My Discovery-Based Math Elective: Exhibitions (Part 9)

This is part 9, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe how I held quarterly presentation-based projects.

• My school had a strong positive tradition around authentic alternative assessment. In particular, we had a culture of "Exhibitions." These were kind of like big projects in some classes, where there was some kind of performance or presentation component. The 9th grade ELA department did poetry readings. 10th grade history did a formal debate. 12th grade science did presentations on climate change.
• My own experience is that math classes often leave themselves out of these presentations. I think this is because live formal presentation seems like such a small part of what many math teachers believe to be authentic mathematical work. An exception here is much of statistics, where there are many rigorous applications in social science. Similarly, there are other fields of applied math that have some authentic presentation contexts. One year, the Financial Literacy teacher had students pretend to be financial advisors, and make a "portfolio" of services, like tax filing, budgeting, etc. They would then present to community members and students, to try and convince us we should hire them. Lots of fun.
• But I found examples of really great "pure math" presentations to be rarer. I experienced a decent example with the 2nd year experience of PROMYS for Teachers, which helped. The example that helped me to understand best what mathematical presentation could be, are the Numberphile videos. They're a really great example of taking classic topics and problems in (mostly) pure math, and making them accessible, while surfacing deep mathematical richness.
• All that experience in mind, in the context of this course, I have two types of exhibitions that I have asked students to do, both of which went well enough. I tried to sandwich these between units.
• Here are the slides I used when talking about exhibitions with students.
• Exhibition 1: Presenting for Depth
• Objectives
• Help students learn how to draw a Big Idea out of a Big Problem and its PSet
• Highlight the purpose of the *Probably* Not Related problems.
• During this kind of exhibition, students are asked to go back to one of the PSets they've already done, which they will return to. The first time around, students tend to focus more on the Big Problem. This time, students are expected to do a few things:
• make sure they totally understand a solution to the Big Problem
• do most/all of the *Probably* Not Related problems
• articulate the Big Idea and any relevant content threads
• show how the Big Idea is realized throughout all of the problems in the PSet, showing that they actually *Are* Related
• Logistics
• Groups of 2-3. I tried to go one-to-one, matching groups to PSets, without duplicating any PSets, because it'd be boring to see the same presentation twice in a row.
• Product: group presentation, with slides
• ~2.5 weeks total: 1 week doing the chosen PSet + 1 week making the presentation + 0.5 weeks for presentations.
• I would do this around the end of the 1st and 3rd quarter, and let them pick from any of the PSets we've done since the last exhibition.
• Exhibition 2: Presenting for Access
• Objectives: 
• Provide an opportunity for students transfer their more developed mathematical content and practices to a novel problem
• Build student capacity to share interesting math problems with others
• During this kind of exhibition, I have all the students sample a bunch of new Big Problems, for which I have made individual one-off PSets. They then pick one, get the PSet, and try to solve it. Here, they are more focused on the Big Problem, using the other problems to support/enrich their investigation. In general, their presentation will focus on a couple things:
• talk about their discovery process, what they tried, what worked and didn't work
• make sure they totally understand at least one solution to the Big Problem
• The PSets I wrote for these exhibition problems were just like the regular PSets I wrote, only they didn't have any explicit threads that connected to the other PSets. But occasionally, if beneficial, they would reference material from earlier in the class, (re)building prior knowledge as helpful.
• How to have students sample the problems, before choosing:
• I would run Stations, and at each station was a different Big Problem. The design principles were the same as any other Big Problem, which I've written about here [insert link]. Students would spend 10 minutes at each station, before rotating. We usually needed two days to get through all the stations.
• Some Big Problems are better experienced with a demonstration. For example, last year Nim was an option, so I played Nim a few times against students, had them play with each other at the front board, and then again with each other. All took about 10 minutes.
• The second time I did this, I had to be absent from school the next day. So instead I made one big packet of the Big Problems available for exhibition, and gave them two days to try them all out. This was much easier, though resulted in less real exposure to all the problems.
• This exhibition is also more generally focused on inviting the audience into the problem. Before diving into the process and solution, students are asked to have a kind of Do Now or something, where they are posing the problem (or a related one), to the audience. The idea here is that unlike the other kind of exhibition, nobody else has really spent any time doing their problem. It's also non-trivial to try to repackage a problem for maximum accessibility, and teaches you about how to consider your audience when planning a presentation.
• Logistics:
• Groups of 2-3. I tried to go one-to-one, matching groups to PSets, without duplicating any PSets, because it'd be boring to see the same presentation twice in a row.
• Product: group presentation, with slides, interactive component
• 3+ weeks total: 0.5 weeks for sampling + 1-2 weeks doing the chosen PSet + 1 week making the presentation + 0.5 weeks for presentations.
• I would do this around the end of the 2nd and 4th quarter. Since my school broke this course into two semesters, these were realized as end-of-semester projects.
• Assessing Exhibitions
• I would assess students using two rubrics. I would use the same rubric for math practices that I used for regular PSets. I would also use a second "Communicating Clearly" rubric (second page) that was pretty common throughout my school. Both rubrics would reflect both the work they presented, and the work they did in the weeks leading up to the presentation. I would keep a circulation tracker for the duration of the exhibition, to support in the final rubric completion.
• Regular PSets constituted 75% of a student's grade, so exhibitions were the other 25%. I made the first exhibition of the semester (Exhibition 1: Presenting for Depth) worth 10%, and the second one (Exhibition 2: Presenting for Access) worth 15%. It made sense to make the second exhibition a little heavier, because not only did it take longer, but it was experienced as a "final project" for each semester.
• Even before I was a math teacher, I coached high schoolers in presentation making and presentation skills. Maybe it's just because I'm teaching out of my content when I do it, but I always found it refreshing to coach students on the non-mathematical parts of presentation. (I am very passionate about title case and the Oxford Comma.) Here is a handout I gave students to help them think about making an effective presentation.
• The process of preparing a presentation is a huge skill in and of itself, and it always took longer than I originally expected. This is definitely because I've made so many presentations that I underestimate the layers and layers of necessary skills and demand. But I was also pretty clear within myself that, when possible, it's usually a good call to give students extra time when making a presentation. Presentation is a very high-leverage, high-transfer skill, and if I can support it in my math classes, I should.