We often ask our students to perform some reflective exercises as part of the process of changing their mindset about mathematics and more generally about education. I think it is just as valuable to do such exercises as part of a culture of growth and improvement in teaching. I have been using this blog as one of my outlets for these reflective exercises and I think it has been a very valuable experience. I hope to have a few more people write on this blog even if just for a single post as part of discussing the reflective part of teaching. I am also thankful for the people who have responded and the productive discussions that have ensued.
I wanted to finish off my posts on the fall semester with a few things that went well, a few things that I can improve on, and what is perhaps a crazy idea that I am considering for the spring.
With no further ado, one of the things that went well in all of my classes was that students were challenged in their mathematical ability as well as other areas that will be vital to their future success, like persistence in problem solving and ability to work on problems that are not mechanical calculations or problems that have no immediate path to a solution. Additionally, some students had enough “Ah-ha!” moments to realize that math should make sense to them and that math was not just some tricks given to them by someone at the front of the room.
About three quarters of the way through the semester, my linear algebra class was discussing some homework that had been presented when a student asked, “Do all linear algebra classes talk about all of this stuff?” Another student quickly offered the response that they had taken this course with another professor and “while the other class talked about more topics, this class is more… profound.” I did not have a stoic response to these comments, in fact one student asked why I “was smiling like that”. After several other students agreed with that statement about this course being more profound, I replied that I was very happy to hear that students had what I think is a wonderful response to what can be a very difficult learning experience.
One of the biggest things I will look to improve on is getting a better set of opening problems to mitigate the frustration students have when they are not simply shown how to do examples. As I wrote about earlier, I think the problems (and probably the students) get better as the semester goes on and in reviewing my questions, I expected to much of the students. It was not that they were not capable of doing what I asked but that the mechanisms for getting unstuck were not fully in place, so students did not always respond well to adversity.
One of the things I am looking to improve is to have earlier reflective exercises in order for them to embrace the growth mindset and help them make more lasting changes. I have seriously reworked some of the opening material in my classes to backup these earlier reflective exercises.
Something else that I have talked about and is really guiding my approach to classes in the spring is that I can talk in the classroom. In fact I should talk, but I should not be the dominate voice in the room . Further what is done and said in class should be guided by students. Toward this end, I am trying to make sure that I have brief demonstrations or 5 minutes of introducing new material for each class meeting. I have been building a lot of things on Desmos since that makes dynamic presentations easy. An example that built for hyperbolas is here.
While students will identify that they need to do more than get the answer, making the changes to sense-making as the primary response to math is a long process. It is my job to get them as far in that process as I can in one semester. Sometimes I can help and sometimes I can’t.
I will end this post with a crazy idea that I have been thinking about especially for my 100-level geometry course. What if I gave student’s the answers to the problems? Not solutions including process but just something like x=2 and y=4. I’m not sure if this would properly alleviate any of their anxiety about getting the answer and allow them to focus on process. Has anyone ever just given answers willingly to students in order to change the conversation?
As always, I welcome your feedback and ideas in the comment section or by email at longne at sfasu dot edu.