functions graphing quadratics

Introducing Quadratics

Quadratics feel important. This impression is no doubt influenced by the boated importance placed on calculus in secondary school. They represent the giant leap from linearity and pave the way for more elaborate functions; therefore, I often find myself musing on ways to have students meaningfully interact with the topic. Once the structure of the function is established, I’ve played around with interesting ways to help students visualize quadratic growth, connect that growth to the Cartesian plane, and build these functions to specifications; however, my introduction to quadratics in vertex-graphing form has always been a series of “What happens to the graph when I change the ___ value?” questions. These aren’t bad questions (and a quick setup of Desmos sliders helps visualize the effects), but they don’t exactly build up understanding from experience. Such was my introductory quadratics lesson for years, lukewarm but lacking the epiphany to address it.

“If you want to kill flies, you don’t need bazookas”

– Ben Orlin, Math with Bad Drawings, p. 44
data analysis estimation fractions play reflection

COVID Math Fair

In 2015, my students and I founded an annual math fair in my school division. Inspired by mathematical play, the fair grew from humble beginnings into a staple of my mathematical calendar. Like nearly everything about this school year, the fair was jeopardized by the pandemic; however, with a touch of innovation and the ongoing support from my school administration, the teams of educators in our five feeder elementary schools, our trustee, and the school community council, I managed to pull together three math invitation carts that could be disassembled, transported, and reassembled in the elementary schools.

integers technology visual

The Bucket of Zero

Over the last year, Dr. Lisa Lunney Borden and I have been working on a model for integer operations that she introduced me to a while back. Our goal is to amplify her research for classroom mathematics teachers. Right now, the idea consists of three pieces, each at varying stages of development.

  • A paper
  • A platform
  • A set of plans
quadratics technology vine

Animating Quadratic Patterns

My first attempt at animating patterns was published on this blog in 2013. I suppose you can consider this post a long-overdue extension of the thinking there, however with a much-needed bump in production quality. In those old days, I hunched over a whiteboard with a collection of square tiles, creating six-second loops on the (now defunct) social media platform, Vine. Now, thanks largely to Berkeley Everett and his crash course on how to make animations in Keynote, the process has become much more streamlined.

classroom structure factors games play projects reflection technology

Report on a Math Tournament

**This post contains the materials and advice you’ll need to run a distanced math tournament with your district, division, school, province, state, classroom, family, coworkers, neighbours, etc., etc., etc.**

Honestly, the more math love, the better! (IMHO)

In mid-October, I designed a math provincial math tournament open to all middle school teachers in my home province of Saskatchewan, Canada. After writing up a blog post that served as a formal invitation, the tournament (which I affectionately called the Saskatchewan Mathematics Invitational Tournament–or #SMIT2020 for short) has been running for just over a month with over 80 classrooms from across the province playing Federico Chialvo’s delightful game MULTI. (see here for more information).

reflection whiteboards

Thoughts on Thinking Classrooms

“Mathematics is equipment for thinking”

Francis Su, Mathematics for Human Flourishing, p. 110.

The sun sets around 5:30PM this time of year in my little prairie slice of paradise. Yesterday, well after dark, there was a ring of the doorbell and a package delivery: My copy of Building Thinking Classrooms by Peter Liljedahl. Over the last couple weeks, I have watched as tweeps1 sent messages of exhilaration having received their own copies. The, now familiar, orange cover adorned with the beautiful illustrations of Laura Wheeler is a welcomed sight on my Twitter feed, each time accompanied with excited messages you’d expect to hear from children anticipating a visit from the Tooth Fairy.

Honestly, holding the book felt weird. I say that as a testament to Peter’s work: It draws you into participation to the point where it feels like it’s a part of your history. In my case, that’s because this book is a part of my history. Receiving the book sponsored a sort of nostalgia, as I’m sure it did for so many who have followed the ideas as they’ve developed over the years. This feeling surprised me, because, despite the real feeling of connection to the physical copy of the book and the brand of teaching it represents:

I don’t run a Thinking Classroom.

classroom structure factors games numerical flexibility play technology

A Math Tournament

**Update: Nov 23, 2020: Follow along on Twitter with some of the thinking at the hashtag #SMIT2020

COVID has created a global (and now chronic) pressure on all teachers in all classrooms, and the shifting, local realities have made teacher collaboration a precious commodity. It’s hard enough to find time to confer with colleagues under the best of situations, and now our major professional muster points are not currently viable–adding further value to any sense of connection that can be generated.

Bummer, right?

assessment probability

Probability Quizzes with No Questions

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.1


A Lesson from a Washing Machine

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 laundry1, 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.

pattern play projects

Project: QuaranTiles

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.