Generating a Genuine Mathematical Discussion

One of the most difficult tasks of a math teacher is fostering an authentic discussion about math. Every now and then it comes back momentarily in small groups, but I have trouble generating a real math discussion. I know there ideas out there in the internet ether, but I have found that as long as students are given prompting worksheets, think-pair-shares, they will always want to know what answers to put down so that they get the highest grade. When I ask a class to discuss for the sake of discussion, most of students will give me, at best, lip service, since the discussion won’t have any immediate impact on their grades.

I want my students to discuss math. I want them to discuss math because it is the most effective form of mathematical learning that I have encountered. In math teacher land there is often debate about finding the right balance between practicing procedural fluency and developing conceptual understanding. The procedural fluency camp usually follows a dogma of basic skills and will lament the “fuzzy” math of the 1980s and 1990s. The conceptualists worry about cookbook math and creating math zombies. Myself, I lean towards the conceptualist. However, I do rely on a lot of drill and kill during class. Procedures are great for immediate impact, but if I want long-term, flexible learning, I need to have high quality discussions.

In the past I have had one class where discussion has flourished. That has been my Caclulus I class. My Calc classes have always been small and have always been with students that I have had in previous classes. Because of this familiarity, I was able to make a bargain with my Calc I students. I would give up my power, in the form of grades, if they would give up their expectation of the reliance on examples. It worked beautifully for three years. There was absolutely no structure to the learning. When we would learn, we would just open the book and start reading and working. Some days math didn’t happen because, well, we didn’t want to. Some days we talked about other stuff, like college essays or homework assignments from other classes. Instead of viewing me as the authoritarian, or even authoritative teacher, my Calc students started to view me as more of first among equals, as more of a peer with extra experience. So, when we decided to math we did it because we wanted to, not because we had to.

Anything that was learned in that environment I really feel is more impactful, more powerful, and more portable than what is learned in a regular classroom. There is one story that I can think of that perfectly illustrates what I mean.

A student in Pre-Calc asks me, “Did you hear about Alex?” (Former Calc I student, name changed, who was then a freshman in college.)

“Umm….no. What happened?”

“He failed his Calc quiz.”

“Okay.” (I really think this student wanted me to make some sort scene in class, but I didn’t. Inside though, I was screaming WTF?!!!)

While my Calc I class is not for college credit or an AP class, I feel that I do enough that Calc I should be mostly review for my students when they get to college. Fortunately I ran into Alex around Christmas break and I felt compelled to ask about the failed quiz.

“So, I hear you failed one of your first quizzes.”

“Yeah, that was stupid. The quiz was about finding derivatives using the limit process, but I just used the power reduction rule.”

“Okay, whew. I was worried that I had really screwed up, but really it is about your inability to read directions.”

“Yeah. I met with the professor during his office hours and talked to him. I explained what happened and then talked to him about what I should be doing.”

It was reassuring to hear that he didn’t ask for extra credit, to redo the quiz, or fix his mistakes. He felt comfortable enough with the math I had taught him to go discuss it with his professor. Not only did he feel comfortable enough with math to discuss math, and not just demonstrate procedures, he felt that his knowledge granted him the authority to approach the professor. (I have wondered if this is a skill I was implicitly teaching during Calc and does it apply to subjects outside of math.)

That is what I want out of my Calc class, but this year my Calc and Pre-Calc classes are combined. I have figured out how to approach the topics so that I can teach both groups without giving too much subject material up, but I wasn’t sure how I was going to grade my Calc students compared to the Pre-Calc students. My Calc students know what my Calc classes in the past were like and have been wondering if they would get the grading leniency that I have shown in the past. I kept telling them I wasn’t sure, since they will be covering the same material as the Pre-Calc students.

This past Friday I gave my first quiz. I have already noticed a couple of interactions with my Calc I students that make them different than most of the Pre-Calc kids, but when the quiz was given they were the last ones working. Their approach to the problems were different than all but a few of the Pre-Calc students. Everything about how Friday went tells me that they are ready for how I run Calc I, but I know I can’t run my Pre-Calc class of 23 like I have run my Calc classes in the past.

I don’t know what to do.

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