Do the Math

I have spent the better part of a year working with my amazing teaching teammates exploring new technologies with our middle school students. We have put in the extra hours and energy it takes to try these new methods of communication because we have seen the impact these methods have had on our own learning. We have explored blogging, wikis, ed.voicethreads, and many other wonderful tools. As a team, we failed with a few of these tools but we were very successful with a few others. One of the tools that has been most effective is online discussions.

Back in November, our amazing Literacy teacher created a unique online environment in which some insightful commenting and fierce, but respectful, debates have taken place over the last eight months. It was an inspiring experience to read our students’ insights and feelings about the ending of Of Mice and Men. Our team has since held a variety of successful online discussions about many different topics, but I have had one question that I have struggled to answer, “How can these online discussions be effective in my mathematics classroom?”

I have since continued to rack my brain, trying to figure out effective ways of discussing mathematics online, and until this evening at 10:00pm, I had no good ideas and no one to turn to.

10:00 pm is when I got home and checked my RSS reader to find a link to a blog post that my Literacy colleague had sent to me. The title of the post was “Mathematics En Masse” written by Jason Dyer on February 28, 2008.

In the post, Dyer describes different kinds of online discussions that can work in mathematics. He mentions discussing questions that have multiple solutions or varied answers such as the area of your bedroom floor or estimate of linear functions that vary student to student.

These ideas sparked my imagination, but then I read the comments left on this post and I really got rejuvenated. I loved two comments, in particular, because they seemed very reasonable, engaging, and particularly useful and new. I have included them below.

Tomorrow is the last day before a long-awaited Spring Break, but I think I just found a new project to plan over the “break.”

So here’s to my wonderful colleague who will never let me quit and has always helped me find ways to become a better teacher and lifelong learner.

Jason – I have a solution for this one called the “Elaboration Method” – in short. You assign a problem and everyone solves it. But the problem doesn’t end there. Then students are required to elaborate on the solution. They might show a) how to check the answer, b) how to use a graph to verify the answer c) what happens to the solution if the problem changes slightly, d) write a word problem that results in the equation or expression, e) anything that elaborates on the problem and demonstrates that they not only understood the problem, but something else that had to do with it. I’ve used this in many traditional classes and often I will get at least 10 different elaborations for the same problem. Sure, some are repeats, but many are different and the elaborations let students explore the concepts that they are learning in a very creative way.

It would not be unreasonable to post a problem set of 3-5 problems and ask students to complete an elaboration on one of the three problems. Of course, the sooner they do their elaboration, the easier it is.

I always awarded truly insightful or original elaborations with an extra point. Thus, students are rewarded for work above and beyond what is asked for.

I haven’t tried it with a wiki, but I’ve been thinking about doing it on the message boards in my calc classes. It’s a good way to emphasize conceptual understanding, notation, and the myriad of relationships between mathematical concepts.

I have a grading rubric and examples to illustrate the technique, but I have to proof 350 pages of algebra text tonight and probably won’t be able to post it today. Hopefully that will be enough to get you started…

Posted by: Maria H. Andersen | February 28, 2008 at 04:14 PM

I suggest that students work together to develop a trickiest problem possible for each chapter/lesson. Individually, students could try to develop their own “tricky” problems (you know, like the ones near the end of the set in the book; they can be solved with the same formula, but they’ve got some kind of twist or extra step). The group could then discuss each other’s problems and work on ways to combine what makes them tricky into one crazy problem. It seems like it could scale well — classes could each come up with their own and vote, or you could have smaller groups within a single class. The problems could be brought back during reviews or tests. Students who are struggling in the class would be able to give strong contributions by pointing out the specific formulas and solution steps they struggle with.

There’s something fun about students deliberately trying to make math difficult, it sort of lets them turn things around.

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