Activity Types within the Contextualized Problem solving Category

The activity types presented to this point describe important conceptual distinctions that can be used to analyze and understand instruction emphasizing contextualized problems. But, to this point, the framework does not provide detailed language for classifying different types of activity within the category of contextualized problem solving. To address this issue and to gain further resolution in this category, I used conceptual distinctions present in literature on mathematical modeling (including RME and MMP) and empirical observation to identify five subcategories of activity possible within the contextualized problem solving category (see Figure 6). I describe the relationship between these subcategories using an adaptation of the common representation of modeling shown in Figure 1.

To capture the distinction between translation and organization approaches to modeling as described by Gravemeijer (1997), I distinguish between activity focused on informal models and activity featuring more conventional or formal mathematical models or representations. The dotted line is meant to signify the possibility of activity in the informal level rising to formal activity through the process of vertical mathematization, as described by Gravemeijer (1994).

Figure 6. Sub-categories within contextualized problem solving

The Focus on context box at the bottom of Figure 6 represents activity that focuses particularly on the situation that the problem refers to rather than on a

mathematical model. This category includes activity wherein students and their teacher discuss a context before a particular problem is posed, assess which data are important to a particular problem, or collect data. An example from Core-Plus occurs on page 198: “Do you or someone you know use the Internet? For what purposes?” This category is also where the warm-up activities in MMP model-development sequences would be classified.

The acts of translating and organizing elements from a problem context into a mathematical model comprise the Produce model category. These tasks or questions are focused on the act of producing a mathematical model, given a particular situation or data

set. An example of this type of activity from Core-Plus occurs on page 102: “Use the words NOW and NEXT to write a rule that shows how to use the price of the item in one

year to find the price of the item in the next year.” This would be an example of what Gravemeijer (1997) would describe as a translating activity; one could distinguish between a translating activity and an organizing activity more in line with an RME approach within this category.

Activity in the focus on model category pertains to one or more informal or formal mathematical models and is not as concerned with the relationship between the model and the context. This category includes questions and tasks that ask students to compare solution strategies, translate from one type of representation to another, match different types of representations, compare or explain the relationship between two different types of representations, or work within one model to produce a model of a particular

mathematical relationship (e.g. find the mean, standard deviation, slope or rate of change, y-intercept, etc.). The importance of this category cannot be overstated: this is the key domain for activity related to vertical mathematization in RME, as this is where informal models are organized, combined and formalized. This shift to the focus on the model aligns with the general level of RME and the model exploration activities in MMP. In reference to the RBC+C model, this is where activity involving reflection on building- with lower level abstractions (Dreyfus & Tsamir, 2004) could lead to consolidation. This is also where formal language for describing mathematical concepts could be introduced (Monaghan & Ozmantar, 2006).

Tasks in the Interpret model category involve the use of a mathematical model to answer a contextualized question of some sort. This category includes questions or tasks

that (a) ask students to summarize what a particular model says about the context, (b) answer a specific question about the context by looking at a single model or comparing a number of models, (c) explain how to use a model to answer a question, or (d) assess a hypothetical response to a question. An example from Core-Plus occurs on page 174: “Using the linear model, estimate the median income of women in 1983 and 2007.”

These four subcategories describe tasks that specifically prompt students to engage with specific steps within the modeling process. However, to solve complex modeling tasks like those described in MMP literature, students would engage with the modeling process in its entirety, and often more than once, in response to a single task.

Now that the contextualized problem solving category has been further decomposed, these four subcategories can be placed into the broader organizational scheme, as shown in Figure 7, and the framework is complete.

Figure 7. The Contextualized Problems in Mathematics Instruction (CPMI) framework

I have included both informal and formal models in the non-contextualized frame to emphasize that students can reason with informal models in non-contextualized settings. This allows for the description of activities that leverage students’ informal reasoning but do not specifically refer to contextualized problem situations.

With these additions, this framework represents a synthesis of significant concepts from two prominent modeling-based instructional theories (RME and MMP model development sequences) and important activity types that emerged from empirical observation of the implementation of two units from the Core-Plus curriculum. The importance of the various categories toward abstracting mathematical ideas is supported by research on empirical and theoretical abstraction. I have added the additional category of reflecting across contextualized and non-contextualized examples, neither of which is explicitly identified in RME or MMP instructional sequences. This category emerged as a potential solution to a significant problem practice: a lack of coherence between students’ contextualized problem solving and non-contextualized problem solving.