Here we are with GeT: The News! It’s May of 2020, and what’s new? Since the last Newsletter, COVID-19 has brought challenges and changes to everybody in the world, including those of us in higher education. Lives have been lost to COVID-19, including those whose work was close to the work of our community. Among them was Judah Schwartz, who (with Michal Yerushalmy) created The Geometric Supposer, the first in a genre of applications that are now described generically as dynamic geometry software. This terrible disease has touched every one of us in one way or another. Our thoughts are with those who have lost loved ones.

Even for those of us who have not suffered physically, our lives have been disrupted in countless ways. As I write this, I know some of you are finishing grading courses that started face to face and ended online, while others have recently started teaching classes that they know will be completely online. We look into the Fall with uncertainty as to whether classes will or will not meet on our campuses. I imagine that most of us have been trying to cope with this new reality and that in most cases any desire to innovate has been put momentarily to the side. I also imagine that, in teaching as in many other aspects of our lives, once the pandemic hopefully begins to recede we will start reflecting on what we have learned from the experience. I wonder if the experiences of teaching online during the COVID-19 pandemic will elicit both problems and ideas that could have an impact on the teaching of geometry for future teachers in the long term.

One idea that we have been considering is using video annotation as a way to structure students’ activity online. Suppose, for example, that you had a video of students discussing the solution to the question, “we know that the angle bisectors of a triangle meet at a point, but what about a quadrilateral?” As one can appreciate, the question is not very clearly stated, which suggests that a discussion of it might include figuring out what mathematical question would be worth asking. The question could be “do the angle bisectors of a quadrilateral meet at a point?”; “on what conditions do the angle bisectors of a quadrilateral meet at a point?”; or perhaps “how could we describe the intersection of the angle bisectors of a quadrilateral?” Would it be useful to you to have your students view and comment on a video that presented a discussion of how to make that question more precise? What goals would you have? How would making comments on such a video prepare your students to pose and solve geometric problems themselves or to teach geometry?

Several years ago we developed a set of animations of classroom scenarios, many of which were situated in high school geometry classrooms and represented how a class might approach a geometry problem. Our goal was not to show exemplary instruction but rather to depict things that could happen in classrooms; we were using those animations as triggers for discussions among high school teachers. One of these scenarios, The Square, started with the teacher presenting the question cited above. A student immediately came to the board and drew a square and its diagonals, claiming that they met at a point. A second student said that if one just extended a pair of parallel sides in the square they could see that the angle bisectors and the diagonals were not the same things. At the insistence of the first student that his claim was only about the square, the class spent some time showing that the diagonals of a square bisect its angles. The discussion was, of course, much more lively and messy than how I am describing it. But, hopefully, my rendition will pique your curiosity and you will be interested in seeing it. You can find it online here.

Some years after we created that animation, our colleague Emina Alibegovic (now teaching in Salt Lake City) was teaching geometry for teachers in the mathematics department at Michigan. Emina decided to show the animation to her students and asked them to tap their desks whenever they saw students doing a mathematical action. The latter expression is rather ill-defined, and purposefully so—it was up to the students to recognize what might be mathematical in the story: a restatement of the question, the assignment of notation, a claim, an argument, a counterexample, etc. After Emina’s students brought up a variety of mathematical issues from the animation, they touched on precision questions (e.g., can one really say that the diagonals and the angle bisectors are the same in the square?), and spent much of the time figuring out whether the proof offered was valid and what was the statement that the proof proved. Watching that discussion in Emina’s class made me think that the animations could be used to teach mathematics to teachers: The animation could immerse future teachers not only in the classroom context, but also in the mathematical content that they needed to learn in order to be able to teach in the future.

Fast forward a few years and now we also have developed a piece of software, Anotemos (www.anotemos.com), that allows users to collaboratively annotate videos. I wonder whether a task similar to what Emina posed to her class could be done online using Anotemos. The advantage of doing it online and mediated by software is that annotations can be done asynchronously: students can play and annotate the video at their own leisure. Anotemos also has a lot of functionalities, not only to enable users to attach comments to moments in the timeline but also to places on the video screen; it also allows instructors to decide when the users can see each others’ comments. If this is something you might be interested in trying in your GeT course in the Fall, we’d be very interested in supporting you. Please be in touch!

Our next newsletter will come out in the Fall. I can’t begin to imagine what the world and higher education will look like then, but I hope that our GeT: A Pencil community can continue to support you no matter what form your courses take. Until then, best wishes for the health and safety of you and your loved ones.