“In math, the answer is either right or wrong.”
This sentence is frequently offered up by math haters and math lovers alike as a defense of their respective sentiments toward the subject. The lovers experience the right-or-wrong-ness of mathematics as comforting and satisfying; the haters find it constricting and stultifying. No matter the light in which it’s viewed, this perception of mathematics as The Land of Right and Wrong can actually have a negative impact on students, their learning, and their view of themselves as being capable of mathematical reasoning.
If students are given the impression that there is a right answer to every question, and that they are expected to know what it is, then those students will, by and large, become more anxious about, and less interested in, engaging in class discussions. Sometimes this attitude is reinforced by a very common model of questioning sometimes referred to as IRE, whereby a teacher Initiates an interaction by posing a question, a student Responds to the question, and the teacher Evaluates their response. In math classes, this often looks like the teacher is hunting for the right answer, and hoping that a student will offer it. But more often than not, this model only reinforces the dynamics already present in the classroom; the more adept students run the show, and those who might simply need more processing time are discouraged from participating. So what is an alternative for teachers who want their classroom to be a “safe space” where students can feel comfortable sharing their ideas?
In 2015, Max Ray-Riek gave a five-minute talk at the NCTM Annual Conference entitled, “Why 2 > 4.” This was quite the hook for a room full of math teachers! He gave away the trick right away, though, saying that what he meant was that “to” is greater than “for.” That is, when interacting with students in class, listening to students’ ideas, and taking them seriously, is far better than listening for specific content that one may be hoping to hear. This hit me like a ton of bricks; it was a near perfect encapsulation of things I had been thinking about on my own for some time, and it forever changed how I conduct my classes.
So how does one truly listen to student ideas, and take them seriously? First, it takes a lot of self-awareness. I have to ask myself: When I throw out a particular question to the class, am I expecting, or hoping, to hear something specific? Am I open to hearing things I’m not expecting? How will I respond if that happens?
Inevitably, you start to change the kinds of questions you ask. You eschew “What…?” questions, which typically do have one correct answer, for more “How…?” and “Why…?” questions, whose answers depend almost entirely on the answerer. And if a student gives a response that you were not expecting, you have to be willing to take that detour, and help the student flesh out their idea with more questions.
In my Advanced Topics class, one of the collective projects we undertake is the creation of our own list of axioms, from which we try to prove simple, fundamental things about the set of integers. (For example, you may rest more easily now that we’ve proven that 0 and 1 are not the same number.) Many, many others have gone before us in such a task over the centuries (Such as Bertrand Russell & Alfred North Whitehead in their Principia Mathematica), but as I emphasize to the students, I have absolutely no interest in the thinking that others have done. We are not going to view the thinking of others as “correct!" I want to know what they, from their many years of experience, think the fundamental properties of the integers are. It’s a challenging exercise, akin to asking fish to describe the chemical composition of water. But we stagger through it, and in the process we begin to see why definitions are important, why it’s desirable to keep our axiom list short, how all of these fundamental properties are related to each other, and many other things about the foundations of mathematics.
But most importantly, we develop some ownership of our work. These ideas did not come from a textbook; they came from student minds! The finished product (though such work is never really finished) looks slightly different every year, a snapshot of those minds, in that room. It’s the ultimate exercise in valuing student voice, and it’s one of my favorite projects every year.
But “right” and “wrong” have little place here. Their definition of “less than,” for example, is not right or wrong; it is something that corresponds to their understanding of the concept, and that fits in with the other statements in their list. Same with their definition of “prime”: In its current version, 1 is prime, as are infinitely many negative numbers. This does not correspond to the more common definition of “prime,” but for now, that’s okay. We will refine our definition as the need arises, and even if we wind up keeping our current definition, we will eventually see what might bring one to make a different choice.
Students are often initially uncomfortable with this degree of uncertainty, this shaky foundation upon which they’ve been placed. But once they recognize that it is because their voices and ideas are being given greater value and weight, they soon buy into the new paradigm. Math class feels less like something that is being done to them, and more like something that they have some degree of control over. And these are the conditions that foster healthy student perceptions of both mathematics and themselves.