IBL Algebraic Geometry: Focus on Student Presentations
Written by: Dr. Volker Ecke
This past semester, Chrissi and I co-taught an advanced mathematics major course focused on Algebraic Geometry. Our goal was to strengthen our students' preparation in reading and making sense of, presenting, and writing about mathematical ideas. We used a new (for us) format for the class, relying heavily on readings outside of class and in-class graded presentations based on Focus Questions related to those readings.
Overall Structure of the Class:
The class met once a week for 2.5h. In the week leading up to class, students worked through a significant portion of a formal mathematics textbook (Cox, Little, O'Shea: Ideals, Varieties, and Algorithms) and prepared written responses to a list of Focus Questions with the expectation that they would be ready to present these on the board to the whole class (Sample Focus Questions). Typically, we would chose a subset of "interesting" problems to be presented in class. We learned from Dana Ernst (who credits Clark Dollard) to give the students "Purple Pens" so they can revise their own drafts based on the presenters' work. This worked really well!! Finally, as part of their homework for the next class, students were asked to write up a "good version" of several of the Focus Questions along with their annotated first draft (sometimes these problems were presented, sometimes not).
A few observations for us:
In summary, here are a few of the observations we made in teaching Algebraic Geometry course using this approach. More details about some of these observations are included further down in this blog, or in related blogs elsewhere on our site.
- Reading formal mathematics is challenging for our students: we're now wondering how we can better support students in building this skill throughout our major program.
- Student writing improved a lot, in the sense that students included more information about their process and their thinking, so we could really get into students' heads and see 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 