Presentations

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.

Abstract/Description: Covariation and covariational reasoning are key themes in mathematics education research. Recently, these ideas have been expanded to include cases where more than two variables relate to each other, in what is termed multivariation. Building on the theoretical work that has identified different types of multivariation structures, this study explores students’ reasoning about these structures. Our initial assumption that multivariational reasoning would be built on covariational reasoning appeared validated, and there were also several other aspects of reasoning employed in making sense of these structures. There were important similarities in reasoning about the different types of multivariation, as well as some nuances between them.

Abstract/Description: The sampling distribution (SD) is a foundational concept in statistics, and simulations of repeated sampling can be helpful to understanding them. However, it is possible for simulations to be misleading and it is important for research to identify possible pitfalls in order to use simulations most effectively. In this study, we report on a key misconception students had about SDs that we call the “multi-sample distribution.” In this misconception, students came to believe that a SD was composed of multiple samples, instead of all possible samples, and that the SD must be constructed by literally taking multiple samples, instead of existing theoretically. We also discuss possible origins of this misconception in connection with simulations, as well as how some students appeared to resolve this misconception.

Abstract/Description: The activities that teacher educators prepare for preservice teachers should be intentional in their purpose for improving teaching practices. We report on a video database activity that our preservice teachers engaged in and their improvement in attending to student mathematics.