Indicator P.2.2. Build Mathematical Practices and Processes

Mathematics includes more than learning appropriate content. Effective mathematics teacher preparation programs ensure that candidates are immersed in mathematical practices and processes of reasoning, sense making, and problem solving. Candidates’ mathematical experiences include continued emphasis on reasoning abstractly and quantitatively, explaining their thinking, and analyzing the thinking of others. When the content lends itself to such practices and processes, opportunities to learn include seeing structure and generalizing. Mathematical modeling—using mathematics to analyze real-world situations—receives continued attention throughout the program. Programs also provide explicit opportunities for candidates to understand statistical ways of thinking and to understand the range of habits of mind in both mathematical thinking and statistical thinking (Chance, 2002). Programs provide opportunities for candidates to make mathematical connections between various approaches to solving problems and opportunities for candidates to make connections between mathematics and other disciplines. Effective programs provide specific opportunities for candidates to use technology to engage in mathematics and statistics. For example, visualizations of a data set can help students better understand patterns within a data set, and spreadsheets can help students create models of mathematical situations, thus supporting the development of mathematical practices and processes. “Mathematical action technologies” used to “perform mathematical tasks" or "respond to the user’s actions in mathematically defined ways” (Dick & Hollebrands, 2011, p. xii) support the development of candidates’ deep conceptual understanding through use of mathematical practices and processes.

In effective programs, mathematics content is taught using mathematics teaching methods that serve as models of effective mathematics teaching for candidates. Professional organizations representing the mathematics community recognize the benefits of active learning in mathematics (e.g., CBMS, 2012; MAA, 2015/2016). Active learning in STEM disciplines improves learning (Freeman et al., 2014). In such settings, learners are typically provided challenging tasks that promote mathematical problem solving and are provided opportunities to discuss their thinking in small and full-group discourse, thus promoting important mathematical practices (Webb, 2016). Effective programs structure opportunities so that candidates learn mathematics by using such active learning strategies. In this way, candidates experience learning mathematics using methods that are consistent with the methods they should use as teachers.

The instructor’s choices about how to address mathematical practices and processes are transparent in effective programs. Often undergraduate students (in this case teacher candidates) focus only on the mathematical products without considering the underlying mathematical practices leading to those products. Effective mathematics instructors emphasize the development of those mathematical practices. In effective programs, mathematics teacher educators explicitly identify and address these mathematical practices for those learning to teach mathematics.

Effective programs focus on building mathematical practices and processes in a manner that honors the social context of teaching and learning mathematics. Those who teach mathematics courses in effective programs model equitable practices, including making apparent underlying beliefs about the role of each individual in the classroom community. For example, in mathematics as a discipline, argument is valued as the way to find mathematical truth, but society does not always value argument from boys and girls equally. Without explicit classroom attention to mathematical argumentation, effective programs risk diminishing the opportunities candidates have to engage in the mathematical practice of constructing viable arguments. Effective programs help teacher candidates who have been successful in the prevailing mathematical culture understand that though mathematics seems to be objective, bias is inherent because it is a human endeavor.