Presentations

Abstract/Description: Productive use of student mathematical thinking is a critical aspect of effective teaching that is not yet fully understood. We have previously conceptualized the teaching practice of building on student mathematical thinking and the four elements that comprise it. In this paper we begin to unpack this complex practice by looking closely at its third element, Conduct. Based on an analysis of secondary mathematics teachers’ enactments of building, we describe the critical aspects of conducting a whole-class discussion that is focused on making sense of a high-leverage student contribution.

Abstract/Description: Research suggests that students’ difficulties in studying algebraic topics in middle school can be remedied at least in part by teaching students to use a relational meaning for the equal sign to reason about equations. However, little empirical research has been done to investigate what meanings for the equal sign and equation types are common in middle school mathematics. This study examines two series of 7th and 8th grade mathematics textbooks to identify what equal sign meanings and equation types are being used in middle school mathematics. Three meanings for the equal sign were used in all four textbooks, and each equation type was typically associated with only one meaning of the equal sign. The results imply that students need to develop three different meanings for the equal sign to succeed in middle school mathematics, and that recognizing equation types can help indicate which meaning of the equal sign is being used.

Abstract/Description: While scholars have noted that variables are used in multiple ways during algebraic activity, little empirical research has been conducted to study which variable types middle school students typically encounter in their mathematics classes. To address this need, we present a study that examined the different types of variables used in three 7th-8th grade mathematics curricula. Using qualitative methods, we identified 8 main variable types. These 8 variable types were present in every year of each curriculum. Most lessons required students to distinguish between 2-5 different variable types. Our findings imply that students need to develop sophisticated and nuanced understandings of variables to meaningfully participate in middle school mathematics.

Abstract/Description: Approaches to integration based on quantitative reasoning have largely developed along two parallel lines. One focuses on continuous accumulation from rate, with accumulation functions as the primary object. The other focuses on summing infinitesimal bits of a quantity, with definite integrals as the primary object. No work has put these two approaches in direct conversation with each other, which is the purpose and contribution of this theoretical paper. In this paper, we unpack both approaches in terms of meanings and reasoning. Because modeling is a key motive for using quantitatively-grounded approaches in the first place, we then analyze and discuss each approach’s method of modeling two example contexts.

Abstract/Description: Intellectual need (IN) is a powerful way to support learning by engaging students and helping them view mathematics as less arbitrary. While IN has been developed theoretically, much less has been done to build frameworks for how to actually create IN provoking tasks – both in terms of what a task designer might attend to and how to attend to those things. In this theoretical paper, we review key premises in IN, from which we extract several components that should be taken up in IN task design. We then describe a process one can use to address these components systematically in constructing a task specifically meant to provoke IN.

Abstract/Description: "Work on the teaching and learning of definite integrals has expanded significantly in recent years, with the specific conceptualization "adding up pieces" being a promising foundational meaning for integrals. Yet, work in this area has largely focused on student understanding and reasoning, with only small attention to the detailed work of how task-based learning over an entire unit of integration could support students in coming to develop these understandings. This poster presents an outline of a learning trajectory for a unit on integration based on adding up pieces and how students come to learn the individual parts that make up the larger integral concept.

Abstract/Description: Four groups of mathematics teacher educators share the ways they are exploring the creation, organization, and use of public records of student mathematical thinking--physical and visual representations of student mathematics that are publicly accessible to all participants within a classroom.

Abstract/Description: Student teaching has long been plagued by a lack of coherence and sustained institutional and research support. In this working group participants will identify challenges and share strategies to support efforts to improve secondary mathematics clinical practice.

Abstract/Description: Abstract: Studies have shown a correlation between religious ideologies and racist beliefs. Less is known about how people are bringing their religious ideologies to make sense of and participate in discussions about racism and other systems of oppression. In this paper, we analyze the discourse used a small set of White prospective and practicing mathematics teachers in response to a question about how their religious beliefs influenced their perspectives on race. This analysis reveals that current White discursive frameworks may not reveal important components of discourse used by Christians discussing race, racial differences, and racism. Therefore, we turn to Bakhtin’s theoretical perspective on language and discourse to make sense of the participant data. We explore the religious teachings of The Church of Jesus Christ of Latter-day Saints focusing on how the meaning of “we are all children of God” is developed through church curricula, the scriptures used by its members, and the teachings of the Church’s leadership. In other words, we unfold the possible dialogic relationships among contexts, speakers, and words associated with derivations of the phrase “we are all children of God” in The Church and its members. We use this unfolding to show how the ventriloquation of “we are all children of God” operated in contexts about racism to illuminate messiness in prospective and practicing teachers’ use of their religious ideologies in making sense of racism.

Abstract/Description: Students who are learning to work with equations in algebra need to understand the structural conventions for equations, i.e., the norms for writing, organizing, and interpreting equations in a problem solution. Little research has been done to identify the structural conventions for equations that are prominent in middle school mathematics. In this study, we examined two middle school curricula to identify the structural conventions used in the materials. We found two main structure—lists of equations and strings of equations—and identifiied conventions for reading and writing these structures.

Abstract/Description: Several researchers have promoted reimagining functions and graphs more quantitatively. One part of this research has examined graphing “conventions” that can at times conflict with quantitative reasoning about graphs. In this theoretical paper, we build on this work by considering a widespread convention in mathematics teaching: putting related, derived graphical objects (e.g., the graphs of a function and its inverse or the graphs of a function and its derivative) on the same set of axes. We show problems that arise from this convention in different mathematical content areas when considering contextualized functions and graphs. We discuss teaching implications about introducing such related graphical objects through context on separate axes, and eventually building the convention of placing them on the same axis in a way that this convention and its purposes become more transparent to students.

Abstract/Description: We describe a framework for characterizing students’ graphical reasoning, focusing on providing an empirically-based list of students’ graphical resources. The graphical forms framework builds on the knowledge-in-pieces perspective of cognitive structure to describe the intuitive ideas, called “graphical forms”, that are activated and used to interpret and construct graphs. As part of the framing for this work, we provide theoretical clarity for what constitutes a graphical form. Based on data involving pairs of students interpreting and constructing graphs we present a list of empirically documented graphical forms, and organize them according to similarity. We end with implications regarding graphical forms’ utility in understanding how students construct graphical meanings and how instructors can support students in graphical reasoning.

Abstract/Description: Productive use of student mathematical thinking is a critical yet incompletely understood dimension of effective teaching practice. We have previously conceptualized the teaching practice of building on student mathematical thinking and the four elements that comprise it. In this paper we begin to unpack this complex practice by looking closely at its first element, establish. Based on an analysis of secondary mathematics teachers’ enactments of building, we describe two critical aspects of establish—establish precision and establish an object—and the actions teachers take in association with these aspects.

Abstract/Description: This poster reports our findings of how teachers used public records of student mathematical thinking throughout whole class discussions. In our work, we consider a public record to be a physical and visual representation of a student contribution that is accessible to all classroom members. We will share how teachers' explicit referencing of public records helped teachers to establish student thinking and engage students with each other’s ideas in whole class discussions.