Indicator C.3.1. Anticipate and Attend to Students’ Thinking About Mathematics Content
Well-prepared beginning teachers of mathematics anticipate and attend to students’ mathematical thinking and mathematical learning progressions.
Understanding how students’ mathematical ideas develop and connect is at the core of mathematics teaching. Such understanding rests upon knowledge of the mathematics that comes before and after a given mathematics topic (see Standard C.1.4) as well as understanding of students’ informal knowledge, common conceptions, and, for topics for which the research knowledge exists, progressions of students’ thinking and learning within the content domain. Well-prepared beginning teachers of mathematics have developed strong understandings of students’ mathematical thinking in at least one, and preferably more, well-defined content domain(s) (e.g., within number and operations). They are committed to, and know how to, continue their learning about students’ mathematical thinking (e.g., by listening to children and their families, through continued education and professional learning, by using print or online research/resources).
Students come to classrooms with unique mathematical perspectives and experiences. Well-prepared beginners know that the quality and focus of their teaching is affected by the depth and detail of their insight into each student’s mathematical thinking. They need to learn about students’ informal ideas and invented approaches and students’ formal knowledge and understandings as well as how these two types of knowledge influence each other. Understanding the ways that students may think about that mathematical content enables teachers to honor how students’ work makes sense to them and to think about instructional moves to use to best extend students’ thinking (e.g., students come to question whether their mathematical ideas and procedures make sense in all cases or over different contexts; Jacobs, Lamb, & Philipp, 2010). Well-prepared beginning teachers therefore first try to see mathematical situations through their students’ eyes rather than immediately correcting mathematical errors or demonstrating their approaches. This ability to decenter and understand students' thinking is not only useful in planning for instruction but also provides a resource for making sense of moment-to-moment interactions with students.
To understand students' thinking, well-prepared beginners have two competencies. First, they enter classrooms with knowledge of progressions of students’ thinking—levels of thinking through which students advance while they learn a specific mathematical topic. They can also access available research-based perspectives on student learning. For example, at early levels of thinking about whole and rational number, students may believe that no numbers exist between two counting numbers or that “multiplication always makes numbers bigger.” For beginners, such knowledge is particularly important in content domains prevalent in the curriculum and domains crucial for students’ later mathematical success. Well-prepared beginners are disposed to—and have skills that enable them to—learn in an ongoing way about students’ ways of thinking related to the mathematical domains at their grade levels, including what comes prior and what will follow.
Second, well-prepared beginning teachers of mathematics can assess and analyze students' thinking. They examine their students’ varied approaches to mathematical work and respond appropriately. They gather and use information available through daily classroom interactions, routine formative assessments, summaries documenting students’ engagement with computer software or tablet applications, summative assessments, and standardized tests. Well-prepared beginners know the affordances and limitations of these sources of data for understanding student thinking and look for patterns across data sources that provide a sound basis for instructional next steps. They have the mathematical knowledge and the inclination to analyze written and oral student productions, looking for each student’s mathematical reasoning even when that reasoning may be different from that of the teacher or the student’s peers. They also enhance their own observations by deliberately drawing on the insights of families, professional colleagues, and sources of information from beyond the classroom.