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Now that the framework has been described, the question remains: what is it good for? What can examining mathematics curriculum or classroom practice through the lens of this framework tell us? There are three primary ways in which the framework can be used to interpret mathematics curriculum. First, the framework can be used to analyze written and enacted curriculum for the presence or omission of each of the types of activity described in the framework and particularly those that research tells us are important for student learning. For instance, Boaler (2002) as well as Jackson, Shahan, Gibbons and Cobb (2012) argue that one key to increasing the potential of instruction based on contextualized problems is to make sure that students understand the contextual features of contextualized problem situations. To do this, they recommend that teachers

devote a portion of instructional time to identifying important contextual elements, clarifying vocabulary words, and verbalizing relationships that might be obvious only to those who are familiar with the setting. In terms of the CPMI framework, this sort of discussion would include statements or questions that would be classified in the Focus on context subdomain within the contextualized problem solving category. A different category of instructional activity emphasized in the RME and MMP literature falls within the Focus on model subdomain within the contextualized problem solving category: a shift of attention from the problem situation at hand to the mathematical models that students used to solve the problems. Finally, White and Mitchelmore (2010) and Lesh et. al (2003), argue that activity that would be classified as Reflection across contextualized examples represents a critical step in the process of abstracting fundamental mathematical concepts from contextualized problem solving. By analyzing textbooks or enacted

lessons using the framework, teachers, instructional leaders, curriculum developers or researchers could detect the presence or absence of each of these recommended varieties of statements or questions in instruction.

In addition to being used as a tool for detecting the presence or absence of particular categories of instructional activity, the framework can be used to characterize the sequencing of instructional activity. In many classrooms mathematical procedures are first explained through a non-contextualized problem, the steps for the procedure are generalized (reflection on general mathematical principles), and students practice the procedure on similar examples (a return to non-contextualized problem solving).

Towards the end of the lesson, students are often asked to apply the procedure to a word problem involving some extra-mathematical setting (contextualized problem solving).

Streefland (1991) terms this a mechanistic approach to mathematics instruction. This traditional positioning of contextualized problems within math lessons can be described using the framework as shown in Figure 8; arrows describe the progression of activity.

Figure 8. The traditional role of contextualized problems in mathematics instruction

Lessons aligned with the RME and MMP instructional sequences would tend to look very different. A common sequence from these approaches is represented in Figure 9.

Figure 9. RME and MMP-aligned instruction emphasizing problem solving in “real- world” settings

RME and MMP approaches argue that students’ understanding of the physical world can often be used as the starting point for instructional sequences. Lessons from a curriculum programs aligning with these perspectives would often begin with students working on tasks in the Contextualized problem solving domain. After time to work in groups, students might be asked to share their strategies. Next, the teacher might facilitate a discussion in which strategies, procedures and/or ideas are generalized (reflection on general mathematical principles). Following this, the students might apply this procedure to a different contextualized problem and eventually to tasks situated in the non-

contextualized problem solving domain. In this form of instruction, contextualized problems do not serve only as settings for the application of known math; instead, students construct new math knowledge through contextualized problem solving. By characterizing the sequencing of lessons in this way, practitioners and researchers would

gain insight into the role that contextualized problems play in particular written or enacted lessons.

The sub-categories within contextualized problem solving domain in Figure 6 could even be used to identify distinctions between different approaches that begin with contextualized problem solving tasks. For instance, RME instructional sequences are designed so that students create informal models, which subsequently are connected to conventional, formal models through vertical mathematization. This vertical

mathematization would show up as a transition from informal to formal activity within the focus on model sub-category. RME sequences also would contain a transition to non- contextualized problem solving. MMP instructional sequences, on the other hand are meant to develop foundational understanding of important mathematical concepts through multiple trips around the modeling cycle; these sequences not explicitly

concerned with progressing to formal models, so activity at the more formal level would be less emphasized. Multiple trips around the modeling cycle within a single problem situation would be observed. Also, non-contextualized problem solving is not described explicitly in MMP model development sequences.

The framework can also be used to clarify differences between Core-Plus

instructional sequences observed for this study and RME or MMP inspired instructional sequences. Rather than emphasizing the creation of informal models through tasks that would be classified as produce model activities, the Core-Plus sequences in the units analyzed here frequently begin from formal models, which students are asked to interpret or translate to another formal models. For instance, in the unit featuring Barry described throughout this article, students are first presented with a formal graph and a symbolic

function rule, which they are asked to interpret in terms of the extra-mathematical situation, a task that would be classified in the interpret category. This type of activity would not show up at the beginning of RME or MMP sequences because the students did not produce these models on their own.

In addition to providing a way to consider the presence, absence and sequencing of various activity types, the framework can be used to understand the extent to which instructional activity in the various categories is explicitly connected. Instruction could be analyzed, for instance, to determine the extent to which students, the teacher, or curriculum materials explicitly reference prior contextualized problem solving during the discussion of tasks or questions categorized as reflection on generalizable mathematics.

Connecting across instructional tasks in these ways would likely increase the potential that students’ understanding of formal, mathematical connections would be connected to their common sense understandings of the world around them, as described by

Freudenthal (1973) and other RME theorists.