(this post is also published in the MTA of Nova Scotia Newsletter)

Great teaching resources push ambitious teaching forward. As my career has gone along, names for these sorts of things have oscillated in and out of vogue, but lesson (and unit) preparation still begins with finding (or building) a task about the topic at hand.

There is, however, a tension between A) granting students the freedom to make sense while working with a task and B) preordaining what that specific task is to be about. Stated differently: Any space that encourages student sensemaking pressures the relationship between the teacher’s purpose for posing the activity and the generative possibility of student actions with the activity. To illustrate that tension, here is a task, along with very brief recreations of some recent thinking that my students did with it.

This is a task about equations.

Many students bring forth the notion of a “tie” as an equality. Perfect. These equations emerge all along the continuum of abstraction—some draw pictures of farm animals, some use geometric shapes like squares and circles, and others use letters as variables. (After these notations emerge, it is a nice time to talk about where variables come from and what notation is useful for).

The dust typically settles at something like:

3p = 2g

3c = 4g

I’ve seen two common courses of action from here:

First, groups, possibly triggered by the “friendly” number of goats in each equation, make the claim that if 3p=2g then 6p=4g. I might call the problem they are solving the “make the goats match” problem. They recognize that having an animal in both tug-of-wars is a powerful moment of connection. (This recognition can later be formalized into work with simultaneous equations). From there, 3c = 6p and then you can divide each team of animals in thirds in the same vein that you doubled the original numbers in the first equation.

Second, groups, possibly triggered by the goats being in both matches, use the 3p=2g equation to determine that goats are stronger than pigs, actually they are one-and-a-half times as strong. (Often, this understanding doesn’t come from “dividing both sides by 2,” but it is a cool moment to show how the equations tell the story.) I might call the problem they are solving the “how strong is a single goat?” problem. Because each single got is one-and-a-half times as strong as a pig, the 4 goats in the second equation are exactly as strong as 6 pigs, and this fact is substituted to create a new equation: 3c=6p.

This is a task about proportions.

Many students comment that they need 3 pigs for every two goats and 3 cows for every 4 goats. Perfect. The first ratio is twice as big as the second—it takes twice as many goats to tie the cows; therefore, the cows are twice as strong as the pigs. I might call the problem they are solving the “how strong is a herd of three?” problem. They don’t key on making the number of goats equal (multiplying it from 2 to 4) or on determining the strength of a single goat; they key on the idea that both pigs and cows have three individuals, so the number of opponents creates a ratio of their strength. This problem is powered by proportional reasoning.

This is a task.

I’m not convinced that this particular task is a task about any specific mathematical topic, but I do know that if students are given the space to act mathematically with it, they often use equations to make and maintain sense. (and less frequently use the ideas of proportional reasoning). This rephrasing may feel like splitting hairs, but taking it seriously has been one of the most important exercises of my near two decades of activity in the math education world.

Every task contains certain possibilities—like landscapes encouraging certain types of actions in order to maintain sense. These potentials (alongside with our histories of interactions with mathematics) push solvers in specific directions. But, naturally, learners of math (with a less robust history with the discipline as formally conceived) are much less predictable. This is both a curse and a blessing. A curse to the teacher looking for a tidy entrance into, and a streamlined journey through, the topic or idea; a blessing to the teacher looking for access to the students’ current understanding. A student will show what they know through the mathematical activity they feel is relevant to resolving their current context.

Episodes like the thinking detailed here consistently remind me of two things:

1. Problems don’t exist separate from solvers; they are posed (and re-posed ) by solvers as they act with the context provided to them. This might initially feel like this leaves us with absolutely no way to predict what students might do, and, therefore, no real way to plan lessons that ask our students to act earnestly in a mathematical environment. Prompts do, however, contain specific opportunities to act, ones that become more predictable as the amount of shared history between learner, teacher, and mathematical context increases. Paying attention to these places where students might make mathematical decisions is a key piece of preparing to teach in spaces that grant students increased freedom.

2. Teaching needs to be sensitive to the problems that emerge from student activity . Before rushing to have an interaction about the problem you anticipate to be at the heart of their activity, take the time to observe what problem is truly driving their work, a stance described by Davis (1996) as one of “imaginative participation” (p. 53). What the task is about becomes, first, an act of observation and, second, an act of influence on the part of the teacher. Sometimes this means the teacher’s interactions provide a subtle gradient in the landscape toward a desired topic, and other times the teacher must stand as a mountain, blocking specific pathways of activity and orienting attention toward new tributaries. Your teaching moves are always subject to your judgements in these unfolding moments, and good teaching becomes less about being perfect than it is about being appropriate.

NatBanting

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1. Banting, N. (2017). What problem are they posing? Viewing group problem solving through an enactivist lens. delta-K: Journal of the Mathematics Council of the Alberta Teachers’ Association, 54(1), 11-15.

2. Banting, N., & Simmt, E. (2017). From (observing) problem-solving to (observing) problem-posing: Fronting the teacher as observer. Constructivist Foundations, 13(1), 177-179. Retrieved from http://constructivist.info/13/1/177