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”
There are a lot of quotes (maybe as many as there are teachers) that begin something like this: “The goal of education is…” For example,
“The one real goal of education is to leave a person asking questions.” –Max Beerhohm
Undoubtedly, many of these are spoken in a hyperbolic manner, and should be interpreted as such. (Myself, I would be wary of anyone who believed that there are precisely N goals of education, true for all places and at all times.) Nevertheless, this sentiment came to my mind last week, the first week of school. Continue reading “Love the questions”
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”
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”
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”
Every being cries out in silence to be read differently. Do not be indifferent to these cries. – Simone Weil
I was substitute teaching today for a group of students I had taught last semester. I noticed that one of the girls had new glasses, and I told her that they looked nice. She replied, surprised, “You’re the only teacher that’s noticed, and you don’t even teach me.”
Not pretending to be a hero. There have undoubtedly been countless occasions that I’ve been blind when I should have seen. Just tying a knot in my mind to notice, amidst the chaos, the students I’m not seeing in my own classroom.
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 approach—that 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”
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”
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”
At some point last semester, I remember feeling apprehensive about using balance scales as a model to introduce linear equations. I think I was worried about students’ ability to move beyond the model and consider things like negative and decimal coefficients. Would they be lost when they no longer had their crutch?
I should have trusted them.
Nat Banting has written about balance scales here. Although the metaphor of balance scales is referred to often in math classrooms – during the linear equations unit, students must hear the mantra “whatever you do to one side, you do to the other” at least 1000 times a class – the model really becomes powerful when students actually work with balance scale problems, and that they do so right from the start. Nat writes:
My Grade 9 students don’t see an equation for the first two weeks of their unit of solving linear equations. That is because I think students get all bogged down in the notation, and lose their problem solving intuition.
I decided to follow Nat’s advice. For a few days, my students worked through sequences of problems such as these, increasing in difficulty, where the goal is to find the “weight” of the square: Continue reading “On balance scales and trusting your students”