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.
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.
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.
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.
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.
> 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.
----
By the way, expect an email regarding what you said about the IMO.
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.
> 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.
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.
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.
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".
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.
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.
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.
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.
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.
> 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.
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.
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.
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.
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!)
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.
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.