Radical Cheat and Playful Mathematics

One of my favourite professors in university taught a second-year course on solving differential equations. He would bounce into class, silver hair flying, making us feel like it was our lucky day to be studying math on a beautiful day in that stuffy old classroom. I still remember how he would preface the problems we would solve with: “Alright, so that’s the game! Let’s play the game.” And off we would go. After all these years, this phrase stuck with me – admittedly, more than the mathematics I learned during the course. Maybe he was referring to the fact that we were working with “toy problems,” as other professors would say, suggesting that they had little resemblance to the problems mathematicians actually work on day to day; I like to think, though, that he was talking about how mathematics itself can be a form of play. Continue reading “Radical Cheat and Playful Mathematics”

The time #MTBoS went bananas

The other day, I tweeted out the following photo and link:

As advertised, the survey (click here if you’d like to respond) consisted of one question: “In your opinion, what’s the perfect banana ripeness? (For eating, NOT for making banana bread.)” The options were “More green than 1,” “1,” “2,” “3,” … and so on until “15,” and “More brown than 15.”

Here’s what I learned: Continue reading “The time #MTBoS went bananas”

The Price is Right (& other tasks to foster reasoning about fractions)

As I mentioned in my previous post, the start of the school year in my Grade 9 class is dedicated to reviewing some basic concepts from elementary school – in particular, integer operations, fractions, and fraction operations. I try to embed these skills as components of tasks that ask students to make decisions, generalize, problem solve, and/or engage with novel or less-familiar mathematical ideas – or, as Nat Banting (who does this so well) describes it, “embedding atomic skills into tasks so that the basic skills are developed and used as tools of mathematics, rather than the ultimate goal of mathematics.” At the beginning of the year, these kinds of activities allow me to simultaneously identify and assess the needs of students who are still struggling with basic concepts and to challenge students who are ready to learn something new. Continue reading “The Price is Right (& other tasks to foster reasoning about fractions)”

Integer Bingo

During the first week or so of the school year, I take time to review some basic concepts from previous years with my Grade 9 students – in particular, integer operations, fractions, and fraction operations. The challenge is designing tasks that embed review and practice within problems or activities that also call for generalization, problem solving, or engagement with novel or less-familiar mathematical ideas. I deem this necessary because, of course, not all students come to high school with the same understandings and strengths, and while some do need the opportunity to review basic concepts, others are ready to move on. For the latter group, review activities are at a high risk of being perceived as baby-ish and not worth the time, which means that I need to design these tasks very carefully – that is, in such a way that I can simultaneously assess the needs of students who are still struggling with basic concepts and challenge students who are ready to learn something new.

I’d like to share one task that, I think, had something to offer to all of my students: Integer Bingo, which is based on a task developed by and discussed in Serradó (2016). I chose this task for several reasons: Continue reading “Integer Bingo”

Pop Box Project: Part 3 – Trade Show and Reflections

This post is the last in a series of posts detailing my take on the Pop Box Design Project. Previously: Part 1, Part 2. Click here for the inspiration.

At last, students’ prototypes were coming to life. The penultimate lesson was a (productive) mess of paper, tape, and running to other classrooms to borrow meter sticks.

I wanted to give students an opportunity to share their hard work with other teachers and students, so earlier in the week, I began organizing a trade show. In terms of physical set-up, preparation was minimal; the biggest hurdle was finding volunteers for students to interact with. I sent out an email inviting other staff members to join, and ended up with about 6 staff participants; another math teacher also very kindly volunteered his Grade 11 class to join in. Continue reading “Pop Box Project: Part 3 – Trade Show and Reflections”

Pop Box Design: Part 2 – Brainstorming, Design, Construction

This post is the second in a series of posts detailing my take on the Pop Box Design Project. Click here for Part 1, and click here for the inspiration.

Last day, after a few lessons of stirring up ideas related to packaging design and strengthening understanding of the concepts of surface area and volume,  students were introduced to the unit project: Design a more effective pop box. Although we had previously focused on comparing the efficiency of different packages in terms of amount of material used and percentage of wasted space, for their design projects students could choose to focus instead on creating a more unique and interesting package if they felt it would increase sales. Students left the classroom buzzing as ideas already started to emerge. Continue reading “Pop Box Design: Part 2 – Brainstorming, Design, Construction”

Pop Box Project: Part 1 – Introduction

For our last unit of Foundations of Mathematics 10, I decided to tackle something that scared me: a project. The outcomes – determining surface area and volume – were well-suited to practical application, and I had been eyeing Nat Banting‘s soft drink project for a while now; moreover, I felt that a strong collaborative culture was finally taking root among this group of students, and that they were prepared to work on a larger, more complex problem together. (Actually, I am sure that they were prepared for this much earlier – I am still in the process of learning to trust my students, and this was a major test of the water for me). In short, it was time to take the plunge. This post is the first in a series of three detailing the experience. (Click here for Part 2.)

Continue reading “Pop Box Project: Part 1 – Introduction”

Zodiac review: In which there was mad flow

Peter Liljedahl speaks and writes often about the concept of flow in the context of mathematics education, building on the earlier work of Mihály Csíkszentmihályi. According to Liljedahl, flow is “that moment where we are so focused and so absorbed in an activity that we lose all track of time, we are un-distractible, and we are consumed by the enjoyment of the activity. As educators we have glimpses of this in our teaching and value it when we see it” (Liljedahl, 2016b). The concept of flow thus offers a productive way to talk about student engagement, and offers a way to think about environments that may occasion engagement in our classrooms. In particular, flow arises in an environment “wherein there are clear goals, immediate feedback, and there is a balance between the challenge of the activity and the abilities of the doer” (Liljedahl, 2016b).

In order to occasion and nurture flow, Liljedahl argues, teachers should be intentional in providing hints and extensions that keep students in balance between their abilities and the challenge of the current task (2016a). Although I typically opt for this approachthat is, offer the same task to all student groups, and provide hints and extensions as necessary—, today, I found success in offering students choice of tasks (coupled with hints and extensions from me, as necessary). Continue reading “Zodiac review: In which there was mad flow”

Reframing a classroom culture problem

tl;dr: Sometimes, a “problem” in the classroom is really a symptom of a bigger issue.

Here’s the backstory. Lately, I’ve been struggling with a lack of engagement in whole-group discussions after small-group work. I was growing increasingly frustrated that the majority of the students – especially the students who had a good grasp on the concepts – didn’t volunteer to share their thinking, and that they had no qualms about chatting with their friends when a student (or me) did choose to put forth a question or share an idea. Increasingly, I was feeling like I was losing them, and searched for explanations ranging from summer being around the corner, to deficiencies on the part of my students (kids these days!), to the possibility that I am a terrible teacher for whom there is, frankly, no hope. (During a recent conversation, Jimmy Pai described teaching as a balancing game between overconfidence and despair, and these days I’ve been leaning heavily to the right.) Continue reading “Reframing a classroom culture problem”

A problem worthy of attack: Friend groups

Quick post to share a really interesting problem a student of mine (G.) posed the other day.

Here’s the context: Last week, our school held a Box Lunch Auction, which involved homerooms preparing lunches of various sizes (e.g., for 6 people, for 10 people, for 30 people) and groups of students bidding on said lunches. The money raised was donated to various local charities.

Reflecting on the experience, G. wondered if the sizes of the lunches might be determined in a way that better suited the needs of the student body, rather than more or less randomly (truthfully, based on how much work a homeroom was prepared to do). This is the question he posed:

Let’s imagine that any two people in the school have a 50% chance of being friends. With n people, what’s the most likely size friend group?

I was hooked immediately. Continue reading “A problem worthy of attack: Friend groups”