Thinking Upstream with a Quadratics Menu

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.  I am imagining the reasons for this are three-fold: Instructional time. Developing these models is messy and takes time, especially with class sizes in the mid-to-high 30s. Accessibility. Developing these models isn’t always possible (i.e. can’t run trials and collect data on every …

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Desmosification: Building Custom Parabolas

After an emoji was named 2015 Oxford Dictionary word of the year, I am holding out hope for next years’ candidate: des-mo-si-fy /dez-MOH-suh-fahy/ verb 1.   to transform the condition, nature, or character of a classroom activity using Desmos. Starting with a Dan Meyer post, the art of infusing dynamic software into student activities changes the ways that students encounter abstract, functional relationships in mathematics. Desmos’ activity builder gives teachers an extremely user friendly platform to create tasks that move students through semi-structured lines of inquiry.  I decided to start with a task that I already liked. Before: I like to …

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Connecting Quadratic Representations

I always introduce linear functions with the idea of a growing pattern. Students are asked to describe growth in patterns of coloured squares, predict the values of future stages, and design their own patterns that grow linearly. Fawn’s VisualPatterns is a perfect tool for this.While stumbling around Visual Patterns with my Grade 9s, we happened upon a pattern that was quadratic. The students asked to give it a try, but we couldn’t quite find a rule that worked at every stage. While I knew this would happen, the students showed a large amount of staying power with the task. The …

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On Collective Consciousness and Individual Epiphanies

I would like to begin with a conjecture: The amount of collective action in a learning system is inversely related to the possible degree of curricular specificity.  The mathematical action of a group of learners centred on a particular task gives rise to a unique way of being with the problem, but also reinvents the problem.In short, what emerges from collectivity is not tidy. How can I justify curating a collective of learners, when school is so interested in individuals?Learners commerce on a central path of mathematical learning while acting on a problem, but each take away personal, enacted knowings from …

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