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?
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.
My students and I are currently in the middle of a unit on linear systems, which follows a unit on linear relations. We had worked a lot with linear patterns during the previous unit, which offer a great foundation for developing understanding of multiple representations of linear relations. Turning to linear systems, a natural transition was to consider two or more patterns side by side, and to consider when, if ever, they would have the same number of objects. However, the model is limited in that it’s not continuous. Consider, for example, the following pair of patterns:
As several of my students noticed during their work on the task, there will always be an odd number of squares in the first pattern and always an even number of squares in the second pattern. Therefore, there would never be a step where the two patterns had the same number of squares (i.e., the system didn’t have a solution). Of course, this is true for the discrete model, and an astute observation. However, the system y = 4x, y = 2x+ 9 does have a solution: namely, (4.5, 18). We could, theoretically, conceive of a step 4.5, or even continuous growth, with strange steps like 4.9143, but at this point, the model may be being stretched beyond its usefulness. Continue reading “In which we Trashketball”
Quelle est une chose qui devrait être interdite à l’école ?
I often start French class with a question. The students reflect and discuss in their table groups, then share their responses with the rest of the class during the whole-group discussion. Yesterday, I opened with the above. In English: “What is one thing that should be banned at school?”
“Littering!” “Homework!” “Vaping in the bathrooms!”, a few students volunteered (in French). Then, a student offered, “Cell phones!” I’m sure you can imagine the disbelief and anger that this response evoked from a class of Grade 9 students, the majority of whom have an iPhone permanently glued into their hand and probably could not imagine a time without them. Once the others died down, I gave the student the floor, who argued with sincerity: “Cell phones are a huge distraction. We’re paid to be here by tax payers, which means we have a responsibility to learn and not waste time on our phones.” Once again, the other students erupted. It took a minute to calm them down.
Finally, a student responded bitterly: “If we’re paid to be here, where’s my money?”
One of my favorite conversations to have with my students (Grades 9-11) has quickly become centered around the concept of zero.
It first arose naturally last semester, when a Grade 9 student blurted out: “But 0’s not a number, it’s the absence of number!” (I’ve forgotten the context of this remark.) I remember a few other students laughing at this, and I know that this was unfortunately a negative formative experience for her, because she wrote about it in her end-of-year reflection. And yet, when I prodded further at a later point in time, I found that many other students, if not most, held the same belief. We had a whole-group conversation about it, talked about how 0 behaves in a similar way as other numbers in many ways (you can add it, subtract it, multiply it…), but still, I think a lot of students held on to this conception. This was especially clear when the students were solving equations and came to a problem such as this: Continue reading “The absence of number”
Exactly where [a lesson] moves depends on such complex factors as the structures of those present, the context, and what has been anticipated. It may move toward more formulated understandings, if such formulation is relevant to the play space or if it becomes part of a further exploration. It may simply move to other sorts of activities. This, of course, is not to say that we should just allow whatever might happen to happen, thus abandoning our responsibilities as teachers. Rather, it is to say that we cannot make others think the way we think or know what we know, but we can create those openings where we can interactively and jointly move toward deeper understandings of a shared situation.
(Davis, 1996, p. 238-39)
My Grade 9 students are currently working on recognizing, analyzing, graphing, and solving problems involving linear relations. Linear relations lend themselves so naturally to describing patterns à la www.visualpatterns.org, and this is precisely how we got our toes wet in the topic: For several days, my students had been analyzing, extending, and (productively) arguing about a variety of linear and non-linear patterns. The intention of these first few lessons was to have students develop (or, in some cases, refine) an understanding of constant and non-constant change and to connect patterns in pictures to patterns in tables of values.
“What you do speaks so loud that I cannot hear what you say.” ― Ralph Waldo Emerson
Since the first day of school, I’ve been working hard to try to establish a classroom culture where students feel comfortable taking risks, asking questions, sharing and building on each others’ fully-formed and partial ideas, and acknowledging and correcting their mistakes; where all students feel that their contributions and questions are valuable and worthy of consideration. I have tried to do so by pointing out (less often than I should) when a student or a group is exemplifying one of these norms, by waiting (again, less often than I should) after questions and contributions to give more students the time they need formulate and share their ideas, by giving tasks that are accessible to a wide range of students and can be tackled with a variety of strategies, by eliciting and celebrating different solution paths, by highlighting different kinds of mathematical smartness (h/t Ilana Horn)…
In our division, classes on the first day of school are only 15 minutes long. By the time students settle in and introductions are made, there is hardly enough time to wrestle and play with an interesting math problem. I saved that for the second day. Instead of going through the syllabus, however, I gave the time over to my students to reflect on the following questions: