Five years ago, I had lunch with a 13-year-old who was thinking about attending my school.
He spent much of the lunch telling me about string theory. As one does, when one is 13, and obsessed with string theory.
I don’t remember much about string theory, but I do remember this part of the conversation:
ROBERT: Of course, we can’t prove any of this yet.
ME (a bit disappointed): Oh. What would we need to prove it?
ROBERT (blandly): We need to invent a new kind of math.
His tone said it all. Inventing a new kind of math struck him as entirely unremarkable. We’ve got this cool scientific theory, and we need some better math to prove it — so let’s get hopping and create that better math.
Why are we waiting? Chop chop!
“Mathematical Mindsets”: Inventing a New Math
Robert’s comment sprang immediately to mind when I read Jo Boaler’s book Mathematical Mindsets.
As she champions applying growth mindset teaching to math, she defines a “mathematical mindset” this way:
When students see math as a series of short questions, they cannot see the role for their own inner growth and learning. They think that math is a fixed set of methods that either they get or they don’t.
When students see math as a broad landscape of unexplored puzzles in which they can wander around, asking questions and thinking about relationships, they understand that their role is thinking, sense making, and growing.
When students see mathematics as a set of ideas and relationships and their role as one of thinking about the ideas, and making sense of them, they have a mathematical mindset.
On the one hand, that seems like a splendid goal. Math teachers everywhere want more students to emulate Robert.
That is, we want students to perceive math NOT as an established set of facts and procedures, but as a field to explore AND EXPAND with their efforts.
Expanding Mathematical Mindsets
Although the idea of “mathematical mindsets” initially sounds splendid, it does — I think — require some essential revision.
For fun, let’s repeat Boaler’s definition; this time, however, I’m going to change two words. (See if you can spot them.)
When students see history as a series of short questions, they cannot see the role for their own inner growth and learning. They think that history is a fixed set of methods that either they get or they don’t.
When students see history as a broad landscape of unexplored puzzles in which they can wander around, asking questions and thinking about relationships, they understand that their role is thinking, sense making, and growing.
When students see history as a set of ideas and relationships and their role as one of thinking about the ideas, and making sense of them, they have an historical mindset.
Isn’t that what history teachers want? Don’t we want our students to perceive history NOT as an established set of facts and procedures, but as a field to explore AND EXPAND with their efforts.
Come to think of it: isn’t that what science teachers want?
Isn’t that what music teachers, and English teachers, and theology teachers want?
Isn’t that what TEACHERS want?
In other words: there is nothing especially mathematical about “mathematical mindsets.” That’s not the goal of math education; it’s the goal of education, period.
Here’s the paradox: I don’t think Boaler’s term needs revision because it’s wrong, but because she hasn’t made her claim bold enough.
At present, it’s too narrowly focused on math.
I think, instead, we can understand her idea as a special addition to Dweck’s work on Mindset.
Dweck’s work, after all, focuses on a student’s perception of her own ability. If I have a fixed mindset, I believe my ability can’t change. If I have a growth mindset, I believe it can.
Boaler — quite wonderfully — switches the mindset away from malleability of self to malleability of academic discipline.
Students who have a fixed disciplinary mindset believe that a particular field — math, Spanish, lacrosse, sculpture, chemistry — won’t change. Everything to know about these things is already known, and our goal is to absorb that current store of knowledge.
On the other hand, students who have a growth disciplinary mindset believe that these fields can indeed change. In fact, they believe that their efforts might change them:
Robert thought that someone will invent a new math. (Perhaps he has already done so.)
My student Serena believes she can not only analyze literature, but can add to the field of literature by writing a novel.
My friend’s daughter Hannah believes she’ll come up with a new chess combination to defeat the London System.
These students have growth mindsets not only about themselves, but about the very fields they are studying.
We do need a better term for this perspective. (When I’m feeling pun-ful, I think we need to make Boaler’s terminology even “boalder.”)
“Mathematical mindset” is — as I’ve explained — too limited.
“Disciplinary mindset” is broader, but sounds a bit like punishment.
So: “inventive mindset”? “Expansive mindset”? “Exploratory mindset”?
Send me your suggestions. Let’s spread the word.