This frustration is largely due to the fact that such a useful lesson for life is generally neglected, skimmed-over or outright deliberately denied in order to satisfy the ideologues of 'modern education'.
Thus many people are way too much surprised that being right is what matters, not the amount of effort it took to get there.
For me, algebra is easy, but writing out steps is very difficult, and sometimes I prefer to use a process other than the one taught in class. Should the goal of math education be to solve problems or to learn processes?
In the real world, it’s solving problems that matters—I wouldn’t say “being right”, necessarily—but it’s also important to be able to weigh different potential solutions to find the best or most efficient. Programming is a decent example of this—don’t go for the O(n²) algorithm when the O(n log n) is just as easy to implement.
But even better, consider a startup. If you want to build an online store application in 1995, and doing it in Lisp means you expend less effort than your Lispless competitors to get to the same “right answer”, then you should take the advantage. The amount of effort does matter.
That is a false dichotomy. The goal of math education is to "learn processes" to "solve problems". You can't skip the processes and jump straight to solving. Before you can pick a proper algorithm, you have to already know a few -- that's the learning processes part in programming. This holds for any skill you choose to learn; before you can do something well, you have to be able to do it in the first place.
The 'procedural' approach to maths is a good thing, though the right initial description is not to be underestimated. It boils down to good teaching.
There are many similarities between maths and computing and often the keen game-players are quite good at maths too, so perhaps the problem is how to motivate those who are not in this category?
I agree that reducing the effort does matter. Especially in maths, where it often leads to a better method. I was trying to argue against giving credit for increasing the effort leading to the wrong answer.
Perhaps reverse the application of the initial observation, that many games use maths, and start with a game without any figures, then, show the numbers behind it, in phases. This would create a link between something kids already grasp, and something new (numbers), which is a great way of learning. It would give the experience of a deeper world being revealed. And if using the numbers also made it easier to beat the game, it would give them a motivation; an example of useful application; and a sense of mastery.
For the game designer, that initial version with no maths would also make it clear whether the game was any fun in its own right. (It seems educational aims often obscure this.)
- Specify 18 (or a multiple thereof with minor number changes) equations of varying difficulty.
- Allow free choice of standard computer calculated operations, and a give-up button once you reach, say, 5 strokes.
- Keep a highscore.
Could be extended by viewing Gauss Jordan class of problems as a golf course and adding other courses.
The emphasis on a low stroke count as well as removing the obstacle of arithmetic makes the experience more like a game, just like Tao is suggesting. This assumes a low stroke count is a good metric, as well as something necessarily desirable - something that might not be true for all classes of elementary mechanical problems.
Still a cool idea though!
Unfortunately, this doesn't work in many classrooms because the teacher has to manage too many different students learning at different rates and can't tailor the lessons appropriately to all of them at once.
The part of gamification that works is dripping out little bursts of serotonin based on intermittent reward schedules.
Farmville has proved you could addict people to successful algebra manipulation, but it won't be in an interface like this, in fact, there isn't a need for the cute dragons and fairies and stuff at all -- just levels, achievements, cool rare rewards, sound and a little bling all around doing the algebra problem correctly.
One thing I think the scratch demo does get right is the per-step nature of solving these problems. I would imagine that you'd have a directed graph of possible intermediate steps for solving and reward along the way for a harder problem if you were serious about teaching algebra this way.
(I say "under the impression" because the first comment on his post implies that games which break things down this way already exist. I don't actually know the truth of it.)