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.



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