Episode 4: Illustrating Inconsistent Reference to Previous Contextualized Work

Each Investigation in the Core-Plus textbook contained a section called

Summarize the Mathematics; through the tasks in this section, students are meant to reflect on the mathematical ideas underlying the tasks they encountered in the

investigation to “construct a shared understanding of important concepts, methods and approaches” (Hirsch et al., 2011, p. 12). Ms. Spence assigned the Summarize the Mathematics section in Investigations 1 and 2 and created her own summative activity targeting general principles for Investigation 3. To determine the extent to which the discourse during these tasks was explicitly connected to prior contextualized work, I analyzed student and teacher interactions around these sections for the presence of explicit references to previous contextualized problems. The following excerpt is from a class discussion about the Summarize the Mathematics section in Investigation 1. In the excerpt, there is no reference to previous, contextualized problem solving, which is representative of discourse that occurred around summative tasks addressing general mathematical principles in this unit.

Question b of the Summarize the Mathematics section of Investigation 1 asks the following:

How can the rate of change or the slope of the graph for a linear function be found from a

i. table of (x, y) values? ii. graph of the function?

iii. Symbolic rule relating y to x? iv. NOW-NEXT rule. (p. 156)

Ms. Spence: Now part b we’ve done several times today. I'm not gonna do it another fifty times but I am gonna show you and talk about it real quick. It says how can you find the rate of change or slope by looking at a table? How can I find the rate of change by looking at a table? Marcus explained it very nicely a couple times today. How do you find a rate of change when you look at a table? Javier, do you know?

Javier: Huh?

Ms. Spence: How do you find the rate of change by looking at a table? Javier: You have to... [trails off]

Ms. Spence: TJ?

TJ: by looking at the y and x?

Ms. Spence: Okay so what would I do with the y's? TJ: You would um... I forgot. Saby?

Saby: find the difference in the numbers of the y’s and x.

Ms. Spence: Find the difference in the y's find the difference in the x's. And in this case it would give me the change in y would be three over the change in x, which would be negative one. So the rate of change would be negative three. [writes delta y over delta x equals 3 over negative 1 equals negative 3]

This example is typical of discussion of the summative tasks for two reasons. First, in this episode, both Javier and TJ struggled to articulate the desired generalization; students tended to demonstrate difficulty understanding and answering questions aimed at having them articulate generalizations. Second, neither Ms. Spence nor the students referenced prior work on contextualized problems.

As shown in Table 1, Ms. Spence referred students back to a previous

tasks. Students never referred back to previous contextualized examples during these summative tasks, although on one occasion a student created a contextualized example involving hourly pay (similar, but not identical to Barry’s situation). In comparison, on three occasions, students referred to non-contextualized examples worked during the same class period in order to make sense of tasks focused on general mathematical principles.

There were a number of reasons why explicit references to previous

contextualized activity might not have occurred during these portions. First, because the task progression went from contextualized to non-contextualized work, these summary activities did not tend to take place on the same day as students’ work with

contextualized examples. There is evidence that proximity matters: students referenced non-contextualized work during these summaries only when the non-contextualized work was done on the same day. Secondly, neither the textbook nor the teacher explicitly pointed students to recall their previous contextualized work during this section.

Although Ms. Spence did not explicitly encourage students to reflect back on their previously contextualized work while discussing tasks that asked students to articulate generalizations, there were thirteen instances in which she referenced contextualized examples during other instructional activity focused on general principles, as shown in Table 1. Eight of these instances occurred during the creation of the graphic organizer around the Barry example, mentioned in the previous subsection. Although this activity was summative, like the Summarize the Mathematics section, this activity was different in that it was teacher directed instruction rather than a task. Students were not meant to articulate the generalizations on their own.

Other examples of explicit connections between activity focused on general principles and previous work on contextualized examples occurred at earlier points in the instructional sequence. As described in the previous section, Ms. Spence had students read the paragraph in Figure 20 and prompted them to reflect back over their

contextualized work to make sense of the formula for slope. On three other occasions, she pointed to previous contextualized work when asking students questions about general mathematical principles during her class introductions.

In sum, when activity focused on generalized mathematical principles occurred in non-summative portions of the instructional sequence, Ms. Spence encouraged students to make sense of mathematical generalizations by recalling the contextual work from which it emerged. When discussing summative tasks in which students were asked to articulate generalizations, however, neither Ms. Spence nor the students tended to refer to contextualized examples. Similarly, the textbook did not contain references to prior contextualized examples either in these portions. During the push for mathematical closure at the end of the investigations, the contextual anchors were not explicitly leveraged.