Matt Switzer

In this study, we build on the prior work of researchers in the field of noticing. Specifically, we build on Jacobs, Lamb, and Philipp's (2010) conceptualization of professional noticing to motivate the field to question the meaning of interrelated skills of attending, interpreting and deciding how to respond to students rather than focus on the individual skills (e.g., attending, interpreting, deciding how to respond). We introduce nine subtypes of professional noticing to connect attending to and interpreting student mathematical thinking. We then use these subtypes to report on the analysis of a collection of 60 written justifications from eight secondary mathematics student teachers who used the Practice of Probing Student Thinking Framework (Author et al., 2016) to complete focused video analysis (FVA) of lessons taught during student teaching. Results indicate that an analysis of the individual skills of professional noticing can mask key details about student teachers' professional noticing that are only seen through the analysis of the interrelated skills. We also found that student teachers with prior FVA experiences were highly effective with attending to student mathematical thinking; however, they were less successful in interpreting the student mathematical thinking. We recommend that research deliberately move to investigate and analyze the interrelatedness of the professional noticing skills to identify how preservice and inservice teachers are connecting these individual skills.

Teachers utilizing student mathematical thinking is important when teaching, yet many inservice teachers find it difficult to implement. The Standards for Preparing Teachers of Mathematics (AMTE, 2017) outline the knowledge, skills, and dispositions that beginning teachers should have after graduating including the importance of attending to and interpreting student mathematical thinking. In this paper, we present results from two focused video analysis assignments that our pre-service teachers engaged in to identify their images of student mathematical thinking and their ability to attend to and interpret student mathematical thinking

The purpose of this paper is to describe secondary mathematics student teachers’ types of noticing while teaching. We discuss the importance of focusing on the interrelatedness of the noticing skills rather than reporting on individual aspects separately. We apply the types of noticing to videos of student teachers by identifying their ability to elicit and interpret student mathematical thinking in-the-moment while teaching. Results suggest that our student teachers are eliciting student mathematical thinking while teaching and are attending to student mathematical thinking while teaching that student thinking, but how they are interpreting the elicited student thinking varies at a general level. We hypothesize identify three reasons for why student teachers may have interpreted student mathematical thinking at a general level.

Research findings have established common student misconceptions for literal symbolic representations of variables but lack corresponding findings of when or how these misconceptions arise. This article reports findings from an exploratory study of U.S. grade 4–6 students’ conception(s) for various representations of unknown addends commonly found in U.S. elementary mathematics textbooks. Thirty-six U.S. grade 4–6 students par- ticipated in two semistructured task-based interviews designed to explore their conception(s) of conventional and nonconventional representations of unknown addends as revealed by their number substitutions across task types and core mathematical tasks. Results showed that participants initially demonstrated a bias toward positive integers, upon further questioning a bias toward non-negative integers, potential conflict factors related to rational-number substitutions, and did not demonstrate common difficul- ties with literal symbols exhibited by students in algebra and higher-level mathematics courses.

Number and operations serve as the “cornerstone” of the K-12 mathematics curriculum in many countries. Solving problems in the mathematical domains of algebra, geometry, measurement, and statistics is often closely connected to student knowledge of number and operation (Griffin, 2005). Although considerable knowledge exists regarding the development of number and operation for typically developing children, less is known about the development of children who struggle in mathematics. Moreover, children enter school with considerable differences in their understanding of number and operation. While most children, through exposure to various informal and formal tasks, develop a deeper understanding of number and operation, this development is delayed for some children. These children do not achieve levels of proficiency required for higher mathematics . . . . Therefore it is critical that difficulties in mathematics are addressed before they become chronic, pervasive, severe, and difficult to remediate” (Fuchs, 2005, p. 351) . . . In some studies, concerns regarding student retention of learned concepts, and success generalizing and transferring mathematical ideas to other mathematical situations or domains were noted. Yet it is unclear why these mixed results occurred . . . . Therefore, they strongly recommend that, “future research should be directed at the role of individual differences in the development of early numeracy and the characteristics of children’s learning responsible for these differences” (Van Luit & Schopmann, 2000 p. 35).

In this month’s Problem Solvers Solutions, readers are provided a window into students’ number and operation sense in the early elementary grades. Second and third graders were presented with problem-solving tasks using the Hundreds Chart consisting of two number cards and a challenge card aligned to an addition or subtraction structure. Drawing on the structure of the Hundreds Chart and prior-knowledge, the students were able to articulate their solution strategies.