Our school operates on a trimester schedule. The fall term ends as the Thanksgiving vacation begins and the winter term ends when our spring break begins. Before our fall finals I have been taking some time to review and reorganize some ideas with my students. I have to say that the past two days with my Geometry students have been filled with a combination of two strong feelings.

The positive feeling is that I am really proud of how my students have been working together. My class is set up in three ‘pods’ of six desks each. I am spoiled and my largest class has 16 students. At the beginning of the week I uploaded five review problem sets for my Geometry students. My plan for this week is to have one problem set each day and to circulate through class during the week, putting some burden on my students to be actively reviewing ideas rather than sitting and listening to me. The students have really been going at it and I have enjoyed listening to them. A couple of good hearted heated debates have popped up and I get called in to moderate these debates. There was one particular problem that I asked that I am proud of.

The points M(3, -1), N (4,3), and P (0, 5) are the midpoints of the sides of a triangle. Find the coordinates of the vertices of this triangle.

I am not certain that I can remember the origins of this question. It is likely that I found this somewhere and if anyone recognizes where it came from, please remind me. What really pleased me about this problem was the response of one of my students. I overheard a number of students discussing this problem, so I took the opportunity to gather up the class and lead a conversation here. I drew a triangle on the board, not on a coordinate plane, and joined the midpoints of that triangle (rough sketches here) then I asked them to make some guesses about the relationships between this ‘midpoint triangle’ and the ‘parent triangle’. It was very quickly apparent that the students expect that the midpoint segments are parallel to sides of the original triangle and the guess that lengths are related came out as well. No questions were raised during this part of the conversation. This group chat convinced me that the question I asked is much more interesting than if I had asked the question in the opposite direction where I gave the parent triangle and asked for the midpoint one.

Even with an agreement in class about the relationships here, the work to translate this to the coordinate plane is a tough leap. Early in the year I gave a few questions where I would give a segments endpoint and a midpoint and ask for the other endpoint. Not a creative problem, but one that the students remember. Since I knew that I wanted to introduce vector notation early in the year, I would frame these midpoint questions by talking about this problem in terms of taking half of the journey here. So the movement from one endpoint to the midpoint is half of the journey and we want to repeat this path. This way, I try to get the students thinking in terms of component movement and not simply in terms of distance. So, I took this approach with the class on this midpoint problem. Time did not allow a conversation that was as deep as I wanted it to be and I plan on starting class today by revisiting this problem. I am unsure whether I will use the same coordinates or not, I am inclined to think I will not. I also know that I will start class today by simply trying to get a few students to explain the thought process we outlined, not to focus directly on the necessary calculations. I have been working hard at getting my Geometry students to frame a process of problem solving for complex problems. We recently worked on finding the distance between two parallel lines and I asked my students to outline their approach to the problem rather than calculating the distance, so framing a question like this in the terms of describing their problem-solving approach will make sense to many of my students. After we discussed this problem at the end of class yesterday, one of my students remarked out loud that this problem was really hard but really interesting. I was pretty pleased to hear her think out loud like that.

i started off by mentioning that I had two strong feelings so far this week and I started with the positive feeling. The not so positive feeling is a deep sense of frustration about the overwhelming lack of recall of some simple facts. At least five of my students were stymied by a simple question of finding the midpoint of a segment. They consistently wanted to apply a simple formula and they could not (would not?) stop and think about what kind of formula might even make sense. For any teacher who has taught this idea, it will be no surprise to you that the debate centered on whether to add endpoint coordinates or to subtract them. I reply by asking what midpoint means and every student quickly says middle. Then I ask what number is in the middle of 2 and 10. I try to convince them that they do not need a formula if they are willing to stop and think. I wish that I had gone to the coordinate plane quickly to try and tie together some physical sense of these points. Instead, I was visibly frustrated and asked a few students what their average would be if they earned a 90 on one test and a 100 on a second. Would their average be 5 or 95? However, thinking back on this exchange I realize that I was not effectively making a point here, I was simply showing my frustration. This phase of the year where students are trying to tie ideas together should be a time when I am happy to see the growth in my students understanding. More often, this ends up being a time of frustration with finding myself repeating ideas that I thought we had successfully conquered early in the year. I understand the pressure that students feel when they are faced with a week of long exams and stress related to trying to show mastery of twelve weeks of knowledge all at once. I have been at this for a long time now so I feel that I ought to have a better idea about how to navigate this challenging time of year. I have worked at four schools now and they all believe in final exams, so I have tried to work within that context. I also believe in the principal behind cumulative displays of knowledge. I just know that the way we do it creates such stress for most of our students that they do not feel like this is a show of knowledge, they feel it is simply more of a test of their endurance.

I would love to hear how any of you navigate this challenge.