Conceptualizing Mathematically Significant Pedagogical Opportunities to Build on Student Thinking

Abstract

The mathematics education community values using student thinking to develop mathematical concepts, but the nuances of this practice are not clearly understood. We conceptualize an important group of instances in classroom lessons that occur at the intersection of student thinking, significant mathematics, and pedagogical opportunities—what we call Mathematically Significant Pedagogical Opportunities to Build on Student Thinking. We analyze dialogue to illustrate a process for determining whether a classroom instance offers such an opportunity and to demonstrate the usefulness of the construct in examining classroom discourse. This construct contributes to research and professional development related to teachers' mathematically productive use of student thinking by providing a lens and generating a common language for recognizing and agreeing on a critical core of student mathematical thinking that researchers can attend to as they study classroom practice and that teachers can aspire to notice and build upon when it occurs in their classrooms.

Investigating Patterns of Pre-service Teachers’ Written Feedback on Procedure-based Mathematics Assessment Items

“…In this vein, researchers also recognize the importance of students' perceptions of the feedback they receive as a way to glimpse the teacher's point of view (Bansilal, James, & Naidoo, 2009). Because how students view feedback is critical to their learning, much emphasis has been placed on the quality of teachers' feedback (Sadler, 1998) as well as a way for them to understand the state of each student's knowledge by employing strategies to elicit student understanding and methods to interpret students' products (Bishop et al, 2014;Leatham et al, 2015). Thus, formative assessment feedback is related to the effectiveness of teachers' questioning and more generally to the teacher's role as a facilitator of learning, which depends on his/her ability to observe students' learning and attend to their thinking (Lee, 2018;Lee & Cross Francis, 2018;Peterson & Leatham, 2009;Stockero, Rupnow, & Pascoe, 2015).…”

Further Investigation into the Quality of Teachers’ Noticing Expertise: A Proposed Framework for Evaluating Teachers’ Models of Students’ Mathematical Thinking

“…Accounts-of indicates recording an event as it would be seen by other observers without including emotion or judgment, which is similar to Jacobs et al (2010)'s notion of attending to, while account-for refers to providing "interpretation, explanation, value-judgment, justification, or criticism" (p. 40), which is comparable to Jacobs et al (2010)'s concept of interpreting. Similar to the third component of Jacobs et al (2010)'s framework of professional noticing, making instructional decisions, Leatham, Peterson, Stockero, and van Zoest (2015) conceptualized a framework they titled Mathematically significant pedagogical Opportunities to build on Student Thinking (MOST), which can be described as the intersection of three characteristics of important components in teaching mathematics: mathematically significant content, pedagogical opportunity, and students' mathematical thinking. They recommended first attending to students' mathematical thinking, then focusing on whether the students' thinking is mathematically significant so that it can advance students' development of critical mathematical ideas, and finally considering, at the moment of teaching, pedagogical ways to build on students' significant mathematical ideas to support their further progress.…”

“…Several researchers have found that effectively noticing students' mathematical thinking requires both the ability to focus on important events and knowledge of what is mathematically significant in students' messy strategies (Jacobs, Lamb, Philipp, & Schapelle, 2011;Leatham et al, 2015). Schoenfeld (2010) also emphasized the importance of teachers' knowledge, resources, and orientations in noticing students' thinking and making decisions to support the students' learning.…”

Programming, mathematical reasoning and sense-making

“…Weick et al (2005) used sense-making to refer to understanding through retrospective reflection on the decisions and actions of individuals, while Klein et al (2006) argued that sense-making is a continuous effort to understand connections in order to anticipate their trajectories and act effectively. Researchers have demonstrated that sense-making is supported in the classroom environment by: interaction, negotiation, and discussion, which create meaningfulness and understanding (Boaler, 2014;Schoenfeld, 2006); using student thinking through involving verbal interactions (Leatham et al, 2015); and using tasks that elicit and extend students' ideas and encourage explanations and justifications of mathematical concepts (Mueller et al, 2014). Many researchers in mathematics education have emphasized the importance of task features, specifying that tasks open to multiple representations and multiple solution strategies promote reasoning and understanding (e.g.…”