It’s true: math education doesn’t give a ton of attention to Sweller and cognitive load theory. Math education researchers who are aware of Sweller are most familiar with his attack on problem-based, experiential, discovery and constructivist learning (“An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential and Inquiry-Based Learning“). As Raymond mentioned on twitter, those within math education who are likely to recognize Sweller are equally likely to dismiss him and his work.
Part of this, I think, has to do with focusing on the wrong aspects of Sweller’s work. Ask 100 people what the key idea of Sweller’s work is, and I bet 99 would say: it’s easy to overload the working memory of students. For learning, it’s important not to. So, don’t. An important but limited insight. (We’re trying not to overload anyone!)
The last 1 person out of the 100 is me. As far as math education is concerned, I think the key idea of Sweller’s work is about problem solving, not cognitive load. Here is that key idea: problem solving often forces a person into answer-getting mode, and answer-getting mode is incompatible with learning something new.
(“Answer-getting” mode also has to do with expectations that students have about math class and the sorts of activities they think are valued in mathematics. Sweller shows it has a cognitive element too.)
Sweller’s early work was with number puzzles. Participants in his studies solved the puzzle successfully, but never came to notice a fairly simple pattern which was sort of the “key” to finding any solution. Why? There were two reasons:
- When you’re looking for the solution to a problem, your attention is massively restricted to those things that are directly relevant to finding the solution. Lots of important details of the scenario or environment get ignored.
- Attention is a zero-sum game. There’s only so much that a person can notice. A person focused on finding the solution is unable to focus on much else.
(For more, read this part of my essay.)
I have found this to be absolutely true and deeply insightful. The first time the idea really hit me was during Christopher Danielson’s talk, titled “What’s the Difference Between Solving A Problem and Learning Mathematics?” There is a difference. Sweller helps us get specific about some of the reasons why.
These limitations of problem solving guide my daily classroom work. My 8th Graders are wrapping up their study of linear functions and moving on to exponential functions. Yesterday, I found myself wanting my students to start thinking about the differences between linear and exponential graphs and patterns. I took this image from David Wees’ project and displayed it on the board:
In the past, my first instinct would have been to pose the problem as quickly as possible. “What are the coordinates of point B? of point A?” I would then give my students time to think, and I would have expected some learning to have occurred.
Now I know that this could be a particularly bad way to ask my students to begin their work. They probably wouldn’t notice what I want them to notice. Instead, they’d probably go into that answer-getting mode that focuses all their resources in an unproductive way:
Another key insight of Sweller has to do with how to avoid ensnaring students in this unproductive struggle. One suggestion of Sweller’s is to ask less-specific questions. These nonspecific questions don’t funnel attention in the way specific questions do, and they therefore don’t overload students in quite the same way.
Sweller first described the power of nonspecific questions with regards to angle problems. Rather than asking students to find a particular angle, he asked “Calculate the value of as many variables as you can.”
With my 8th Graders, yesterday I began class with two nonspecific questions. I asked these questions so that they’d notice as much about the diagram as possible and start putting together some of the pieces about exponential relationships.
My first question: “What do you notice?” I waited for lots of hands to go up, and then I quickly called on three students. (I find it’s important to move quickly here — not so interesting to rattle through everyone’s noticings.)
My second question: “Study the diagram and find something to figure out.” I asked students to do this in their heads, alone. Then, “Talk to your partner — come up with at least two different things to figure out, then as many as you can.” (What counts as something “figured out”? We’ve done this routine many times, so my students know from experience.)
Here is an incomplete list of what my students calculated/figured out from the exponential graph:
- The y-coordinates are doubling
- The y-axis is going up by 4
- The slopes are changing between each pair of points
- The graph is non-proportional
- The next coordinate would be (6, 64)
If my students had mentioned, at this phase, that the coordinates of B were (2,4) we would have moved on. Since they hadn’t, and since they were saying so many smart things, I decided that this would be a great time to ask a third question:
“What are the coordinates of point B? point A?”
My students were able to answer these specific questions, but that’s hardly the point. Sweller’s research suggests that you can’t use problem-solving success as a gauge of whether kids have learned something or not.
I do think, though, that the reasons my students gave for their correct answers are revealing. Some students, in justifying their answers, mentioned that you could be sure that point A was at (0,1) because the y-coordinate seems to be 1/4 of the way up to 4. Other students then pointed out that (0, 1) fits the general pattern. What’s interesting is that this first observation — the position of point A up the axis — never came up in the first two questions I asked. That makes sense, because that way of looking at the position of point A has nothing to do with the exponential pattern. In fact, it’s the sort of hyper-focused response that you’d only expect to hear when a very specific goal has been set by the teacher — find the coordinates of point A. Otherwise, that’s not the thing that’s worth noticing here (probably). It misses the forest for the trees in the way people do when they are focused on achieving a narrow goal.
The second response, though, showed that some of my students had started making good connections. They justified the coordinates of points A and B based on the general pattern.
All this suggests to me that while some of my students are ready for working on specific problems, many of them aren’t yet there.
Asking more nonspecific problems isn’t the only recommendation that Sweller makes, of course. He’s better known for recommending the heavy use of worked-out examples and explanations in class. We do those too, though probably not as often as Sweller would like. Still, there’s more to Sweller’s theory than worked examples.
The key idea here is that specific questions cause students to chase specific goals. Chasing a goal isn’t always helpful for learning. On the one hand, I think this makes the case for developing a specific question more slowly, asking students to notice before posing a problem. On the other, this calls for us to be more cautious and deliberate about how we use problems in our teaching, especially in the early stages of teaching a new idea.