In a blog post seven years ago, I summarized “Embracing Contraries in the Teaching Process”, an important article by English professor Peter Elbow. In that post, and its sequel, I tried to apply Elbow’s ideas to the teaching of math. I encourage you to read both posts, and Elbow’s article (which I linked to in the first post.)

Elbow’s key point is that there is a tension between our obligation to our students’ education, on the one hand, and our obligation to our discipline, on the other. In many ways, those are “contraries”, as he put it, and yet we must learn to embrace both. If we are not committed to the discipline, we betray the students by offering a watered-down version of the subject. If we are not committed to the students’ learning, we betray the discipline by not promoting it to the next generation. Good teaching is in part the ability to navigate that continuum, sometimes emphasizing one obligation, sometimes the other, and thus avoiding the binary choice of being a “hard” or “soft” teacher. We must be 100% committed to our students’ education, and we must be 100% committed to mathematics. To quote my 2013 post:

My commitment to mathematics requires me to read a student-written proof with a relentlessly critical eye, making sure the logic is tight and the writing clear. But my commitment to the student requires me to be encouraging and supportive, to look for whatever germs of good thinking are there even if the proof is flawed, and to find ways to build on those.

So yes, we have to embrace opposites. But there are some widespread practices which in my view serve neither the students, nor the discipline. Those are the practices that prioritize student compliance.

Some are obviously unrelated to learning math. Penalizing students for placing the staple in the wrong place, not putting a box around the final answer, using the supposedly wrong writing implement, or any similar violation is absurd. Sure, some of those sorts of requirements facilitate our lives as teachers, and there’s no reason not to explain that to the student. But it is ridiculous to have students lose “points” over them.

And then, there is the issue of time.

*The clock*: the quiz needs to be completed by 10:10. Why? If a student needs or takes more time, what difference does it make?*The calendar*: this concept needs to be mastered by Friday. Really? Understanding achieved a week later is not as valuable?

Again, this is an issue of teacher convenience, and should be discussed as such. Within reason, we should do everything we can to reduce time pressure. (To watch my entertaining 5-minute video on that topic, go to my About Teaching page, and scroll down. See also my detailed advice on how to address the reality that students learn at different rates.)

Finally, there are some practices which on the surface appear to be about math, but in fact are about enforcing obedience. Here are some examples:

*Overemphasis on terminology, and unnecessarily complicated notation.*For example, asking kids to learn the words “minuend” and “subtrahend”. (Fortunately that does not seem to have survived in contemporary curricula.) Or insisting on a different notation for a segment and its measurement. (I learned high school geometry in the French system, and we were not burdened with this sort of pickiness.)*Rejecting correct answers that have not been “simplified”*(fractions, radicals), or are in the wrong format. Students make enough real mistakes — let’s not tell them they’re wrong when they’re right! Example: writing polynomials not in descending order.*Memorization without understanding*: if students are asked to memorize some facts or algorithms that mean nothing to them, it’s just an exercise in blind obedience.

Yes, there is a place for all those things. Distinguishing “equal” from “congruent” makes sense for polygons but not so much for line segments. Lowest terms fractions and simple radical form can be useful, and they can help communication, but they should not be elevated beyond that. Memorized shortcuts and formulas are worthwhile if they encapsulate understanding, but not if they are a substitute for it. And so on.

In general, when we make pedagogical and curricular choices, we should ask ourselves: does this choice support the math? does it support my students? or is it more about controlling the students and enforcing compliance? If we are strong in our commitment to our discipline and to our students, we can dramatically decrease our preoccupation and emphasis on demanding obedience.

— Henri

Not unrelated: my article on The Assessment Trap.

Henry, I find this post very relatable, especially when you introduce the idea of time being a restraining factor on all students. I find that I am constantly urging students to get better with their math facts because this is the ultimate reason the majority of students are struggling to complete assessments. The simple fact they cannot recall quickly what “9 x 8” is causes them to take more time to arrive at a solution to a 6th grade level math problem. What are your thoughts on this? Should I allow students multiple days to take assessments? Should I shorten may assessments? I want to encourage creative thinking, but we all know that takes time out of the day which is something teachers are constantly fighting against. What would you suggest would be the best way to balance creative thinking in a 6th grade classroom while also being on a time constraint?

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Yes, it is far better to “just know” 9 x 8, but what do we do about students who don’t “just know” it? If they don’t know it by 6th grade, is there any reason to think you will be successful at changing that state of affairs? If they don’t know 9 x 8 it may be that it was not taught effectively and they don’t have a clear sense of what multiplication means. It may be that they have some sort of learning difference. Who knows. The main thing is to decide to support the students, not penalize them.

You can shorten assessments. You can change the assessments to not emphasize speed. You can have a multiplication table on the bulletin board, or maybe just the few facts that trip kids up so they can just look and see. You can allow calculator use in assessments that are not primarily about calculations. You can schedule assessments at a time that would allow students who need more time to get more time, while those who finish quickly do something else. There are lots of options, and I’m sure you can come up with your own ideas.

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