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 fairly straightforward to collect this information every week before our class. Given that we had realized that interesting gaps and mistakes were actually desirable for our presentations, we slightly modified the rubric as indicated here:
Indicate the degree of your progress on each problem using the following scale:
4 - I am ready to present a solution to this problem, including interesting gaps and possible mistakes. (We don't expect perfect presentations.)
3 - I've made a solid start on this problem and am willing to share my progress.
2 - I have ideas about how to solve the problem, but I'm struggling to consolidate them.
1 - I understand what the problem is asking and could explain it in my own words
0 - I do not understand the problem or I haven't looked at the problem
Using Corrin's suggestions, we downloaded the data into a spreadsheet and then color coded the numerical data. The image shows a color-coded grid for each of the 12 Focus Questions, going across from left to right. Each line corresponds to one student name (not shown here). In the picture you can see the a quick visual overview of where the class is at. We can see who is struggling or under prepared and which problems were particularly challenging. We could use this information to decide what to focus on in class and to choose who to call to the board to present what problem. This image suggests, for instance, that there is no need to present questions 2, 11, or 12--albeit for different reasons. Based on just this data which questions would you be curious about for an engaging presentation that could advance most students' learning in this class?
Where we initially wanted to use the Presentation Rubric to encourage students to put effort into being well-prepared for class, we are using this Preparation Rubric to communicate the importance of student effort in a better way.
Presentations and Giving Questions back to the Class
When questions arose around the presenter's work (or around the bigger mathematical picture), we typically collaborated 