In my last two posts on problem solving, I put forward a definition of problem solving and an idea for how to teach it. I defined problem solving:
A student engages in problem solving when a problem or situation requires that student to develop an approach by transferring prior knowledge to the context at hand.
Based on that definition, I made this claim:
Teaching problem solving is not the teaching of a set of things students should do, it is teaching a set of ways for students to see.
I believe that this idea of transfer — taking what students know and finding a way for them to apply it in a new situation — relies on the student’s insight into the path to a solution, rather than giving them a list of “problem solving strategies” to fall back on when they get stuck. But this all sounds really abstract. I want to get into some concrete classroom actions that I can take to do this. As I put this list together, it seemed more and more to me just a list of good teaching practices, but with a very specific focus — building deep knowledge that students can use to draw connections between situations and solve increasingly difficult problems requiring transfer of that knowledge. I am focused less on what to do when students are working on difficult problems, but instead on some strategies to build the knowledge they need to solve problems independently. I have three ideas for how to do this.
Three Avenues of Thinking
This is inspired by a really incredible Ignite talk by Grace Kelemanik at NCTM. The premise of her talk was pretty simple — the Common Core Standards for Mathematical Practice are not all created equal. First, we need problems that are worth making sense of and solving — Practice 1. Without problems worth solving, everything else falls apart. But students need ways of accessing these problems. Here, Grace turns to Practices 2, 7, and 8.
First, students need to be able to implement Practice 2, reason abstractly and quantitatively. In the classroom, this means contextualizing and decontextualizing — taking quantities and looking at the math behind them, using the math to manipulate the quantities involved, and moving between the concrete and the symbolic to look for a solution path. This could be as simple as realizing that 4 cups of 8 straws is the same as 4×8, or as complex as noticing that if a shop charges $9.00 for a cheese pizza and $1.50 for every topping, the cost would be 9 + 1.50 t
Next, students need to be able to implement Practice 7, look for and make use of structure. In the classroom, this means looking for similarities between a problem and another problem — Grace gives the great example of realizing the 4/5 full is the same as 1/5 empty, and using that connection to facilitate a solution.
Finally, students need to be able to implement Practice 8, look for and express regularity in repeated reasoning. In the classroom, this means looking for patterns and generalizing a rule based on those patterns. For instance, when finding square roots, realizing that the last digit of the square can be used to predict the last digit of the square root, and using this fact to refine a guess-and-check strategy.
These strike me as excellent strategies, but not in the sense that they would be especially useful in solving problems. “Have you tried generalizing based on patterns?” seems to me far too general to be of use for very many students. Instead, I think these are the best avenues for teaching content in a way that promotes problem solving. When students contextualize and decontextualize, they build knowledge of the connections between a context and a broader mathematical idea, setting them up for having another insight later. When students identify similarities between problems and situations, they consolidate and deepen the knowledge they have, and create opportunities to apply it in new situations. When students generalize based on patterns, they make connections between individual examples and a broader rule that governs them and can be used to solve a different problem. As students build this deep, connected knowledge, they are putting themselves in a position to have similar insights in the future. This is much more humble work than much of what I have called problem solving in my class in the past — instead of finding lots of really hard, non-routine problems, I am using a variety of related problems and constantly looking for opportunities to push student thinking down these three avenues. This gives me three specific ways to build deep knowledge in students, and I think it is exactly that humble work that sets students up to be effective and successful problem solvers.
From Concrete to Abstract
I often hear references to the “ladder of abstraction” — the idea that students’ understanding begins with the concrete, and climbs a metaphorical ladder as it becomes more and more abstract. I think this is a useful metaphor, but is also incomplete. My thinking here is influenced by Daniel Willingham’s Why Don’t Students Like School, which devotes a chapter to the difficulty students have in understanding abstract ideas. His basic premise is that the mind prefers to avoid abstract ideas — we naturally try to understand things in terms of what we already know, and what we already know is mostly concrete. Abstract ideas can’t just be poured into a student’s head. Instead, students build deep, abstract understanding through the accumulation of lots of concrete examples, analysis of the similarities and differences between them, and a focus on the underlying structure that connects all of the ideas.
Based on this perspective, I think the metaphor of a ladder of abstraction would be better replaced by a pyramid of abstraction. It’s a subtle difference, but I think a significant one. If I want students to build a deep understanding of functions, I need to give them as many concrete examples of functions as possible, and spend time analyzing the features of those examples that make them functions — moving toward abstraction, but doing so through a variety of examples, rather than jumping up in abstraction each time. If I want students to understand what the derivative means, I need to give them as many concrete examples of derivatives as I can, with and without context, as they build a flexible understanding of rate of change and its relationship with functions.
I worry that the ladder of abstraction metaphor leads me to believe that, once a student understands one concrete example of a function, they are ready for a more abstract example. While some students may be, I want to focus on building a broad base first, and then moving up the pyramid after we have spent time analyzing the connections between the examples and the underlying structure. If the goal of the abstract idea is to be able to problem solve with it — to transfer that understanding to new situations and new problems — my students need a large number of examples to work from to build the knowledge they need to transfer.
This is also an important example of a time when I want to do some of the cognitive work for students. It’s fashionable as a math teacher to brag about how little help we’re giving our students, and how we’re making them do the hard intellectual work. I’m all for that, but not as dogma that supersedes everything else. In this case, when I’m at the bottom of the pyramid of abstraction, I want to put students in positions to make connections between concrete examples — and if students don’t make those connections themselves or in small groups, I want to explicitly teach them to do so. If I am introducing concrete examples without building the connections to the deeper concepts, there is no pyramid of abstraction, just some scattered and disconnected bits of knowledge. These connections are the building blocks of knowledge transfer, and it is my responsibility to make sure students make them.
I wrote recently about hints, and proposed that hints can promote learning in five ways:
- Redirect attention to features of the problem (What is it saying?)
- Redirect attention to student knowledge (What do I know?)
- Redirect attention to student cognition (How am I approaching it?)
- Promote students’ beliefs in their mathematical efficacy (I think I can solve it)
- Provide missing information (I know what I need to solve it)
I think one critical goal of hints has to be to build a student’s belief that they can solve problems themselves — that they are sense-makers, and capable mathematicians. This perspective is important, and is a rational reason to use restraint when giving students hints. That said, I think there are important times when we need to do a bit of the work for students, in order to build the knowledge they need to actually be the effective problem solvers we are looking for. I have two criteria for hints that do some of the work for students, but I think build critical knowledge students need to be problem solvers.
First, if a student is missing a piece of mathematical structure or context, my hint should do some of the work of making that connection for them. If they don’t see that a situation corresponds to a mathematical tool, pushing the student to make that association rather than struggling blindly sets them up to make more associations with that tool in the future. If they don’t see how a problem is similar to another problem they can solve, my hint should point them in that direction. Fundamentally, the goal of hints needs to be to promote learning. If, without a hint, the student is struggling blindly, they aren’t learning. If, without the hint, they are likely to just copy the answer down during a discussion, they aren’t learning. If, without a hint, they aren’t making the mathematical connections they need, they aren’t learning as much as they could be. Hints that serve these purposes can be subtle and push students to do a great deal of the work — but a broad fear of robbing students of thinking can lead just as easily to confused students who aren’t learning, and I want to make deliberate decisions around exactly how much help to give.
Second, if a student can’t figure a problem out, but knows the information they would need to solve it, I need to give it to them. If a student is close, and I can push them to articulate exactly what they would need to solve it, I need to give it to them. If a student is solving a geometry problem, and just needs to figure out the length of one missing side of a right triangle, and can articulate that if they just knew how to do that they could solve it, I should give them the Pythagorean Theorem in that moment. That’s a bit of a silly example and rarely happens in real life, but I think the principle is an important one. I can push students to articulate what they would need to know to solve it. At that point, they have built the mathematical structure they need for that missing piece to be a durable part of their knowledge, and it is my responsibility as their teacher to fill it in for them. And, as students build a habit of figuring out what they need to know to solve a problem, they build for themselves tools that will allow them to use their resources effectively to solve new problems in the future.
Looking back on this post, and my thinking on problem solving over the last few weeks, I feel like I’ve moved away from almost everything I’ve previously thought. I’m focusing much less on some specific things that I do while students are working to teach them problem solving. I think that the quiet work of building knowledge is at the center of effective student problem solving. I want my students to be mathematicians who look at a problem, draw carefully on their knowledge, and see a path to a solution. I’ve used a few words and phrases over and over again in this post — connections, deep knowledge, transfer. These all come back to the same idea. If I want students to be problem solvers, I need to pick the types of problems I want them to be able to solve. I need to teach that content, and teach it in a way that builds abstract concepts that students can apply in new contexts. I need to select problems deliberately, moving up the pyramid of abstraction. And I need to provide appropriate support — not doing too much work for the students, but not leaving them struggling blindly for fear of doing any work for them at all. And, over a whole lot of years of education, students will build expertise in math, connect concepts into broader and broader mathematical ideas, and become expert mathematicians.
I think, as a final note, it’s important to mention that many students do this themselves. Some students are sense-makers, incessantly questioning, clarifying, sorting new ideas and making connections with old ones. A happy, engaged classroom makes all of that much more likely. These are the students who impress teachers with their insights year in and year out. I’m not writing this post, or clarifying my thinking, or changing my teaching, for those students. They will be fine. But there is a significant subset of students who struggle to make sense of math, struggle with their beliefs in themselves, and don’t have that positive feedback loop of mathematical success. These are strategies that I think are critical for working with and teaching these students — building their knowledge deliberately and giving them a chance to become problem solvers as well.