## A Function of Freedom and Constraints

In June, a colleague invited me into his classroom to teach a Desmos modelling task — Predicting Movie Ticket Prices — in his Math 12 class. Students experienced exponential functions earlier in the course. We were curious about whether his students would apply what they knew about exponential functions to a task situated outside of an exponential functions unit — a task not having to do with textbook contexts of half-life, bacteria, or compound interest. They did. And they deepened their understanding of how change by a common ratio appears in exponential equations (vs. change by a common difference in linear equations). They did this within 45 minutes of a 75-minute class. So my colleague let me try out another, less sexy, task — one adapted from MARS. This task, like much of Math 12, is about naked functions; no real-world context here. Nat Banting’s closing keynote at #NWmath reminded me of it. Watch Nat’s talk; view his slides.

The original MARS task above is closed: two functions, one linear and one quadratic, each passing through four points. I wanted to open it up so I changed the prompt: “A set of functions pass through the points shown. What could the equations for the functions be?” Also, I removed one of the points — (5, 3) — to allow for different solutions of two functions. The thinking is that open questions encourage a variety of approaches. And then, from fifteen pairs of students:

I anticipated this. The points scream linear and quadratic. They are sources of coherence. I had lowered the floor but no Rileys entered y = 5, y = 7, y = 8, y = 9. The problem wasn’t problematic. I had raised the ceiling but no one wrestled with equations for sinusoidal or polynomial or radical or rational functions. The freedom within my open question didn’t bring about new and diverse ideas. To support creativity — mathematical creativity! — I had to introduce a source of disruption, a constraint“A set of nonlinear functions pass through the points shown. What could the equations for the functions be?”

A student could have used the linear nature of absolute value functions to get around my nonlinear constraint — a bit of a Riley move? — but no one did.

Instead, some students picked up on the symmetry of two new possible parabolas:

Writing the equation of the second parabola — finding the parameters a and q — presented more of a problem.

Others bent the line; they saw the middle of its three points as the vertex of a cubic function that had been vertically stretched and reflected:

Some saw four compass points and wrote an equation of a circle. This led to a function vs. not a function conversation: “Does that count?” Others saw a sine function that passed through three of these four points. There were “close enough” solutions — great for Coin Capture but not quite passing through the given points:

I didn’t anticipate this. Students weren’t as constrained by “pass through” as I was. Also, they were motivated to capture the points using only two functions, as before.

With more time, I could have shifted constraints again: “A set of functions pass through the points shown. What could the equations for the functions be? (P.S. The graph of at least one of them has an asymptote.)” This would have triggered exponential and logarithmic or rational functions. (Even without introducing this constraint, we noticed at least one student playing with rational functions at the end of class.)

Above, there’s evidence to support Nat’s #NWmath conjecture: “Shifting constraints triggered new mathematical possibilities.” My (more) open question didn’t cut it. The student thinking — and conversations — that I had hoped for only emerged when freedom “sloshed against” constraints.

Desmos activity

# Artsy

A few years ago, I completed a questionnaire to determine my personal operating style. I’m green. Creativity. At first, I questioned the validity of the assessment. I didn’t see myself as creative. I’m not… artsy.

But taking a closer look, the results made sense. I scored very highly in the four strategies that made up creative thinking in this system: brainstorm ideas, challenge assumptions, reframe problems into opportunities, and envision possibilities. To be clear, this was an assessment of preferences, not proficiencies. Also, there are trade-offs; to choose one thing is to reject another. For example, my 98 in reframe and 91 in envision meant zero — zero! — in tune-in to feelings and ten in empathize with others. These results did not suggest that I can’t tune-in and empathize; they did suggest that I don’t want to. Preferences, not proficiencies.

More important to this post, this assessment tool offered a different definition of creativity: “the generation of a wide variety of options, ideas, alternatives and fresh ways of approaching difficult situations and everyday challenges.” BC’s Ministry of Education defines creative thinking, one of the core competencies, as “the generation of new ideas and concepts that have value to the individual or others, and the development of these ideas and concepts from thought to reality.” There are similarities between these two definitions: both talk of the generation of novel ideas; neither talk of art.

# Broccoli with Cheese Sauce

The MoE also has this to say: “Core competencies are evident in every area of learning; however, they manifest themselves uniquely in each discipline.

Over the last few years, I’ve sat through many presentations where examples of creative thinking across subject areas have been shared. The examples from mathematics almost always make me cringe. The math song is a common offender. (Usually the topic tips towards the procedural — BEDMAS, the quadratic formula, etc. — but that’s a different post.) Here, creative thinking manifests itself outside of mathematics. It happens in language/fine arts. (Maybe. Talk to a language/fine arts teacher.) You can substitute provincial capitals for divisibility rules and the nature of creative thinking within the task remains unchanged. Math is merely the context.

Worse, the message is that math is unappetizing in and of itself. Broccoli. The cheese sauce that is the math song (or poster, or skit, or diorama, or …) comes at a cost. Limited time means tension — time spent on products versus time spent solving interesting problems and having interesting conversations. Note: in my mind, the opportunity cost isn’t coverage of content; it is engaging students in the “doing” of mathematics.

Yesterday, I attended a meeting where the MoE repeated the message: “By doing the curricular competencies, students will be developing their core competencies.” The math song attempts to have students develop a core competency without doing the curricular competencies.

# Et tu, Desmos?

The connection between creativity and art is strong:

I’d like to suggest a better title:

Creative Math is clearly evident. Just click on one of the staff picks and look to the left. Focus not on the equations themselves, but on the thinking behind them. Not on “front mathematics,” but on “mathematics in back.” (A lovely metaphor from Reuben Hersh that I first came across in Tracy Zager’s Becoming The Math Teacher You Wish You’d Had.)

To most math teachers, this title makes no difference. Just me nitpicking. But it matters where teaching includes designing curriculum/learning experiences. If teachers think of creativity in terms of art, they may look to Pinterest when planning; if they think of creativity in terms of ideas, they may dive deeper into Desmos.

Last year, one of my highlights was being invited into a classroom to observe Marbleslides: Lines in action.

I observed students experimenting with new ideas by changing the variables one at a time. They asked “what if…” questions. They made — and checked — predictions. “New ideas” here means new to the students themselves. These new ideas had value, evident in cheers and high fives. “Right here, right now” value, not “real-world,” career, or “when you take Calculus” value.

(The Desmos Teaching Faculty designed the activity with students in mind who were familiar with equations for lines in slope-intercept form and the idea of domain. In the classroom that I visited, the students were not. We worried that introducing restrictions on the domain at the same time as slope-intercept form would overcomplicate things. It didn’t.)

# More Mathematical Manifestations

I don’t fault my fellow educators for associating creativity with art. It’s a natural thing to do. We in mathematics education need to articulate better what creative thinking looks like in mathematics. I’ve had some success in asking teachers to sort curricular competencies by core competency. (Here they are, in random order. Venn diagrams work nicely; I let that idea come from teachers themselves.)

There’s still the leap required to go from making these connections to designing curriculum/planning learning experiences with these connections in mind. Rather than listing activities that elicit creativity, like Marbleslides above, it may be helpful to think about the attributes of these tasks.

Marbleslides is immediately accessible and highly extendable (“low floor, high ceiling”). It invites a wide range of responses (multiple solutions). (The teacher can view novel solutions at a glance on the dashboard.) Open questions, like Which one doesn’t belong?, share these attributes, as does Quarter the Cross.

A rich task can have a single solution, but invite a wide range of approaches (multiple strategies). To me, this has less to do with the task/problem itself and more to do with pedagogy. A curriculum that values creative thinking has pedagogical implications. Consider a typical What’s the best deal? task. A step-by-step-worked-examples-now-you-try-one approach to teaching leaves little room for creativity. The strategy — calculate and compare unit prices — is predetermined. What if students were presented with the problem before the strategy? The class would generate several different ways to solve the same problem. They’d see and discuss a wide range of ideas. Note: this doesn’t preclude the teacher from later bringing a particular strategy (e.g., compare unit prices) to the conversation, if necessary. Ask “Why does this strategy make sense?” or “What’s the best strategy?” and students develop another core competency: critical thinking.

# Mathematics is creative. Is math class?

In her ShadowCon talk, Tracy Zager shared a word cloud generated from the language mathematicians use to describe their work. Creative sticks out. And invent, curiosity, play, imagination, wonder, etc. The image generated from the language society/teachers use to describe math… not so much.

But I know that there are places where school math is creative. In Surrey Schools (#sd36learn), in the “MathTwitterBlogoSphere” (#MTBoS), and beyond. When I wrote “We in mathematics education need to articulate better what creative thinking looks like in mathematics” above, I really meant “I need to articulate better…” So, I need your help. Did I get it right in this post? Artsy or not, what does mathematical creativity look like in your classroom?