A few years ago, I came across the following multiple choice question:
Argue about the solution all you’d like (oh, and people argue about the solution), the beautiful part of this, for me, is that the question is not really the question. The point of the exercise is not to complete the exercise, it’s to dwell a while in the complexities it offers. By constructing the argument, you interact with notions of odds, randomness, probability, and the like. This is similar to the idea of #SandwichChat, where the point is not to define what a sandwich actually is, but, rather, to play with emerging definitions and consider their consequences. I love these sorts of activities, because they, almost unexpectedly, turn our own thinking upon ourselves. They have a way of snapping us out from the familiar ebb and flow of the mathematics classroom, whereby prompts are passed to solvers who manufacture resolutions and, in turn, re-sell them back to teachers at increased costs. Teachers cover this inflation by remunerating the students with a most precious commodity–grades. And so the classroom economy ticks forward.
Continue reading Probability Quizzes with No Questions
My mind has been wandering back to the math class lately. I’ve missed it, and, given the current health concerns associated with the re-opening of schools, I may not be getting it back anytime soon. (At least in the form that I feel the most comfortable operating in). Perhaps it is the pendulum between anticipation and dread that has teaching and learning at the forefront of my awareness lately. Although this is not uncommon for me, absence does, as they say, make the heart grow fonder. It is, therefore, possible that this post represents my final descent into pandemic-induced psychosis; maybe this strained analogy symbolizes just how much I need the classroom back, and serves as a sort of Warshak test–math education style–where ink blot after ink blot of everyday experience suddenly holds latent lessons about the mathematics classroom. Maybe it’s just a way to air my dirty laundry, to simply stop some thoughts from rattling around in my skull by writing them down. Tabling the discussion of my sanity for the time being, what follows is a quick story about my Saturday afternoon.
Our washing machine was broken for a week. Like any dutiful husband, my initial reaction was to let the violent banging work itself out, hoping the problem would come out in the wash–so to speak. Today, we (my wife and I) finally settled into the task, and it was over in less than 3o minutes. That is, it would have been over if it were not attended to by an obsessive educationalist. After we were through, there was a moment of relief and accomplishment, but it was quickly replaced with a curiosity. How was it that two people with really no experience fixing washing machines were able to navigate the task?
Initially, I really only knew three things:
- The experience was rewarding. Not just the resolution of the chore, but the entire process of resolving it.
- We really had no blueprint to approach the task, and that made it more interesting.
- This is what I want students to feel like when they are being mathematical in my classroom.
After reflecting on what, in particular, about the process might have granted this, typically mundane, chore its educative potency, I arrived at the following:
- We began with tinkering. We moved the drum with our hands, trying to recreate the noise. We rocked the machine to see if it was unstable. We checked to see if any bolts or screws were loose or if any part of the shell could be pulled apart. All “aimless” tinkering, getting a sense of what we might be dealing with.
- We only stuck to plans as long as they were useful. Because there was to recipe for fixing the problem, plans emerged through our tinkering. We felt something hard under the rubber seal, and so we pried it away, only to find that it was an electrical connection. This was immediately dismissed as inconsequential to our problem, and we looked elsewhere. No questions asked. No catastrophic melt down; no immediate need for assistance. Soon we noticed some markings on a plastic piece near the front, and, once pulled on, the bolt holding it securely in place quickly slid out of its socket. A new, useful plan emerged.
- We focused on the good enough. Having isolated the problem, we noticed that the plastic washer that previously held the bolt in place was nowhere to be found. We found a spare washer from a jar filled with mismatched hardware, and installed. It wasn’t the exact part, but it adequately accomplished the same function–it was good enough.
Maybe this is the pandemic speaking, but I can’t shake the feeling that these same three features that I learned from my washing machine are also key features of potent mathematical experiences–both inside and outside the mathematics classroom.
If you are like me, your workload hasn’t exactly petered out during these recent weeks of quarantine. Within this new normal, I have found it incredibly beneficial to play. That play is freeform; you could categorize it as aimless, but it is far from mindless. The need to step away from the computer for a few precious moments has allowed me to finish up a couple math projects that have been brewing for a while. The first was the creation of Upscale Pattern Blocks. The second was really an unintended one, born from the influence of Christopher Danielson’s new Truchet Cubes. I affectionately call them QuaranTiles.
Continue reading Project: QuaranTiles
[Updated April 9th, 2020]
First off, I hope you are well. This post represents a portion of my attempt to remain “well enough” in the midst of tremendous uncertainty. Most of my time is spent talking about the teaching and learning of mathematics, something that seems to have ground to a necessary halt in recent days. Given our collective circumstance, the time feels as good as ever to talk about a little project I’ve been working on, and ask for a smidge of help.
Recent access to a laser cutter and a kindergartener got me wondering. I began to play with a few possibilities. One of the fun things that fell out was a set of scaled pattern blocks I’m calling, “Upscale Pattern Blocks”. Essentially, they are pattern blocks scaled in three different sizes. The sizes interacted in some very interesting ways, and after some test cutting and multiple trips to the craft supply store, I ended up with a really fun result.
Continue reading Upscale Pattern Blocks
[Update Mar 16, 2020: If you read the following post and feel inspired to spread the geometric love, #FreeScalene products are now available here. Proceeds of the Math Positive store support teachers through professional development book giveaways.]
This past weekend I was invited to Toronto to give the 2019 Margaret Sinclair Memorial Award Lecture at the Fields Math Ed. Forum at the Fields Institute for Research in Mathematical Sciences. While the layers of the organizational hierarchy can be a mouthful, the bottom line is that I was given the great honour of presenting my thoughts on the teaching and learning of mathematics–as they are formulated at this time of writing. I broke the day into three distinct sections: The recipient’s lecture, a poetic provocation about hotdogs and mathematics education, and a gallery walk composed of some of my favourite invitations from my career to date.
(Link to the video archive of the invited lecture.)
Continue reading #FreeScalene
In early December, I found out that my submission had been selected as the winner of the 2019 Rosenthal Prize for Inspiration and Innovation in Math Teaching. At the time, I had zero reference point to understand what that meant, but have since experienced first hand the hospitality of the international math education community. This post is not a summary of the winning submission; that detailed lesson plan has been posted on the MoMath website. Here, I want to take the time to reflect aloud on how this whole thing happened–a process, I think, might be of value for math teachers. I’ve attempted to distill my take-aways into four categories, but, in actuality, they still exist (for me) as a tangled heap composed of equal parts disbelief, gratitude, and empowerment to pursue the next challenge.
Continue reading An Improbable Run at the Rosenthal Prize
The best thing about online communities (IMO), is the emergence of artefacts from the collected actions of many people. The online math education community (known as the MTBoS) has seen many of these collections throughout the years, most of which are aimed at supporting imaginative mathematics instruction in grade school. Personally, I have felt the community around Fraction Talks explode right under my nose, and it has been a joy to see how the prompts have sponsored amazing student reasoning. A few months ago, I had another idea for a task structure–that I dubbed #MenuMath–and began to collect examples from engaged math teachers. Since then, the collection has grown and become bilingual thanks to the translation work of Joce Dagenais. I love hearing about student and teacher creations, and you are encouraged to submit menus via my contact page if you feel inspired to do so.
Continue reading Menu Math Easter Eggs
There is too much to like about Desmos. Really, though. The pace of innovation is gross. I am the first to admit that my sophistication with the platform is lagging behind the possibilities. I have never dabbled in Computation Layer, and I haven’t played with the Geometry. Part of my problem is the core team and the army of fellows are so darn accommodating with any questions.
One of my favourite activities remains the Marbleslides. They set a beautiful stage for students to stretch their imagination, and I have not yet met an activity that sponsors a need domain and range in a more organic fashion. I have used them with all secondary grade levels, and they will be a regular part of the weekly work for my undergraduate students in their mathematics methods course this Winter.
Continue reading All-But-One Desmos
A huge piece of my identity is invested in being a mathematics teacher. This week I began a new and interesting challenge as a university faculty member preparing pre-service elementary and secondary mathematics teachers. This provides me more time to think deeply and openly about the entirety of the mathematics education enterprise, and put some of those ideas into public circulation through speaking and writing opportunities. I am really looking forward to that.
It also means that I am charged with orchestrating the formative experiences with mathematics teaching for about half of my province’s new teachers. That fact is terrifying. I am given just thirteen days in each course with which to shape the impressions, experiences, and ambitions of the future teachers of my province, city, school division, and (quite possibly) my own children. Thirteen days.
Continue reading Thirteen Days
For the first time in a decade, I am not reconvening with a high school staff to begin preparations for the school year. (I’m preparing to work with pre-service teachers on a university campus). It feels weird–very weird. It is a day that I look forward to because optimism is a constant across the building. Staff feels fresh, materials are crisp, and possibilities are endless. It sadly belies what’s to come.
Continue reading Re-Constructing Shapes