Abstract:

The level of cognitive demand, or the mental effort necessary to solve a mathematical task, can vary within a mathematics textbook. The design of many curricula offer a collection of low-level tasks that emphasize procedural thinking coupled with high-level tasks that emphasize conceptual thinking and applications. However, textbooks often disproportionately favor one level of task over another. Research has shown that the level of cognitive demand of tasks in textbooks has helped explain student learning outcomes in mathematics. For instance, textbooks that emphasized low-level tasks have been shown to be ineffective at supporting student meaning-making since there are few opportunities to grapple with complex mathematical ideas. This is further supported by many reform efforts, including the Common Core State Standards (CCSS) in the United States, which emphasizes a balance of low- and high-level cognitively demanding tasks in order to support the diverse needs of mathematics learners.

In this study, tasks of two textbooks were analyzed and compared for their levels of cognitive demand. These two textbooks were: Glencoe Pre-algebra (GP) (2012), a popular textbook in the United States that claims CCSS-alignment, and Beast Academy (BA) (2017), an innovative and specialized textbook catered towards students who participate in mathematics competitions. The square-roots and exponents chapters were selected for analysis in both GP and BA. These chapters were chosen because they are typically introduced in a Pre-algebra course and are emphasized in the CCSS Expressions and Equations content areas.

A modified version of Stein and Smith’s (1998) cognitive demand framework was used to analyze the cognitive demand for tasks in GP and BA. The original framework categorizes tasks based on their level of cognitive demand. Low-level cognitive demand tasks emphasize memorization or procedures without connections objectives; such tasks are primarily algorithmic with a set of steps students can follow. High-level cognitive demand tasks emphasize both procedures with connections or “doing mathematics” objectives; such tasks often extend into applications, using multiple representations, and require non-algorithmic thinking. The framework was modified to include a mid-level cognitive demand task which emphasizes procedures with potential connections. In these mid-level tasks, students can find solutions similar to those in procedures without connections, yet procedures with potential connections tasks have other approaches towards their solutions in which students can engage with the underlying concepts of the properties.

The results showed that GP included a greater percentage of low-level cognitive demand tasks compared to BA, 70% to 39%, respectively. Also, BA included a greater percentage of mid-level cognitive demand tasks compared to GP, 20% to 4%, respectively. Furthermore, BA included a greater percentage of high-level cognitive demand tasks compared to GP, 41% to 26%, respectively. These results imply higher-level cognitive demand expectations in the BA chapters than in the GP chapters, with a more specific focus on procedures with potential for connections, procedures with connections, and “doing mathematics.” This suggests that textbooks with expectations like BA offer a balance of cognitively demanding tasks conducive for the diverse needs of mathematics learners, and should inform the design of reform-based curricula.

Keywords:

Mathematics education, curriculum, cognitive demand.