Potential Obstacles to Student Learning from CP-based Instruction

What is meant by formal mathematics? In this paper, I define formal mathematics as established, generalizable, abstract mathematical principles including conventional algorithms, procedures, representations, relationships, concepts, language, symbols, and ways of reasoning and arguing—important outcomes of what Wu (1997) calls

“mathematical closure” (p. 194). Proponents of the Standards argue that students can abstract or generalize many of these mathematical principles through work in problem situations and that teachers and the curriculum can provide established conventions like symbols and vocabulary.

Research on generalization in mathematics education suggests that the process of generalizing mathematical principles from work on CPs might not be straightforward. Davis (2007), for instance, describes students’ difficulties in understanding the meaning of a y-intercept while using a context-based Standards-based curriculum He found that students invented terminology to describe contextual instances of y-intercept; but, as instruction proceeded from work on contextualized problems to the formal introduction of the term “y-intercept”, the instructor failed to connect students’ invented terminology to the formal term. As a result, many students demonstrated an incomplete understanding of the concept. Lubienski (2000) in a study focused on variation between students being taught from a Standards-based curriculum program, found that students from families with low-SES tended to have greater difficulty generalizing from their work on contextualized problems than students from families with higher SES. She also found that students from families with low-SES more often complained that they did not see the connection between different contextualized tasks and did not understand what they were supposed to be learning.

Lubienksi’s (2000) findings point to an issue particular to contextualized-problem based instruction: if multiple problems set in multiple contexts are used, students may not necessarily see the mathematical similarities linking the problems. Laboratory studies directly targeting this issue provide additional evidence that students struggle to make the desired connections between different problem situations. Herbert & Pierce (2011), for example, found that students did not necessarily transfer successful solutions strategies developed in one context to problems set in other contexts. Gick and Holyoak (1983) found that when different problem contexts are used, students are unlikely to use solution

strategies from a previous context unless they are explicitly reminded of the previous problem.

In addition to difficulties in transferring strategies developed in one context to another contextualized problem situation, research also suggests that students might have difficulty transferring strategies that they developed in response to CPs to non-

contextualized problems, or mathematical tasks that are not situated in any contextual setting. Walkington, Sherman and Petrosino (2012) found that students were more likely to attempt story problems compared to analogous, non-contextualized symbolic problems in a laboratory setting; they also pointed out that students brought informal, arithmetic strategies to solve story problems but did not easily coordinate these strategies with more formal, algebraic strategies involving symbolic manipulation. The authors concluded that, “if contexts are to promote access to central concepts, they ultimately should give meaning to abstract representational systems. Whether this can be achieved by traditional story problems within the system of school algebra, and whether it is likely without strong support for such coordination, is more complex than everyday notions of the benefits of contextualization would suggest” (Walkington et al., 2012, p. 198). Lobato and Ellis (2002) noted classroom disconnects between contextualized and non-

contextualized problems in a class using Standards-based curriculum; in that case, the teacher failed to maintain a focus on the relationship between two variables in a linear function once the instruction moved from contextualized to non-contextualized problems.

To mitigate difficulties students have with generalizing intended mathematical ideas and seeing inherent connections between instructional tasks, researchers suggest practices that might prompt students to make the intended generalizations from

contextualized tasks. Jurow (2004) documents students’ generalizations working from a

Standards-based, contextualized curriculum sequence and concludes that, in addition to opportunities to work on CPs, students “need guided reflection and multiple scaffolded opportunities to talk about, write about, and otherwise represent what is general in and across situations” (p. 296). Davis (2007) recommends that teachers and curriculum designers need to carefully attend to possible disconnects between informal language and strategies that might arise from contextualized problems and analogous formal language and experiences with non-contextualized problems.

Together, these studies highlight an instructional practice that is an essential for facilitating the development of mathematical generalizations from work on particular examples. This practice is what Mason (1996) describes as supporting students’ “shift of attention” from the particular to what is general about the particular. Teachers, Mason notes, are of prime importance: “The presence of someone whose attention is differently structured, whose awareness is broader and multiply-leveled, who can direct or attract pupil attention appropriately to important features, is essential” (p. 71). There is reason to believe, though, that this shift of attention might not be occurring to the extent

necessary for productive learning in many classrooms, particularly in the US. Boaler and Brodie (2004) found a great deal of variance in the frequency with which teachers asked questions that engage students in attending to underlying relationships and meanings, i.e. the formal mathematical generalizations that are the goals of lessons. They found these questions were used frequently in only one of the four classrooms they observed using

Standards-based curriculum. Furthermore, international comparative studies have consistently found that US teachers were are less likely to engage in statements of

mathematical summary than their counterparts in nations that perform higher on comparative assessments (Hiebert et al., 2003; Stigler, Gonzales, Kawanaka, Knoll, & Serrano, 1999).

It follows from research described above that one way teachers can promote generalization from contextualized tasks is to prompt students to see and articulate what is similar among various CPs and their analogous non-contextualized analogs. The degree to which prior and future tasks are referenced during instruction is one component of a construct called instructional coherence in mathematics education research, another focus of international comparative studies (Cai, Ding, & Wang, 2014; Stigler & Hiebert, 1999). Here again, there is reason to believe that in some countries, including the US, high levels of coherence may not be the norm. After comparing lessons taught in three countries, researchers found that in Japan, references to past and future activity were made significantly more frequently than in the US or Germany (Stigler et al., 1999). Like researchers who study generalization, scholars describing instructional coherence stress that the connections within lessons should be made explicit, to ensure that students have an understanding of the sequence of instruction and how they can leverage prior work toward learning.