Posted by: Gary Ernest Davis on: February 23, 2011

Jean Lave and Etienne Wenger wrote an influential book: Situated Learning: Legitimate Peripheral Participation (Learning in Doing: Social, Cognitive and Computational Perspectives).

As a budding mathematics educator this book had a big influence on me and, over the years since first reading it, I have often reflected how different a typical mathematics classroomÂ is from a community of potters.

Potters get together to share techniques, to learn from each other, to discuss projects and progress, because they are interested in making pots, or other objects from clay.

Students in a mathematics class are, in the main, doing things they do not want to be doing, because someone told them they need to know it, and generally have to listen to large amounts of instruction to do those things they did not want to do in the first place.

There’s nothing terribly “legitimate” about most mathematics classrooms.

Could it be otherwise?

To a large extent a mathematics class is going to be focused on what the teacher thinks the class is about.

There are times when what the mathematics class is about is just doing mathematics.

For example I have taught an undergraduate class called “Mathematical Inquiry” and what that class is about, in my mind, is solving mathematical problems and looking as deep as we can into mathematical questions. Which questions we look intoÂ is a moot point – sometimes questions i come up with (because I’m more experienced, and so the teacher) or sometimes questions the students come up with.

One of the courses my colleagues and I run is simply a workshop/seminar course in which students choose a topic in conjunction with an adviser and dig as deeply as they can into this topic over the course of a semester. That class runs more like a pottery session. Students are doing what they want to do, sharing progress with each other, learning from each other, and the more experience people in the class.

Currently I am teaching statistics and differential equations. I have written something about the differential equations class here. Despite that class being project based there is nothing “legitimate” about the class: most students would simply stop coming if I offered them a B grade right now. After Spring break I hope to focus on applied modeling, particularly on differential equation models for cancer growth. I am hoping this will engage the students, yet it still only makes the class interesting, not legitimate.

The statistics class is based on the book StatLabs by Deborah Nolan and Terry Speed (my former teacher).

I like this book a lot because it bases the learning of statistics on applied case studies, and helps students get a feel for what a mathematical scientist does.

After Spring break I want students to focus on their own data sets, so as to get more invested in the analysis. Maybe this will help the class become more legitimate.

Already 3 students are focusing on their own data: one is interested in how many steps different people take to walk a mile, another in autism and population rates in the various US states, and another in comparisons for quarterback completion rates.

So the statistics class is heading toward legitimacy, in my view.

But this is generally hard work, I find.

It’s not hard helping students learn the techniques they need to analyze data. What is hard is getting them to be interested in the first place: it takes time and effort to draw out their interests.

Form a “curriculum” point of view there is, of course, always the “danger” that some supposedly vital aspect of the curriculum will not be “covered”.

I say, as clearly as I can: screw the curriculum. I do not cover material. I engage with students in legitimate mathematical activity, through doing and participation, as best I can.

Most of the time that’s not good enough, but I’m going to keep trying.

1 | Blair

March 3, 2011 at 2:46 pm

“Itâ€™s not hard helping students learn the techniques they need to analyze data. What is hard is getting them to be interested in the first place: it takes time and effort to draw out their interests.”

I think you hit the nail squarely on the head here. The challenge in turning math class a collaborative workshop is not posing interesting questions, but getting students to develop their own interesting questions. Once a person knows the properties of clay (mouldable, dries hard, can be stained) it is easy to conceive of an end product that can be made of clay. It then becomes a process of developing the techniques to get from the clay to the end product.

If we think of this comparison in a mathematics context, students must understand the “properties of mathematics” and perceive a connection between the end product (situation/case/problem) and a mathematical understanding of it. I think this is difficult for students because in K-12 math is usually introduced as a method to solve a particular problem or situation rather than as a flexible material or tool similar to clay. In K-12, we (I am guilty of this) don’t help students look at end problems and make mathematical connections then learn or choose skills. We use/create problems that have one answer and particulary suited to one solution. No wonder those skills get associated with a limited situation.

Because of this approach, students are conditioned to be taught a specific approach to each problem they are presented. There is no development of the concept of what approaches might be available or why would one be more suitable than the other because they expect from experience to look for the one approach. Consequently, it isn’t surprising that students aren’t inclined to view situations in their world as being understandable through mathematical approaches. It takes a really interesting subject where the student has experience to then develop the questions and subsequent search for tools to answer those questions.

Changing this will require a dramatic shift in the way math is taught from K-12 and in the way curriculum is developed. Curriculum constrains the process of exploration by limiting to a very specific set of problems and methods. For example, one of the learning objectives is “Solve one-step linear equations of the form ax = b”. It takes a creative teacher with a good understanding of mathematical relationships to interpret and translate that into an exploratory approach where the students seek out situations that encourage them to ask the questions that would be solved in this method.