Math Wizard

I just read the following online article: Why Use IBL? and wanted to note the link for later use. This is a great summary of the history, use and effects of Inquiry Based Learning (IBL) in the classroom, and its undergraduate-level cousin, the Modified Moore Method.

My approach to teaching has evolved along with the use of IBL. My work is to develop material to guide my students’ exploration, and then to facilitate their mathematical investigations.

In my current work, I teach two graduate level courses. One is an Abstract Algebra course where I guide my students to reconstruct and understand an unusual proof of the Fundamental Theorem of Algebra first developed by Euler and others. This course is a delightful collection of topics connected by several themes, including problem solving, the connections between graduate level mathematics, use of historical mathematical documents and the school curriculum, and of course, the Fundamental Theorem of Algebra. The other course is a Real Analysis course, which makes use of my modification of Mahavier and Mahavier’s Analysis problem sequence. We don’t have much time, so for this course my personal goal is that the students prove the Intermediate Value Theorem from the ground up.

I liked reading the Why Use IBL? article I mentioned in the first paragraph, so now I intend to make it required reading for my students as they start the Analysis course.

One of my favorite sketches in the Monty Python film, The Meaning of Life, is of a middle-aged couple (completely lacking any intellectual curiosity) who have an awkward and scripted conversation.

The waiter tries to interest them in conversations about minorities, football, baseball, and finally manages to start them on a conversation about philosophy.

You can read the conversation here at the movie script site www.intriguing.com.

This sketch came to mind last week when my MAT students and I discussed conversations in class. They told me why they value conversations in math classes, and we considered two math ed articles:

Chazan, D. & Ball, D. L. (1999). Beyond being told not to tell. For the Learning of Mathematics, 9, 2-10.

This article which is based on two case studies, seems to be a reaction against the (then) recently released NCTM Principles and Standards which admonished teachers “not to tell”. Chazan and Ball are eager to describe many classroom circumstances where it is completely appropriate for teachers to take the lead and actively direct the class.

Chazan and Ball leave with three considerations:

Mathematical Value in Relation to Students – does the current student discussion have significant value to the students’ understanding of mathematics? Not all conversations are valuable.

Direction and Momentum – is the conversation at the right pace and level? The waiter in the Monty Python sketch recognized that the couple’s conversation was going nowhere, and helped them get started. The results were, well… let’s say better than before the intervention. A teacher can keep a conversation going at the right pace and at the right level of intellectual challenge for the students.

Social and Emotional Tone – part of the classroom culture is how students treat and respect each other. I’ve seen classes where the students are downright mean to each other, and nobody is willing to take intellectual risks. Luckily, the opposite is true in most classes I visit, and a lot has to do with how the teacher intentionally created the supportive and cooperative atmosphere.

The second article we looked at was more recent, and directed towards teachers taking their first steps at facilitating rich mathematical discussions in class.

Orchestrating Discussions, Margaret S. Smith, Elizabeth K. Hughes, Randi A. Engle, Mary Kay Stein, May 2009, Volume 14, Issue 9, Page 548.

This article presents a framework for teachers in the context of a discussion that follows a student activity on proportional reasoning. The five steps to a successful discussion are:

Anticipating – as part of your planning, imagine different ways that your students could successfully approach the task, and what misconceptions or difficulties they might have. Write these down and have it with you during class.

Monitoring – as you circulate around class during the activity, intentionally monitor your students’ work and identify which students are using the strategies that you’ve anticipated. This frees you up from having to think too much in the moment, since you’re prepared for most of what will happen. It also means that you’ll have more energy to focus things that you have not anticipated.

Selecting – the authors recommend against a show-and-tell style discussion. Instead, carefully select which student work you’d like to highlight, and have a reason to do so.

Sequencing – now that you’ve selected which student work to highlight, the order that the class shares it in is just as important. One possible trajectory is from the simplest strategies to the most abstract, but it’s important to plan a sequence.

Connecting – as the class discusses their solutions, the teacher can make explicit mathematical connections for the students and highlight certain aspects of the material. After the activity, the students are primed for assimilating the new knowledge and connecting it to what they already know, so here is where the teacher can maximize the benefit of the activity.

Favorite Quote

My favorite student comment is that all of this is just plain common sense. That’s how I see it too – that once you’ve read through this article, it’s pretty much self-evident that this is a good way to plan for the discussion after an activity. No controversy here.

Common sense may be a revelation to some, but it might be too often overlooked, and it is useful to hear it once in a while, especially as we get our start in teaching! What is your favorite common sense advice in teaching?

One of the many special things about the Bard MAT Program is just how much time students spend out in the schools, allowing for a gentle development of their skills and understandings as teachers.

Now that September is nearing an end, the MAT students will soon scatter around the neighborhood, to the nearby classrooms at some of our partner schools. Here’s a map I made of the placements:

View Welcome to the Bard MAT Neighborhood in a larger map

Where did you do your student teaching? What would make for the ideal student teaching experience?