Defining mathematics is the motivation behind Professor Eva Thanheiser's recent paper, "What Is the Mathematics in Mathematics Education."
Thanheiser, a professor of mathematics education at Portland State University, discovered that a clear consensus among other mathematicians on what defines mathematics proved to be both a struggle and open to debate.
Thanheiser concluded that during these discussions, while some educators view mathematics as an abstract, politically neutral area of study, others feel the way mathematics teachers deliver curriculum in schools doesn't provide for social and political equity, leaving many students struggling, discouraged, and left on the margins.
"One of the things that became clear to me is that I don't think we should teach math that is only this thing that grants you access," Thanheiser notes. "[Or] as a tool for getting into college. Why don't we teach it as a way to view the world and help us understand it?"
Thanheiser's paper, published in the Journal of Mathematics Behavior, attempts to define–and elevate the conception of–mathematics by providing a three-frame approach.
The first frame, Thanheiser notes in her paper, is the standard mathematics curriculum in K-12 classrooms and academia; rote understanding and comprehension of mathematical concepts, operations, and relations, as well as the retention of procedural fluency to comprehend and solve math equations.
"The issues with frame one," says Thanheiser. "it's exclusive. It's rooted in the European White supremacist culture, and we see that by the results; by who gets access and who doesn't."
In Frame 2, Thanheiser draws on Monash University Emeritus Professor Alan Bishop's six general forms that mathematics can take, each steeped in context: counting (how many?), locating (where?), measuring (how much?), designing (what?), playing (how to?), and explaining (why?). Mathematics, in this context, is a tool for exploring the world, viewing and understanding cultures apart from those that are immediately familiar to us.
Almost everybody can pinpoint an experience in school where they started disliking math, multiplication, times tables, or fractions. And so why, if this is how we experienced it, why are we so set on our kids having the same experience? Meaningful math instruction might look really different in different communities, right?
From Bishop's six general forms, Thanheiser extrapolates that math can be a tool for solving formulations and becoming questioners and identifiers of injustices. "It starts with understanding," she explains. "'My life's different from yours. My life is different from other people's.' Let's use math to help us understand ourselves and each other. Let's approach the world with, 'Oh, I wonder if this is true. I wonder what does this actually mean.'"
The shift in perceiving mathematics as a noun (a stable thing one needs to learn about and understand, existing without human activity) and seeing it as a verb (something created and enacted by humans) comprises Frame 3. This shift in understanding, Thanheiser observes, allows teachers to work with students as the authors of their mathematical comprehension. Mathematics, in this sense, shifts the context from practices involving quantitative thinking outside of the learner to an activity that individuals perform innately, inextricably linked to identity and making sense of the world around them.
"If how we look at the world is (and must be) informed by mathematics," Thanheiser writes. "then mathematics is part of our identity. Mathematical identity is who we are."
Reframing mathematics as a verb, Thanheiser postulates, teaches students to become questioners, making them better and more active participants. Students who engage in thought-compelling math activities that speak to their interests and culture give them equity and representation that nationally administered and rigidly structured mathematics testing, ranking and scoring may not.
"Almost everybody can pinpoint an experience in school where they started disliking math, multiplication, times tables, or fractions," Thanheiser says. "And so why, if this is how we experienced it, why are we so set on our kids having the same experience? Meaningful math instruction might look really different in different communities, right?"
Taken together, Frames 1 ("I hear and see math"), 2 ("I hear and see the world with math") and 3 ("I do math, I can use math to change the world, I am math") form an overlapping Venn diagram illustrating the relationship among all or some of the three Frames. "I don't want to have my view of how I view things imposed on others," Thanheiser explains. "Other people might see differently. So here's three different views of math, and how I see them interrelating, and where I find myself."
Thanheiser sees the three frames outlined in her paper as markers in her development as a mathematician and critical thinker. So much so she envisions her framing system as a continuum between things she's learned and ideas she hasn't even thought of yet. "I like to keep learning," she says. "So maybe there's going to be a Frame 4 coming out at some point, but I'm just not there yet."