[ctl]

I love this article, but: what can a practicing math teacher take away from it? How can you apply this stuff if you still have to teach a standard curriculum?

I'm really asking -- my friend is about to start as a high-school math teacher.

I guess the first recommendation would be: motivate every new technique by starting with one or more problems that the technique helps to solve. (Here "problems" is meant in the Lockhart sense -- real puzzles, not exercises.)

But how often are "techniques" actually taught in high school math, especially algebra and precalculus? A lot of high school math consists of digesting new definitions, or the generalization of old definitions. A fair amount of it consists of learning theorems that go unproven, or that are proven (by the teacher) too quickly for students to understand where they come from -- and in general it isn't satisfying to solve a puzzle with a theorem that one doesn't actually understand.

On top of that... students have to spend time with problems before they become genuinely interested in their solutions, so progress would be slower with this method. It's not clear that you could teach a whole year's curriculum in one year like this. (And if you fail to do that you'll eventually get fired.)

Any insight? I believe that it's possible to teach math, even standard high school curriculum, in such a way that students are at all times intrinsically interested in what's presented. But it would be awfully hard to do at scale, at the standard pace, as a high school teacher would have to. How might a teacher start in that direction?

[jfarmer]

I co-founded Dev Bootcamp and while I was still there one of my not-so-secret missions was to make mathematics less alienating. I only say that because it was incredibly difficult, even in an environment where I had complete autonomy and authority to make whatever curricular and pedagogical decisions I wanted. The problem becomes combinatorially more complex in a public school where teachers have much less autonomy, have to teach to a common set of state-wide standards, and have students of varying levels of interest.

Here are my scattered thoughts, though. I'm going to try to not suggest a pie-in-the-sky solution like "new curriculum!"

First, I majored in mathematics at the University of Chicago, but I hate, hate, hated mathematics in high school. Take something you'd see in Algebra II like matrix multiplication, matrix inverses, and solving systems of linear equations. You're presented with these things called matrices and taught a bunch of rules. Where did these rules come from? Why are we calling this "multiplication" when it doesn't look or act anything like multiplication?

And sure, I see that when I go through the steps you tell me to go through like a monkey I get an answer that works, but how do we know there aren't more correct answers? How did anyone even come up with these steps in the first place? It's not like someone sat down and tried a trillion random combinations of symbols and steps until one of them happened to work.

Augh. In that world the only recourse for students is to memorize, usually just enough to do the homework or pass the test, and then promptly forget. The only experience they associate with math is the utterly humiliating feeling of being terrible at it.

So, I think that's one of the root problems. People remember what they feel and most people remember feeling stupid, humiliated, and possibly ashamed when it comes to mathematics. It's only a matter of time before that becomes part of their identity. "Oh, I'm terrible at math. Oh, I'm not smart enough to do math." and so on.

If I were a HS math teacher my top priority would be to watch out for when those counterproductive, self-defeating beliefs were forming and do whatever I could to preempt them.

Second, I think the way math is taught is overly symbolic. What most non-mathematicians don't realize is that when most mathematicians look at a set of abstract symbols they don't "see" the symbols per se, they see what those symbols are meant to represent. They freely move between a geometric and algebraic picture of the world, but the algebraic picture is usually incredibly compressed.

I think the key thing is not to pick a side -- algebra vs. geometry -- but to show the relationship between the two. Geometric objects admit a symbolic representation and vice versa.

Third, students have this idea that math is all about being "right" or "wrong", that it's "black" or "white", that there's some universe of Proper Math that is insisting on certain rules for no rhyme or reason

Here's a silly but illustrative example that I think students would cover in 6th or 7th grade: order of operations.

Hey class! Look at this expression: 45+6. What does it equal?

A bad teacher says "It's 26 and any other answer is wrong." An ok teacher says, "Remember the order of operations. If we apply those rules we get 26, so that's the right answer."

A great teacher shows their students that some things are necessarily true and other things are definitionally (or conventionally) true. This teacher would do something more like...

Who got 26? Who got 44? Students who said the answer was 26, how did the students who got 44 arrive at their answer? Students who said the answer was 44, how did the students who said 26 arrive at their answer? Neither of you are wrong per se. We could have chosen to live in either world, but we have to choose one consistent set of rules.

These rules lead us to 26. If we chose the other set of rules, we'd get at 44. We only do this because we don't want to have to write down parentheses all the time, but without them it's unclear what order we're supposed to apply + and . So we need to agree on a set of rules so that two people looking at the same expression both understand how to make sense of it.

It's like traffic laws. There's nothing stopping people from driving on the left side of the road. In fact, there are countries where everyone does drive on the left side of the road. The important thing is that everyone agrees on a convention -- left-side or right-side. It works as long as everyone agrees and breaks if people don't.

I could go on, but I'll stop here. Like I said, these are my scattered thoughts. :)

[nileshtrivedi]

> when most mathematicians look at a set of abstract symbols they don't "see" the symbols per se, they see what those symbols are meant to represent.

Might we benefit from a different set of symbols that actually convey the geometrical meaning behind them? If instead of π, we used a glyph that shows a circle over a diameter, instead of x for a variable, we show an empty rectangle that shows that it's a placeholder for a value?

> Students who said the answer was 44, how did the students who said 26 arrive at their answer?

I came across a great example of this approach in Chess: The Complete Self-Tutor by Edward Lasker. Instead of just showing the right answer for a chess puzzle, he tells you what you did right in your answer and what you missed in getting an even better answer. This was a printed book that was completely interactive.

[calibraxis]

People should note that * are turned into italics, when text is between them. So, the original equation is 4 * 5 + 6.

(Here, I use spaces to "escape" the * . '\' doesn't work as an escape character.)

[gizmo686]

I was a student from a small private school that literally wrote their own math book, so I have no idea how generally applicable this is. As you suggest, a common technique my teachers employed is setting us loose on problems we did not yet have the tools to easily solve (but which were within reach). We would typically work in small groups, and if necessary the teacher could speed up progress by dropping us hints. We inevitably (in the beginning) would come up with week/non-rigourous solutions, which would often lead to debate as a class, pushing us to formalisation. As far as learning new techniques/generalizations/ETC, we would almost always 'learn' them after we have already been using them.

One thing I noticed during in math classes is that I don't really need to know anything. For me, and most of my classmates, most of formulas could be easily derived from simple and intuitive principles. For example, almost no one in my class actually 'knew' the quadratic equation, or the common trig values (ie. sin(30)). What we did know was how to quickly find those if we needed them.

As to your question of a standard pace, groups tend synchronize themselves. If every comes in at a similar place, and you have alot of group work and full class collaboration, then the slower students will gennerally still be able to follow the groups discovery trail, even if they do not contribute as much. The important thing here is you make sure that students are comfortable to ask questions, and that you do not have a few students dominate the discussion such that they loose the rest of the class.

Again, that comes from the perspective of a student at a school where the teachers had a lot of leeway in how and what to teach.

[nileshtrivedi]

Would it be possible to get that math book they wrote and perhaps the teachers' notes to go with it?

[nullc]

One thing you could do is take an interesting piece of subject matter from later curriculum (like next years or later in the current year) and present it as a puzzle for the students to explore with the inquisitive techniques presented in the paper.

Award participation credit, etc as relevant to help keep people engaged who need it. The fact that it's "future" material can also help students who need the extra goals pay attention.

Most likely you can't actually cover the required curriculum like this— as you note, a lot of things do not lend themselves to compact discovery. (It's all ashame, it's not like the students actually retain into adulthood all those procedures that they don't really understand in any case :( ) But maybe you can still inspire people with a few things which do lend themselves to compact discovery, and that inspiration may also make the rest of the subject more accessible to them. "This things have a reason and a pattern to them, even if I don't know what it is right now."

I had some challenges in math in school because I studied calculus, analysis, linear algebra, discrete math, etc. on my own and would derive solutions— sometimes the same as they wanted me to memorize, sometimes not— on my own instead of memorizing the fixed routines, and this was unwelcome. It would be nice if more teachers made an effort to at least not penalize students that were independently interested.

[jfarmer]

Also, your friend should read Mindstorms: http://www.amazon.com/Mindstorms-Children-Computers-Powerful...

One of the major themes is the relationship children have with mathematics and ways teachers can change it.