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.

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Faking Excellence: The Art of Milking Mediocrity for all its Worth

(Note: This is a guest post by my high schooler, an excellent student. It came out of a chat with some of her high-achieving friends.)

An Informative Guide

Part I: In the classroom

In order to uphold the image of “dedicated student” in the eyes of one’s educators, it is important to maintain a certain level of pseudoattentive behavior. Always have a notebook and a writing utensil out on your desk. Try to sit towards the front of the classroom, and make eye contact when teachers are lecturing. Take notes. Ask questions when you have them. (This practice both elevates the teacher’s opinion of you and helps to further the image of you as caring.) Greet your teacher upon entrance to the classroom. Converse with him or her on the finer points of their subject that you have diligently researched (see below). Bid them farewell upon your leave. Have books and supplies with you at all times. Participate in classroom activities. Make yourself feel actively present, and your teacher will take note.

Part II: Active Procrastination

In a survey of 2 high school students, both agreed that most to all of their homework is “boring.” As a result, we can conclude that one’s homework may not always come first in their lives. So if you don’t want to do it, does that mean you should binge watch Parks and Rec on the floor of your bedroom in shame? No! Procrastinate actively. Use homework time to expand your mind in more interesting ways. Read articles that are somewhat vaguely related to classroom materials (see above). Talk to your friends about how much you would prefer to do nearly anything but said assignment. Live the life of an overworked student while only spending a fraction of your time acting like one.

Part III: Completing Work

Close your eyes. Take yourself back to the last time you put off an assignment until 11:30pm the night before it was due. Get a good, long look at this mental image of last night, and open your eyes. Sure, you know how to put off work. But do you know how to cram it? The first lesson to be learned when attempting to do three weeks of work in one night is that you never outright admit this weakness. When dating the paper, always think back to when it was originally assigned. Then, count forward to the due date. Take one third of that number. Count that many days ahead from the original assignment date. There you have it: a believable but still respectable starting date. Exceptions may apply, but this is a good rule of thumb until you are a more seasoned procrastinator. The next lesson to be learned is the art of rephrasing. Many, many teachers steal each other’s work sheets. It is in their nature. So many, many foolish students at schools without honor codes (or with flagrant disregards to them) post the exact wording of these questions onto Yahoo Answers. And many, many Good Samaritans spend their time answering these questions. Learn to rephrase the work of these kind souls and make it sound like your own. Chop up sentences. Rearrange. Use synonyms. Expand on ideas. Cut down ideas. The Best Answer on Yahoo Answers is your marble, and you are Michelangelo. Now get on it, before you switch into complete sleep deprivation mode.

Part IV: Emulating Those More Well Rested than Yourself

The ideal model of a student is one who is not only well educated, but bright eyed and bushy tailed each and every morning. Now, on days when your eyes are more shadowed than bright and your tail is a deflated balloon, what is there to do? Worry not. The first step is hydration. Cold, cold water can jerk anyone out of dreamland, as can some nice old fashioned caffeine. Another tip is to remember the saying “dress for the sleep you wished for, not the sleep you got.” Wear clothes that make you look alive. Dead zombie clothes will turn you into a dead zombie. It’s science. Smile at things so that you do not appear to be a sleepy lump. And heaven forbid you fall asleep in class.

Part V: Eloquent BS

When completing various forms of free response questions, it is important to master the art of key term dropping. Sometimes, a question or prompt will only evoke a 404. message from your brain, and in that moment, do not be afraid. Recall the overall subject matter being assessed. Bring to mind the key terms of the section (often found alongside textbook passages) and think about whether you have any recollection whatsoever of how to use them. If so, you’re in luck! Teachers do not always read all 120 essays they have to grade, (and so especially for a class that isn’t a language class) they sometimes just skim to make sure that you have captured the general essence of the subject matter. Term dropping will not hurt, especially if you can bulk it up with any other somewhat related content. The author has personal experience of herself and very close friends getting 100s on answers for simply using the phrases “Christian-based society,” “complex gender issues,” “King John,” and “high death rates” in a paragraph with hardly any other coherence. Miracles do happen, but sometimes you have to help them along.

Part VI: Tying it All Together

In our short time together, you have learned how to become a more deceptively talented student. This skill, however, can only take you so far. Without a deep commitment to maintaining your facade of greatness, it will collapse like the Berlin Wall in 1989 and your lies will become obvious. Treat your mediocrity like a channel for something greater. Believe.