Past Presentations

Presenters: Steven Jones, Leilani Fonbuena, Michelle Chambers and Spencer Young from Brigham Young University
Location: RUME in Omaha Nebraska
Abstract/Description:
Intensive quantities result from quantitative operations on two or more extensive quantities. As such, their units of measure consist of “compound units.” Students regularly encounter symbolically-written compound unit structures that are directly given to them, rather than constructed or developed, such as m/s 2 , ft-lbs, or kg∙m/s. It is consequently important to understand how students might try to reason about such symbolically-presented compound unit structures, which is the focus of this study. We examined “ways of reasoning” students used to make sense of such units, and describe in this paper five themes that emerged during analysis: (1) decomposing into separate units, (2) treating units as variables, (3) using covariational/ multivariation reasoning, (4) posing a quantification, and (5) bringing in pure math concepts.

Presenters: Steven Jones, Brigham Young University and Jon-Marc Rodriquez, University of Wisconsin - Madison
Location:
Abstract/Description:
In broad terms, much of the research on graphical reasoning can be characterized as focusing on misconceptions, covariational and quantitative reasoning, and graphing as a social practice. In contrast, other research has focused on graphing as a cognitive process, emphasizing the fine-grained knowledge elements related to graphing, with a focus on characterizing ideas students associate with graphical patterns (i.e., graphical forms). This paper moves beyond graphical forms to characterize other categories of fine-grained knowledge – “graphical resources” – that are activated and used in concert when constructing and interpreting graphs. In this study, we identified six categories of graphical resources: graphical forms resources, framing resources, ontological resources, convention resources, quantitative resources, and function resources. We posit that holistically considering different categories of fine-grained graph-related knowledge resources can connect various bodies of research on graphing.

Presenters: Aaron Weinberg, Ithaca College; Michael Tallman, Oklahoma State University; and Steven Jones, Brigham Young University
Location:
Abstract/Description:
The idea of intellectual need (IN) has received much interest from instructors in trying to design tasks that engage students in impasse-driven learning. However, we argue that the literature on IN is currently insufficient for supporting the careful design and implementation of tasks meant to provoke IN. In this paper, we examine two particular shortcomings: (1) What exactly IN can be created for, and (2) How an instructor might support students in navigating the experience of resolving the confusion and constructing the targeted meanings. For the first of these, we describe the category error of thinking of producing IN for a “topic”, and use the idea of conceptual analysis to suggest a way to address this shortcoming. For the second, we bring in control-value theory to explain what an instructor might attend to in order to ensure that the disequilibrium stays productive and does not lead to frustration and disengagement.

Presenters: Shari Stockero, Michigan Technological University; Ben Freeburn, Western Michigan University; Jessica Postma, Western Michigan University; Nishat Alam, Michigan Technological University; Keith Leatham, Brigham Young University; Blake Peterson, Brigham Young University; and Laura Van Zoest, Western Michigan University
Location: AMTE in Nashville, Tenneessee
Abstract/Description:
We use rehearsal debrief discussion excerpts to consider how rehearsals with experienced teachers might be planned and structured to position the debrief as a mechanism for mathematics teacher educators and teachers to negotiate an understanding of a complex teaching practice.

Presenters: AnnaMarie Conner, University of Georgia; Keith Leatham, Brigham Young University; Laura Singletary, Lee University; Laura Van Zoest, Western Michigan University, Jonathan Foster, University of Virginia; Shari Stockero, Michigan Technological University; Hyejin Park, Drake University; Blake Peterson, Brigham Young University; and Yuling Zhuang, Emporia State University
Location: AMTE in Nashville, Tenneessee
Abstract/Description:
We explore teachers’ facilitation of whole class discussions by comparing and contrasting the analysis of such discussions through two different lenses: 1) teachers’ support of collective argumentation; and 2) teachers’ productive use of student mathematical contributions.

Presenters: Doug Corey, Brigham Young University
Location: Joint Mathematical Meetings, Boston, MA
Abstract/Description:
John Dewey pointed out that one of the problems with the US K-12 educational system is that when a teacher retires, they take all of their accumulated knowledge with them out of the educational system. This is largely the case with undergraduate mathematics education as well. The available instructional resources for undergraduate mathematics instructors lack key features required to build a robust knowledge base for teaching. Few resources address the everyday work of teaching undergraduate mathematics by exploring the details of teaching specific content in a specific context, how to reason through various possible instructional decisions, and how the instructional decisions connect with or help to deepen student mathematical thinking. In this talk I discuss the idea of Lesson Analysis (LA), a process for generating instructional knowledge, and the closely associated written genre, Lesson Analysis Manuscripts (LAMs), to store and share important instructional knowledge largely absent in current resources. LAMs are a type of detailed lesson plan developed to solve a particular problem of practice. However, the emphasis is on understanding the reasoning behind the instructional decisions, usually justified through student mathematical thinking, not on the particular instructional choices of the lesson. I discuss how LA fits into a broad SoTL umbrella, the key features of a LAM, and explain where to publish LAMs for the undergraduate mathematics teaching community.

Presenters: Navy Dixon, Sariah Stevenson, and Dawn Teuscher - Brigham Young University and Shannon Dingman - University of Arkansas, Fayetteville
Location: PMENA in Nashville, Tennessee
Abstract/Description:
With the adoption of the Common Core State Standards for Mathematics 12 years ago, the topic of geometric transformations was shifted from high school to grade 8. In our research with middle grades teachers, they often discussed their difficulty in teaching geometric rotations. Therefore, we analyzed 444 middle grade students’ responses, across four states, to eight rotation questions from the SMART assessment. The results corroborate teachers’ challenges with teaching and student learning of rotations. Results indicate that students have a rigid understanding of angle measure that may be impacting their understanding of geometric rotations. Although angle measure is introduced in grade 4, we hypothesize that teachers need to provide additional opportunities for students to expand their rigid understanding of angle measure.

Presenters: Ben Freeburn - Western Michigan University, Keith Leatham - Brigham Young University, Sini Graff - Brigham Young University, Nitchada Kamlue - Western Michigan University, Shari Stockero - Michigan Tech University, Blake Peterson - Brigham Young University, and Laura Van Zoest - Western Michigan University
Location: PMENA in Nashville, Tennessee
Abstract/Description:
The more researchers understand the subtleties of teaching practices that productively use student thinking, the better we can support teachers to develop these teaching practices. In this paper, we report the results of an exploration into how secondary mathematics teachers’ use of public records appeared to support or inhibit their efforts to conduct a sense-making discussion around a particular student contribution. We use cognitive load theory to frame two broad ways teachers used public records - manipulating and referencing - to support establishing and maintaining students’ thinking as objects in sense-making discussions.

Presenters: Shari Stockero - Michigan Tech University, Blake Peterson - Brigham Young University, Keith Leatham - Brigham Young University, and Laura Van Zoest - Western Michigan University
Location: PMENA in Nashville, Tennessee
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.

Presenters: Daniel Siebert and Chelsea Dickson, Brigham Young University
Location: PMENA in Nashville, Tennessee
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.

Presenters: Daniel Siebert and Ashlyn Rounds, Brigham Young University
Location: PMENA in Nashville, Tennessee
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.

Presenters: Konda Luckau, Mount Nebo Middle School and Daniel Siebert, Brigham Young University
Location: UAMTE 2022 in Kaysville, Utah
Abstract/Description:

Presenters: Steven Jones, Brigham Young University and Rob Ely, University of Idaho
Location: SIGMAA on RUME 2022 in Boston
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.

Presenters: Steven Jones and Brinley Stevens, Brigham Young University
Location: SIGMAA on RUME 2022 in Boston
Abstract/Description:
Previous calculus education work on integrals, including definite integrals and accumulation functions, has created useful theoretical frameworks that decompose the “integrals with bounds” concept into constituent parts. Yet, each framework focuses on distinct aspects of the integral and leaves certain parts implicit. Further, definitions and operationalizations are absent in many of these frameworks. This theoretical paper contributes by: (a) pulling together the various pieces in these frameworks into a comprehensive decomposition of the integral concept, (b) explicitly defining the processes and objects within it, and (c) operationalizing the processes and objects within each of the numeric, graphical, and symbolic representations. This comprehensive framework is useful for researchers, curriculum or task writers, and instructors alike to have a more complete picture of the elements that make up the integral concept.

Presenters: Aaron Weinberg, Ithaca College and Steven Jones, Brigham Young University
Location: SIGMAA on RUME 2022 in Boston
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.

Presenters: Brinley Stevens and Steven Jones, Brigham Young University
Location: SIGMAA on RUME 2022 in Boston
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.

Presenters: Rob Wieman, Rowan University; Jill Perry, Rowan University; Keith Leatham, Brigham Young University; Basil Conway, Columbus State University; Marilyn Strutchens, Auburn University; Cathy Liebars, The College of New Jersey; and James Beyers, The College of New Jersey
Location: AMTE 2022 in Las Vegas, Nevada
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.

Presenters: Shannon Dingman, University of Arkansas; Dawn Teuscher, Brigham Young University; and Travis A. Olson, University of Nevada, Las Vegas
Location: AMTE 2022 in Las Vegas, Nevada
Abstract/Description:

Presenters: J. Matt Switzer, Texas Christian University and Dawn Teuscher, Brigham Young University
Location: AMTE 2022 in Las Vegas, Nevada
Abstract/Description:

Presenters: Kate Johnson and Emma Holdaway, Brigham Young University and Amy Saunders Ross, Montana State University - Billings
Location: Mormon Social Science Association of the Society for the Scientific Study of Religion and the Religious Research Association in Portland Oregon
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.

Presenters: Chelsea Dixon and Daniel K. Siebert, Brigham Young University
Location: PMENA in Philadelphia Pennsylvania
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.

Presenters: Steven Jones, Doug Corey and Dawn Teuscher, Brigham Young University
Location: PME-NA 43 in Philadelphia, Pennsylvania
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.

Presenters: Jon-Marc Rodriquez, University of Iowa and Steven Jones, Brigham Young University
Location: PME-NA 43 in Philadelphia, Pennsylvania
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.

Presenters: Keith R. Leatham, Brigham Young University; Laura R. Van Zoest, Western Michigan University; Ben Freeburn, Western Michigan University; Blake E. Peterson, Brigham Young University; and Shari L. Stockero, Michigan Tech University
Location: PME-NA 43 in Philadelphia, Pennsylvania
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.

Presenters: Ben Freeburn, Western Michigan University; Sini Graff, Brigham Young University; Nitchada Kamlue, Western Michigan University; Zeynep Arslan, Trabzon University; Lincoln J. Sorensen, Michigan Tech University; and Keith R. Leatham, Brigham Young University
Location: PME-NA 43 in Philadelphia, Pennsylvania
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.

Presenters: Steven Jones and Haley Jeppson, Brigham Young University
Location: PMENA in Mazatlan, Mexico and virtually
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.

Presenters: Steven Jones and Kiya Eliason, Brigham Young University
Location: PMENA in Mazatlan, Mexico and virtually
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.

Presenters: Laura R. Van Zoest, Western Michigan University; Shari Stockero, Michigan Tech University; Keith R. Leatham and Blake E. Peterson, Brigham Young University; and Joshua M. Ruk, Western Michigan University
Location: PMENA in Mazatlan, Mexico and virtually
Abstract/Description:
We draw on our experiences researching teachers’ use of student thinking to theoretically unpack the work of attending to student contributions in order to articulate the student mathematics (SM) of those contribution. We propose four articulation-related categories of student contributions that occur in mathematics classrooms and require different teacher actions:(a) Stand Alone, which requires no inference to determine the SM; (b) Inference-Needed, which requires inferring from the context to determine the SM; (c) Clarification-Needed, which requires student clarification to determine the SM; and (d) Non-Mathematical, which has no SM. Experience articulating the SM of student contributions has the potential to increase teachers’ abilities to notice and productively use student mathematical thinking during instruction.

Presenters: Dawn Teuscher, Brigham Young University and J. Matt Switzer, TCU
Location: AMTE 2021, Virtual
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.

Presenters: Porter Nielsen and Dawn Teuscher, Brigham Young University
Location: AMTE 2021, Virtual
Abstract/Description:
Teachers’ instructional decisions are important to students’ mathematics learning as they determine the learning opportunities for all students. We will discuss 8th-grade teachers’ reasoning for their instructional decisions in the context of geometric reflections and orientation of figures.

Presenters: Blake E. Peterson, Brigham Young University; Shari Stockero, Michigan Tech University, Laura R. Van Zoest, Western Michigan University, and Keith R. Leatham, Brigham Young University
Location: AMTE 2021, Virtual
Abstract/Description:
To productively use student mathematical thinking, it must be 1) made clear and 2) established as the object of discussion. The nuances of these two aspects of the teaching subpractice, Make Precise, will be discussed through examples from the data.

Presenters: Laura R. Van Zoest, Western Michigan University; Carlee E. Madis, Western Michigan University, Blake E. Peterson and Keith R. Leatham, Brigham Young University and Shari Stockery, Michigan Tech University
Location: AMTE 2021, Virtual
Abstract/Description:
We share our research on uses of a public record to support whole-class discussions, show examples of revising a public record in real-time to support the discussion, and consider how this information can be used in developing well-prepared beginning teachers.

Presenters: Navy Dixon, Erin Carroll, and Dawn Teuscher, Brigham Young University
Location: RUME 2020, Boston, MA
Abstract/Description:
The Pathways to College Algebra curriculum aims to build concepts that cohere with the big ideas in Calculus, and initial results suggest improved readiness for Calculus by students who use the curriculum. Our study examines similarities and differences of Pathways and non-Pathways students understanding and reasoning about the calculus concept of the limit. We compare students’ understanding of limits at the beginning and at the end of the unit. Our findings suggest that (1) students reliance on procedures, combined, or quantitative reasoning was dependent on the calculus instructors’ emphasis in the class; (2) students who begin their Calculus class with high covariational reasoning gain a more sophisticated understanding of limits; and (3) when curriculum is coherent students will identify mathematical connections.

Presenters: Porter Nielsen, Janessa Cloward, and Dawn Teuscher, Brigham Young University
Location: AMTE 2020, Phoenix, Arizona
Abstract/Description:
In this session, we examine multiple middle grades teachers’ mathematical goals and how much time students spend during class working towards these goals. We also examine if the teachers’ goals are aligned or not aligned with research-based learning trajectories.

Presenters: Travis Olson, University of Nevada Las Vegas; Dawn Teuscher, Brigham Young University; and Shannon Dingman, University of Arkansas
Location: AMTE 2020, Phoenix, Arizona
Abstract/Description:
We focus on data collected within a larger study into middle grades mathematics teachers’ curricular reasoning. Specifically, we present data from three teachers’ integration of Desmos’ Transformation Golf: Rigid Motion activity into their 8th grade unit on geometric transformations.

Presenters: Dawn Teuscher and Porter Nielsen, Brigham Young University
Location: NCTM 2019 Regional Conference & Exposition in Salt Lake City, Utah
Abstract/Description:
This session will examine the relationships among the three geometric transformations-reflections, translations and rotations. We will also examine how the sequence of transformations can be used to connect the three transformations. Finally, we will discuss how understanding these relationships will help students make sense of transformations.

Presenters: Porter Nielsen, Brigham Young University
Location: NCTM 2019 Regional Conference in Salt Lake City, Utah
Abstract/Description:
In order to leverage the power behind the mathematical definition of reflections as a transformation, we examine how curriculum, teacher decisions, and student interpretations of reflections relate to one another.

Presenters: Blake E. Peterson, Brigham Young University; Keith R. Leatham, Brigham Young University; and Laura Van Zoest, Western Michigan University
Location: NCTM 2019 in San Diego, California
Abstract/Description:
Incorporating student mathematical thinking into classroom instruction is a best practice, but not all student thinking provides the same leverage for accomplishing mathematical goals. Learn about characteristics of student thinking that can be used to determine which thinking has significant potential to support students’ learning of mathematics.

Presenters: Steven Jones, Haley Jeppson, and Doug Corey, Brigham Young University
Location: 22nd Annual Conference on Research on Undergraduate Mathematics Education, Oklahoma City, Oklahoma
Abstract/Description:
Unfortunately, students far too often have little or no intellectual need for learning the second semester calculus topic of Taylor and power series. In this study, we examine the “potential intellectual needs” (PINs) provided by commonly used textbooks. While the textbooks used different approaches, they both often lacked problems developing intellectual need, suggesting that instructors must incorporate intellectual need by themselves. To assist in this endeavor, we focus part of the paper on a discussion of including PINs for this content. We found that it may be difficult to incorporate genuine problems for first-year students through an approach based on a “family of series” meaning for Taylor/power series, but that stronger problems could be incorporated through an approach based on an “extension of linear approximation” meaning.

Presenters: George Kuster, Christopher Newport University and Steven Jones, Brigham Young University
Location: 22nd Annual Conference on Research on Undergraduate Mathematics Education, Oklahoma City, Oklahoma
Abstract/Description:
In this study we investigated how a small sample of students used variational reasoning while discussing ordinary differential equations. We found that students had flexibility in thinking of rate as an object, while simultaneously unpacking it in the same reasoning instance. We also saw that many elements of covariational reasoning and multivariational reasoning already discussed in the literature were used by the students. However, and importantly, new aspects of variational reasoning were identified in this study, including: (a) a type of variational reasoning not yet reported in the literature that we call “feedback variation” and (b) new types of objects, different from numeric-quantities, that the students covaried.

Presenters: Brianna Levia, Navy Borrowman, Dawn Teuscher, and Steven Jones, Brigham Young University
Location: 22nd Annual Conference on Research on Undergraduate Mathematics Education, Oklahoma City, Oklahoma
Abstract/Description:
The Pathways to College Algebra curriculum aims to build concepts that cohere with the big ideas in Calculus, and initial results suggest improved readiness for Calculus by students who have taken a Pathways class. However, less is known about how Pathways might influence students’ initial understanding and reasoning about calculus concepts. Our study examines similarities and differences in how Pathways and non-Pathways students initially understand and reason about the calculus concept of the limit. Our findings suggest that Pathways students may engage a little more in quantitative reasoning and in higher covariational reasoning, and have more correct and consistent initial understandings. Further, the Pathways students were explicitly aware of how their Pathways class may have benefited their understanding of limits.

Presenters: Shari L. Stockero, Michigan Technological University; Keith R. Leatham, Brigham Young University; and Blake E. Peterson, Brigham Young University
Location: AMTE 2019 in Orlando, Florida
Abstract/Description:
We explore issues related to responsive teaching by presenting excerpts of whole-class discussion and considering the degree of responsiveness within each excerpt as it relates to the collection of instances of student thinking that had been shared thus far.

Presenters: Shannon Dingman, University of Arkansas; Dawn Teuscher, Brigham Young University; Travis Olson, University of Nevada, Las Vegas
Location: AMTE 2019 Orlando, Florida
Abstract/Description:
We present six aspects of curricular reasoning and illustrate the interactions among teachers, students, mathematics, and curriculum materials using data from Grade 8 teachers as they planned and enacted geometric transformation lessons. We discuss differences across teachers with varying backgrounds and consider how teachers’ curricular reasoning can influence students’ opportunity to learn mathematics.

Presenters: Scott Hendrickson, Brigham Young University
Location: NCTM Western Regional Conference, Seattle, Washington
Abstract/Description:
The transition from the static perspective of right triangle trig ratios to the dynamic perspective of circular trig functions, and from measuring angles in degrees to measuring angles in radians, can generate roadblocks and misconceptions. In this session we will examine a sequence of tasks that reveal, rather than obscure, trigonometric ideas. Participants will engage in tasks that develop the following CCSSM concepts for students: (1) defining radians as a proportionality constant, prior to defining radians as an arc length on a unit circle; (2) using the unit circle to generalize the definitions of the trigonometric functions and to establish trig identities, rather than focusing on memorizing "special angles" which can hinder students' understanding; (3) modeling contexts with trig functions, rather than just sketching graphs.

Presenters: Scott Hendrickson, Brigham Young University
Location: North Carolina Council of Teachers of Mathematics, Greensboro, North Carolina
Abstract/Description:
The Comprehensive Mathematics Instruction Framework (CMI), developed by the Brigham Young University Public School Partnership, captures the research and best practices of the CCSSM standards and the NCTM Principles to Actions and makes these ideas accessible to practicing and preservice mathematics teachers. By 
organizing these principles and practices into a Teaching Cycle, a Learning Cycle and a Continuum of Mathematical Understanding, teachers can attend to learning progressions, formative assessment and teaching practices that support construction of conceptual understanding and procedural fluency on a foundation of student thinking. Mathematics Vision Project (MVP) is an example of a curriculum created using the CMI Framework. In this session, participants will be introduced to the CMI framework and its implementation through classroom vignettes where student thinking is elicited by tasks from the MVP curriculum designed to promote conceptual, procedural and representational understanding.

Presenters: Kelly Campbell, University of Delaware and Daniel K. Siebert, Brigham Young University
Location: NCTM in Washington DC
Abstract/Description:
The ways students are positioned influence what students come to learn. The purpose of this report is to illustrate the value of analyzing student and teacher interactions through the lens of positioning. We found that a student struggled because she was following the storyline of "doing school mathematics" while the teacher was following the storyline of "doing mathematics." Teachers need support in learning to help students take on new positions within the storyline of "doing mathematics."

Presenters: Daniel K. Siebert and Kelly Eddington, Brigham Young University
Location: NCTM Conference, Washington DC
Abstract/Description:
Conceptually-oriented mathematics explanations (CMEs) are understudied even though they support students' mathematical reasoning and learning. In this study, we examine the CMEs written in a university mathematics education course to identify the components and structure of CMEs. We found that CMEs are comprised of constructions and equivalences, and that students use templates of class-sanctioned definitions and processes to build validity for their CMEs.

Presenters: Laura R. Van Zoest, Western Michican University; Blake E. Peterson, Brigham Young University; Shari L. Stockero, Michigan Technological University, and Keith R. Leatham, Brigham Young University
Location: AERA Conference, New York City, NY
Abstract/Description:
This session focuses on developing clarity about issues related to analyzing teachers’ responses to student mathematical contributions during whole-class interactions. We do this by examining and juxtaposing the approaches that two different research groups have taken to investigating teacher responses. Each group will share their goal for their research, the grainsize of their units of analysis, and the coding scheme they have developed. The third presentation focuses on applying both coding schemes to the same excerpts of whole-class interactions. The discussant will consider relationships between the two approaches, advantages and disadvantages of each, and what this work means for research on facilitating productive discourse around student mathematical thinking. Attendees will discuss issues raised by the discussant and presenters.

Presenters: Steven Jones, Brigham Young University
Location: RUME Conference, San Diego, California
Abstract/Description:
Covariation and covariational reasoning have become key themes in mathematics education research. In this theoretical paper, I build on the construct of covariation by considering cases where more than two variables relate to each other, in what can be called “multivariation.” I share the results of a conceptual analysis that led to the identification of four distinct types of multivariation: independent, dependent, nested, and vector. I also describe a second conceptual analysis in which I took the mental actions of relationship, increase/decrease, and amount from the covariational reasoning framework, and imagined what analogous mental actions might be for each of these types of multivariation. These conceptual analyses are useful in order to scaffold future empirical work in creating a complete multivariational reasoning framework.

Presenters: Steven Jones and Mitchell Probst, Brigham Young University
Location: RUME Conference, San Diego, California
Abstract/Description:
This study was motivated by practical issues we have encountered as second-semester calculus instructors, where students struggle to make sense of the various series convergence tests, including the divergence, integral, direct comparison, limit comparison, ratio, and root tests. To begin an exploration of how students might reason about these tests, we examined the visual imagery used by students when asked to describe what these tests are and why they provide the conclusions they do. It appeared that each test had certain types of visual imagery associated with it, which were at times productive and at times a hindrance. We describe how the visual imagery used by students seemed to impact their reasoning about the convergence tests.

Presenters: Steven Jones and Omar Naranjo, Brigham Young University
Location: RUME Conference, San Diego, California
Abstract/Description:
Ordinary differential equations (ODEs) comprise an important tool for mathematical modelling in science and engineering. This study focuses on how students in an engineering system dynamics course organized the act of setting up ODEs for complex engineering contexts. Through the lens of ODEs as a “coordination class” concept, we examined the strategies that seemed to guide the students’ interpretations of problem tasks and their activation of knowledge elements during the tasks, as the students worked to produce ODEs for those tasks. This led to our uncovering of three main strategies guiding the students’ work, and the finding that being able to flexibly draw on all of these strategies may be beneficial for student success.

Presenters: Steven Jones, Brigham Young University and Peter Thembinkosi, Miami University of Ohio
Location: RUME Conference, San Diego, California
Abstract/Description:
Contributing to the growing body of research on students’ understanding of related rates of change problems, this study reports on the analysis of solution strategies used by five calculus students when solving three related rates of change problems where the underlying independent variable in each problem was time. Contrary to findings of previous research on students’ understanding of related rate of change problems, all the students in this study were able to translate prose to algebraic symbols. All the students had a common benchmark to guide their overall work in one of the tasks but no benchmark to guide their overall work in the other two tasks. Three students exhibited weaker calculational knowledge of the product rule of differentiation. Directions for future research and implications for instruction are included.

Presenters: Dawn Teuscher, Brigham Young University and J. Matt Switzer, Texas Christian University
Location: AMTE 2018 in Houston, Texas
Abstract/Description:
We share findings from an analysis of eight pre-service secondary mathematics teachers’ ability to notice student mathematical thinking while student teaching and discuss differences among student teachers who had varying degrees of exposure to analyzing video during their undergraduate program.

Presenters: Shari L. Stockero, Michigan Technological University; Laura R. Van Zoest, Western Michigan University; Blake E. Peterson and Keith R. Leatham, Brigham Young University; and Annick O. T. Rougee, University of Michigan
Location: PMENA Conference, Indianapolis, Indiana
Abstract/Description:
This study investigates teacher responses to a common set of high potential instances of student mathematical thinking to better understand the role of the teacher in shaping meaningful mathematical discourse in their classrooms. Teacher responses were coded using a scheme that disentangles the teacher move from other aspects of the teacher response, including who the response is directed to and the degree to which the student thinking is honored. Teachers tended to direct their response to the student who had shared their thinking and to explicitly incorporate ideas core to the student thinking in their response. We consider the nature of these responses in relation to principles of productive use of student mathematical thinking.

Presenters: Blake E. Peterson, Brigham Young University; Laura R. Van Zoest, Western Michigan University; Annick O.T. Rougee, University of Michigan; Ben Freeburn, Western Michigan University; Shari L. Stockero, Michigan Technological University; and Keith R. Leatham, Brigham Young University
Location: International Group for the Psychology of Mathematics Education Conference, Singapore
Abstract/Description:
To contribute to the field’s understanding of the teachers’ role in using student thinking to shape classroom mathematical discourse, we developed the Teacher Response Coding Scheme (TRC). The TRC provides a means to analyze teachers’ in-the-moment responses to student thinking during instruction. The TRC differs from existing schemes in that it disentangles the teacher move from the actor (the person publically asked to consider the student thinking), the recognition (the extent to which the student recognizes their idea in the teacher move), and the mathematics (the alignment of the mathematics in the teacher move to the mathematics in the student thinking). This disentanglement makes the TRC less value-laden and more useful across a broad range of settings.