It is what we know already that often prevents us from learning.
- Claude Bernard
I'm confident most people would agree that life is a never-ending process of learning. Sometimes what is learned is significant and salient. Sometimes what is learned is menial and unnoticeable. We're constantly presented with situations that make us think and respond, and the required adaptations require learning.
I'm confident most (all?) people in the academy of mathematics education would agree that their study and scholarship is a never-ending process of learning. I certainly agree. But, I believe we, as a field, often overlook...underplay...ignore...are not aware of...I'm not sure of the right word or phrase...a form of learning that is the foundation of our field and purpose.
The learning of mathematics.
Here, I do not refer to the study of traditional mathematics. Although I firmly believe that is a critical exercise for members of the field to participate in, I'm referring to a different mathematics. The form we often overlook...or maybe I have the phrase now...the form we often lack precision and clarity about is the mathematics of others. There is no doubt our field champions the mathematics of others, but rarely does a researcher accomplish this in a way that simultaneously honors the mathematics of others and challenges their personal mathematics. One need look no further than the pervasiveness of researchers applying their own mathematical knowledge and structures to others. This often occurs implicitly and despite the researcher's intentions suggesting otherwise. Methodologies rarely afford a researcher the critical pursuit of another's reasoning, thus positioning a researcher in a place where only their extant operations are used to make sense of another's observable products and behaviors.
I challenge all of my advisees to think about the questions: What mathematics are you learning from your students? What mathematics are you learning from your research participants? What are you doing to intentionally learn from others and force a critique of your own mathematics?
In working with students and research participants, one should—as best as possible—put themselves in a position to investigate a new mathematical reality. That investigation and constructed reality of another should shape and mold their mathematical reality. Most, if not all, of the mathematical ideas that trace through my scholarship are far from my own. Most, if not all, of those mathematical ideas were co-constructed through interactions with students and attempting to formulate mathematical realities that I can attribute to them. My mathematical reality developed so I could provide a viable model of my participants' mathematical realities. Said another way, I was not their teacher, they were my teachers. (as an aside: this underscores Piaget's stance that social interactions are one of the most important drivers of learning)
Returning to Bernard's quote, this form of learning requires a researcher to act in ways that abandon their personal mathematics in favor of understanding their participant's mathematics; what one already knows can stand in the way of learning their students' mathematics. Relying on what one already knows leads to characterizing students and participants as "not knowing things" and "having difficulty", and looking to remedy that. Or, relying on what one already knows leads to over-attributing a mathematical reality to others even when trying to honor their personal narratives, culture, and context. On the other hand, whether a researcher or teacher, relying on what a student knows opens the doors to all sorts of possibilities and a mathematical world one would never reach without learning through their student.