For the first time in a decade, I am not reconvening with a high school staff to begin preparations for the school year. (I’m preparing to work with pre-service teachers on a university campus). It feels weird–very weird. It is a day that I look forward to because optimism is a constant across the building. Staff feels fresh, materials are crisp, and possibilities are endless. It sadly belies what’s to come.
Continue reading Re-Constructing Shapes
Much of what appears in mathematics textbooks is what I like to call, downstream thinking. Downstream thinking usually involves two features that set the stage for learners. First, it provides a context (however doctored or engineered–often referred to as “pseudo-context”). Second, the problem provides a pre-packaged algebraic model that is assumed to have arisen from that context.
Continue reading Thinking Upstream with a Quadratics Menu
Exactly one month ago, fellow Saskatchewan mathematics teacher Ilona Vashchyshyn tweeted about an area task that she used in her class. Long story short, it captured the imagination of Math Ed Twitter like elegant tasks have a tendency of doing.
Continue reading A Viral Area Task
One of the great parts of my job as a split classroom teacher and division consultant is that I get to spend time in classrooms from grades six to twelve. This means I often need to be in one head space to teach my own Grade 12s and then switch gears to act with younger mathematicians. It also means that the classroom experiences are sporadic and involve teachers working in several different places in several different curricula.
On this particular occasion, I was working with a 7/8 split class that had just finished a unit on perfect squares and divisibility rules, and we wanted an activity that could serve as a sort of review of divisibility rules but also reveal something cool about perfect squares. I thought about the locker problem, but it doesn’t require students to factor in order to see the pattern. Instead, I took some of the underlying mathematical principles (namely: that perfect squares have an odd number of factors) and wrapped it in a structure suited for a Friday afternoon.
Continue reading Counting Factors with Grade 7/8s
Every time I teach a unit on fractions, there are many students who insist that they’d rather use decimals, and I don’t blame them. The obvious parallels to the whole numbers make decimals a “friendly” extension from the integers into the rational numbers. Many of the things school math asks kids to do with rational numbers can be easily transferred into decimals with minimal stress on the algorithms. Such is not the case with fractions. Take addition for example.
Continue reading A Twist on Ordering Decimals
I have mixed feelings about student questions.
We (as teachers) act like we want students to ask questions; however, there are plenty of implicit messages about teaching that tell us that good teachers don’t need students to ask questions. One of the oldest pillars of teaching tells us to provide adequate wait time for students to formulate and ask questions, but there is a sense of relief when time passes without the need for clarification. This feeling essentially equates clarity with quality. Wait time becomes an emergency procedure to be used when we feel an awkward imbalance in the room.
Continue reading Expecting Questions
You can count me among the folk that believe that there is a real possibility to teach mathematics (among many other things) through coding. I do not claim to have any expertise in the area aside from a handful of undergraduate credits and the odd project that has grabbed my attention over the years; however, the intuitive nature of Scratch provides a novice entry point for anyone interested in giving it a shot. This post describes my initial foray into using coding technology in the classroom. Like all things, the structure of school provided certain constraints, but in the end, it was a very positive experiment for both myself and the students.
Continue reading Estimating with InO-Bot
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.
Continue reading Using Visible Random Groups in Assessments
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). 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).
Continue reading Math Storytime App: Talking Math with Your Kids
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 ).
Continue reading Problem(s) with Triangles