It is a curious and unsettling process, the business of changing your mind on a subject about which you had very positive convictions.
– John Holt, “How Children Fail”
I made a significant change in my teaching this spring. About four months ago, I committed myself to a new weekly system, where each Friday I would come to class with 3-5 worthwhile problems for students to solve. Students worked individually and in groups, solved, constructed an argument, critiqued each others’ work, and revised. We usually got through about 3 problems in a 50-minute period. And I made a deliberate effort for these problems to be non-routine — that is, a range of topics from our course, from problem-solving scenarios drawing a variety of math, and puzzles that provided opportunities for mathematical argument and discussion. I drew these problems from all kinds of sources — Play With Your Math, Dan Meyer’s Three-Acts, problems I wrote, and others found scouring the internet. I really enjoyed teaching these classes — and a number of my students did too.
Since then I have done a great deal of thinking and reading about learning, and have formed some new opinions. This is a rough overview of how some of my opinions have changed these last four months.
Make It Stick – Spaced Practice & Distributed Practice
Make It Stick is a great book about the structures and routines that help learning. The first point that was really salient for me was the importance of spaced practice and distributed practice. The wisdom of spaced practice seems self-evident — we all know that cramming leads to less retention than spreading studying out over a period of time, and there is a great deal of research that supports this. The research on distributed practice was much more interesting to me. One example comes from the baseball team at Cal Polytechnic. They split their batters, randomly, into two groups for six weeks. The first group did batting practice according to their usual regimen — each hit fifteen curveballs, then fifteen fastballs, then fifteen changeups. Each round, they got better at timing their swings and making contact, and learning felt easy. Players in the second group were given forty-five pitches, where the three types of pitches were interspersed randomly. At the end of the round of practice, these batters were still having trouble making contact, and practice felt more difficult and less successful. However, at the end of the six weeks, those who had practiced without knowing which type of pitch was coming hit significantly better (80). The authors of Make It Stick attribute this improvement to the cognitive demand on the batters during practice — practicing without knowing what pitch is coming is harder and may not be pleasant, because of the increased cognitive demand, but it is exactly that increased cognitive demand that leads to durable learning. This is what the authors call a “desirable difficulty” — that when things are difficult, as long as they are still achievable, learning is stronger.
This seems to support non-routine problems in math class on a regular basis. Cool. Let’s cut to the next piece of Make It Stick that really resonated with me, and that’s the idea of structure building.
Make It Stick – Structure Building
[Structure building is] the act, as we encounter new material, of extracting the salient ideas and constructing a coherent mental framework out of them … High structure-builders learn new material better than low structure-builders. The latter have difficulty setting aside irrelevant or competing information, and as a result they tend to hang on to too many concepts to be condensed into a workable model (or overall structure) that can serve as a foundation for further learning (153).
This gets at the heart of learning math for me. If students are to go beyond producing in math class to gain mathematical expertise, that expertise comes from seeing connections between concepts, and using them to solve problems. That structure-building is what I want students to do. This obviously can’t happen if we take the assembly-line approach to math learning — today we are learning multiplication. Instruction, practice. Then we do division. Instruction, practice. If students are to build structure, to reinforce mathematical connections in their minds and gain deep, flexible, mathematical knowledge, we must be constantly putting students in the position of creating these connections.
Some measure of distributed practice is absolutely necessary to make this happen. But here is actually where I started to fall off of the non-routine problem train. I’ll come back to that point in a moment. Next up, problem solving.
Problem Solving and Modeling (Lesh & Zawojewski, 2007)
This is an absolutely amazing article surveying the literature on problem solving in math class, and draws some fascinating conclusions that I am still sorting out. There are a huge number of insights here that have influenced me, but I’m going to focus on two in particular.
The labels we have for problem solving may not be as useful as we thought
When math folks talk about teaching problem solving, they usually talk about problem solving strategies. And most problem solving strategies have as their antecedent Polya’s work in How To Solve It:
However, as Lesh & Zawojewski write,
…research has not linked direct instruction in these strategies to improved problem-solving performance … Perhaps one reason is because using a strategy, such as draw a picture, assumes that a student would know what pictures to draw when, under what circumstances, and for which type of problems … Our interpretation of Polya’s heuristics is that the strategies are intended to help problem solvers think about, reflect on, and interpret problem situations, more than they are intended to help them decide what to do when “stuck” during a solution attempt” (768).
I think this critique is right on. These problem solving strategies are not useful in the sense that “teaching” students a strategy will make them a better problem solver. They are useful in that they provide a framework — a structure, if you will — for students to make connections between mathematical problems that consolidate understanding and better prepare them to solve similar classes of problems in the future.
Michael Pershan would go a step further, and I agree with him, in that strategies like Polya’s are too general to be of use to school students. We can’t teach “Solve a Simpler Problem” — students lack the knowledge and structure to use this as a connection to build while problem solving. Even narrowing it to geometry, and specifying that students “Chop a Shape Into Simpler Ones”, may still be too broad for students to make worthwhile connections — they must first build up examples of the strategy in trigonometry, area, auxiliary lines, and more, before they have a structure that can reach the generality of the strategy “Chop a Shape Into Simpler Ones”.
I’m getting close to my realization, but I have one more quick reference, this from Elstein, Shulman, and Sprafka (1978) on expert medical diagnosis — in my opinion, one of the best parallels in the professional world for problem solving:
The most startling and controversial aspect of our results have always been the finding of case specificity and the lack of intraindividual consistency over problems, with the accompanying implication that knowledge of content is more critical than mastery of a generic problem-solving process (292).
This blew my mind a little bit. Doctors making diagnoses — which has to be one of our best examples of expertise in a field — don’t show a significant domain-general problem solving ability, and it instead seems to be a function of their content knowledge for that specific case. If something similar is the case in mathematics, it suggests that any goal of teaching domain-general problem solving strategies that may apply to any mathematical problem is a fruitless exercise, and we should instead be focusing on deep content knowledge in different areas of mathematics, and the strategies and perspectives directly relevant to that content area.
Lesh & Zawojeski note that research suggests two other areas that influence problem solving. The first is metacognition — students actively monitoring their problem solving processes, in particular thinking about why they are employing the strategies they choose, and noting when they are following an incorrect path and reversing course. The second is student beliefs toward math, for instance having a growth mindset toward their mathematical ability. While these two may be relevant, they were not my original intention in teaching through non-routine problems, and I am going to put them aside for now.
Let’s try to bring this all together
Non-routine problems. Right. Here is my argument:
Non-routine problems are useful if
1) They are being used deliberately for distributed practice, to promote long-term retention
2) They are situated in a sequence where meaningful connections are being made between content areas to deepen understanding.
Non-routine problems are not useful “per se” — they do not teach math in and of themselves, but as a tool to reach other ends.
I came to this conclusion from the perspective that math learning is all about the connections and the structure that students build in their minds. If math is a set of isolated skills, students will struggle to apply concepts and consolidate new knowledge. Solving non-routine problems may present opportunities to make connections across content areas, and these connections may be really tempting for teachers, but students’ knowledge must be build from the ground up. A teacher may look at two disparate problems and see what they have in common — maybe a math teacher sees the two problems below as two connected examples of problems where it is useful to work backwards.
But I’m skeptical. I think that students who are still novices mathematically lack the knowledge structure to make this connection. Instead, students need to encounter problems that make connections at a smaller level, and build knowledge systematically — deep content knowledge first, and strategies coming alongside and after as student knowledge develops.
I am rethinking what I want to do with non-routine problems next year. Providing students opportunities for distributed practice will absolutely be a priority — but this will be distributed practice that is deliberate and focused on content knowledge first, rather than prioritizing “non-routine” as a factor in choosing problems. I will also continue presenting non-routine problems to my students, but with a focus on problems directly and concretely related to the content we are learning at the time, whether through an adjacent problem type, a contrast with the previous day’s focus, or an analogous solution method. Non-examples and new problem types that build connections and distinctions between content areas are critical, but I will plan these in the context of a unit, choosing problems so that the connections I am asking students to build are revisited and reinforced over a period of time, rather than coming up haphazardly in a way doesn’t encourage structure-building. This will not look so radically different. But the major change is that these problems will be interspersed day by day, rather than concentrated all at once, and that I will choose problems for their value in teaching content first, rather than an abstract idea of the usefulness of non-routine problems.
Things I Don’t Know
I love the work of Mary Bourassa, Alex Overwijk and others on a fully spiraled curriculum. I admire them for this, but I don’t feel like my curriculum knowledge is there. A fully spiraled curriculum provides different opportunities and different challenges that may make this argument inapplicable.
I don’t know enough about how to teach students metacognitive strategies to monitor their problem solving, or how to effectively teach growth mindset. I think non-routine problems may have a place here, but I also think there are valuable opportunities to address these goals through content, and I don’t want to make non-routine problems a regular and significant part of my class solely for the purpose of metacognition and growth mindset.
I don’t know what the impact is on student engagement. A number of my students — although mostly high achievers (I might say high structure-builders) enjoyed my Friday non-routine problem days. Maybe this engagement spilled over, and was worth the time spent on it.
I don’t know how to teach problem solving. I’m puzzling through this at the moment. I think there are some important connections, and this post is helping me put my ideas in order.
I have some definitional problems here. “Non-routine”, “problem solving”, “content knowledge”. I think I have a good idea of what I mean, but is there a shared understanding among math teachers on these ideas? I doubt it. Maybe a future project.
That’s what I got. I’m curious for thoughts — what am I missing? What other research is relevant? What are other experiences with these types of problems?