# Group Presentations: Mathematical Flow

There are many things to consider when having students work in cooperative groups and make group presentations. One of the major decisions is how to group the students. Another is how to choose the order of student presentations.

Mathematics educator Jo Boaler and mathematics teacher Cathy Humphreys hope to challenge the way you pick your groups. They have assembled a set of video cases from Humphreys' classroom in order to examine the complexities of teaching mathematics in an inquiry-oriented manner. Throughout the school year, students in Ms. Humphreys' classroom were challenged to work on tasks like "Hikers Beware" and form mathematical understandings based on their work and subsequent discussions.

So what did they discover? In their analysis of the lessons they studied, it became apparent that an important feature at the end of each collaborative problem solving session in Ms. Humphreys' class was the sharing of solutions and strategies. To optimize learning at this critical time, Ms. Humphreys had to consider the *mathematical flow* of the students' ideas (Boaler & Humphreys, 2005, pp. 86-88). She would purposefully sequence group presentations such that the ideas escalated from simple to complex to ensure that opportunities to make important connections would arise. This required her to take notes about students' work as she monitored each group's progress during their collaborative work.

The following two clips from Ms. England's "Hikers Beware" lesson demonstrate this idea of **mathematical flow**. Let's start with the first group, who have written equations to represent the families' water supply in relation to days in the form y = mx + b (specifically, y = -5.8x + 40). A key item to note: since each slope is negative, the coefficient of x is negative for each of their equations.

The group in the second clip was selected to present last. Their equations are written in the form of y = b + mx (specifically, y = 40 - 5.8x). The first term is no longer negative and we now have a subtraction operation. The teacher takes a moment to highlight this and asks her students to consider whether this is equivalent to the earlier equation given by other groups. Once this is confirmed, she asks them to think about which form of writing the equation makes more sense in relation to the problem context.

The decision by Ms. England to have this group present last was an important one. She had noticed that only one group wrote their equations in this manner-- a fantastic opportunity to have the whole class think about whether these were equivalent forms, and whether one equation made more sense than the other in relation to the actual scenario being represented.

**Classroom Clip Reflections:**

- What considerations did Ms. England make as she determined order of presentations? What evidence would she have looked for to determine order?
- What strategies could be used to help the whole class engage with the group presenting?
- How does she deal with the last group who had a different equation than the other groups? Were the students able to explain whether or not their equation was equivalent? How could you find out whether or not they understood this concept?
- What are the implications in your work with students?