programmers... on math for programmers.
I dunno. I am trying hard to refrain from saying X is not important for C.S. because the nature of this site is that you are going to find a ton of C.S. people who point out that they would be crippled with the lack of X :-)
Having said that, I read Steve's article many many years ago and upon reflection, I have realized that Steve is way better in Math than he lets on. Some of the techniques there are designed not for you to "get" Math but to get some sort of cultural awareness of Math (which of course is a fun pursuit on its own). The reason why casual foraging of a ton of math has never worked for me when I started learning math is because one of the reasons Math is incredibly hard is that it is inherently a personal journey. Getting math sometimes requires working at it patiently for weeks before you finally understand how to wrap your brain (which really means making connections with how you think about something that you previously learned and this new math) around it. This requires hard work mostly in the form of rather mind numbing problems which at first glance can be boring and feel like you are being told to "wash on wash off" (to use a Karate Kid reference). However, the subconscious is a powerful tool and helps by developing weird connections. As I got more and more trained in math, I realized these connections became much much easier. I suspect this is why math researchers are able to look at paper and quickly grasp the intuition behind it.
The Princeton Companion to Mathematics is the best tool I know for this job. This is a bedtime reading, no problem solving needed. A bit like reading, say, Python Modules of the Week. You'd need to actually use that module in your code to "get" it, but just knowing that some module exists is very valuable. Same goes for math.
That said, it's absurd to make a dichotomy between the two. Both are essential. Maybe not at first glance at a simple CRUD app, but both get surprisingly handy once you want a bit more out of it.
On the other hand calculus is also used in machine learning...
Plus optimisation via calculus is sometimes a better choice then other cs specific methods.
Is Math of a college level necessary for everyone? Certainly not. Should algebra be necessary for everyone - I think so (I think the same way about calculus).
Nothing is necessary to perform your day-to-day duties - not even arithmetic - people have been getting by and been leading fruitful lives for thousands of years without that sort of stuff. So when people talk about learning Math as a life-skill they're dead wrong.
But if you want to anything more that just get by - if you want to ask questions and support your arguments, analyze problems and come up with answers you can trust you gotta, oughta, shoulda learn Math.
The political science professor in question might consider the fact that while many college students struggle with basic math, they struggle just as much with basic literacy (most undergraduates can't write grammatical sentences, use punctuation, don't know how to write an essay, and have a hazy grasp of the idea of citation). Extending his argument we might consider removing History and English from the curriculum.
It's easy to make up real-world examples of when each could be an advantage, but it's hard to say anything is necessary. It is what you make it.
Our economies are huge, number driven in almost every aspect. To ignore that or turn out graduates are less well rounded would not be more productive as a whole.
Ironic, since in our modern computerized world algebra is important and long division is useless. The writer himself probably carries a powerful computer in his pocket.
when i was a 10 yr old kid in 1982 i wanted to create email and newsgroup type systems and my computer science teacher was like that's a big waste of time like video games, get thinking about data structures and algorithms instead
10 years later I'm in college and the guy who runs the computer center got a source code license for SunOS and fubar-ed our whole system because he wanted to block our access to the internet, which would stop our academic pursuits.
hell we were just trying to be matt zuckerberg a decade before his time.
or there's this guy who was the one bright kid who remembered what he learned in organic chemistry class. all the other muggles sold their textbooks and forgot about it, but he applied what he learned to invent a way to make a new drug with stuff you can buy at the grocery store.
this grad school dropout got his work cited in the "journal of emergency medicine" because some kid made a batch of this vile brew and chucked all over his bed and the doc in the ER is like "what the f'uh?", "i mean you read this on the f'en internet and really did it?"
And all this time this guy and his crew are worried they're going to get called to testify about this in Congress and they were just so happy when the statue of limitations came up.
If students studied science and engineering and got both the ability and the will to confront technological society, we'd have a revolution.