This is an excerpt from JMTE 2018’s article by Sarah Quebec Fuentes and Jingjing Ma entitled “Promoting teacher learning: a framework for evaluating the educative features of mathematics curriculum materials”.
Connecting to my research: Video studies deepen teachers’ knowledge about how students reason about mathematics. Russell (2007) promote that teachers must continue to augment their understanding of student conceptions throughout their career (Russell 2007). How can peer video coaching support this continuous growth? Can video peer coaching allow for teachers to customize their learning and create self initiated goals for enhancing their practice?
Fuentes & Ma, 2018, p. 360-361
Teacher knowledge of student thinking in mathematics The curriculum materials support the development of teacher knowledge in anticipating and understanding student thinking.
- What supports are provided for anticipating and understanding student mathematical thinking (e.g., common perceptions and misconceptions)?
- To what extent are rationales for addressing the significance of student thinking in relation to understanding the mathematics content present, clear, and appropriate?
- To what extent is implementation guidance for anticipating and understanding student mathematical thinking present, clear, and appropriate?
Reasoning is a critical component in learning mathematics with understanding (NCTM 2000). In Adding It Up, a National Research Council (2001) report, literature is synthesized to describe the various, codependent components (conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition) constituting learning mathematics with understanding, or mathematical proficiency. Student reasoning, or thinking, cuts across all components from understanding of (conceptual understanding) to representing (strategic competence) to explaining (adaptive reasoning) mathematical ideas. In order to foster student thinking, teachers must have knowledge about how students reason about mathematics. In particular, knowledge of content and students, a subdomain of PCK, represents an interaction between knowledge about students as learners of mathematics and knowledge of mathematics content (Ball et al. 2008). Teachers need to be aware of student reasoning and common misconceptions with respect to particular content as well as instructional approaches that support student thinking (National Research Council 2001). For instance, the use of representations, one of NCTM’s (2000) process standards, provides objects around which to reason mathematically and subsequently helps students develop an understanding of mathematical concepts (Leinwand et al. 2014). Further, different representations of a mathematical idea afford different conceptions about that idea (e.g., various interpretations of fractions such as part of a whole or quotient) (National Research Council 2001). Throughout their career, teachers must continue to augment their understanding of student conceptions (Russell 2007). Educative curriculum materials could support teacher learning in this realm by giving examples of how students typically respond to a task, including common difficulties, explaining how students’ responses relate to their understanding of a concept, and providing recommendations for ways in which to address students’ responses (Arias et al. 2015a; Ball and Cohen 1996; Beyer et al. 2009; Davis and Krajcik 2005; Dietz and Davis 2009; Duncan et al. 2011; Doerr and Chandler-Olcott 2009; Heaton 2000; Remillard 2000; Stein and Kim 2009). Collopy (2003) described a teacher’s approach to using curriculum materials, which provided examples of common student errors. She read the information about student misconceptions prior to teaching the lesson, observed student thinking during the lesson, and reread the information after the lesson to prepare for the next lesson. Teachers also have opportunities to learn content through examining student thinking presented in curriculum materials (Schneider and Krajcik 2002; Tyminski et al. 2013) and during the implementation of lessons (Remillard 2000), demonstrating a reciprocal relationship between subject matter knowledge and understanding student thinking.”
(p. 361)Teacher knowledge of disciplinary discourse in mathematics The curriculum materials support the development of teacher knowledge in fostering disciplinary discourse in mathematics.
- What supports are provided for fostering disciplinary discourse in mathematics (e.g., focus and direction or step-by-step script)?
- To what extent are rationales for the inclusion of a discussion and its relationship to the development of student understanding of the mathematics content present, clear, and appropriate?
- To what extent is implementation guidance for fostering disciplinary discourse in mathematics present, clear, and appropriate?
One aspect of learning a discipline is developing an understanding of the specific ways of communicating ideas within the field, a component of literacy (Moje 2008; NCTM 2000). Moje (2008) and Siebert and Draper (2008) argue that instruction needs to take into consideration the discipline-specific forms of literacy. With respect to communication, students need to learn the norms of discourse for a particular discipline, what Gee (2008) terms Discourse. Yackel and Cobb (1996) describe norms of discussions, which apply across subject areas, such as those stated in the NCTM (2000) process standard of Communication. Specifically, instruction should support students in sharing their ideas with their classmates and teachers, expressing and solidifying their thinking, and considering and assessing the thinking of others. However, Yackel and Cobb further elaborate that there are norms of discussions specific to the field of mathematics, sociomathematical norms, including what makes various explanations mathematically different, sophisticated, efficient and/or acceptable. In Principles to Actions: Ensuring Mathematical Success for All, Leinwand et al. (2014) articulate eight research-informed mathematics teaching practices, one of which is facilitat[ing] meaningful mathematical discourse. Content knowledge and knowledge of student thinking contribute to the facilitation of discussions, and teachers often need support to direct and participate in discussions centered on developing particular mathematical ideas as well as the disciplinary Discourse of mathematics (Moje 2008; Russell 2007; Stein and Kaufman 2010). Educative curriculum materials can provide guidance for conducting a productive classroom discussion by identifying its mathematical and pedagogical purpose, explicating its contribution to the mathematical focus of a unit, suggesting approaches to structure it, providing possible initial and follow-up questions to promote interaction about student ideas, recommending means to foster student participation, sharing possible directions to pursue, and including examples of discussions (Beyer et al. 2009; Davis and Krajcik 2005; Grant et al. 2009; Remillard 2000; Russell 2007). One risk of providing sample conversations is that they are perceived as a script. For instance, in a case study of two teachers’ use of educative curriculum materials, Collopy (2003) found that one teacher thought that the sample dialogues were supposed to be read word-for-word as a class. In comparison, the other teacher used them to plan for class by learning how students might reason about and discuss the mathematics. Similarly, Grant et al. (2009) reported that teachers used sample questions and dialogues, provided by the Investigations curriculum materials, in lesson preparation. To avoid the faulty perception of a fixed interaction, example discussions could be supplemented with information about the teacher’s thought process in the midst of facilitating the conversation (Grant et al. 2009; Remillard 2000). Teachers need support in orchestrating discourse by considering and responding to student thinking in the moment with the goal of the discussion and norms of the disciplinary Discourse in mind (Russell 2007).