Baiting the Hook with the Absurd
While commiserating with a colleague about the need for stretchy work pants on days when the ice cream truck stops by, we started chatting about whether or not there is a good math lesson to be discovered. Thinking on it a bit more, Tensile Strength is definitely an interesting topic and seems like it is ripe for creating a scenario that baits the hook quite nicely (a Dan Meyer-ism worth adopting)…especially when centered around stretchy pants and other elastics. But is it enough to engage students in a way that causes them to pursue an exploration that leads to some substantial mathematical experience?
While real-world, authentic, personalized-learning and other such popular themes for improved educational experiences contribute (meaningfully) to the educational theory that drives instruction, I believe there are complimentary concepts that also may contribute certain beneficial characteristics of the learning experience. Perhaps these complimentary concepts are a bit more unreasonable, but within the absurd, or variations of, students might endeavor to engage in whatever mathematics is at hand instead of lamenting the otherwise dull and expected mathematics classroom experience.
It is absurd to define how “mullet-y” a particular mullet may be; and yet, middle schoolers rejoice as they explore the incredible difficult and essential understanding of ratios.
Perhaps my favorite example of such a lesson is from Matt Vaudrey’s Mullet Ratio Lesson. This lesson achieves something I believe cannot be achieved if real-world, authentic, and personalized-learning are the aim. It is absurd to define how “mullet-y” a particular mullet may be; and yet, middle schoolers rejoice as they explore the incredible difficult and essential understanding of ratios. The absurdity is a key ingredient that is too often left out of the mathematics explorations I had as a student and employed as a teacher. For whatever reasons, I fell into the belief system that mathematics was too formal to explore such things…I could still talk for days on the value of Teaching Mathematics Through Social Justice. I now believe that there must be a balanced approach in the development of themes that shape our instruction. Nonetheless, there are worthwhile arguments, for example, that imaginary, impossible, and social learning (see here and here) are of equal, if not greater, importance.
Building a Task That Is Worth Doing
Back to my original thought: how might we use the elastic waste band as a hook that might engage students in the absurd, imaginary, impossible, and social aspects of mathematics? The need to hook the students is not nearly as critical as the complexity the problem presents and how the discussion is framed. Too much scaffolding and we’re back to the starting point; too little and we’re floating off into the unknown with students bouncing off the walls.
The question I really wonder about is whether a students will actually care at all about the elastic waste band. This is the essential question, right? I can’t assume that just because the topic is interesting, that the students will be willing to play along with me and do boring math. If I could reference Dan Meyer once more, there is actually something critical worth reflecting on at this point:
[The question is] real world, so students are disarmed of their usual question, “When will I ever use this?” But the questions are still boring.
Sure, any question about an elastic band might be just interesting (real or fake) enough to do, but how do we do it in a non-boring way? Let me start with a standard elasticity question that might be familiar-ish. This is based on Hooke’s Law:
What is the force required to stretch a spring whose constant value is 100 N/m by an amount of 0.50 m?
The solution is rather straight forward if you happen to know Hooke’s Law, which states that the force needed to extend or compress a spring by some distance X is proportional to that distance. That is: F=kX, where k is a constant factor characteristic of the spring, its stiffness. (Source: Wikipedia)
So, the boring problem plays out: F=kX, where k is given as 100 N/m and X is given as 0.50 m. Thus F = (100)(0.50) Nm/m = 50 N
But, how can we turn this problem into something that is worth doing? I’ll give it a shot…you be the critic.
Everyone knows zombies cannot lift their legs to step over obstacles and when they set their rotting brains to doing something, they don’t give up…ever. The zombie mob is creeping closer and closer to town, dead-set on making their way down Main Street. Luckily, the throng of zombies is far enough away that there is time to stretch out the Community Rubber Band around the Main Street entrance to town. The question on everyone’s mind: How long do we have until the zombies break the Community Rubber Band?
A few things worth pointing out:
- It is obviously not real.
- It doesn’t provide a formula or any real guidance.
- The entire solution pathway for the students is open.
- There is no correct answer.
- Assuming the appropriate conditions maintain so that the zombies don’t climb over the rubber band, the prompt requires some serious work to develop a defensible answer.
- It is obviously contrived.
- Some students might not care about zombies…and I can’t kick them out of class if that’s the case. (right?)
- The students are responsible for thinking quite deeply about the distance the rubber band would have to stretch for it to break, what the stiffness of the rubber band is (k), and how much one zombie walking into the rubber band contributes to the overall force (F) applied to the rubber band. Then, they have to make some argument for the numbers of zombies that are walking into the rubber band per unit of time.
- Also, the answer is in Newtons, which I have almost no reference for understanding. This is no simple or straightforward task.
I’m not saying this problem is perfect, but I am arguing that it is less boring than one that is really just a plug-and-play problem. I do think a problem like this allows for students to explore an imaginary world while still concerning themselves with real physics and mathematics.
It boils down to a few principles in lesson design I tend to take as necessary, but not necessarily sufficient:
- Math problems shouldn’t be boring to assign, solve, discuss or evaluate.
- Math problems should cause students to be inquisitive.
Additionally, we have to be thoughtful about how we go about supporting students as they struggle to persevere. We certainly have to work to ensure the work on the task doesn’t devolve into our own zombie riot; Mary Kay Stein and Peg Smith’s book on the 5 Practices for Orchestrating Productive Mathematics Discussions is my go-to resource for this.
I’ll happily suggest that not every problem needs an absurd context, but shouldn’t it be in our toolbox anyway? Why not let students work on understanding proportional relationships while thinking about zombies and elastic bands?
One argument I hear is that the students need more experiences with questions that are analogous to the ones on the high-stakes tests they must pass. I get this at a basic level, but I will almost always disagree with it. Passing a big exam requires two skills: 1) Thinking flexibly about the content and processes being assessed and 2) Knowing how the game works. I attribute less than 10% of a student’s success to the second skill and do not believe there is a real need to train students on this skill. Rather, it is something that improves over the course of the year if the learning experience is founded on meaningful mathematics, whether the context of that math is real, imaginary, absurd, or otherwise.
Ultimately, it comes down to seeing mathematics as playful. As educators attempting to create a learning environment in which all students feel valued and engaged, let’s let humor and bits of the absurd find its way into our classroom every once in a while. Who knows, maybe the absurd will happen… all kids will love math!