The Devil Is in the Details
My skepticism about productive struggle, and one example of what I do instead
I’ve ended up in a few conversations online recently about teaching that make me want to give a really specific example of how I teach something. Conversations about teaching are hard because lots of stuff sounds good in the abstract, but in real teaching the devil is in the details.
The conversations started because I’ve expressed skepticism about productive struggle in math class. A number of people have defended asking students to figure out mathematical ideas themselves. Some concerns people share are that teaching without productive struggle can lead to shallow understanding, that students become dependent on the teacher, and that students struggle to apply what they have learned in flexible ways.
A Concrete Example
Let’s get specific. One topic I teach is multiplying negative numbers. Here’s how I approach that topic.
I start pretty much every lesson by asking students a bunch of questions on mini whiteboards. My goal is to activate prior knowledge and help students connect what they know with what I want them to learn. Those questions might look like:
Here I step in. I leave those last few problems on the board, and I say hey, when we say 2 x (-3), what we mean is -3 + (-3), so 2 x (-3) is -6.
When we say 3 x (-5), we mean -5 + (-5) + (-5), so 3 x (-5) is -15.
I’ll give a few more examples, this time juxtaposing multiplying positives and negatives. What is 2 x (4)? Ok, what is 2 x (-4)? I’ll be clear about the meaning of those parentheses. Then I check for understanding and we practice a bit. This includes mixed practice — if every question is a multiplication question where the answer is negative, students aren’t thinking much, so I mix in positives, addition with negatives, etc.
I’ll do the something similar when we get to questions like -4 x 2. I start with questions like:
Then I step in. 10 x (-6) is -60, so -6 x 10 is also -60. Again, I give a few more examples, check for understanding, and we practice a bit, again with some mixed practice. I do something similar for a negative times a negative, activating prior knowledge and looking at patterns to help students see why a negative times a negative makes a positive.
Why I Choose This Approach
In each of these examples, some teachers would ask students to figure out the big idea on their own. I can imagine that working well in some classrooms. I do use that for approach some topics. Some topics lend themselves more to that type of small-step discovery, others less so. It’s not an approach I use often because what I’ve observed is some students consistently end up frustrated or discouraged when trying to make that leap. It works for some students, while for others it feels overwhelming. Using that approach day after day is likely to teach those students that they just aren’t very good at math.
This might have to do with my context. My school has persistently low math achievement. I teach a lot of students who, for years, have felt unsuccessful in math class. It’s not all of my students, but it’s a critical mass where that lack of confidence means students often tune out when they feel like the teacher isn’t supporting them, and tuning out is contagious.
Another reason I avoid having students figure new ideas out themselves is to save time. I have limited time, and I find a good way to get students solving harder problems is to build confidence early on when learning a new topic. Once students have that confidence, I ask them to apply their knowledge lots of different ways. One example is to have students look at the patterns in negatives when multiplying more than two numbers, and to figure out the pattern that three negatives multiply to a negative, four to a positive, five to a negative, and so on. Here I’m much more likely to use an exploratory approach. Students have built some confidence, and even if they don’t figure out the pattern they still get practice with their multiplication skills.
Another example is a little puzzle I stole from a Desmos activity that I now can’t find. I give students a bunch of numbers. Maybe the list is 2, -3, -2, 4, 3, 5, 1, -6. The goal is to find two subsets of numbers that have the same product. A simple solution might be 2 x (-3) and (1 x -6). Then, the challenge is to use as many numbers as possible. Can you find two groups of three numbers with the same product? This is a great way to get students practicing while also reasoning more flexibly.
The point of math class is for students to be able to apply their knowledge in lots of different ways. Taking an efficient approach to initial instruction leaves more time for this type of application.
The Criticisms
Let’s go back to those criticisms from before. My approach focuses on connecting new learning to prior knowledge, and my experience is that this helps students understand the math. I don’t think learning through productive struggle is the only way to develop deep mathematical understanding. While I give students plenty of straightforward practice to start, as soon as they’re ready I vary the questions I’m asking so students don’t become dependent on the teacher or believe they can only solve a problem if they’ve been explicitly taught a question exactly like it. I ask students to apply what they know in lots of different ways and try to ask plenty of non-routine questions.
I’m not saying I’m the perfect teacher. I’m trying to be thoughtful about lots of different pieces at the same time. I’m navigating my students’ confidence in math, their procedural skill, and their ability to apply that knowledge flexibly.
If I was tutoring a student one-on-one, I would be much more likely to ask the student to figure new ideas out on their own. I can keep a close eye on their confidence, adjust if they’re having a hard time, and make sure the process is productive. With a class of 25 students, getting that right for every student is much harder. I think it’s absolutely possible, but doing it day after day is challenging and risks discouraging students who already lack confidence. I choose to do it occasionally but it’s not my default pedagogy. Where I become even more skeptical is when curricula try to design that type of productive struggle on a daily basis. Not only is it hard to get right, but following the recommendations of a curriculum writer who doesn’t know my students, doesn’t know their strengths and weaknesses, makes success even less likely.
A reasonable person could disagree here. I’m happy to hash it out in the comments. My point is that it’s much easier to have these conversations by talking about a specific piece of teaching, rather than abstract arguments disconnected from real classrooms. There’s a stereotype that the alternative to productive struggle is just lecture and rote practice — “sit and get, then drill and kill” if you want to use the popular pejoratives. Whatever your approach to teaching, I hope this post illuminates the space between productive struggle and lecture.
The instruction that you describe here seems very much like productive struggle to me! The problem seems like the equation of productive struggle with students figuring things out “on their own.” I see this a lot with science instruction (my subject) and inquiry, where there is the assumption that inquiry = students just figuring things out themselves. What you’ve described here is, in my view, a great example of (guided) inquiry.
This post feels like a lot of bluster over very little substance.
Productive struggle means that you ask students to do some thinking, and to work through moments when they are stuck. When students are stuck on a problem, do you tell them what to do so they can just write down what you say? Or do you give them a hint that helps them continue to work independently? That is the difference between creating dependent students and allowing for productive struggle.
Productive struggle is NOT the same as inquiry, though inquiry also requires productive struggle. Who is telling you that they are one and the same?
Also, you are correct to realize that when struggle is not productive, it can be a bad experience for kids. You describe times when you have students try to figure out a rule or procedure for themselves, and some students are discouraged. That is real! You should not let those students just struggle unproductively. Knowing who, when and how much to help, that is the very challenging art of teaching. It sounds like you do let students explore sometimes, which is great. Some topics lend themselves much more to this than other topics. Multiplying negatives does NOT lend itself nicely to a discovery method. Other ideas like, say, the effect of a scale factor on the area of a shape, do lend themselves better. And it sounds like you do more explorations when it is appropriate.
I don't think I'd agree with Jacob that the lesson you describe qualifies as guided inquiry. Maybe I'd call it "Engaging discussion to develop new ideas." But your lesson is a far cry from the teacher who walks in at the start of the class and tells the students that today, we are going to learn 6 new rules, lists them on the board (neg x pos = neg, neg x neg = pos, etc) and then has the students practice. And if a student is stuck on a problem, the teacher repeats the appropriate rule and shows how it applies here, giving the student the correct answer. The truth is that many, many teachers will do just that, and little more. Those are the teachers who people like me are trying to reform.
In your lesson, you encourage all students to make connections between multiplying integers and what they just learned about adding integers, as well as what they know about the meaning of multiplication. You give room for some students to jump ahead of you with their logical reasoning if they can, by laying it out in such a thoughtful way. At the same time, other students will just wait for you to make it clear for them, but they will still be asked to figure out some small pieces on their own when they do the practice, because the practice is thoughtfully curated to keep them productively struggling. Beautiful.