I agree with the comment that math is satisfying but most often it is satisfying for math teachers and this goes to the empathy position. Students often don’t feel the lack of resolution that we feel that compels us to keep thinking about the problem. When we can get them to care about the answer in a way that leaves them unsatisfied until they solve it, then we have really addressed the portion of resilience and persistence that will lead them to hate the blank response.

And for the most part that probably could have worked with no pushback up until maybe 15 years ago. But the fact is that now students have more available to them in the way of information, entertainment, and savviness, that we cannot give them what they have always got and expect them to just take it.

I’m actually criticizing your recent work for being too much like traditional instruction, not for being too crazy or creative.

Explicit instruction is the world’s easiest pedagogical act for the majority of teachers, I’m convinced. So when explicit instruction isn’t incompatible with a technique I’m describing, I try not to foreclose it, or foreclose the teachers who find it easy.

I don’t love the idea that a teacher would just yammer instructions at her students in tight little IRE loops. But just because I believe there are better ways to help students learn, I don’t want to send her away from this series emptyhanded.

Finally, I just think you misinterpret cognitive load theory as saying that we should only show kids worked examples and drill problems.

I use the terms “speaking clearly about mathematics and assigning spiraled practice sets” as a level-best attempt at being fair to my critics. These are their words — and they adhere tightly to CLT — and one of them is a grad student of Sweller.

So we have to be really careful that the moment of correct discovery is followed by a decrease in cognitive load, with appropriate repetition and rehearsal, so that the discovery actually gets remembered. Is that really objectionable?

Definitely not. I find CLT very useful. It shares an interesting parallel with constructivism, though, in that its most extreme adherents extract from a useful theory of learning a useless theory of teaching.

I know I could use better hooks, so again, I appreciate this summer series, but I’m mostly concerned that students (especially those with big gaps in prior knowledge) can only understand the conclusions of the lecture that follows but don’t really understand the concepts leading to those conclusions. Like a student who never realizes that a graph shows the locus of points that are solutions to an equation, but remembers the procedure for graphing from y=mx+b. In the last couple of years, your work seems not to have addressed this issue as much as I would have liked…or when it did, I felt it overloaded kids’ brains in a way I fear could lead them to the correct conclusion without the time, practice, and discussion needed to consolidate that conclusion and the *reasons* for it in their long-term memories. So although you know I think cognitive load theory is a good guide for instruction, I’m actually criticizing your recent work for being too much like traditional instruction, not for being too crazy or creative.

How do you get kids to understand that in 2x/5x, you can simplify the x’s, but in (2+x)/(5x), you can’t? And how do you structure a lesson that addresses that kind of thing without being dreadfully boring? Your recent approach would help me find a flashy hook to show kids they need to know when you can and can’t simplify the x, but how do you get them to stay tuned to a long debate about what it really means to simplify a fraction, especially when they have already given up on math meaning anything?

Finally, I just think you misinterpret cognitive load theory as saying that we should only show kids worked examples and drill problems. There’s no reason you couldn’t do a Schwartz/Harel-inspired hook and then use worked examples in the lecture & notes that follow. And the worked examples could demonstrate both correct examples and common mistakes, and could ask students to explain the mistakes conceptually. All of that would be consistent with cognitive load theory, and as far as I can tell, would also be consistent with you like to do. The one thing cognitive load theory tells you not to do, which has been a big revelation for me, is to lead kids through a sequence of discoveries without giving them time to consolidate each discovery in long-term memory before moving on to the next one. That’s what I was criticizing here and then here.

It turns out that the process of encoding something in long-term memory actually consumes working memory. So if a student makes a discovery (“Aha!”) and it’s bouncing around in their working memory, they can lose it the next minute unless we give their working memory a break from thinking about new things and give it a task that reprocesses that thought a few times until it gets encoded in long-term storage. It’s simply not true, despite what many teachers think, that we tend to remember the things we discovered for ourselves. Instead, as Daniel Willingham says, memory is the residue of thought, so we tend to remember whatever we spent the most time thinking about. If we spent most of our time thinking something wrong, and only a little bit of time realizing that it was a mistake, we might remember the misconception more strongly that the fact that it’s not right! So we have to be really careful that the moment of correct discovery is followed by a decrease in cognitive load, with appropriate repetition and rehearsal, so that the discovery actually gets remembered. Is that really objectionable?

Second, a blog lends itself well to discuss one topic at a time, so it makes sense that some readers may feel disjointed. Motivation–like every other challenge to learning–requires a variety of strategies, none of which work unilaterally for every teacher nor every student. This discussion is an edge-piece of a large puzzle.

Having heard you speak, I’m familiar with the urgent tone you use for a room full of math teachers, and I hear it in your writing, too. It’s good.

For a start a non-patient Maths teacher is going to die early of stress. You are daily given to watch people do slowly and badly what you find easy. The very first lesson I learned about being a good teacher is the ability to wait for someone slow do something at their own pace.

I’ve also as a teacher learned many different ways of doing the same problem. (As a trick to keep me occupied while students do hard calculus problems, I often solve the same problems the easy way without calculus.) If anything it is students who want to focus on the “one right way”. It’s the teachers who try to get the students to think graphically, numerically and algebraically to solve the same problem.

It’s true that I stop students doing things by a method that isn’t correct, even though the student thinks it is acceptable (most often “solving” by arithmetic methods). I also shun complicated techniques that are difficult to learn and easy to get wrong, which can annoy the more formula focused students. But a correct method that works is always OK.

So, actually, I call BS on Brian. It seems much more like a personal animus than an actual statement of truth. I’ve known people say how all PE teachers were bullies and idiots, and that’s not true either.

My firm belief is that the extensive study of math literally reconfigures your brains so that you are no longer capable of appreciating anybody else’ solutions, methods or modes of thought, and are, coincidentally, rendered incapable of clearly expressing your own.

Nearly every math professor I’ve had was impatient, arrogant, demeaning and condescending… Right up to the point that they graded my tests.
I have had no such problems with ANY of my other professors, and I have noticed PRECISELY the same behaviours from my own childrens’ teachers.

I see similar qualities in the participants of the MTT2k series of videos. Indeed, the videos demonstrate EXACTLY what I’ve learned to loathe about stepping into a mathematics class of any sort. Namely, there is a strange propensity to consider methods that are not yours to be “wrong”, regardless of the quality of results.

I love the subject. I can not STAND the “professionals” who teach it.

Note: Below this box into which I am typing is a series of checkboxes. The first two say the same thing. Why are there two of them?