Coevolution of Qualitative and Symbolic Reasoning in Invention Activities

• In Event: On the Design, Implementation, and Outcomes of Using Contrasts in Learning

Abstract

Invention Activities (IA) ask students to develop mathematical methods that capture target properties of given data, prior to receiving instruction on canonical solutions. For example, students may be asked to develop methods that capture the variability of data prior to learning how to calculate standard deviation (Roll, Aleven, & Koedinger, 2009; Schwartz & Martin, 2004). As described by Chase and Shemwell (this session), invention activities use contrasting cases to direct students’ attention to the deep features of the domain. Contrasting cases are carefully designed examples that vary on one target feature ceteris paribus. For example, the contrasting cases in Figure 1 emphasize spread while fixing range, mean, and number of points. By contrasting different sets of examples students notice different features of the domain. This helps students set requirements from the general solution. For example, the contrasting cases in Figure 1 help students realize that a valid method should include all available data.

In this talk I will present two other forms of contrast that are facilitated by invention activities (see Table 1). Invention activities encourage students to compare between their invented methods and the canonical solutions. This comparison helps students identify the shortcomings of their invented methods, gain a better understanding of the limitations of their prior knowledge, and identify valid mathematical ways to achieve the same goals.

The third form of contrast is a coordination process between two forms of reasoning: qualitative and symbolic. Students identify qualitative relationships between properties of the data and the target concept (e.g., large N reduces variability; large spread increases variability). In addition, students search for symbolic manipulations that capture these relationships (e.g., dividing by N controls for sample size). In two studies (N = 226) we found that students who received support for both forms of reasoning showed superior gains compared with students who received support only for the qualitative or the symbolic reasoning. Analysis of students’ transcripts suggests that learning from invention activities is essentially a process of coordination between these two forms of reasoning: Students use qualitative reasoning to set goals for their symbolic inventions and evaluate these. Students use the symbolic inventions to develop a more integrative understanding of the qualitative relationships.

This talk demonstrates the richness and complexity of using contrasts in invention activities, a process that improves understanding of the target concept and its symbolic representation.

Author

• Ido Roll, The University of British Columbia