Archive for Uncategorized

Tuesday, February 18th, 2020

Reflecting on Knowledge of Student Thinking and Knowledge of Disciplinary Discourse

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).

Friday, February 7th, 2020

Math Modeling Social Issues

NO more recycling?

Not Enough Supplies

Teacher Spending on School Supplies: A State-by-State Breakdown


Tuesday, December 10th, 2019

Being Intentional about Equitable Teaching Practices

Download (PDF, 83KB)

Thursday, December 5th, 2019

Supporting “Pedagogical Courage” through Instantiating One’s Practice

Connecting Research to Practice: In the Spring of 2019 and Summer of 2019, we designed a professional development course focused on developing teachers’ mathematics teaching practices. Using NCTM’s Principles to Action (2014), we designed the Spring course as a 15 week course. We spend each week focused on a teaching practice where teachers selected a rich task and vetted tasks in peer groups. On alternating weeks, teachers did not meet in class. Instead they enacted the vetted task and brought intentionality in one’s practice and video taped and posted about 5 minutes focused on one of the practices to share with their peers. With the vignette, the teacher and peers annotated and marked instances of the teacher enacting the teaching practices. The peer coaches validated and encouraged growth as the teacher opened up his or her practice. Meanwhile, the teacher developed what we called “pedagogical courage” to continue to try enhancing his/her practice.

Thursday, December 5th, 2019

Math Podcasts

Math Podcasts


Making Math Moments That Matter Podcast


Sunday, December 1st, 2019


Monday, November 18th, 2019

2019 PMENA Working Group on Early Math Modeling

Thursday, September 12th, 2019

Routines to support math talk

Love how my friends at Howard County share all their wonderful math resources.

Look fors by my friend John San Giovanni gets at the heart of meaningful math teaching!!!



Wednesday, September 11th, 2019

Math Modeling Networked Improvement Communities (MMNIC)

Lewis, C. (2015). What is improvement science? Do we need it in education? Educational Researcher, 44(1), 54–61.

process of improving practice through systematic inquiry


Education researchers engage in improving practice, but their findings are seldom shared beyond their school, district, or state (Zeichner, 2001). Thus, promising practices often are not implemented in new contexts or are implemented on a large scale without the necessary capacity to do so and without careful attention to the challenges of implementation (Bryk et al., 2015; Coburn & Stein, 2010)

Thursday, August 29th, 2019

Motivated-COMPUTE Equity

As I begin a new semester with excited new pre-service teachers, I am inspired by Ilana Horn’s book @ilana_horn, Motivated- Designing Math Classrooms Where Students Want to Join In! She identifies five features of a motivational classroom: Students’ sense of belongingness, the meaningfulness of learning, students’ competence, structures for accountability and students’ autonomy. Dr. Horn shared how the mathematics classrooms are socially risky places and how we need to decrease that social risk to increase student participation and math talk. Thank you to stellar teachers like @pegcagle, Rafranz Davis, Sadie Estrella, Chris Luniak, Fawn Ngyuen, Elizabeth Statmore who open up their practices and routines that motivate student participation. I believe all their effort builds each and every student to have math power. M-power “empower” as I like to call it.

This powerful message aligns with 7 best equitable teaching practices that I call COMPUTE to provide equity in the math classroom that I will share with my pre-service teachers and with teachers I work with Lesson Study.

COMPUTE for Equitable Teaching and Learning in the Math Classroom

  • Caring, celebrating and connecting to cultural diversity, cultural contexts and the world we live in to engage in mathematics.
  • Owning the math-  Allow students to share their mathematical thinking and author math ideas that builds on collective knowledge
  • Motivating student to learn by providing experience that taps into learners curiosity and interest where they find challenge and academic success.
  • Problem Posing and Problem Solving as the core math activities to develop metacognition.
  • Understanding with competence and confidence that builds  students’ math identity
  • Targeted feedback to math learning for individual needs and accountable learning
  • Emotionally supportive learning environment where learners feel safe, valued and cared for where mistakes are embraced as steps to learning.

(See blog connecting COMPUTE Equity with Math Modeling Activities that Connect to Students Lived Experiences