mixed ability teaching – In Pursuit of Great Mathematics Teaching

Better Group Work

I just finished reading an article called “When Smart Groups Fail” (Barron, 2003). It’s a study of groups of three sixth graders solving maths problems. Barron divides the groups into less successful and more successful at solving the problems and then analyses what features of their group work were significant.

Surprisingly to me, the success of a group of students was not influenced by:

the amount of talk
the average achievement level of the students
whether anyone in the group had a correct idea

Instead, Barron found that group success was marked by:

accepting or discussing correct ideas rather than dismissing them
correct ideas were brought up when they related to ideas that were being discussed at the moment
group members paid attention to each other and if the others were paying attention to them
group members physically showed their togetherness by eye contact and standing around or pointing to a common workbook

This article’s findings indicate to me that my classroom culture can lead to better group work. I can help students learn to discuss a mathematical idea by building on each other’s thoughts. A well-managed whole group discussion can lead to more cooperative group work sessions. I want my students to learn to listen carefully to what is said and then agree or disagree or ask questions. I can request these responses in whole class time. Then students will see they are the norms that also should guide their group work.

Furthermore, the groups in Barron’s study were of mixed ability and their previous achievement levels did not correlate with their success as a group. This adds to my feelings about the benefits of mixed ability teaching. If students of differing abilities can be helped to communicate well, they can all achieve well.

The study also found that students in successful group went on to be more successful in individual tasks. Interestingly, students in less successful groups did as well as if they had worked on their own. Thus poor group work neither helped nor hindered their achievement. However, good group work improved the success of individual students in to a significant degree.

The school year is just about to start and I am looking forward to inculcating a culture of social, mathematically focused talk.

Barron, Brigid. “When Smart Groups Fail.” The Journal of Learning Sciences 12.3 (2003): 307-359.

A Pythagoras’ Theorem Open Question

An open question like this one:

is easy to pose,
is easy to understand,
can lead to lots of practice (if that’s your goal for the students),
can be kept simple or extended to suit lots of students,
can lead to an interesting discussion,
can lead to a generalisation.

I used this with a year 11 class that are revising for their Foundation IGCSE exam. I asked students to come up to the board and sketch some triangles that fit the criteria. Some triangles had 10 cm as the length of one of the shorter sides. Others had 10 cm as the length of the hypotenuse. Some students had chosen to draw isosceles triangles; other students drew scalene triangles.

This task can be made more simple by asking students to draw accurate triangles and measure the sides. Then they can extend this to checking using Pythagoras’ theorem.

On the other hand, this task can be extended by asking students to generalise what they have found. If 10 cm is length of the hypotenuse, what can be said about the other two sides? If the triangle is isosceles, there is only one answer. But if it is scalene, perhaps students will call the length of the second side x. Then they can come up with a formula for the third side in terms of x. And how does this change if 10 cm is not the length of the hypotenuse but of one of the other sides?

Wisdom of Crowds

At the 21C Learning Conference this past weekend, I heard James Surowiecki speak. He wrote a book called The Wisdom of Crowds. He shared many examples of circumstances in which a diverse crowd can give more accurate answers than any individual in the group. For example, a large crowd’s mean guess for the number of jelly beans in a large jar is often within 5% of the correct answer. Frequently the mean guess is better than any individual guess.

Surowiecki said that the key to a wise crowd is that it is diverse. When there are lots of differing guesses, the mean is more effective. But Surowiecki took this idea further and said that when a group needs to make decisions, a diverse group does better at this, too. On the other hand, a group of experts often all think the same as each other, and so do not always come up with creative solutions.

Surowiecki tried to make connections from his ideas and to issues facing the teachers at the conference. One useful idea is that a personal learning network (PLN) is more useful if it contains many different types of people. For me, this means I need to follow (on Twitter, for example) and talk to teachers from many different backgrounds. People with diverse skills will teach in different ways – and I should tap into this. Beginning teachers, for example, may not be experts but they have lots of new ideas to share.

But I was more struck by an idea that concerns teaching diverse groups of students. Surowiecki said that non-correlated mistakes don’t affect the wisdom of a group. In other words, as long as everyone is not making the same (type of) mistake, their work together will be greater than any individual contribution. I draw from this that I need to encourage a classroom atmosphere where everyone feels free to share ideas with none censored and all taken as valid. If students can share vastly differing ideas, then they are valuable even if they are wrong.

This made me think further about teaching groups of homogeneous students. A popular strategy in maths is to stream or set children by their ability. I can see that Suroweicki’s opinions can be used to argue against the wisdom of this. A more diverse group of students should be able to help each other learn better. Suroweicki didn’t go as far as saying this, but I believe this is a reason to teach in mixed ability groups.

Mixed ability teaching has been much on my mind lately – since arriving at West Island School I have had to teach more mixed ability classes than previously. Our year 7s are in mixed ability classes and our year 10s and 11s are in broadly mixed ability groups. I’ve (usually) enjoyed coming up with ways to differentiate work for these groups. But now I see that not only can students work together effectively in mixed groups, but perhaps they can even spur each other on to greater accomplishments because of their diversity. I can help this along by allowing (and requiring) discussion of ideas and methods among students of differing abilities.

Where do you stand on mixed ability teaching?