At the start of the opening chapter of our new book (There Is No One Way to Teach Math), Robin Pemantle and I argue that teachers should learn to embrace contraries. The idea is that instead of choosing one or another option in a binary pedagogical choice, and instead of seeking a supposedly happy medium between them, one should learn to navigate the whole spectrum of choices between the extremes. I had made that point in a 2013 post and its 2015 sequel. To help math teachers and departments think about this, I listed dozens of apparently contradictory options in a worksheet (The Art of Teaching), and encouraged teachers to think about where they stood on each, and what they could gain by being more flexible.
In today’s post, I’d like to discuss productive struggle. Its proponents point out that having to struggle with problems can be productive for student learning. Another way to put it is that to learn anything important and/or difficult requires intellectual engagement on the part of the learner. I completely agree with this idea, and it has been part of my teaching and curriculum development practice over my entire career. But implementing it successfully is not easy, as the obstacles are many.
Culture: If students’ past experience has been that learning math consists of memorizing and deploying techniques without understanding, it can be difficult to transition to a problem-solving culture.
Curriculum: If the curriculum is based on the memorize-and-deploy paradigm, there is no opportunity for thinking or engaging intellectually. Likewise if the content is too easy.
Challenge: If on the other hand the problems are too difficult, students may turn off and not even try.
So, as ever, the ball is in the teacher’s court.
Changing classroom and department culture is doable, but it takes time and patience. It requires a step-by-step process, gradually moving from exclusively direct instruction to inquiry, from implementing algorithms to creating them, from plug-and-play to student-centered thinking, and so on.
Even if a problem-solving culture has been established, and even if a problem-rich curriculum is at an appropriate level for a class, a specific topic may be too easy or too hard for an individual student, and will require some degree of differentiation. (We discuss differentiation by time, not content, in Chapters 9-11 of our book. See also Reaching the Full Range on my website.)
What does this have to do with embracing contraries? Everything!
- In the reality of “coverage” and time pressure, it may not be possible to do everything by way of interesting problems. You may have to use some shortcuts — particularly when addressing necessary review, or trying to keep sections of a course in sync.
- You may strongly believe in cooperative learning, like I do. But almost every day there are times where you should work with the whole class, and times where you need to zero in on a specific student or group.
- All classes are heterogeneous. Some students need more challenge. Some need more support.
As a teacher, you need to constantly navigate between these extremes. The hard-core “I never give hints” stance of some fans of productive struggle is counterproductive. (See my posts About Hints.) If a student is stuck, refusing to help them so that they can struggle productively does work in some cases. But it often has unintended and possibly catastrophic consequences. The student may think you are not doing your job (which they see as explaining things, so they can do the work.) They may decide that you hate them, or just that you don’t care. Worst of all, they may conclude that they are “not a math person”. None of these outcomes are what you were hoping for.
Also note that if a student remains stuck because you refuse to help them, they have not engaged in productive struggle! So you do not achieve what you are trying to accomplish by withholding your help. I will quote myself (from About Hints, the sequel): trust your intuition, avoid dogma, be flexible, be kind.
Students all different, and you need to tune into those differences. Embrace opposites! Learn to navigate across the whole spectrum from hand-holding (100% support) to “sink-or-swim!” (100% challenge). Neither extreme is always right. You may sometimes hint too much, or too little, but over time your intuition about the right level of hintage will grow as you get better at sensing where a student is and how you can best help them without stealing their opportunity to think.