Nick: Trying for Earlier Ah-ha Moments

A few years ago, my wife bought me a yoga mat and a membership at an awesome studio in town. While I had done some yoga-ish things before, I had never really had a yoga practice. After doing a foundations course, I regularly attended different classes at different levels. One of the things I enjoyed about yoga was how different it was from almost any other athletic activity. In other words, the difficulty was the draw. About six months later, I was in a more advanced class. I had done the poses many times and heard the instructions about how to move into the positions just as many. I don’t remember what I was working on in particular but I finally heard what my instructor was saying about the movements. It was a revelatory moment; I finally understood the poses and motions were more than just lift your leg and extend your arm type things. I also understood that those other little modifications weren’t always apparent to someone watching from across the room. They were very personal modifications. I know I had an excellent teacher when after I started to try to find those personal adjustments, she gave me even more comments during class. I’m pretty sure she saw things she wanted me to change for those six months (and sometimes she did comment), but I wasn’t ready for those small but personal changes yet.

Those revelatory moments are impossible to predict but monumental in their effect. One of the big things I am trying for this semester is get more of those moments as early as possible for my students. This spring I have three different courses: a freshman level analytic geometry class, a junior level linear algebra, and a section of junior level differential equations.

Since I have done the geometry class and linear algebra several times before, I am making relatively minor changes to both classes. The biggest changes I am trying to make in my geometry class is a shift in the early problems. I am spending a couple of weeks doing some problems that I would expect students to know before coming into this course. Since many of the students know how to find the answers (and thus have an early success), I get to use this time to acclimate students to focusing on the whys of the problem. One of the important algebra skills in the course is taking the algebraic expression you are given and making it fit a very specific form (think of skills like completing the square). Students rarely have an idea of why they do the algebra they do. In fact, they are usually told by a teacher that the algebra will make the problem easier, but they don’t often see it that way. Students have had about a dozen Ah-ha moments already with the following two problems:

Question 14.

a) For what values of A and q will the expression 3x+2 be of the form A(x+q)?
b) For what values of A and q will the expression 3+2x be of the form A(x􀀀-q)?
Question 15.

a) Expand (x􀀀-a)2.
b) What value should ♦ be so that x􀀀2 -4x+ ♦ is of the form (x􀀀-a)2?
What is a for your expression?
c) What about for x􀀀2 +9x+ ♦?
d) Or 2x􀀀2 -2/3 x+ ♦?
e) For what value of ♦ will 2x􀀀2 -2/3 x+ ♦ be of the form B(x􀀀-a)2?

I don’t really know why we have had so many Ah-ha moments with what seems to me to basic problems, but I’m thrilled to have them.

In my linear algebra class, I am using my own set of problems that have worked out pretty well for students the last couple of semesters. One difference that I think will make this semester interesting is that I only have 1 student in the class who has seen this material. In previous semesters, there were at least three students repeating this course. I’m not sure how this will effect the level of work the students present but I’m interested to see how this is different than previous iterations.

I last taught our junior level differential equations course three years ago and only found out that I would be teaching this class the Friday before classes began. I have to thank everyone who sent me information about active learning activities in an ODE class. I am using a great traditional text (Blanchard, Devaney, and Hall) but I didn’t feel like I had time to prepare much in the way of activities. As of now I am introducing new topics and having students work on problems as a class or in small groups. I’m hoping that I will incentivize reading before class enough that we can work up to a minimal discussion lead by me followed by students working in class with homework presentations at the start of the next class.

As always, I welcome your feedback and ideas in the comment section or by email at longne at sfasu dot edu.



Liza: Which One Doesn’t Belong

I was first introduced to Which One Doesn’t Belong (WODB) at a talk given by Dr. Skip Fennell at NCTM Regional. I was very impressed with the engaging discussion that WODB inspired among the participants of Dr. Fennell’s session. I also loved how the focus was not on the correct answer, but rather developing participants ability to explain their thinking and justify their answers. Unfortunately, I was unable to use WODB with my students after the conference, because the semester was over (except for final exams). However, ironically a couple of weeks after the conference at a meeting of our local Math Teachers’ Circle one of the participants shared WODB . The circle members (all math teachers) absolutely loved it! Now that the spring semester has started, I am looking forward to incorporating WODB into my classes. I think that it will fit well with IBL methods. More specifically, I plan on assigning WODB items to students and having them present their problem solving processes and answers to the class. I also think it would be beneficial to ask students (particularly my preservice teachers) to create their own WODB. I am looking forward to using WODB this semester and will be sure to follow up on the blog with how things go.

Nick: How Things Went This Semester

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.