Here is Frank Quinn’s article “A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today.” Appearing in Notices of the AMS, January 2012
This article articulates clearly, (in a way I wish I could!), some of the confusion/friction I have encountered between the fields of Mathematics and MathEd. It seems to me that so often that when a professional mathematician says “mathematics” they mean something entirely different than when a mathematics educator says “mathematics”. Quinn points to the mathematical revolution occurring in the late 19th and early 20th centuries, a revolution that in his estimation reached the math research community but did not reach the MathEd community as the source of this disconnect. I have been mostly trained as a proto-research mathematician (what Quinn might call a “core user”), not as a math educator. And as such, this part of Quinn’s article resonated in particular for me:
“The new methodology is less accessible to non-users. Old-style definitions, for instance, usually related things to physical experience so many people could connect with them in some way. Users found these connections dysfunctional, and they can derive effective intuition much faster from precise definitions. But modern definitions have to be used to be understood, so they are opaque to nonusers. The drawback here is that nonusers only saw a loss: the old dysfunctionality was invisible, whereas the new opacity is obvious.”
As a proto-research mathematician, mathematical objects do not have any existence outside of our definitions of them, let alone any sort of empirical existence. They are figments of our collective imaginations (that we can use to predict the future, how cool is that?). So trying to “explore”, or “discover”, or develop Vinner’s concept image, or develop an “intuitive understanding” outside of a definition is literally impossible. One cannot understand a thing that does not exist! One may as well seek to learn facts about unicorns. (There are no facts about unicorns. My apologies to Hannah Gadsby). One develops intuition about a mathematical object by winding up the definition and letting it go to see what it does (aka proving theorems about the object). Proof is the method by which we come to know in mathematics. (Though, like “mathematics”, “proof” seems to mean two different things to the two communities). One can certainly motivate a definition empirically or otherwise, but a motivation does not necessarily constitute mathematical knowledge or learning.
This stands in sharp contrast to the mathematics education perspective, in which one tries to develop a concept image almost exclusively through experiment and intuition, rather than deduction. In my estimation there exist certain areas of math that this approach is better suited to than others. Euclidean geometry comes to mind. Which is ironic, considering this is the area of the school curriculum in which precise definitions and deductive proof are most emphasized.
Anyway. These are some of my initial thoughts. Pelican Brief forthcoming. I hope I’ve piqued somebody’s interest, I’d be curious to hear your thoughts in the comments or on social media.