Modeling Mathematical Aesthetics

Fractal-Geometry-HQ-Desktop-Wallpaper-24806Note: This post was written by my two doctoral students, Lara Jasien and Nadav Ehrenfeld, as part of the Virtual Conference on Mathematical Flavors. This essay responds to the prompt  “How do teachers move the needle on what their kids think about the doing of math?” It is also part of a strand inside that conference, inspired by an essay by Tim Gowers,Two Cultures in Mathematics.

Before we begin responding to Gowers’ essay, we’d like to share a little bit about what draws us to this conversation. As budding researchers of mathematics teaching and learning, we spend our days watching teachers go about their daily routines with their students. We look for the ways teachers support their students to engage in meaningful learning and position them as capable, curious thinkers. Our work is fundamentally concerned with the ways classroom culture shapes what it means to teach, learn, and do mathematics. Gowers’ essay provides us with an interesting new lens on the role of culture in mathematics. We want to share (what we think is) a problem-of-practice worth considering and then point to an often overlooked teaching move that we recently saw a teacher use in ways that counteracted this problem-of-practice.

As educators and learners of mathematics, our experiences usually involve engaging students in correctly solving mathematical problems that are predetermined, handed down to us over generations through textbooks and pacing guides (with slight variations). This means that students have few opportunities to engage in a core element of mathematics — finding and articulating problems that are interesting to solve. We think this is intimately connected to a missing aspect of mathematics culture in typical math education: the mathematical aesthetic. Mathematicians’ aesthetic tastes and values lead them to pursue some problems, solution strategies, and forms of proof write-ups over others. When mathematicians’ inquiry is driven by their aesthetics, they engage in exploration, noticing, wondering, and problem-posing.

The mathematical aesthetic is the mechanism by which mathematicians distinguish between what they experience as meaningful, interesting mathematics and trivial, boring mathematics. In his essay, The Two Cultures of Mathematics, W.T. Gowers identified two groups of mathematicians who find each other’s work equally distasteful (with a little dramatic flare): problem solving mathematicians and theoretical mathematicians. Typically, school instruction exposes students to problems that fit both cultures of mathematics: Some school mathematics is done for mathematics sake, some is done for the purpose of real life or pseudo real life (word problems) problem solving. Yet, when do students have the opportunity to develop aesthetic preferences for different ways of engaging with and thinking about mathematics?

In our work, we have seen classrooms with cultures that support students in posing questions to their peers — questions like, I wonder if there is a reason for that? or What’s your hunch?. In these classrooms, we see students begin to be interested in and passionate about mathematics. In our minds, when students develop such passionate tastes about meaningful mathematics, we are on a good track for empowering our students for success.

The questions we just mentioned are actually questions we recently overheard a teacher asking her students as her last statement before exiting small group conversations. We consider her enthusiastic questions to be a form of modelling mathematical aesthetics, prompting students to be curious, explore, wonder, and use their intuition. While ideally the classroom culture would eventually lead to students asking themselves and each other these kinds of aesthetic questions, we know that our own authentic intellectual curiosity as educators does not go unnoticed by our students. Importantly, this teacher did not ask these questions and then hang around and wait for student answers. She left the students with juicy questions that they could investigate together.

As teachers, we rarely get feedback on how our exit moves from small group conversations affect their conversation or the classroom culture. Of course, some exit strategies ­–– such as telling students the answer or funneling them towards it –– will clearly lead to cultures where students see mathematics as a discipline of quick-and-correct answer finding. This view of mathematics can preclude opportunities for students to develop as autonomous doers and thinkers of mathematics. Fortunately, options for productive exit strategies and modelling of intellectual interests are many. These options also present new decision-making challenges to teachers as what happens when we exit the conversation becomes far less predictable. Our students do not need to have the same mathematical tastes as we do, but we do want them to feel empowered and intellectually curious in our classrooms. By foregrounding noticing, wondering, and problem-posing as authentic mathematical practices, we can support students in developing their own mathematical aesthetic. Of course, doing so requires us to model genuine intellectual curiosity, make room for uncertainty and ambiguity in our tasks (groupworthy!), create access to multiple resources for pursuing mathematical questions (Google is acceptable!), and scaffold for conversation rather than bottom-lines (exit moves!). Leaving students with a juicy, natural question is a start.

Auditing Your Classrooms for Competence and Status

This past weekend, I had the great pleasure of giving a keynote address at the Mathematics Council of Alberta Teachers (MCATA) Conference.

First things first: @minaclark did sketch notes of my talk!  I am delighted because I have always wanted somebody to do that. She did a fantastic job too.

During the breakout session afterwards, I talked about how we can audit our classrooms to support better interactions. In particular, we need to pay attention to issues of mathematical competence and student status. (I have written a lot on these topics since they are critical to fostering positive relationships between students and the subject. You can read earlier posts here, here, and here.)

Here are my audit questions.

Competence audit:

  • What kinds of competencies are valued in your classroom? Where do students have a chance to show them?
  • Consider the last few activities you have done in your class. Did they provide multiple entry points toward a rich mathematical idea? If not, can you use the table below to adapt them to become a low ceiling/high floor question?
  • When you look at your class roster, can you identify at least one way that every student is mathematically smart?
  • When you think of students who struggle, do they have competencies that you might better support by redesigning some of your class activities?
  • When you think of students who have a history of high achievement, do they value other ways to be smart aside from quick and accurate calculation? Do they value other competencies in themselves? In others?
table

Some low floor-high ceiling question types. (Adapted from Will Stafford’s “Create Debate” Handout)

Status audit:

  • When you think of the students you worry about, how much of their challenge stems from lack of confidence?
  • How much do students recognize the value and contributions of their peers?
  • What small changes could you make to address status problems and support more students in experiencing a sense of competence?

Please feel free to add others or offer your thoughts in the comment section.

First, Do No Harm

I have often wondered if teachers should have some form of a Hippocratic Oath, reminding themselves each day to first, do no harm.

Since the network of relationships in classrooms is so complex, it is often difficult to discern what we may do that causes children harm. Most of us have experienced the uncertainty of teaching, those dilemmas endemic to the classroom. Was it the right decision to stay firm on an assignment deadline for the child who always seems to misplace things, after giving several extensions? Or was there something more going on outside of the classroom that would alter that decision? Why did a student, who is usually amendable to playful teasing, suddenly storm out of the room today in the wake of such an interaction?

What I have arrived at is that there are levels of harm. The harm I describe in the previous examples can be recovered if teachers have relational competence — that is, the lines of communication are open with their students so that children can share and speak up if a teacher missteps.

What I am coming to realize is that mathematics teachers have a particular responsibility when it comes to doing no harm. Mathematics, for better or worse, is our culture’s stand-in subject for being smart. That is, if you are good at math, you must be smart. If you are not good at math, you are not truly smart.

I am not saying I believe that, but it is a popular message. I meet accomplished adults all the time who confess their insecurities stemming from their poor performance in mathematics classes.

Here is an incomplete list of common instructional practices that, in my view, do harm to students’ sense of competence:

1. Timed math tests

Our assessments communicate to students what we value. Jo Boaler recently wrote about the problems with these in terms of mathematical learning. Students who do well on these tend to see connections across the facts, while students who struggle tend not to. But if timed tests are the primary mode of assessment, then the students who struggle do not get many opportunities to develop those connections.

2. Not giving partial credit

Silly mistakes are par for the course in the course of demanding problem solving. Teachers who only use multiple choice tests or auto-grading do not get an opportunity to see students’ thinking. A wrong answer does not always indicate entirely wrong thinking. Students who are prone to getting the big idea and missing the details are regularly demoralized in mathematics classes.

Even worse, however, is …

3. Arbitrary grading that discounts sensemaking

Recently, a student I know had a construction quiz in a geometry class. The teacher marked her construction as “wrong” because she made her arcs below the line instead of above it, as the teacher had demonstrated. This teacher also counts answers as incorrect if the SAS Theorem is written as the SAS Postulate in proofs. Since different textbooks often name triangle congruence properties differently, this is an arbitrary distinction. This practice harms students by valuing imitation over sensemaking.

4. Moving the lesson along the path of “right answers”

Picture the following interaction:

Teacher:    “Can anyone tell me which is the vertical angle here?”

Layla:        “Angle C?”

Teacher:     “No. Robbie?”
Robbie:       “Angle D?”

Teacher: Yes. So now we know that Angle D also equals 38˚…

That type of interaction, called initiation-response-evaluation, is the most common format of mathematical talk in classrooms. Why is it potentially harmful? Let’s think about what Layla learned. She learned that she was wrong and, if she was listening, she learned that Angle D was the correct answer. However, she never got explicit instruction on why Angle C was incorrect. Over time, students like Layla often withdraw their participation from classroom discussions.

On the other hand, teachers who work with Layla’s incorrect answer –– or even better yet, value it as a good “non-example” to develop the class’s understanding of vertical angles –– increase student participation and mathematical confidence. And, they are doing more to grow everybody’s understanding.

What are other kinds of teaching practices that stand to “harm” students?

Asking The Right Question

John Dewey . . . asked a class, “What would you find if you dug a hole in the earth?” Getting no response, he repeated the question; again he obtained nothing but silence. The teacher chided Dr. Dewey, “You’re asking the wrong question.” Turning to the class, she asked, “What is the state of the center of the earth?” The class replied in unison, “Igneous fusion.” (Bloom, Engelhart, Furst, Hill, & Krathwohl, 1956, p. 29)

Did the class know about the center of the earth? The point of this little parable, of course, is not really. Being able to correctly recite the words ‘igneous fusion’ on cue does not make evident that these words had meaning to the students.

Meaning. An ultimate goal of learning, to enrich our lives with meaning.

From this perspective, Dewey’s question was the right one. It connected school learning to meaning in the world.

How do we do that in math class, the place with the most deep-seated rituals of recitation and mindless calculation? How do we move from what English mathematician and philosopher Alfred North Whitehead called ‘inert ideas’ — those which have been received and not utilized or tested or ‘thrown into fresh combinations’?

I am about to go teach a class of pre-service teachers to engage this very set of issues. We are looking in math class at the nature of the tasks being used through the lens of Margaret Smith and Mary Kay Stein’s work on this topic.

I already anticipate some of the arguments that will come up about the richness of particular tasks. Over the years of using this framework with both new and experienced teachers, I can boil the issues down to these questions:

1. What kind of thinking will students do as they engage in this activity?

2. Is there sufficient ambiguity that they can approach it in different ways or re-think familiar ideas in new ways?

I would love to hear from others about your ways of judging good mathematical tasks, as well as your favorite resources for finding them.