Supplementing with Connecting Activities

LR: Is there anything else that they had trouble with that surprised you.. or not surprise you but you noticed they had trouble with?

Ms. Spence: It's not so much something that surprises me, it's something that I've grown to expect that they don't really make connections well. LR: You mentioned that last time I was here.

Ms. Spence: They don't make connections well even in terms of the work that they just did.

(10/27/11)

Ms. Spence frequently noted that students had trouble making connections from one activity to the next. To promote these connections, she supplemented with three types of connective instructional activity that were not present in the Core-Plus unit. Although she did not supplement with each type described below on a consistent basis, taken cumulatively, these insertions represent a pattern of inserting activities that offered the potential to bridge between various tasks presented by the curriculum developers.

4.3.1 Reflecting across contextualized examples

On three occasions during the unit, Ms. Spence planned to supplement the written curriculum with tasks that asked students to reflect across contexts. This type of activity is not present in the Core-Plus investigations, a pattern illustrated by the absence of circles in the “reflect across contextualized examples” in the lesson map in Figure 10 and in the other lesson maps I created. Ms. Spence supplemented with this connective type of activity twice in the first investigation of the unit. After the terms dependent and

independent variable were introduced, Ms. Spence planned to ask students to students “look back over the two preceding contextualized examples to identify the independent and dependent variables” (slideshow for 10/27/11). This is indicated by the TCT labeled “look back dependent” and “look back independent” on 10/27 in the map in Figure 12. The next day, Ms. Spence planned an “exit slip” activity asking students to look across three different contextualized examples: “Write down what you notice between questions 1, 3 and 4 in terms of what we talked about today.” In another investigation, she planned an activity where students were to answer the question: “What is the same about

questions 1 and 2?” and “What is different about questions 1 and 2?” (slideshow for 11/8/11). Questions that ask students to identify similarities across contexts have the potential to prompt students to shift their attention from the specific examples to the general mathematical principles that underlie the examples; and, questions targeting differences across contextualized tasks have the potential to prompt students to attend to how mathematical ideas are developing along the instructional sequence.

4.3.2 Reflecting across contextualized and non-contextualized examples

In the Core-Plus curriculum, contextualized examples are meant to provide students access to more abstract, non-contextualized examples. With this in mind, it is notable that, in the lessons analyzed in this study, there were no tasks in the Core-Plus

unit that asked students explicitly consider contextualized and non-contextualized examples simultaneously. The map in Figure 10 is an illustrative example of this trend; there are no circles in the “reflect across contextualized and non-contextualized” row.

The absence of this type of question left the students (and teacher) without support for making connections between these two types of activity. And although Ms. Spence frequently noted students’ struggles connecting these two types of activity, she did not frequently supplement the curriculum with tasks that asked students to simultaneously consider both types of example. However, she did supplement with such a task once, at the end of Lesson 1, Investigation 2. After grading students’ responses to a quiz, she planned to begin class with a review in which she posed two tasks:

 For the linear function below, identify the rate of change and the y-intercept. y = 5 + 4x

 Write a situation that this linear function could represent.

The second task asked students to develop a context around a non-contextualized example. This task stands out amongst all the other tasks in the written and intended curricula as an illustrative example of a type of activity that asks students to reflect across both contextualized and non-contextualized examples.

4.3.3 Connecting general principles to contextualized examples

Ms. Spence also supplemented the curriculum with tasks that involved both general mathematical principles and contextualized examples. This sort of task has the potential to help students make connections between these two different types of

instructional activity involves activity. One instance of this involved a graphic organizer used to show students four ways that linear functions can be represented: as a verbal description, a symbolic rule, a table, and a graph. Ms. Spence planned to have students copy the table and symbolic representation of the function from the example involving

Barry, the representative from the credit card company. When I asked why she referred back to this example, she explained that students understood the example and were comfortable with it:

LR: I'm wondering why you decided to use Barry's situation in that graphic organizer.

Ms. Spence: Because they understood that one. LR: okay

Ms. Spence: it made sense to them. I mean pretty much it was flushed out when we did the notice wonder [on the first day of the unit]... They started with the graph, they were able to read and interpret the graph. Based on that graph they were able… we came up with an equation even though they didn't really understand writing

equations at that point but they did understand the idea that n represented the number of applications and five dollars per

application they got that. And they did understand what did he get twenty dollars or ten dollars or something like that.

LR: yeah it was twenty

Ms. Spence: So they understood that and to me it made it relevant for them because they already had that, the graph. Then we were able to make the table because that was part of question one I think it was, then they had the rule. It was there. And I thought it was

something that was easy enough to understand if they went back to look at their graphic organizer than that was something they were already comfortable with.

(1/17/13)

Through this graphic organizer, Ms. Spence explicitly connected the generalized mathematical principles to students’ previous work with a contextualized example.

An instance of supplemental activity that contained a contextualized example, non-contextualized example, and reflection on general principles is a video that Ms. Spence played as an activating strategy. The video began with an explanation of how to

draw best-fit lines to determine whether the correlation between two variables is positive or negative. Instruction began at the reflection on general mathematical principles level, proceeded to non-contextualized examples of positive and negative correlations, and concluded with a contextualized example: drawing a best fit line to represent age and weight data. Following the video, Ms. Spence planned a series of questions asking students to make generalizations about best-fit lines and correlations.

A third example of connective activity was manifested in a supplemental general activity that Ms. Spence created as a replacement for a Summarize the Mathematics

section of the book. At the end of Lesson 1 Investigation 3, Ms. Spence planned to ask students to answer the essential questions of the lesson as a consolidating activity. Her written prompt asked students to “use examples to explain.” Like the activity asking students to write a context that matches the non-contextualized example, this activity prompted students to develop an example, leaving the possibility for students to use previously encountered contextualized example or developing a new one. The Summarize the Mathematics sections of Core-Plus examined in the study do not contain any explicit cue to provide or consider examples.

A primary factor influencing the supplementation of these sorts of connective activities was Ms. Spence’s assessment that her students had difficulty making the connections that she felt the developers of Core-Plus intended her to make. On sixteen occasions across nine interviews, Ms. Spence noted connections that students struggled to make. This perception, combined with her belief about her duty to supplement the

curriculum to fit the needs of her students, contributed to the pattern of transformation described above.