In my last post, I summarized Peter Liljedahl's paper on "visibly random groups." That research confirmed many things I already knew from experience. Today, I will summarize another one of his papers, this one titled "Building Thinking Classrooms: Conditions for Problem Solving." (It is also available on ResearchGate.) I learned quite a bit from reading it, and recommend it without reservations.

Liljedahl is a math ed prof who works with classroom teachers, trying to promote a problem-solving agenda. Early on, he encountered a major obstacle: the belief among math teachers that their students "could not, or would not, think". I have led hundreds of in-service workshops, and I have heard this expressed countless times in various ways: "I've tried this kind of approach, but my students just want me to tell them the answer;" "This may work at your school, but my students are very low;" "My students are so anxious about standardized tests and grades that they are too impatient to do this." A teacher educator coming across this strongly-held belief repeatedly has some choices.

One possibility is to decide to work just with teachers who are already on a problem-solving path. Such teachers, unfortunately, are a small minority. Another option is to hope against hope that by just working with pre-service teachers, one can affect the next generation. Alas, that does not work so well, because young teachers get jobs in schools where there already is an established anti-intellectual culture within math departments, and good teaching is seen as giving students ways to do well on tests by avoiding thinking and memorizing procedures. The last and worst choice is to sink into cynicism.

Instead of those paths, or in addition to them, Liljedahl decided to explore what concrete changes can be implemented to change classroom culture. This paper summarizes his discoveries. The first is that what he calls "classroom norms" cannot easily be changed. This, of course, is true of all aspects of school culture. To tackle this, he made a list of nine elements of math teaching, such as: type of tasks, how groups are formed, room organization, the use of hints, and so on. He then proceeded to experiment in his own teaching, and in that of collaborating teachers, with varying these elements.

Here are some of his conclusions:

- Working in groups on (vertical) whiteboards while standing up is vastly superior to any other arrangement. Horizontal whiteboards (on desks) were second. This seems to be due to visibility, erasability, and the break with routine. Moreover, teachers who adopted this setup stuck with it, even though the idea was introduced in a single professional development session.

I have no problem believing this. I once visited a private school in San Francisco (the Bay School,) where math classrooms are surrounded with whiteboards, and students are expected to do a lot of exploratory work on the boards. I was blown away by how effective that seemed to be. For one thing, student thinking was easily visible to the teacher, and to students in other groups, which is definitely helpful in developing a thinking culture.

- Lessons need to begin with problem solving tasks. Initially those need to be highly motivating, and over time they can be more and more accurately aimed at the intended content of the lesson. (See this post on Scott Farrand's approach to problem solving warm-ups.)

- Combining visibly random groups, (preferably vertical) erasable surfaces, and problem solving at the outset are three practices that are reasonably easy to implement, easy to maintain, and have an enormous impact.

Liljedahl suggests six additional refinements for a second and third stage in the process of moving towards a thinking classroom. Those are harder to implement, and some may be more debatable. I encourage you to read the whole paper, and see which ones might help you in your classes.

--Henri

"There is no one way"

## Monday, May 11, 2015

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