Making Structure Visible

In my teaching, I often point out to my students the patterns that underlie the work that they are doing, the ways in which math is consistent across problems that appear to be different from each other.  I am not a deep mathematical thinker and these repeated structures are a source of much comfort for me as a practitioner of mathematics–there is a reason why I love algebra and hate stats!

I think my students are often overwhelmed in 7th grade by a sense that they are being swamped with a limitless supply of disparate and unconnected problems.  It makes me sad to see students overwhelmed by thinking that they need a new approach for every problem, rather than being able to see patterns that go across content or even multiples contents.  This is my third blog post about our use of templates and how we create them.  Last week, I wrote about using teacher curriculum materials to make the patterns explicit to students.  In this post, I would like to highlight a different approach, this time for the concept of tree diagrams.

Tree diagrams are brilliant in their flexibility (they can be used with multiple-stage probability experiments) and in their ability to fully capture all possible outcomes.  However, they are pure misery for many 7th graders, particularly those with spatial weaknesses.  Tree diagrams can also be challenging for students conceptually because students often want to put the events (i.e., flipping a coin, then rolling a die) into the tree  diagram itself, rather than realizing that the tree diagram captures the outcomes of the events (i.e., heads and tails, then the numbers 1-6).  In a moment of inspiration, my colleague, Irene Witt, created this tree diagram template for our probability Module this year:

The structure of this template allowed us to separate the events from the outcomes, while giving students a place for each to be represented on the tree.  It’s also flexible, in that it can be use to represent multi-stage probability events as well as probability events with unequal probabilities.  As you think about ways you might generate templates for your content, you might consider if there are structures like this that you, as an advanced and experienced user of content, use in your own practice that you could share with your students to help guide their experiences.