Active learning in entry-level college mathematics classrooms

Researchers, professional organizations, and policymakers increasingly note a need to attend to meaningful student engagement in mathematics classes (CBMS, 2016; Dirks, 2011; Eddy & Hogan, 2014; Freeman et al., 2014; Haak, HilleRisLambers, Pitre, & Freeman, 2011; Hake, 1998; Hmelo-Silver, Duncan, & Chinn, 2007; Kober, 2015;  Laursen, Hassi, Kogan, & Weston, 2014; NRC, 2012; NSTC, CSE, 2013; PCAST, 2012; Porter, Bailey-Lee, & Simon, 2013; Prince, 2004 Saxe & Braddy, 2015; Sokoloff & Thornton, 1997; Svinicki, 2011). Instruction with higher levels of student interaction enhances problem-solving skills, demonstrates greater conceptual gains, and improves retention of information. Classes taught with active-learning strategies also have significantly lower failure rates and are particularly beneficial for underprepared, minority, female, and first-generation students.

Although a preponderance of evidence indicates the benefits of active learning, many mathematics instructors struggle to engage students in active learning experiences effectively. The affordances of active learning do not simply result from students’ active participation during lessons. Active learning strategies realize their potential when an instructor has an understanding of how the actions in which students engage support their mathematics learning. At the level of student engagement, the MIP characterizes conceptual structure as abstracted from reflection on the structure of one’s actions to resolve a problem. The MIP operationalizes this perspective through processes that focus students’ attention on the nature of a problem, selection of appropriate mathematical tools, application of those tools, and the reciprocal influences of the tool applied to, and evaluated against, the problem. The MIP definition of active learning is that

Students engage in active learning when they work to resolve a problematic situation whose resolution requires them to select, perform, and evaluate actions whose structures are equivalent to the structures of the concepts to be learned.

Asking students to make predictions engages them in conceiving and anticipating actions and results, drawing attention to the structure of the activity and supporting its abstraction. The elements that students retain long after instruction are the solutions they constructed to explicitly address problems. In order to abstract the appropriate concepts, the actions students perform to solve those problems must reflect the structure of the targeted concepts. To complete a cycle of inquiry, students must also evaluate the effectiveness of their solution and reflect on their problem-solving process in attaining that solution. Designing this level of student engagement, first requires faculty to detail that conceptual structure. Subsequent activity in MIP designs reinforce concepts by repeated abstraction through which students apply the invented solutions as tools to solve new problems. This is a particularly effective strategy when used to structure an entire course around a small number of powerful ideas that students repeatedly use, refine, and extend.

The MIP promotes activity design that affords multiple approaches that are accessible to students through different starting points and techniques, allowing more students to contribute and providing opportunities to compare approaches. Noticing and reconciling such differences engages students in abstracting the common structure that is central to the targeted mathematical concept(s). Active learning also requires an instructor to inquire into student thinking in which they listen to, interpret, and respond to the interplay between intuitive and formal ways of reasoning. The MIP supports instructors implementing active learning to develop a clear understanding of different approaches, common variations in student reasoning, and techniques for engaging group and class discussions. MIP materials help faculty scaffold this integration in ways that enable students to develop greater responsibility and autonomy over their own learning.