Making Mathematical Practices Explicit in Secondary Mathematics Discussions

• In Event: 33.040 - Revisiting Explicit Instruction in English and Mathematics

Abstract

Purpose and Theoretical Framework

Mathematical practices, like constructing arguments, are central to learning and doing mathematics (Common Core State Standards, 2010). To learn mathematical practices, students need opportunities to participate in them. However, opportunities to participate in mathematical practices may not be sufficient to support all students in learning them (RAND, 2003). Normative ways of participating in discourse may be novel to certain groups of students, raising concerns about equitable access if these practices are not made explicit (Ball, Goffney & Bass, 2005; Boaler, 2002). But explicitly teaching mathematical practices could be problematic if instruction becomes prescriptive, inauthentic, or separates practices from content (Schoenfeld, 1988).

One strategy might be to make mathematical practices explicit is to help students notice (Lobato, Hohensee, & Rhodehamel, 2013) what was important in discourse. Mann, Owens, and Ball (2013) showed how naming and highlighting made mathematical practices visible to elementary students. This study develops a framework for the types of moves used to make mathematical practices explicit, and provides empirical evidence about when teachers do this.

Data Sources and Methods

Data include 26 class discussions from three case study math classes (Yin, 2009) in an urban secondary school. The teachers were purposefully sampled (Marshall, 1996) based on their commitment to cultivating mathematical practices. I segmented the discussions into mathematical practice episodes, a unit allowing for the co-construction of mathematical practices. Episodes were coded using the Common Core Standards for Mathematical Practice (SMP). Next, I identified moves that initiated, sustained, or reprised mathematical practice engagement. Reprising moves are reflective discourse (Cobb et al., 1997) that makes prior discourse the object of discussion. Table 1 shows an example. I coded these moves using a priori and emergent codes for explicitness about mathematical practices, and 15% were double-coded (94% inter-rater agreement).


Results

Analyses revealed a framework of eight moves that made mathematical practices explicit (Table 2). For example, naming involved using conventional or locally meaningful terminology to label mathematical practices; explaining the goal/rationale consisted of making explicit the purpose for a mathematical practice.


The teachers made mathematical practices explicit primarily after students had participated in them, using reprising moves. For example, in a geometry discussion, Alejandra used color (Figure 1) to connect representations (SMP1) and highlight structure (SMP7). Then Ms. Larson reprised, “Color code things. Color code shapes. Now it makes it way easier for me to see the red triangle on the left and the blue triangle on the right”, naming the practice and explaining the goal after the mathematical practice engagement.

Significance

This study has implications for current debates about explicit math instruction. This study extends prior research on making mathematical practices explicit (Mann et al., 2013) and illustrates how instruction could make what is implicit in discourse explicit for all students. This study demonstrates how explicitness about mathematical practices does not have to be prescriptive or inauthentic, since reprising moves refer to student activity that has already happened. This suggests that the math education community might need to expand conceptions of “explicitness” in mathematical practice instruction.

Author

• Sarah Kate Selling, University of Michigan - Ann Arbor