more clearly what they knew, and what they were still struggling with. Read more about these "Homework Stories" in one of Chrissi's blogs.
- A Perfect Presentation has Mistakes.
- Students Revising Draft Work: Purple Pens Rock!
- We found quick, ungraded check-in surveys, or preparation rubrics, helpful in deciding what questions to have presented, who gets to present, and for which questions students may be totally unprepared, etc. Thanks to Corrin Clarkson from Columbia!
- Technology can be a valuable tool in supporting students with inquiry activities.
- Mathematics can be beautiful, and Art can motivate mathematical inquiry.
- Group work can help to digest and clarify presentations.
- One open question about this classroom structure that we're still thinking about is: how to make sure that all students get to work at their learning edge.
Technology supporting Student Inquiry:
Presentations typically involve some written work that supports the audience in making sense of the argument. For example, students can write key ideas or steps on the blackboard as they explain their ideas. Writing in real time can have the advantage of focusing everybody on the same idea, and giving the audience time to digest and reflect on the material. Some of our students also made valuable use of a document camera showing their homework write-up to guide the presentation. This way less time can be spent with writing when ideas are clear. We have also seen models where all the students selected for a presentation in a given meeting simultaneously take 5-10 min at the start of the class to write the key ideas on the board (while the rest of the class is involved in some other activity). This can be especially valuable in an intro to proof course, since students get to be deliberate before class about their choices of how much, and what, to write down.
We opened the class with activities using the beautiful "SURFER" software from Imaginary. Surfer artists can enter a single algebraic formula, which is set equal to zero. The software then renders the geometric object corresponding to the equation (typically a surface), using pretty color schemes. The wish to create beautiful images can motivate the inquiry into the algebra: e.g. how to you create the image of a snow man; how to you show the intersection of two geometric objects.
A project towards the end of the semester asked students to investigate, in the specific context of their "own" surface, a number of questions about the formal mathematical tools we had developed over the semester (e.g. the relationship between the Groebner bases of two varieties and the Groebner basis of their intersection). We have included two beautifully intriguing screenshots of student projects in this blog. In addition to SURFER, students also made extensive use of GeoGebra and WolframAlpha to explore computational aspects of algebraic geometry.
Our plan for the graded presentations in Algebraic Geometry was in part to give students incentive to put effort into the assigned reading and Focus Questions ahead of class. In other classes, we also have students share out their ideas and process with the whole class. Yet this typically happens at a time when they are clearly not done yet. Those "process presentations" are not graded and are meant to spread ideas, catch mistakes and make connections. Chrissi reflects on the evolution of our ideas around a grading rubric for presentations in her blog A Perfect Presentation has Mistakes.
About half-way through the semester, Corrin Clarkson from Columbia shared an interesting teaching strategy on an IBL workshop email list. She asks students to fill out a simple, ungraded survey ahead of class rating their level of understanding for each of the problems they were asked to prepare. In addition, students can ask questions or suggest topics for class discussion. We immediately realized that this could provide us with additional information in deciding who would present what problem. Corrin was kind enough to provide us with a copy of her Google Survey Form, which made it fa