Since the onset of my career, I have been keenly interested in how students work together in the contexts of school. We know that students (and humans in general… actually animals in general) form collectives to accomplish elaborate tasks. These traffic jams of human interaction transcend individuality to the point where the level of activity is so dense that groups begin to synchronize into a sort of group mind. However, we have a school system built on individuality and (unfortunately) competition, and triggering these collective structures is extremely difficult in part because students know that, when push comes to shove, they will be weighed and measured as an individual.
When I started this blog, I had no children of my own but spent lots of time talking math with the children of my friends. This talk began to pop up more frequently on my twitter feed as well in posts. Now that I have children of my own, I am wholly invested in the project of talking mathematics with them (whether they notice it or not)1. This has resulted in many moments of surprise and delight, and continues to fuel my interest in the roots of mathematical learning (far before I get to see them in secondary school).
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Problem(s) with Triangles
My provincial curriculum scatters trigonometry throughout several high school courses. Right-angled trig appears first as an isolated experience at the Grade 10 level. From there, the two pathways in Grade 11 cover the Sine and Cosine laws, but only one stream (Pre-calculus) continues into the idea of the unit circle and eventually the connections between the side ratios of right-angled triangles, the unit circle, the wave functions, and trigonometric identities. Since trig is doled out in piecemeal portions each semester, I often find that the hidden beauty of trig is masked by things like SOH CAH TOA. (Or, if you dare to place special triangles on the unit circle, SYR CXR TYX 1).
Everyone knows that you can’t wish for more wishes, but no one says you can’t wish for more genies.
According to the binding rules of genies (as published by Disney in the 1992 film, Aladdin), there are a few restrictions on what can and cannot be wished for. Probably the most famous restriction is that there is unequivocally no circumstance in which one is permitted to wish for more wishes. This is grouped with three other limitations stating that genies will not kill people, make people fall in love, or revive people from the dead. Other than that, the wishes are limited only by the imagination of the master.
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Second-hand Student-ing
Billy: “Banting, I have a question for you.”
It was 5-minute break between classes and I was trying to reset the random seating plan, open up the electronic attendance system, and load the image that would serve as a starter for the day’s lesson. During this small window of time, questions are usually about missing binders, requests for future work due to mid-semester holiday plans, or updates on my ever-present pile of grading. In short, I usually do not want to deal with them. Begrudgingly, I obliged.
Billy: “I need a piece of paper and a pen”
“How do you assess this?”
This is the question I eventually field during every opportunity I get to share pieces of my classroom with other stakeholders in education–be it teachers, administrators, or pre-service teachers. I don’t mind fielding it; it is a good question, one teeming with complexities and littered with implicit values.
I transferred schools at the end of last year, so for the first time in seven years, every one of my students I meet on the first day of school will be a stranger. This means that the first hour I have with each of the four classes is not only their introduction to the course, but also their introduction to me. It won’t take long for them to make an impression of me, of mathematics, of their classmates, and how I expect us all to co-exist for the next five months or so.
I have written on first day tasks before, but not for several years. In that post I identify four “attributes” of an effective first-day task*. One of the tasks I settled on for this year was The Shoe Sale task from Peter Liljedahl. (Other bloggers have documented work with the problem as well: e.g. Fawn Nguyen and Chris Hunter).
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Prime Climb Puzzles
Let it be known that I am not a huge fan of math board games. That being established, I have tried on multiple occasions to create one that I like because the undeniable engagement factor is there. One of two things always seems to happen to my attempts:
- The game does nothing to change how students interact with the mathematics. Rather, it divulges into an attempt to get students to complete drills in order to win points of some type. Here, the math and the game exist as ostensibly separate entities.
- The game mechanism does not support flexible mathematics without a plethora of complicated rules. In an attempt to ensure that the first problem does not occur, the game soon balloons out of control until the simplistic spirit of gamification is lost.
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Constraining the Two-Column Proof
There is no dedicated course for geometry in Saskatchewan’s secondary curriculum. Instead, the topic is splintered amongst several courses. There are advantages and disadvantages to this, neither of which will be the focus of this post. I just thought that, especially for the non-Canadian crowd, a glimpse of context would be helpful.
The notion of a geometric proof only appears in one course. It is presented as a single unit of study during a Grade 11 course and is preceded by a short unit on the difference between inductive and deductive reasoning. I have taught this course a lot over the past few years, and have always had mixed emotions toward this portion. I love the metacognitive analysis students participate in during the inductive v. deductive reasoning unit. It is a (metric) tonne of fun to teach because it largely entails the completion of games, puzzles, or challenges and a subsequent interrogation of our thinking patterns. This could be my favourite week and a half in the course. After we have experienced the difference between induction and deduction, we spend a couple weeks slogging through angle relationships and parallel lines, triangles, and polygons using the ultimate edifice of deductive reason: The two-column proof.
A colleague and I have often bemoaned our attempts to teach the concept of scale factor in higher dimensions. A topic that has such beautiful and elegant patterns and symmetries between the scale factors consistently seems to sail directly past the experience of our students. I have tried enacting several tasks with the students including some favourites from the #MTBoS (Mathalicious 1600 Pennsylvania and Giant Gummy Bear). Each time, the thinking during the task seems to dissipate when new problems are offered. It just seems like students have a hard time trusting the immense rate that surface area and volume can grow (or shrink). In the past, I had used digital images of cubes growing after having their dimensions scaled by 2, 3, 4… etc.; students seemed to grasp the pattern yet under-appreciate the girth of 8, 27, 64… etc. times as many cubes.