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Vi Hart: Every school should replace calculus with recreational math (newscientist.com)
112 points by RiderOfGiraffes 1496 days ago | 102 comments



I'm a Math and CS student and still wonder how was it even possible that I did not start to hate math in high school. Things done there are dull and boring, exercises are repetitive and do not teach anything substantial other than applying another algorithm shown by a teacher. I found out that almost everyone did not really understand not only the problems that were told to solve, but also the solutions they gave -- and it was hard to blame them, for it was mainly teacher's and education system's fault not to give them thorough understanding of what is math really about.

My idea? Dump all that algorithmic crap and teach real problem solving. Do not punish for bad performance, because it is really hard to actually come up with a solution to a problem, and this skill is not something that can be learned by doing the same thing over and over again. Problem solving also require less concepts to be introduced and internalized, so more time can be spent on actual thinking and discussing, and less on preparations.

This kind of math course may seem not to prepare to good to "university level" math, but in fact it is quite opposite -- universities have to start almost from scratch anyway, and it is easier to teach new concepts a person who has a good understanding what math really is, than a person who only know tens of algorithms, used in high school to grind all kind of "problems" given there.

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Lockhart's Lament: www.maa.org/devlin/LockhartsLament.pdf

A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory....

Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music ... are considered very advanced topics and are generally put off until college, and more often graduate school.

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The thing is, students already think of math as difficult.

If we switched from algorithms to actual creative problem solving, it would become a lot more intimidating. I'm pretty confident of this. I'm in high school, and the other students have noted how much more they would 'hate' math if it were like the creative problem solving math competition that we have here in the US each year. http://en.wikipedia.org/wiki/American_Mathematics_Competitio...

What many -- perhaps even most -- kids really want is a course where they do only simple mechanical work, and get a boost to their GPA. At least, that's what it seems to me to be.

Also, we have to consider that unfortunately, not all teachers care. Now that teaching to the test is rampant(at least in the US), I think many classes would fall apart if the added structure and 'accountability' disappeared, like it would if we made that switch.

I don't disagree with you. The world would be a better place if we could get rid of this; if we didn't have all of this baggage to deal with.

EDIT: RiderOfGiraffes seems to have said the same thing in a different way in this thread; check it out: http://news.ycombinator.com/item?id=2390960.

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But that's math competition stuff, which is seriously more daunting, and practising mathematicians will often claim is even less like "real, proper math." It's occasionally suggested that some real, long-standing unsolved problems should be set as IMO questions, because there's a non-zero chance they'd get solved.

  > What many -- perhaps even most -- kids really want is a
  > course where they do only simple mechanical work, and
  > get a boost to their GPA.
"Most people would sooner die than think. And most of them do." -- G.B.Shaw.

  > ... not all teachers care. 
And not all teachers are capable.

And largely all of your points are exactly right, except for your misconception about what kind of math we want kids to be doing.

Going back to that point, the IMO is like playing at Carnegie Hall. What we want is like messing about with instruments to see what they do, and getting hooked on the curious things that are possible.

I'm not offering solutions here, I'm just helping define the problem, and explore directions.

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Going back to that point, the IMO is like playing at Carnegie Hall. What we want is like messing about with instruments to see what they do, and getting hooked on the curious things that are possible.

I'm torn on this.

I personally got hooked on math while trying to figure out the probability of getting various scores after rolling 5 dice and taking the sum of the top 3. (I wound up suggesting this as a Project Euler problem - http://projecteuler.net/index.php?section=problems&id=24... was the result.) So I know full well the value of messing around.

However I wouldn't trust the educational establishment with a task like this. Right now we're caught in a tension between opposing groups. One is apt to recite drill and kill and learning to learn and then proceed to avoiding teaching things that they think are too hard. (Which is apparently everything.) The other is fond of standardized testing at every opportunity as a way to force the first group to actually teach something. (Usually with pretty ridiculous tests.) At the moment in the USA, the testers have the upper hand. But if history is an indication, this won't last forever.

I would trust neither group with what you are trying to do. The first group would be quick to agree with you, grab a slogan, and then head off to do the wrong thing. The latter group would not see the point, and would either start talking about the 3 Rs, or would add to their tests some random, out of context, facts.

I'm not just saying this out of pessimism. We've seen this particular movie before in the New Math movement. Early experiments, with actual mathematicians involved, went well. The mathematicians presented interesting material, and kids enjoyed it. But when the educational establishment tried to imitate that success and get not particularly mathematically inclined teachers to repeat that, it was a disaster. (I was after the main movement, but there was still some of it going on. And I experienced first hand how bad it was when a teacher who didn't understand the material taught his misunderstandings rather than the material.) In the end outraged parents forced teachers back to the 3 Rs, and New Math became nothing more than a bad memory.

I would highly recommend studying that particular episode with the goal of figuring out what went wrong, where. Because what you would like to do has the potential to do the same thing.

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You apparently think that

> One is apt to recite drill and kill and learning to learn and then proceed to avoiding teaching things that they think are too hard. (Which is apparently everything.)

is somehow comparable to

> The other is fond of standardized testing at every opportunity as a way to force the first group to actually teach something. (Usually with pretty ridiculous tests.)

I don't. The former is, crudely put, evil, while the latter is, at the same granularity, stupid.

These stupid are not a huge problem. They're skeptical of being conned because that's what everyone tries to do to them, but they're open to what works so long as it actually works.

> In the end outraged parents forced teachers back to the 3 Rs, and New Math became nothing more than a bad memory.

And they were absolutely correct to do so because New Math, as delivered, was a sham.

Intentions matter far less than results.

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I was attempting to describe, not compare.

And, of course, I have described both sides as caricatures of complex truths. Both groups have plenty of intelligent, well-meaning, members.

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> Both groups have plenty of intelligent, well-meaning, members.

You do know that "he meant well" is an insult, right?

Almost all evil that happens claims "good intentions".

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I'm very familiar with the "New Math" debacle/fiasco, and keep it in mind whenever I think of new ways to teach "proper" math. But equally, I give over 100 talks a year on math to kids, and I can see some of them light-up with enthusiasm at the idea that math isn't sums, equations, arithmetic, formula, and mindless manipulation.

But the delivery mechanism is part of the challenge - most high school teachers aren't equipped to deliver the sort of thing we "serious math graduates" would love to see delivered, and certainly most primary school teachers aren't. Getting the delivery right has to be part of the deal.

"Hurray for New Math" - "The idea's the important thing" ...

Yes, Tom Lehrer has something to teach us as well about the dangers of radical ideas.

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Maybe I should take up egg-sucking. I'm sure that there is a grandmother somewhere who hasn't learned how to do it.

You understand my concerns, and are better informed on this topic than I am. I'll watch what you do with interest, and wish you good luck.

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You're pretty much right on the math competitions front, but the first few problems on the AMC are pretty simple IMO(pardon the pun). They are barely above the level of mechanical problem solving. It does get harder, though, and I agree that that is more daunting.

You've reminded me, however --

I remembered thinking that the Phillips Exeter Academy model for math education (the "Harkness" method) is amazing. http://en.wikipedia.org/wiki/Phillips_Exeter_Academy#Harknes...

  > Exeter does not teach math with traditional textbooks. Instead,
  > math teachers assign problems from workbooks that have been
  > written collectively by the Academy's math department. From these
  > custom workbooks, students are assigned word problems as
  > homework. In class, students then present their solutions at the
  > blackboard. This means that in math class at Exeter, students are not
  > given theorems, model problems, or principles beforehand. Instead,
  > theorems and principles emerge more organically, as students work
  > through the word problems."*
This is what we want. Discovering math makes it fun for everyone. Honesty and actual learning with that method(my whole would be not be too difficult, because everyone would inevitably have their own personal spin on the origins of the theorems et cetera. Plus, it'd be embarrassing to go to the blackboard and say that you haven't done the work.

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By the way, expect an email regarding what you said about the IMO.

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    What we want is like messing about with instruments to see what they do,
    and getting hooked on the curious things that are possible.
In high school, I wanted girls to have sex with me and adults to leave me alone (so I could get better at football and maybe girls would have sex with me). I got good at math to effect the latter, and it so happens that I got quite good. Until calculus senior year, I don't ever remember having any notion of "the curious things that are possible".

Now I work in scientific computing.

Anyways, I don't know if student engagement is as important as you seem to think.

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"Most people would sooner die than think. And most of them do." -- G.B.Shaw.

That's actually from Bertrand Russell.

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I'm probably one of the few, but I didn't have any problem with the paradigm of high school math education, just the pace.

As far as my later career, it's been my experience that rote manipulation without conceptual understanding can solve your problem, and that the converse isn't true. Sometimes, if you do the math right, it will contradict your intuition.

I'm hesitant to endorse any paradigm that aims to make math more intuitive, since at the end of the day that's a lie. One of the take-home messages of mathematics education should be that these mechanical processes are better than your own "problem-solving skills".

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I guess I've rarely had situations where I had to manually execute most of that machinery. I agree the rote machinery is quite useful to apply, but the question for me usually is what to apply and how. Applying it is a job for computers.

For example, it's quite important for me to know when I might need to compute an integral, and what I'd do with it. But when it comes to banging out the symbolic manipulations, remembering tables of what integrates to what, recalling integration strategies for common kinds of integrals, etc., Mathematica is more skilled than I am, so I defer to its expertise, and haven't tried to keep any of that stuff in my head since high school.

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One case I have trouble doing in Mathematica, etc. are variable transformations to increase numerical stability. A lot of times if you're clever you can do a variable change on an improper integral with (integrable) singularities to get an integral of a continuous, bounded, function over (0, 1). Working out this change of variables is something I find easier to do by hand.

I also generally have trouble getting Mathematica to do all of the work I need to do to come up with error bounds on numerical schemes.

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As far as my later career, it's been my experience that rote manipulation without conceptual understanding can solve your problem, and that the converse isn't true.

Appealing to symmetry or applying Green's theorem (e.g.) will save time over "mechanical processes". Also avert mistakes and be more comprehensible.

{if you don't know what I mean: what's ∫sin x · cos^55 x · abs|x|^131 · x^444 · exp(-x^2) ?}

Moreover only conceptual understanding allows the development or use of certain things, like SVD, wavelets, eigendecomposition, game theory, theory of distributions, topology, single-crossing curves...

I'm hesitant to endorse any paradigm that aims to make math more intuitive, since at the end of the day that's a lie.

Is it a lie? It sounds like maybe you just didn't experience the intuition personally.

math ... will contradict your intuition

To me the point is to re-ground intuitions in true mathematical facts. For example I intuitively think of the reals as transcendentals ∪ algebraics or as the completion of the rationals, since I learned counter-intuitive things about the reals that disfavour the "possibly nonterminating decimal" view.

paradigm ... pace

Great distinction. Grand changes are often proposed when small changes might do.

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In some sense, yes. Mathematics is the science of how to solve problems without thinking. Progress in mathematics means elimination of thought needed.

But actually inventing that time saving machinery is different from applying it.

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Mathematics is most definitley not that. The fact that your definition probably sounds reasonable to most people who made it all the way through calculus (and even excelled in those courses) probably supports what Vi is saying.

On the other hand, maybe it's okay that people don't really understand what math is (as mathematicians understand it). I don't see any evidence that society is falling apart due to lack of appreciation for pure math.

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I think "mathematics is a mechanical process for solving problems without an intuitive grasp of their nature" is a very good definition, with which many professional mathematicians would agree. Sometimes you're surprised by what pops out when you turn the crank. (Witness: every non-trivial probability problem ever)

Einstein's special theory of relativity was mainly an intuitive interpretation of Lorenz's discovery of Maxwell's equations' invariance under Lorenz transformation. The insight "c is constant in every frame of reference", would not have been possible without Lorenz mechanically working out what sort of transformation would leave Maxwell's equations invariant.

Dirac predicted the existence of the positron solely based on a mechanical process of finding out what sort of equation satisfied the symmetries observed in nature.

My point being that many great intellectual advances have been made by people who trusted the process more than themselves, and that's one of the cornerstones of mathematical thinking: trust the process more than yourself.

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  > I think "mathematics is a mechanical process for solving
  > problems without an intuitive grasp of their nature"
  > is a very good definition, with which many professional
  > mathematicians would agree. 
I am astonished.

No professional mathematician of my acquaintance (and there are many, including three winners of the Fields medal) would agree with that. Every professional mathematician I know would say that mathematics is a creative subject requiring insight, intuition, rigorous logic and occasional luck.

Blindly turning handles just doesn't get results - the search space is way too big to chance across stuff regularly unless guided by some feel for what's going on. Listen to Wiles talk about his proof of FLT, or Gowers talk about the process of doing math.

I'm amazed that you make the claim you do, and am intrigued to know what there is in your background that has led you to that conclusion.

For reference, I'm a PhD in Pure Math, have an Erdos number of 2 (of the second type of 3), and regularly meet with groups of professional mathematicians. I don't tell you this to create a "Proof by Authority" argument, but to give you some background as to my personal experience.

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Maybe it's a pure/applied split?

When I'm investigating a physical system, doing the mathematics often tells me something qualitatively different from what I was expecting, and I find that my expectations were wrong more often than I screwed up calculations. I don't mean to trivialize what goes into doing the calculations - what I mean is that I'm constantly solving math problems that force me to revise a flawed understanding of a system.

That's what I mean by "mechanical" - I have to remain disciplined and resist the temptation to reason by analogy to something I may not even understand completely.

If you care, I came to applied mathematics via nuclear engineering.

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You're not doing math, you're using mathematical tools. Calculations are effectively arithmetic, and that's not doing math any more than typing code is doing programming.

It's a lot more complicated than that, but setting up the equations is the doing of math - solving the equations is just manipulation, and that's using, not doing. The difference is important - conflating the two leads to many misunderstandings.

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Perhaps we are in violent agreement.

I don't want to say that doing math is not creative. Far from it. But mathematicians strive to make themselves unemployed. They prove theorems once and for all, so you don't have to for each right triangle why it has this curious properties about the sum of squares.

Eliminating the need for creativity takes a lot of creativity.

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Perhaps I did not express myself clearly. My definition is explicitly aimed at doing mathematics at a high level, i.e. professional mathematics. (I dropped out of professional mathematics, but since I'm now working in Functional Programming I did not stray too far.)

What I want to say is, that the process of doing mathematics is solving and understanding problems. And since we are rarely interested in concrete solutions to concrete problems, we build up theories and algorithms to solve whole classes of problems. Building up those theories is a highly creative process, but ultimately, a problem can only be seen as `solved' if we find a mechanical process for eliminating the need for creativity.

Let me give you an example: There are lots of interesting questions you can ask about linear recurrence sequences (e.g. the Fibonacci sequence), like what happens if we add to sequences? Or when we interleave them? Or when we only pick every n-th element. Or when we want to find out the i-th, without having to calculate every element that comes before.

Solving those kinds of problems requires lots of thinking.

But if you apply even more thinking, you can come up with generating functions. They are a tool that will enable you to solve all those problems really easily. (And enable you to spare your creativity for much harder problems. That's progress!)

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You only need rote manipulation to help your conceptual understanding, and maybe learn some algorithms.

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The incentives should be around changes in performance, +ve or -ve, especially if it was as a result of the kid's own actions.

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From talking to my professors, there really was once a time when university math (calc 101 or w/e) was not starting from scratch (from the perspective of engineering anyways). Spend maybe a month over derivatives and limits, and then off to integrals and everything else you'll need so that mechanics actually makes some sense.

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I'm really not convinced by this. Sure, math education is far from ideal. But the solution cannot be to make it more "touchy-feely" and make everybody feel good.

Courant, in his book "What is mathematics?", says the field historically seems to go through alternating phases, from wild, creative, adventurous phases in which new things are discovered -- often inspired by "real-world" problems in physics or engineering -- to more sober, rigorous phases in which another generation takes over and basically cleans up the findings of their predecessors (making proper definitions, constructing and refining the proofs etc.).

But this is far from what math _education_ can be about.

The biggest obstacle to math education is the meme that mathematics is "only for intelligent people". Mathematics at the basic, cleaned up level has almost nothing to do with "intelligence" and a lot to do with memory. I know this first-hand because I used to be really bad at maths and then one day decided to learn it anyway (and did). Most often, when someone doesn't get some result or can't figure out the derivation, it's because he doesn't remember or has never learned the relevant facts. [EDIT: Also, if you think you need to be "intelligent" (however defined) to do math, you will give up sooner when you don't understand sth because you think "uh, I'm too dumb" instead of asking "what am I missing".]

Mathematics is a kind of language, and as in any language, most people will actually need to sit down and memorize vocabulary. It's mundane, it's not fancy-creative, it's a pain in your behind, but once you succeed in automating certain things, the process takes on a momentum of its own.

This "dragon mom" stuff I occasionally read about in the NYT is silly IMHO but it's just as silly for a 22 year-old, who by her own admission doesn't even know the standard curriculum of a plain vanilla mathematics education, to want to be "the ambassador of mathematics" (http://www.nytimes.com/2011/01/18/science/18prof.html). First you do the hard work, then you can fly.

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But the solution cannot be to make it more "touchy-feely" and make everybody feel good.

Why? That certainly isn't the type of response that would cut it in mathematics, since it is just an assertion without even a hand-wave to support it.

Also, I'm not sure you really understand her argument. Her point isn't that we should to make math classes easier so that people feel good about math; her point is that students might be more interested in math class if it contained some mathematics.

The primary aim of math class up through calculus is to teach kids to perform the calculations needed to do physics, which is needed to do engineering, which is generally perceived (correctly I think) to be greatly beneficial to society. This is fine, but there is a huge chasm between this and the types of things that actual mathematicians study. (My brother once commented that if people had any idea what was actually going on in the math department, there is no way in hell they'd fund it).

The best counter to Vi's point, I think, is "why would we want to make kids appreciate pure math?" It's really fun and really interesting, and personally I would love for more people to get to experience math, but I'm sure the same could be said of classical music, or poetry, or knitting or pretty much anything else that people find stimulating. There may be some benefit to it, but I don't think that benefit has been quantified.

btw... sitting down and memorizing vocabulary is an awful way to go about learning a language. Communicating in a langauge every day, reading a ton and trying to think in that language are much more effective.

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I like/appreciate your reply a lot (except the first line ;) and would comment more in response if I didn't have to run just now.

But let me at least thank you for this: "The primary aim of math class up through calculus is to teach kids to perform the calculations needed to do physics, which is needed to do engineering, which is generally perceived (correctly I think) to be greatly beneficial to society."

This is the best summary I have yet seen about why math is taught the way it is (in high-school, not in university) -- and by extension why many people, incl. myself, feel or felt uncomfortable with math up to high-school.

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Specially because it should be economics and budgets and financial stuff the focus of math in those formative years. And statistics too!

Not everyone will be an engineer, but almost everyone will have to deal with money in their life. And the world would be 1000 times better if only people could understand a little more statistics.

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Math education should at the very least include a description of what math is and why it's interesting. Current math curricula don't let students know that math can be engaging, and so they're not engaged. The ambassador thing is to tell "normal" people what math is about. That's fundamental, not "touchy-feely".

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> Math education should at the very least include a description of what math is and why it's interesting.

But let's be realistic about this. "Interesting" is an opinion, and the number of students who don't already find it interesting who would if we took the time to explain it would be fairly small.

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I disagree. If you look at the comments left on Vi Hart's YouTube videos, it's clear that there are many, many people who only realize that math can be interesting after watching her videos. http://www.youtube.com/user/Vihart

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It has to do with more than memory, right? At least if you want to come up with new things, whether directly in mathematics itself, or to use it creatively in some other field. The idea is that intelligence is partly indicated by how you can visualize things that cannot be seen, and often when someone cannot figure out a derivation of get at a result, it's more likely because they are not seeing the big picture.

I don't mean to say that people in general cannot be taught to see mathematics, it's just that they need good teachers and textbooks.

Edit: of course, all this comes only after having grasped the basic terminology at your fingertips.

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It doesn't need to be more touchy-feely. It just needs to be less boring.

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(constructive suggestion: Answer the question Why should I care?)

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When people start thinking about math education - as I have at considerable length - they always come up with the idea that since "real math" is about problem solving (which it is) and not about rote algorithms and manipulations (which it isn't) then what we should do is come up with problems to solve, not formulas to manipulate. We should encourage creative thinking, novel solutions, and ways of approaching problems.

This is, actually, all true. Students should indulge in problem solving, analysis, investigation, early proof, ideas of proof, argument, explanation, and all the things that math is really about, and which make math useful.

After all, who except for science and engineering students ever differentiate a polynomial? Knowning qualitatively and intuitively what rate-of-change really means is useful. Differentiating x.sin(x) isn't.

But here's the problem.

To within 20% or so life expectency in the UK is about 70, and there are about 70 million people, so roughly at every age point (up to about 50) there are about a million people.

How do you assess and grade the mathematical standard (whatever that means) of about a million children?

How do you persuade a government full of lawyers and classics graduates that what you're doing is better?

How do you assess the teaching, and the school, if not by the grades they and their students achieve.

It's all about assessment.

So that's the first stumbling block.

The second is the question of getting teachers who are up to the task of teaching in an open and interactive manner, rather than trying to teach rote algorithms and manipulations.

But you don't need to worry about that - schools will never teach proper mathematics. They'll continue to teach arithmetic and manipluation.

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If you don't get the fundamentals down, you can't handle the applications. Wax on, wax off - boring but effective. But you also need to spend time putting those fundamentals to work.

The real question is, why do you need assessment? To test that most kids are getting the fundamentals? Easy. Have a few core tests. To check that class time is also devoted to creative stuff, and not just drills? That's a management problem. You can't test whether or not kids have had (say) 50% of their time doing fun non-core work, but you can regulate that through other means.

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    why do you need assessment?
Are you serious? There are a small number jobs that

1. Are very desirable (lucrative, secure, etc.).

2. Require very high mathematical proficiency.

3. Have drastic consequences if performed poorly.

It's self-evident why assessment is an important function of the education system.

The question becomes: is it more important to correctly identify potential engineers and scientists and to maximize their proficiency at their careers, or more important that Joe the Plumber can balance his checkbook when he goes straight into the workforce after his high school diploma?

I doubt that we can optimize both of these cases simultaneously.

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  > If you don't get the fundamentals down,
  > you can't handle the applications.
Absolutely, which is why I advocatedoing the algebra, doing the manipulation, and doing "the sums" until the act of doing them is almost reflex.

  > The real question is, why do you need assessment?
I'm assuming you're serious. Society demands assessment. Government demands assessment. Employers demand assessment (and then ignore it and apply their own tests). Students themselves are conditioned to demand assessment - there are many instances of teachers or lecturers playing with the idea of not giving grades, and mostly the students hated it. (whether it worked better is another question)

FWIW - I advocate a radical shift in what's done, and am working subversively on a ten year plan to make changes. Just becuase I can identify some problems, don't assume I believe they are real, or insurmountable.

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> Society demands assessment. Government demands assessment. Employers demand assessment.

My college didn't hand out GPAs and all evaluations were narrative. Only on rare occasions has the lack of numbers attached to my performance prevented me from obtaining what I wanted, and even for those cases I remain dubious about the benefits.

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How did they apportion scholarships or determine who received a spot in programs with a limited number of openings? Institutions assess for many reasons, but primarily because educational opportunities are rationed and they need some method for separating the worthy from the unworthy without relying entirely on personal testimonials or politics.

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When you say "real math" do you mean pure maths or applied maths?

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False dochotomy. Pure often leads to applied, applied often leads to pure. What is often regarded today as pure maths has its roots - perhaps long ago - in questions from real life.

Puzzles arise from the real world, and they arise from abstract structures. Regardless, they are puzzles to solve, and the techniques of pure math and applied math can be used.

Thing is - not all students like puzzles, not all students like curiosities, some students actively want stabilty and assurance.

Some people aren't cut out to be entrepreneurs.

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From what I recall of working in academia while the "pure maths" vs "applied maths" is certainly a false dichotomy, there were certainly mathematicians who made it very clear as to which side of that dividing line they were located!

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Vi Harts youtube is pretty entertaining, Here's a video that shows pretty well what her videos are like: http://www.youtube.com/watch?v=Yhlv5Aeuo_k

The whole topic vaguely reminds me of Khan Academy, both seem to recognize that the way math is taught in schools at the moment is fundamentally broken, and doesn't engage students the way it should.

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Freeman Dyson wrote this of Richard Feynman: "The reason Dick's physics was so hard for ordinary people to grasp was that he did not use equations. The usual way theoretical physics was done since the time of Newton was to begin by writing down some equations and then to work hard calculating solutions of the equations ... Dick just wrote down the solutions out of his head without ever writing down the equations. He had a physical picture of the way things happen, and the picture gave him the solutions directly with a minimum of calculation."

edit: That's from Dyson's autobiography 'Disturbing the Universe'.

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Which is interesting and awesome but so not relevant for general public math education... ;-)

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"The usual way theoretical physics was done since the time of Newton was to begin by writing down some equations and then to work hard calculating solutions of the equations"

I cannot believe that Freeman Dyson would claim such false notion that Newton worked with "equations." I checked the quote and http://books.google.com/books?id=RHzoMeU2bxsC&printsec=f... it is true, the quote is in the book.

I would think that Dyson would know that Newton did not write one single equation ever. He worked with proportions.

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Every school should replace calculus with spreadsheets and a basic programming course. Or maybe with statistics. Each is more practical than calculus.

(I don't mean to put down calculus; it's spiritually rewarding and, when paired with a few other subjects, yields crazy awesome higher math.)

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While I would agree that a basic programming course might be helpful as a required course for all schools, I don't understand why it should replace Calculus? There is room for both, and they have different goals and advantages/disadvantages. You say yourself that calculus is awesome for several reasons - why do you think we should replace it?

If it is only because a different course might be more practical, you have entered a slippery slope. College is not the same as trade school - we don't learn things because they are practical, but because there is an advantage simply to learning. I believe that all basic college courses should aim to promote critical thinking above all else, as the practicability of any given subject will vary widely between students.

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why replace

I'm just trying to be charitable to people who don't want music, drama, driver's ed, health class, ........ cut out.

Why should recreational mathematics replace calculus, either? Ideally math would be taught well and written so that people could engage with it throughout life. (But how many adults do you see reading those yellow spiney Kluwer books at coffee shops?)

slippery slope

Thought Vi Hart was talking about high school. Maybe I was wrong. I agree that calculus is the right kind of brain-stretching for college.

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I agree completely. I don't think that recreational mathematics should replace calculus. I think that there are some definite changes that should be made to mathematics education, but that the majority of these should take place at a much younger age, even grade school. Early education is the perfect place for recreational mathematics. Fostering an interest at a young age (rather than force memorization of formulas, algorithms, and tables) can help a lot to keep people engaged throughout their lives.

I think that by the time students reach the college level (which I think starts with Calculus, even if they reach this level in high school), they should be able to deal with some of the more boring parts of mathematics, sustained only by nuggets of usefulness and beauty. After all, not all mathematics is wondrous joy.

I also agree that many of these problems could be fixed simply by teaching better. In calculus, for example, the beautiful thing is how closely it mirrors nature. Calculus states that if we know the position of an object at all times, we also know the velocity of the object at all times. This is a beautiful and fundamental result, one that I feel is not stressed enough during a first-year calculus course. I think that it is amazing not just that we know (in theory) the velocity, but that we can actually calculate it explicitly (in most circumstances) or approximate it numerically (in most circumstances). The parts of calculus on how to do that are somewhat less beautiful and paradigm-shifting, but are necessary, and adult students ought to be able to deal with that.

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What basic programming language would you suggest? I started with Visual Basic, and I found that it really didn't prepare me for programming in general. It seems like the best combination of useful and easy would be either Python or bash, but I'm sure other people can come up with better suggestions.

I think the key is to find something that's teachable to people completely new at programming, but something that will be useful to them. I was thinking of Python due to it just feeling "natural" to me, at least after coming from VB and then Java. Bash is somewhat of a weird choice (especially considering most students aren't going to be using Linux), but I was thinking about that due to it being useful for immediate computer tasks, but it may not be enough of a "real programming language". I wish Windows had a good equivalent scripting language...

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I'd start with Python. Try to figure out how to do one thing that you think would be cool w/ python. Like learning to play the guitar by learning a song you like.

Really it depends on what you want to do...but Python is widely used in science, finance, math, and the web (thru django). So that covers all of what I'd want. And it's a high-level (read: easy), widely used (read: supported) language.

If I were just starting out I'd use zed shaw's book. As it was I found the django tutorial good; and Norman Matloff's tutorial on scientific python.

I'd learn programming in linux. Install Ubuntu if you don't already use linux. You can dual boot and leave the windows side for video games.

HTH

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Scheme http://teach-scheme.org/

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Haskell feels closer to college math than does high school math, actually. Maybe it would help in both math and programming.

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I don't know why people would down vote you. Perhaps they haven't read Seymore Papert "Mindstorms"

Here is the thing. The computer can simulate more or less anything so when dealing with abstract concepts such as math, what better way is there than to combine math with experimentation.

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Monte Carlo, por ejemplo. FEM, por ejemplo.

And once you've struggled to solve the problems that can't be brute forced away, you start wondering if maybe a little linear algebra wouldn't do you good...

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The only reason Calculus seems so dry is that the ones teaching it are not very competent and don't understand the subject quite well.

During my school I'd ask teachers all about limits and why we need it and their attitude was - Don't get too involved - here are the formulas, go learn them and solve the exercises.

I later discovered that there were some amazing applications of limits (calculus) but wasn't taught the way they were meant to. This is the reason why I never found Calculus to be interesting.

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If taught properly, and with the proper illustrative exercises, calculus is recreational. Period.

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Here's the thing. Most of calculus's power comes from its use as a mechanical calculational system. Hence the name. Archimedes's proof of the surface area and volume of a sphere is far more insightful and explanatory than the standard calculational proof using integration that any unthinking freshman can crank through with ease. As a rule, calculation proceeds by reducing a big problem into a sequence of small, easily dispatched problems, but human understanding usually works the opposite way. So, it's not that calculus cannot be taught in a way that highlights human understanding, but the greatest strength of calculus is that it produces answers whether you understand what you are doing or not.

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Great point ! The "recreational" part can be applied to any subject imho.

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I know people here (the top 0.1% of the high school class) love to bash math education in particular. I'd like to ask how you would apply your views on teaching a topic such as History for instance.

Sure, rather than learning the dates of middle-age battles, it would be more fun to interpret a play, in costume, or rebuild a battlefield with legos.

But can you spend your entire school year this way, or could this "recreational" teaching need to be combined with more traditional teaching?

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My opinion on history is that it does not matter which set of dates you learn. But it matters that you have learned some set of dates, so that you have context for facts you encounter. Having that context gives new facts more meaning and ability to shape your world view.

Let me give a random fact to illustrate this principle. Mexican silver mining started in 1531, and by the early 1600s Mexico was the source of about 1/5 of the world's silver production. (This happens to be true. And Peru was an even richer source of silver for Spain.)

Suppose you learned a fair amount of American history, as I hope my children will growing up in the USA. This fact gives context for how late the founding of the early English colonies really was.

Suppose you learned something about English history, as I did growing up in Canada. This fact provides context for the English history of privateering, which was one of the causes of the Spanish Armada.

Suppose you learned about central European history. This fact gives context for Spain's wealth, which played an important role in their involvement in the Spanish Netherlands and the Thirty Years' War.

But if you learned no history, this fact has no connection with anything, and has no possible fate other than being forgotten.

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Use primary sources.

Have individuals or teams look at an original primary source, interpret it, then discuss with the class, who all interpret something from a related source. Students put their heads together, discuss/debate and post some interesting original research to the Net.

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In mathematics?

Primary sources are way above the level of most college mathematics students, much less high school or younger students. Almost anything written in the last 50 years requires very advanced mathematical ideas, while anything written before that uses outdated notation which is very difficult to understand. Moreover, the primary sources for common ideas are often spread out over huge periods of refinements - going from a small idea to a general notion over the course of years, perhaps involving dozens of papers by multiple authors. One notable exception to this: Euler's papers. These might be readable by some students, yet when I see them presented in college courses, they are often presented by a secondary source for clarification.

The other thing I don't agree with is your use of the word "interpret." I don't think that it is common to find a mathematician who believes that mathematics is open to interpretation.

Lastly, original research in mathematics is hard. It's not the interpretation of previous ideas (although it is occasionally relating concepts previously thought to be unrelated - some consider this the most important type of mathematical result, but it is much harder to do than you might think), but the formulation of new ideas. Even to know if an idea is new requires a great deal of mathematical training.

Perhaps I'm wrong about this, but I think that for young mathematics students your ideas are unfeasible. Remember that the original sources for a lot of basic mathematics are hundreds of years old. Even Euler's famous writings on geometry (the exception that I said might be workable) were largely a rewriting of previous ideas into a coherent whole - basically a textbook - and could not really be considered an original source.

Edit: my apologies, I misunderstood. I did not realize that we were talking about teaching history, in which case I defer to someone of greater experience.

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shou,

alain94040 asked what we would change about the way History is taught.

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I wholeheartedly agree with the idea of primary sources, and of studying mathematics historically in general.

The Elements of Euclid is a model of clear, concise, beautiful mathematics which is easily accessible. Archimedes as well. Follow that up with Apollonius, Ptolemy, some Descartes, and then Newton, and you have a junior high and high school curriculum in mathematics that would give students a real advantage over the rote "here is what Disembodied Authority says you should know" learning.

Note -- I am not advocating the study of mathematical history in junior and high school, but rather the study of mathematics historically. It gives access to mathematics as a branch of the humanities, less focused on the answers, and much more focus on the questions and how some really smart people have addressed those questions in the past, which gives a good guide on how to address new problems that will come up in the future.

When it comes to history specifically, as the OP asked, I would still agree with going with some of the great works of history. Historiography can be left for college, but give the kids access to the letters and diaries and personal accounts of people who were at the scene of history, as well as the great works of history that have been written (Thucydides, Herodotus, all the way to Toynbee, Gibbon, and, hell, even Spengler). Drop the mundane and milquetoast textbooks.

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I always missed the historical context in high school, especially with chemistry and to a lesser degree physics.

You always knew that the current model you were being taught was "kinda wrong" or not the current state of knowledge (Rutherford's atomic model, Bohr's atomic model, and so on), but you never heard anything about the motivations of the people developing it.

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At St. John's College in Annapolis, science is taught by reading Darwin, Maxwell, Newton, and so on. It's a mixed bag.

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My high school physics teacher went to St. John's. We discussed one short part of an original source in physics every Friday.

Notable downside: he didn't know whether purple or red has longer wavelength.

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Primary sources can be rather dry/difficult reading. I'm fairly intelligent, but it's just not easy for me to grok older style language.

I have had some experience with reading primary sources though. I had an excellent history teacher in high school who had us read The Autobiography of Benjamin Franklin to prepare us for American History. Now, Benjamin Franklin is a fascinating person, and he did plenty of extremely interesting things, but I had a lot of trouble getting through the style it was written in. I love learning about the things he did, but the language and style of his autobiography just did not work well for me.

Primary sources are an option, but it doesn't work for everyone.

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Strongly agree with that one. I've always hated history going through school. Then I took an american literature class in college where we read primary sources from the early colonies up to the founding of the US. I enjoyed every minute of it. I learned more about history in that one class than probably the last 14 years of education combined. History should all be taught this way.

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primary sources is known to be one of the worst way to study any field, especially when you are not familiar with it, because you lack the context, and because they are rarely written for the profane. This is all the more true for maths: have you ever read a math books from the XIXth century ? This is almost impenetrable, if only because of the formulation

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cdavid,

I am talking about History. Pick up some letters from a soldier. The teacher can still give some context.

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I would have hated that.

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I think there's huge potential for computer games to influence how history is taught.

Imagine an RPG set in a slice of medieval Europe, where you experience daily life (in a variety of roles, from peasant to king) and hear about events occurring around you and confront choices, with every detail as accurate as we know it. Imagine that this game is actually fun to play, and that it's not teaching information via walls of text or memorization games, but making real information an integral part of the game world.

That's the future, IMO. Or part of it, anyway. Talk about the topics after the students have experienced it first-hand.

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It doesn't use the medium's full potential, but Oregon Trail is actually moderately successful imo at imparting basic knowledge about: 1) the existence of this episode in history; 2) some of the geographical information and landmarks along the way; and 3) a few basic pieces of information about period travel.

There's a lot more that could be done, and it's not clear it imparts only accurate information, but for me at least, it's where I learned that "fording a river" was a thing that existed; that Chimney Rock is a landmark; that there was a period in history when wagon trains were regularly heading out west along this lengthy route; etc.

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Have you seen the Victorian Farm, Edwardian Farm, Victorian Pharmacy or WW2 British House shows?

That's how history should be taught.

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From what I can see, history teaching seems to doing pretty well in the UK at the moment (I have a 12 year old son) - they do a lot of visits to pretty interesting places and do a lot of re-enacting of things (e.g. what it was like in a Victorian classroom).

Of course, this approach does rather depend on having a lot of reasonably interesting historical resources within easy reach.

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The head of the math department at my college actually taught one of these classes. He would assign two students each class period to take notes. He said that way, the rest of us could pay attention :) He would stand at the front and ramble about whatever came to mind, and scribble on an overhead transparency. He would start with a topic and take it wherever we asked: multiple dimensions, knot-tying, origami, computational geometry, ray-tracing. The course was called "3D Math" or something, so everything was relevant :) There were no tests, and we got to choose our own final projects. The notes and overheads were duplicated as study guides for the final, and the questions were taken right out of them. It was the most informative math class, if only because we were all doing our own thing and learning from each other.

I did a 3D extension of the Lyapunov fractal images that Mario Markus published in Scientific American in 1992. Markus's 2D: http://charles.vassallo.pagesperso-orange.fr/en/lyap_art/lya... and my 3D http://www.youtube.com/watch?v=-omN5ZM3Jho with a moving cross-section http://www.youtube.com/watch?v=ePRJfF3pwqg

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I had no interest in math until I had Calculus in High School. I then went on to pursue dual degrees in Computer Science (because I <3 Programming) and Pure Mathematics (because Calculus made me <3 Math).

Maybe it was just an exceptionally good teacher, but the more I learned about Calculus the more I feel it was the subject itself. There was something seductive in the idea that by changing the way you think about something, like the area under a curve, you could calculate the previously impossible. The core of Calculus (at least the part you would learn in High School), derivatives and integrals, are easily enough explained.

I'm glad I had Calculus or I wouldn't have gone on to pursue mathematics so doggedly, I wouldn't have gotten to learn the strange and beautiful world of numbers, I may have missed out on one of my great loves.

That said, I'm sure in the hands of a less than stellar teacher, Calculus is a painful thing.

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Incidentally there is a textbook on calculus that Donald Knuth said he read in his first year at university, and after doing all the problems in it he moved from beginner to top of the class (and thereafter, renowned computing scientist). I think the author was named Thompson, but I haven't seen the book myself. Maybe some people here have?

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Maybe it was "Calculus Made Easy", by Silvanus P. Thompson? That's an old book that explains the basic idea of integration and differentiation, but doesn't try to be rigorous at all. http://www.scribd.com/doc/8533492/Calculus-Made-Easy-by-Silv...

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Non scribe D version: http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf

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Well some of the videos look interesting but I don't agree with the title, what is recreational math?

It would be great if Maths could be made more interesting and students make their own unique discoveries etc. However as a former maths teacher I can tell you that structure exists for a reason.

In a class of maybe 30 students for one hour, the teacher can afford to give at best only 2 mins average time to each student. Some students may work very well independently, but others would take advantage of the freedom to mess about.

Even if the students were focused how is a teacher with so little time per student supposed to pick up a random piece of work and then decide if the child is doing something of worth.

The Maths curriculum needs to be rethought for the information age, but the solutions need to be practical.

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I think the Khan academy has a solution to this particular problem. Salman in his TED talk proposed that students view the videos in their own time for the basics, then engage in problem solving in class so that the teacher can devote much more attention to each student.

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Experience with this shows that the kids simply don't watch the videos. It sounds brilliant - let the kids watch videos together, they can eat pizza, drink cola, laugh, text, watch the video again to get the bits they missed, and then come to class to work the problems and get help.

The experiment has been done - the vast majority of the kids don't watch the videos.

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This is basically like how students don't read the textbook, only the textbook is a video.

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Is there a record of the experiment online somewhere?

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I'm pretty sure I read of it on a web site lunk to from here, but I don't have a record of it. I have seen it reported from at least two different people who tried it, but don't have a record of where I read it. Sorry.

EDIT:

I've found one reference to a teacher who's trying it but has some kids who don't watch the videos:

http://www.msteacher2.org/forum/topics/flipping-the-math-cla...

Here's another in the comment stream for the Khan TED talk that suggests the idea:

http://www.joannejacobs.com/2011/03/khan-use-video-to-flip-e...

There are more, but they require filtering/finding:

http://www.google.co.uk/search?q=math+homework+flip+video+%2...

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Thanks. That's somewhat disappointing, I guess.

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When I was in school majoring in math we always thought back to how poorly motivated the subject was growing up. So we decided to explain some pretty basic group theory (permutations, etc) to our friend's dad who taught elementary school.

When he brought in the topic with some basic shapes to illustrate the concepts to his kids (4th grade, I think), they all went crazy for it. We then gave him other concepts from a geometry class, which they loved as well (http://www.gang.umass.edu/~kusner/class/462hw).

I think catching them at an early age -- sparking their natural curiosity and giving them confidence in the subject -- is most important, before they're disheartened by all the years of computation and applying algorithms.

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I have to ask... does the name Vi come from where I think it does? If so... that is awesome, and then I would have to ask is there a improved version of her named Vim?

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If you aren't familiar with Vi Hart, let me suggest some of the more awesome videos that seriously tickled my geeky innards.

http://www.youtube.com/watch?v=DK5Z709J2eo (infinite series through doodling)

http://www.youtube.com/watch?v=heKK95DAKms (graph theory awesomeness, through doodling)

http://www.youtube.com/watch?v=e4MSN6IImpI (some lovely properties of binary trees ... through doodling)

http://www.youtube.com/watch?v=Gx5D09s5X6U (snakes)

Especially that last one, it poses a question as to whether you can use a grammar to show that a snake lives (doesn't bite itself). Unfortunately she doesn't provide a solution to the problem, but me and a few mates went ahead and proved that it is impossible to prove whether the problem is solvable without simulating the snake.

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Everyone here on HN always seems to say the same thing: the way math is being taught in most schools sucks, we need to change it. Math is glorious, it's the kicks, etc.

I'm curious why everyone is so enamored with mathematics here though; I've tried to understand what is so captivating about the subject but have failed to do so for years.

I've never gotten past trigonometry and algebra myself, and this idea of Calculus being more than the tedious integration work that textbooks I tried to self teach from said it was is intriguing.

Enlighten me, how is mathematics not just a bunch of rules and methods to solve problems within constraints, and that some professional practitioners are forced to prove...

How many of you programmers and software engineers here even interact with the Calculus you learned in high school/college on a day-to-day basis in your apps/software projects?

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"You don't actually need to know a lot of the groundwork and the basics to appreciate the exciting bits of mathematics. The more technical things are still awesome as tools and are necessary in some places. But luckily I am usually with other people who do know that stuff."

How about this not being an either/or? Anything that exposes more people to the beauty and power of the maths, and convinces some of them to more of the "technical things" is goodness, IMHO.

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It's so true that the way that maths is taught at schools needs a complete revamp. At secondary school level it should be taught as a creative subject from the start: choose some rules and see what happens.

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Just one question -- does the US have a shortage of engineers because math education isn't touchy-feely enough now?

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