Gamifying Algebra(terrytao.wordpress.com)
terrytao.wordpress.com
Gamifying Algebra
http://terrytao.wordpress.com/2012/04/15/gamifying-algebra/
23 comments
So are you opposed to teachers offering partial credit for problems that are mechanically correct but have clerical or arithmetic errors?
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.
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.
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?
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.
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.
Since you bring up Programming; yes, it is a good example.
It is a good example of the need not to ignore any errors.
Double-guessing how some incorrect maths or code may have worked, given a lot of imagination and goodwill, is the start of the slippery slope. It is not something computers can do, as we all know.
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.
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.
I agree that difficulty is a good lesson; but it doesn't seem to be working for a lot of people in the US. Many students give up on math early. That might not be economically sustainable. I don't think math education is optimized to make the most of intuition and motivation, and I really don't think experimenting with improvements is harmful.
I concur. Hard life lessons should not be taught in the context of learning a new skill. Those can wait until after a degree of proficiency has been attained lest we undermine their efforts completely.
I believe this is at best an unfair comparison between school and a "real world" environment. Out in the working world it is assumed that you are going into the work with the necessary knowledge to complete a task, this is not the case in education. We specifically develop a sandbox environment for kids to develop skills at school. At the end of the day students do not have the fear of losing a job to motivate them. We must instead encourage kids to keep going, even after they make mistakes. I stand firmly against people who believe tests are unfair, because at the end of the day children show they can stand on their own before moving on, but in the time that they are learning new material the focus must be on building them up and encouraging perseverance.
I don't think the lesson is "being right is what matters" (working hard is praiseworthy, even if the results aren't always great), but this encourages students to work carefully and list out their steps instead of trying to do everyhing in their heads.
Playing the demo game, I think the idea has merit. But I'm concerned that even the presence of numbers is a turn off - that aspect certainly looks unappealing.
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.)
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.)
... now I'm also reminded of Alligator Eggs ( http://worrydream.com/AlligatorEggs/ ) - a symbol-free kinda-gamification of lambda calculus.
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Idea for a simple mobile game, Gauss Jordan Golf:
- 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.
- 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.
I agree; the whole thing could be done with pictures rather than numbers. Lay out an actual grid, with objects -- seven ladybugs in the third row, fourth column, etc. When you subtract one row from another, you physically drag the ladybugs from one row over the other row, and when they 'match' they disappear. You can tap into physical intuition.
I'm not sure that gamifying algebra will make it any more desirable to kids. I think that those who want to see it as a game already see it as a game in their heads. I know I certainly did back in high school.
Still a cool idea though!
Still a cool idea though!
I'm reminded of Manufactoria ( http://pleasingfungus.com/Manufactoria/ ), which does a rather nice job of 'gamifying' a concept that's effectively Turing Machines.
It seems that the main improvement here is not anything magical about "gamification" but the idea of presenting students with progressively harder algebra problems in which they get to try out new problem solving methods gradually.
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.
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.
I'd think that a huge benefit "gamification" is that it allows students to learn at their own pace and without much teacher guidance. Kids figure out how to play Portal and Civilization, for instance, without a teacher hovering over them to tailor their lessons.
This is the wrong sort of of 'gamification'.
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.
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.
You're kind of missing the point. Tao is under the impression that most "gamified" math education software is based on rewards for solving entire problems. He believes that rewarding smaller steps on the path to the solution of more complex algebraic problems would be more effective. He's not trying to suggest the specific rewards to use with his demo, only the reward schedule.
(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.)
(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.)
I had a similar idea after watching my kid staring at the blank paper. The idea was to have an interactive white board where you could manipulate an equation. The kid can try several things but the computer will only allow those that are valid. I even made a very crud attempt to show the concept. Please take a look at it. http://algebraexplorer.appspot.com/
I absolutely love this concept. The game mechanics have the potential to make the concepts extremely clear, though we might end up with a generation of people who need "algebra calculators". I suppose that is why he suggest in game points for doing things by hand.
Math Blaster needs to make a comeback!
Can you still edit the post title? It's missing an important question mark.
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.