Past Presentations

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Classifying Curricular Reasoning: Ways for Capturing Teachers’ Curricular Decisions

Presenters: Shannon Dingman, University of Arkansas; Dawn Teuscher, Brigham Young University; Travis Olson, University of Nevada, Las Vegas; and Amy Roth-McDuffie, Washington State University
Location: PMENA in Reno
Abstract/Description:
Mathematics teachers make numerous decisions that form lessons that in turn greatly influence what students learn. In making these decisions, teachers rely on their curricular reasoning (CR) to decide on what mathematics to teach, how to structure their lesson, and what problems or tasks to use to achieve their lesson goals. However, teachers differ with respect to the sophistication of their CR and the diversity of CR aspects used in their reasoning. In this paper, we detail two ways to classify teachers’ CR: a leveled approach to capture the increasing sophistication of teachers’ CR, and a heat map approach that highlights the extent to which teacher use various CR aspects in their planning. These methods provide stakeholders avenues by which CR can be studied and that teachers’ CR abilities can be further developed.

Professional Noticing: The Interrelated Skills of Attending to and Interpreting Student Mathematical Thinking

Presenters: J. Matt Switzer, Texas Christian University and Dawn Teuscher, Brigham Young University
Location: PMENA in Reno
Abstract/Description:
We seek to extend the understanding and application of the interrelatedness of professional noticing (Jacobs et al., 2010) by identifying the student mathematical thinking to which (STs’) ability to attend to and interpret student mathematical thinking while student teaching and the ways they interpret the student mathematical thinking that was available to them. We report findings from STs’ individual professional noticing skills of attending to and interpreting students’ mathematical thinking. We then compare these findings to the combination of the two professional noticing skills (i.e. interrelated skills). In this poster, we answer the following research questions, How do STs’ individual skills of attending to and interpreting student mathematical thinking differ from their interrelated professional noticing?

Developing a Qualitative Data Analysis Process with a Multi-Research Team

Presenters: Tara Heikila, Washington State University; Nicole Stripling, University of Arkansas; Kate Webster and Dawn Teuscher, Brigham Young University; Amy Roth-McDuffie, Washington State University; and Shannon Dingman, University of Arkansas.
Location: PMENA in Reno
Abstract/Description:
This poster presents a multi-researcher team’s process of engaging in qualitative data analysis. Three subgroups, each including an experienced researcher and a graduate student, applied iterative approaches to code and identify data patterns regarding ways middle school mathematics teachers use curricular reasoning (CR) to engage learners. Teachers use CR as they design and enact instruction with their students, curriculum materials, and standards in mind. This poster will present ways each subgroup of researchers analyzed the following CR aspects: analyzing curricular materials, viewing mathematics from the learner perspective, and considering mathematical meaning. The poster will illustrate how we created space for dialogue about data analysis, wove seven researchers' perspectives together, and discussed different approaches to analyzing data. Our process has implications for other researchers as they consider data analysis approaches in their contexts, especially when analyzing complex data sets focused on teaching and learning.

Research Expectations for Mathematics Education Faculty in US Institutions of Higher Education

Presenters: Blake Peterson, Keith Leatham, and Steven Williams, Brigham Young University  
Location: PMENA in Reno
Abstract/Description:
This paper reports the results of a survey of 404 US mathematics education faculty regarding the research expectations for obtaining tenure. Survey questions asked about expected numbers of publications per year, how much different types of publications (e.g., journal articles, book chapters) and scholarly activities (e.g., giving presentations, obtaining funding) were valued. Statistical analyses were used to examine differences in these results across three demographic characteristics (institution type, research commitment, department). We found statistically significant differences related to each of these variables. Research expectations varied substantially across institution type. For example, the average expected number of yearly publications was 2.23, 1.63, and .99 papers at R1, R2, and Other institutions respectively. By contrast, research expectations seldom varied by department.

Students’ Structural Reasoning about Rational Expressions

Presenters: Dan Siebert, Brigham Young University
Location: PMENA in Reno
Abstract/Description:
Scholars suggest that students’ difficulties in making sense of and meaningfully manipulating algebraic expressions is due to their lack of structural reasoning. Research studies have documented that students seldom use expert structural reasoning but give little insight into the nature of students’ non-expert structural reasoning. Our study examines how six AP students identify structure, match structures to rules for manipulation, and evaluate their matches as they solve problems involving rational expressions. We found that students were engaged in structural reasoning throughout the hour-long interviews, and that successful solutions were characterized by students identifying structures by breaking expressions into smaller parts based on the highest level of operation (HLO), matching those structures to valid rules, and evaluating the correctness and progress made by the match they constructed.

How Students Reason about Compound Unit Structures: m/s2, ft-lbs, and (kg*m)/s

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.

Graphical Resources: Different Types of Knowledge Elements Used in Graphical Reasoning

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.

Theoretical Considerations for Designing and Implementing Intellectual Need-Provoking Tasks

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.

Using Rehearsal Debriefs with Experienced Teachers to Negotiate an Understanding of an Ambitious Teaching Practice

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.

Viewing Classroom Mathematics Discourse through Two Complementary Lenses

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.

An Introduction to Lesson Analysis

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.

Geometric Rotations and Angles: How are they Connected?

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.

Using Public Records to Support the Productive Use of Student Mathematical Thinking

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.

Conducting a Whole Class Discussion About an Instance of Student Mathematical Thinking

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.

Uses of the Equal Sign and Equation Types in Middle School Mathematics Textbooks

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.

Variable Types in Middle School Mathematics Curricula

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.

Meanings, Reasoning, and Modeling with Definite Integrals: Comparing Adding Up Pieces and Accumulation from Rate

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.

Combining Sealey, Von Korff & Rebello, Jones, and Swidan & Yerushalmy into a Comprehensive Decomposition of the “Integral with Bounds” Concept

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.

A Framework for Designing Intellectual Need-Provoking Tasks

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.

A Learning Trajectory Based on Adding Up Pieces for an Entire Unit on Integration

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.

Using Public Records to Support Class Discussion

Presenters: Blake Peterson and Keith Leatham, Brigham Young University, Shari Stockero, Laura Van Zoest, Christina Koehne, Eva Thanheiser, Kate Melhuish, Bill Deleeuw, Samuel Otten, Ruveyda Karaman Dundar, Michael Hicks, and Jessica Bishop
Location: AMTE 2022 in Las Vegas, Nevada
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.

Improving the Practice of Secondary Clinical Practice: Collaborating for Support and Alignment

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.

“We Are All Children of God”: White Christian Teachers Discussing Racism

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.

Structural Conventions for Equations in Middle School Mathematics Textbooks

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.

Conventions and Context: Graphing Related Objects Onto the Same Set of Axes

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.

Identifying Graphical Forms Used by Students in Creating and Interpreting Graphs

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.

Establishing Student Mathematical Thinking as an Object of Class Discussion

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.

Teachers’ Referencing of Public Records of Student Mathematical Thinking

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.

Students’ Reasoning About Multivariational Structures

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.

Students’ “Multi-sample Distribution” Misconception about Sampling Distributions

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.

Articulating the Student Mathematics in Student Contributions

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.

Improving Secondary Preservice Mathematics Teachers’ Attention to Student Mathematical Thinking

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.

Examining Teachers’ Reasoning for Their Instructional Decisions

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.

Establishing Student Mathematical Thinking as an Object of Class Discussion

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.

Using a Public Record to Anchor Whole-Class Mathematical Discussion

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.

A Comprehensive Hypothetical Learning Trajectory for the Chain Rule, Implicit Differentiation, and Related Rates: Part I, the Development of the HLT

Presenters: Steven Jones and Haley Jeppson, Brigham Young University
Location: RUME 2020, Boston, MA
Abstract/Description:
Students can learn more deeply when conceptual understanding is at the forefront and connections are made between topics. We hypothesize that such understanding and connections can be achieved for the chain rule, implicit differentiation, and related rates through the construct of nested multivariation (NM). In this first paper, we describe the process of creating a hypothetical learning trajectory (HLT) rooted in NM for this sequence of topics. This theoretical paper contains our conceptual analysis, literature review, and construction of the HLT.

A Comprehensive Hypothetical Learning Trajectory for the Chain Rule, Implicit Differentiation, and Related Rates: Part II, a Small-Scale Teaching Experience

Presenters: Steven Jones and Haley Jeppson, Brigham Young University
Location: RUME 2020, Boston, MA
Abstract/Description:
Students learn more deeply when conceptual understanding is at the forefront and connections are made between topics. While previous work has examined the chain rule, implicit differentiation, and related rates separately, we have created a hypothetical learning trajectory (HLT) for these topics to teach them in a conceptual, connected way. In a previous paper we outlined the creation of the HLT based on the construct of nested multivariation (NM). In this second paper, we describe a small-scale teaching experiment done to test the HLT. Our results suggest NM was an appropriate construct to base the HLT on, and we present the students’ developing understandings as they progressed through the HLT. Based on the results, we made final adjustments to the HLT, in preparation for a full-scale classroom teaching experiment.

Undergraduate Students’ Perspectives on What Makes Problem Contexts Engaging

Presenters: Steven Jones and Tamara Stark, Brigham Young University
Location: RUME 2020, Boston, MA
Abstract/Description:
Using engaging problem contexts is important in instruction, and the literature contains themes of contexts being realistic, worthwhile, or enjoyable, as well as motivating. Yet, the literature largely lacks detailed student perspectives on what helps problem contexts achieve these characteristics. In this study, eleven calculus students were interviewed to identify features of problems that made them engaging. This led to a new top-level characteristic “variety,” and the identification of features that helped contexts have the characteristics described in the literature. In particular, problems that were realistic/motivating contained features including: (a) expansion of awareness, (b) need for math, and/or (c) explicit purpose. Contexts that were enjoyable/motivating contained features including: (a) insertion into problem, (b) teacher’s personal story, or (c) absurd story. At the end, we show the usefulness of these results by critiquing problems from the literature in terms of how engaging they might be to students.

A Theorization of Learning Environments to Support the Design of Intellectual Need-Provoking Tasks in Introductory Calculus

Presenters: Aaron Weinberg, Ithaca College and Steven Jones, Brigham Young University
Location: RUME 2020, Boston, MA
Abstract/Description:
Intellectual need is the need that students feel to understand how and why a particular mathematical idea came to be. We are interested in creating tasks that calculus instructors can use to provoke intellectual need. However, the current suggestions for designing such tasks lack detail and don’t account for several issues specific to undergraduate introductory calculus. In this theoretical paper, we discuss the idea of intellectual need, explore three issues related to the teaching of calculus, and present a theoretical model that task-designers can use to frame important factors that affect the development and use of these tasks.

Exploring the Knowledge Base for College Mathematics Teaching

Presenters: Douglas Corey, Linlea West, and Kamalani Kaluhiokalani, Brigham Young University
Location: RUME 2020, Boston, MA
Abstract/Description:
We explore a wide sample of currently available instructional materials intended for college mathematics instructors (textbooks, magazines, teacher editions, lesson plans, teaching articles, classroom notes for flipped classrooms, books, etc.) in order to assess how available materials are building a knowledge base for teaching. We modify a framework from Hiebert & Morris (2009) to look for key categories of knowledge that are fundamental for a knowledge base for teaching mathematics. We found that few articles contained meaningful amounts of multiple categories. We use the categories to describe the nature of current available materials and argue that a new genre of instructional material and scholarly work to create the missing knowledge is needed.

Influences of Curriculum on College Students’ Understanding and Reasoning about Limits

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.

Understanding and Developing Skills Needed to Build on Student Mathematical Thinking

Presenters: Keith R. Leatham, Brigham Young University; Shari L. Stockero, Michigan Technological University ; Blake E. Peterson, Brigham Young University; and Laura R. Van Zoest, Western Michigan University
Location: AMTE 2020, Phoenix, Arizona
Abstract/Description:
We will discuss the teaching practice of building on student mathematical thinking, unpacking important nuances of this practice. Together we will consider how we as mathematics teacher educators can help teachers to develop skills related to these nuances.

Humanizing Approaches to Grading with Mathematics Pre-Service Teachers: Navigating and Pushing Beyond Systems

Presenters: Mary Raygoza, ; Alyson Lischka, ; Amy Tanner, Brigham Young University; Lorraine Males, ; Jamalee Stone, ; Frances Harper, ; Patrick Sullivan, ; and Marrielle Myers
Location: AMTE 2020, Phoenix, Arizona
Abstract/Description:
This session highlights examples of praxis that challenge traditional grading practices. Drawing on collective insights of participating MTEs, we will identify next steps in our praxis of humanizing grading, brainstorm strategies for systemic change, and develop a shared resource.

Rethinking the Student Teacher Experience: Engaging in a Dialogue about Paired-Placement Student Teaching

Presenters: Kelly Edenfield, University of Georgia and Sharon Christensen, Brigham Young University
Location: AMTE 2020, Phoenix, Arizona
Abstract/Description:
Join us in a discussion around the purposes of student teaching, the design of existing paired-placement student teaching experiences, and how such experiences can be designed to better aligned with current purposes of student teaching.

Geometric Transformations: Alignment of Teachers’ Mathematical Goals to Research-based Learning Trajectories

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.

Using Curriculum to Move Teachers’ Curricular Reasoning from Sequencing to Learning Trajectories

Presenters: Dawn Teuscher, Brigham Young University; Shannon Dingman, University of Arkansas; and Travis Olson, University of Nevada Las Vegas
Location: AMTE 2020, Phoenix, Arizona
Abstract/Description:
Teachers are using more online materials and often modify their existing textbook sequences as they plan and enact lessons. These decisions impact the scope and sequence of the mathematics for students. We will discuss teachers’ reasoning for these decisions.

Geometric Transformations and DESMOS: Reflections from a Study on Curricular Reasoning

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.

An Elaboration of Four Subpractices of the Teaching Practice of Building on Student Thinking

Presenters: Alicia Heninger, Brigham Young University
Location: PMENA in St Louis, Missouri
Abstract/Description:
The teaching practice of building has been conceptualized as a productive way to take advantage of student contributions during whole-class instruction that provide leverage for supporting student learning and accomplishing mathematical goals. In this theoretical paper that is informed by empirical data, we elaborate on four subpractices of building: make precise, grapple toss, orchestrate, and make explicit. We provide illustrations from our efforts to make sense of building through work with teacher-researchers who are enacting the building practice in their classrooms and explore different aspects of the subpractices and relationships between these aspects and principles underlying productive use of student thinking.

Dissecting Curricular Reasoning: An Examination of Middle Grade Teachers’ Reasoning Behind Their Instructional Decisions

Presenters: Shannon Dingman, University of Arkansas; Dawn Teuscher, Brigham Young University; Lisa Kasmer, Grand Valley State University; and Travis Olson, University of Nevada Las Vegas
Location: PMENA in St Louis, Missouri
Abstract/Description:
Mathematics teachers are vital components in determining what mathematics students have the opportunity to learn. There are a vast number of factors and reasons that influence a teacher’s instructional decisions. As such, teachers rely heavily on their curricular reasoning (CR) to make decisions about what content to teach, how that content is taught, and the tasks to use to facilitate student learning. In this paper, we outline five strands of CR gleaned from research with middle grades mathematics teachers as they plan and implement instruction with unfamiliar curricular resources. These strands lay the foundation for our Instructional Pyramid model of CR and provide a lens through which teacher decision-making can be further understood and enhanced.

Secondary Pre-Service Teachers’ Images and Interpretations of Student Mathematical Thinking

Presenters: Dawn Teuscher, Brigham Young University and J Matt Switzer, Texas Christian University
Location: PMENA in St Louis, Missouri
Abstract/Description:
Teachers utilizing student mathematical thinking is important when teaching, yet many inservice teachers find it difficult to implement. The Standards for Preparing Teachers of Mathematics (AMTE, 2017) outline the knowledge, skills, and dispositions that beginning teachers should have after graduating including the importance of attending to and interpreting student mathematical thinking. In this paper, we present results from two focused video analysis assignments that our pre-service teachers engaged in to identify their images of student mathematical thinking and their ability to attend to and interpret student mathematical thinking.

Dissecting Curricular Reasoning: Middle Grades Teachers’ Reasoning for their Decisions

Presenters: Dawn Teuscher, Brigham Young University and Porter Nielsen, Brigham Young University
Location: SSMA 2019 Convention, Salt Lake City, Utah
Abstract/Description:
Curricular reasoning (CR) is the thinking processes that teachers use to make decisions (e.g., what content to teach, the tasks to facilitate student learning). In this presentation, we outline five CR aspects gleaned from research with middle grades mathematics teachers as they planned and implemented instruction with unfamiliar curriculum.

Real-world Modeling Problems in School Mathematics

Presenters: Douglas Corey, Brigham Young University
Location: SSMA Conference, Salt Lake City, Utah
Abstract/Description:
In this session we will work on a real-world task that is appropriate for students from upper middle school to college: predicting when a ceiling fan will stop after watching it slow down for 30 seconds. We will discuss the nature and use of modeling tasks in school.

Reflections, Translations, and Rotations – Is There Any Connections Among These 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.

What You Wish You Could Get From Other Teachers That Would Help Improve Your Teaching

Presenters: Douglas Corey, Brigham Young University
Location: NCTM Regional Salt Lake City, Utah
Abstract/Description:
An analysis of Japanese and U.S. lesson plans show that some are good at helping teachers improve their teaching. The most prominent feature is the use of student mathematical thinking. It is a lot easier to make effective instructional choices when you know how students will respond. Come find out how to write what teachers could really use.

Do You Use the Math You Teach? How to Find Problems That Show the Power of Mathematics in Real Life

Presenters: Douglas Corey, Brigham Young University
Location: NCTM Regional Salt Lake City, Utah
Abstract/Description:
My best problems come from situations where I have actually used math to solve a real problem in my life, from 3D printing loaded dice to wondering about tie-dyed t-shirts. We will work on some of these problems that I use to motivate and apply math, then talk about how to find such problems. Many of these have made great STEM fair projects.

Talking Math with Kids

Presenters: Amy Tanner, Brigham Young University
Location: NCTM Regional Salt Lake City, Utah
Abstract/Description:
This presentation will explore how to engage the children in our life in meaningful mathematical conversations outside the classroom in order to encourage mathematical curiosity and positive dispositions, and how to provide advice and resources for the parents of our students who wish to do the same.

To Pursue or Not to Pursue: Making Decisions about Student Mathematical Thinking

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.

Potential Intellectual Needs for Taylor and Power Series within Textbooks, and Ideas for Improving Them

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.

Variational Reasoning Used by Student While Discussing Differential Equations

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.

Influences from Pathways College Algebra on Students’ Initial Understanding and Reasoning about Calculus Limits

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.

Examining Which Student Thinking is Considered in Responsive Teaching

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.

Secondary Student Teachers’ Ability to Respond to Student Mathematical Thinking

Presenters: Dawn Teuscher, Brigham Young University and J. Matt Switzer, TCU
Location: AMTE 2019, Orlando, Florida
Abstract/Description:
We share findings from an analysis of eight preservice secondary mathematics teachers’ noticing of student mathematical thinking while student teaching. We focus on how they responded to student mathematical thinking and discuss differences among student teachers.

The Role of Curricular Reasoning in Middle Grades Mathematics Teachers’ Instructional Practice

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.

Taking Trig to Task

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.

A Characterization of Student Mathematical Thinking that Emerges During Whole-Class Instruction: An Exploratory Study

Presenters: Laura R. Van Zoest, Western Michigan University; Keith R. Leatham, Brigham Young University; Okan Arslan, Mehmet Akif Ersoy University; Mary A. Ochieng, Western Michigan University; Joshua M. Ruk, Western Michigan University; Blake E. Peterson, Brigham Young University; and Shari L. Stockero, Michigan Technological University
Location: PMENA Conference in Greenville, South Carolina
Abstract/Description:
This exploratory study investigated 164 instances of student mathematical thinking that emerged during whole-class instruction in a high-school geometry course. The MOST Analytic Framework provided a way to categorize these instances according to their Building Potential—that is, the potential for learning to occur if the student thinking of the instance were made the object of consideration by the class. The variations in the building potential of student thinking revealed in the study highlight the complexity of teaching, and the need to support teachers in identifying and appropriately responding to instances with different levels of Building Potential.

Teachers’ Responses to Instances of Student Mathematical Thinking with Varied Potential to Support Student Learning

Presenters: Shari L. Stockero, Michigan Technological University; Ben Freeburn, Western Michigan University; Laura R. Van Zoest, Western Michigan University; Blake E. Peterson, Brigham Young University; and Keith R. Leatham, Brigham Young University
Location: PMENA Conference in Greenville, South Carolina
Abstract/Description:
We investigated teachers’ responses to a common set of varied-potential instances of student mathematical thinking to better understand how a teacher can shape meaningful mathematical discourse. Teacher responses were coded using a scheme that both disentangles and coordinates the teacher move, who it is directed to, and the degree to which student thinking is honored. Teachers tended to direct responses to the same student, use a limited number of moves, and explicitly incorporate students’ thinking. We consider the productivity of teacher responses in relation to frameworks related to the productive use of student mathematical thinking.

Covariation Graphing Practices: The Change Triangle

Presenters: Daniel Siebert and Konda Luckau, Brigham Young University  
Location: PMENA Conference in Greensville, South Carolina
Abstract/Description:
Using a sociocultural lens to study graphing, we investigate the graphing practices of an experienced function-based algebra teacher to see how she uses the change triangle to support students reasoning about covariation and rates of change. We describe the elements of a change triangle and the ways the teacher attends to and reasons with these elements and multiple copies of the change triangle to enact a variety of practices as she completes common tasks related to functions and their graphs.

Cultivating Coherence and Connections on a Foundation of Conceptual Understanding and Students’ Funds of Knowledge

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.

Mira, Mira on the Wall – How Should I Teach Geometric Transformations?

Presenters: Josie Bastian, Brigham Young University
Location: UCTM in Draper, Utah
Abstract/Description:
This presentation addresses the relationship among curriculum, teachers’ interpretations, and student understanding of geometric transformations. We present results from eight teachers in four states across the US on how teachers’ interpretations of grade 8 geometry standards regarding transformations influence students’ understanding. We will share practices and strategies that we have identified as helpful for students to understand rotations, reflections, and translations.

Beginning of a Teaching Career: What We Know About Preservice Secondary Mathematics Teachers’ Curricular Reasoning

Presenters: Kimber Mathis, Brigham Young University
Location: UCTM in Draper, Utah
Abstract/Description:
Researchers have found that a teacher’s decisions affect opportunities students have to learn (Stein, Remillard, & Smith, 2007). Teachers make mathematical decisions as they plan, develop and enact lessons for students, and the reasoning about these decisions is referred to as curricular reasoning (Roth McDuffie & Mather, 2009). Empirical data on preservice teachers’ curricular reasoning will be presented. Data on how these preservice teachers' goals, resources, and orientations influenced their curricular reasoning will also be presented. Implications for educating and training new teachers will be discussed.

How Positioning Affects Student Learning in an Inquiry-based Classroom

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."

The Structure of Conceptually-Oriented Mathematics Explanations

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.

CPR for the Common Core: Using the Comprehensive Mathematics Instruction (CMI) Framework to Unpack Standards Across a Learning Cycle

Presenters: Scott Hendrickson and Sterling Hilton, Brigham Young University
Location: NCSM Conference, Washington DC
Abstract/Description:
The Comprehensive Mathematics Instruction (CMI) Framework developed by the Brigham Young University Public School Partnership informs teachers on how to align CCSSM content standards along a progression from emerging ideas, strategies and representations towards more robust conceptual, procedural, and representational understanding. In this session participants will use the CMI framework to deepen their understanding of a subset of high school standards as they select, sequence and connect these standards across a learning cycle of instruction.

Beyond the “Move”: A Scheme for Coding Teachers’ Responses to Student Mathematical Thinking

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.

Building on Covariation: Making Explicit Four Types of “Multivariation”

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.

Students’ Usage of Visual Imagery to Reason about the Divergence, Integral, Direct Comparison, Limit Comparison, Ratio, and Root Convergence Tests

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.

Students’ Strategies for Setting up Differential Equations in Engineering Contexts

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.

A Study of Calculus Students’ Solution Strategies when Solving Related Rates of Change Problems

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.

Teachers’ Orientations around Using Student Mathematical Thinking as a Resource during Whole-class discussion

Presenters: Keith Leatham, Brigham Young University; Blake Peterson, Brigham Young University; Shari Stockero, Michigan Technological University; Mary Ochieng, Western Michigan University; and Laura Van Zoest, Western Michigan University.
Location: AMTE 2018 Conference in Houston, Texas
Abstract/Description:
We characterize teachers’ orientations related to using student mathematical thinking as a resource during whole-class discussion. We consider the potential these orientations provide to either support or hinder the development of the practice of building on student mathematical thinking.

How Does Video Analysis Influcence Pre-Service Teachers’ Ability to Notice Student Mathematical Thinking While Teaching?

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.

Teachers’ Responses to a Common Set of High Potential Instances of Student Mathematical Thinking

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.

Beyond the “Move”: A Scheme for Coding Teachers’ Responses to 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.

Taking Trig to Task

Presenters: Scott Hendrickson, Brigham Young University
Location: NCTM 2017 Conference, San Antonio, Texas
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 angels in radians, can generate roadblocks and misconceptions.  In this session we will examine a sequence of tasks that reveal, rather than obscure, trigonometric ideas.

Using Technology to Engage in Whole-Class Mathematical Inquiry

Presenters: Keith R. Leatham, Brigham Young University
Location: NCTM 2017 Conference, San Antonio, Texas
Abstract/Description:
Together we will explore strategies for using a variety of technologies to facilitate whole-class mathematics discussions-discussions in which students are motivated and positioned to engage in making sense of mathematics. Bring your laptop, tablet, calculator, smartphone, or just yourself and join in the fun.

What Japanese Lesson Plans Teach us About Sharing Knowledge of Teaching?

Presenters: Doug Corey, Eula Monroe and Michelle Wagner, Brigham Young University
Location: NCTM 2017 Research Conference, San Antonio, Texas
Abstract/Description:
US mathematics education has failed to find a robust way to develop and store a knowledge base for teaching. We explore the use of detailed lesson plans as a solution to the storage problem for a knowledge base for teaching. We gather lesson plans and lesson-plan like documents from seven different sources (2 in Japan, 5 in the US) and analyze them to see which ones tend to best capture the key elements of high quality lessons and, moreover, makes the reasoning behind the instructional decisions explicit. We found that Japanese lesson study lesson plans tended to be the best examples of a knowledge base for teaching, although activity articles from Mathematics Teaching in the Middle School and Teaching Children Mathematics also did very well on a few dimensions and fairly well overall. Lessons from the Chicago School Lesson Study Group also scored high. One feature that was common among the better example lessons plans was that they tied together three elements: (1) specific instructional decisions based on (2) student mathematical thinking around a (3) a particular mathematical topic or idea. The good examples integrated these three things differently, and some specific examples were shared about how these were integrated into the lesson plans or lesson-plan like documents.

A Framework for Thinking Through a Unit: Implications for Task, Instructional Practices and Student Outcomes

Presenters: Scott Hendrickson and Sterling Hilton, Brigham Young University
Location: NCSM Annual Conference, San Antonio, Texas
Abstract/Description:
The Comprehensive Mathematics Instruction Framework developed by the BYU Public School Partnership informs teachers in making decisions regarding the selection and sequencing of tasks, in implementing instructional practices that intentionally align with the nature and purpose of tasks (e.g., level of cognitive demand), and in assessing expected student outcomes. Classroom video and student work will be used to illustrate the Framework.

How Does Focused Video Analysis in Methods Courses Impact Student Teachers’ Attending to Student Thinking?

Presenters: Dawn Teuscher, Brigham Young University and John Switzer, Texas Christian University
Location: AMTE 2017 Conference, Orlando, Florida
Abstract/Description:
We share results from our analysis of our preservice secondary mathematics teachers’ student teaching videos to demonstrate the impact of focused video analysis and discuss differences in the degree to which the student teachers were attentive to probing students’ thinking. 

Barriers to Building on Student Mathematical Thinking

Presenters: Shari L. Stockero, Michigan Technological University; Laura R. Van Zoest, Western Michigan University; Keith R. Leatham and Blake E. Peterson, Brigham Young University
Location: AMTE 2017 Conference, Orlando, Florida
Abstract/Description:
In our work with teachers we have identified barriers that inhibit them from productively implementing the teaching practice of building on student thinking.  We share examples of barriers and ways we have supported teachers to make progress toward overcoming them.

Conceptualizing the Teaching Practice of Building on Student Mathematical Thinking

Presenters: Laura R. Van Zoest, Western Michigan University; Blake E. Peterson and Keith R. Leatham, Brigham Young University; and Shari L. Stockero, Michigan Technological University
Location: PME-NA 2016 Conference, Tucson, Arizona
Abstract/Description:
An important aspect of effective teaching is taking advantage of in-the-moment expressions of student thinking that, by becoming the object of class discussion, can help students better understand important mathematical ideas.  We call these high-potential instances of student thinking MOSTs and the productive use of them building.  The purpose of this paper is to conceptualize the teaching practice of building on MOSTs as a first step toward developing a common language for and an understanding of productive use of high-potential instances of student thinking.  We situate this work with the existing literature, introduce core principles that underlie our conception of building, and present a prototype of the teaching practice of building on MOSTs that include four sub-practices.  We conclude by discussing the need for future research and our research agenda for studying the building prototype.

What Does it Mean to “Understand” Concavity and Inflection Points?

Presenters: Steven Jones, Brigham Young University
Location: PME-NA 2016 Conference, Tucson, Arizona
Abstract/Description:
The calculus concepts of concavity and inflection points are often given meaning through the shape or curvature of a graph.  However, there appear to be deeper core ideas for these two concepts, though the research literature has yet to give explicit attention to what there core ideas might be or what it might mean to “understand” them.  In this paper, I propose a framework for the concavity and inflection point concepts, using the construct of covariation, wherein I propose conceptual (as opposed to mathematical) definitions that can be used for both research and instruction.  I demonstrate that the proposed conceptual definitions in this framework contain important implications for the teaching and learning of these concepts, and that they provide more powerful insight into student difficulties than more traditional graphical interpretations.

Isometries in New US Middle Grades Textbooks: How are Isometries and Congruence Related?

Presenters: Dawn Teuscher, Brigham Young University; Lisa Kasmer, Grand Valley State University; Travis Olson, University of Nevada-Las Vegas; and Shannon Dingman, University of Arkansas
Location: ICME 13 Conference, Hamburg, Germany
Abstract/Description:
In this session we present findings from our analysis of six middle school textbooks purported to align to the Common Core State Standards for Mathematics (CCSSM). We specifically report on the approach and connection of isometries and congruence in grade 8. We found the majority of the curriculum materials to be lacking in three important mathematical ideas related to isometries: properties of isometries, congruence in terms of isometries, and orientation of figures. This lack of connections will impact teachers as they implement the CCSSM and students as their opportunities to learn isometries as outlined in CCSSM will vary depending on their teachers’ understanding of isometries and congruence as well as the textbook that they are using.

The Structure of Student Teaching Can Change the Focus to Students’ Mathematical Thinking

Presenters: Blake E. Peterson and Keith R. Leatham, Brigham Young University
Location: ICME 13 Conference, Hamburg, Germany
Abstract/Description:
This presentation describes our efforts to change the focus of our student teaching experience by altering the structure of that experience. We provide evidence that the restructuring accomplished its purposes. In particular, we achieved less focus on issues of classroom management and student behavior, more focus on students’ mathematics, and substantial opportunity to grapple with the elicitation, interpretation and use of student mathematical thinking during class discussion. Although there is still room for improvement, our experience provides an existence proof that the focus of the student teaching experience can indeed be altered and improved.

How Are New Textbooks Aligned to CCSSM – Geometry Through Transformations

Presenters: Lisa Kasmer, Grand Valley State University; Shannon Dingman, University of Arkansas; Travis Olson, University of Nevada-Las Vegas; and Dawn Teuscher, Brigham Young University
Location: NCTM Conference, San Francisco, California
Abstract/Description:
In this session, we share results of our work that examined how new middle grades textbooks are organizing and presenting transformational geometry concepts aligned to CCSSM. We explore what happens when there is a mismatch and how to identify a mismatch between the mathematical content presented in the books and what CCSSM teachers are held accountable to teach.

I’ve got my Students Sharing Their Mathematical Thinking – Now What?

Presenters: Shari L. Stockero, Michigan Technological University; Laura R. Van Zoest, Western Michigan University; and Keith R. Leatham, Brigham Young University
Location: NCTM Conference, San Francisco, California
Abstract/Description:
Once students share their ideas, creating meaningful mathematics discourse requires that teachers decide which ideas are worth pursuing and how to capitalize on those ideas. We share a framework for determining which student ideas have significant potential to support mathematics learning and discuss how teachers might productively use those ideas.

How We Can “Attend to Precision” in Classroom Mathematics Discussions

Presenters: Keith R. Leatham, Blake E. Peterson, and Lindsay Merrill, Brigham Young University
Location: NCTM Conference, San Francisco, California
Abstract/Description:
Explore examples of teacher and student imprecision in classroom mathematics discourse. Discuss types of imprecision that occur in classrooms, the ramifications of this imprecision, and strategies for addressing that imprecision. Learn how to minimize your own imprecision and to view student imprecision as an opportunity to learn mathematics.

Why and How to let Students Struggle? Thoughts from Research

Presenters: Blake E. Peterson, Brigham Young University
Location: NCTM Conference, San Francisco, California
Abstract/Description:
Principles to Action endorses “Supporting Productive Struggle in Learning Mathematics.” With a common societal belief that student struggle indicates poor teaching, allowing and supporting student struggle seems foreign. We will discuss research on the benefits of this practice and some suggestions to effectively support student productive struggle.

A Framework for Building Conceptual Fluency on a Foundation of Conceptual Understanding

Presenters: Scott Hendrickson and Sterling Hilton, Brigham Young University
Location: NCSM Conference, Oakland, California
Abstract/Description:
The Comprehensive Mathematics Instruction Framework, developed by the Brigham Young University Public School Partnership, highlights the relationship between conceptual, procedural and representational understanding. The three components of the framework: Teaching Cycle, Learning Cycle and Continuum of Understanding will be described and illustrated.

Productive Use of Student Mathematical Thinking is More than a Single Move

Presenters: Blake E. Peterson, Brigham Young University; Laura R. Van Zoest, Western Michigan University; Shari L. Stockero, Michigan Technological University; Keith R. Leatham, Brigham Young University
Location: AMTE Conference, Irvine, California
Abstract/Description:
We will introduce the teaching practice of building and its constituent components as the most productive use of worthwhile student mathematical thinking, analyze teaching examples for evidence of building, and consider how to support teachers’ development of this practice.

Influence of Focused Video Analysis on Preservice Secondary Mathematics Teachers’ Noticing of Student Mathematical Thinking

Presenters: Dawn Teuscher, Keith R. Leatham, Blake E. Peterson, and Allyson Derocher, Brigham Young University
Location: AMTE Conference, Irvine, California
Abstract/Description:
We discuss evidence that preservice secondary mathematics teachers who participated in focused video analysis, watching, analyzing and discussing videos through the lens of a specific theoretical framework, are able to transfer their noticing into the real-time classroom.

Learning to Teach Through Video Analysis: Preservice Teachers Learning and Engaging in Participation Questioning Discourse

Presenters: J. Matt Switzer, Texas Christian University; Dawn Teuscher and Kylie Palsky, Brigham Young Univeristy
Location: AMTE Conference, Irvine, California
Abstract/Description:
We share video learning activities that support preservice secondary mathematics teachers’ implementation of participation questioning discourse that consists of (a) modeling and engaging students in mathematical discourse and activity, and (b) supporting and assessing students’ development of conceptual understanding.

Exploring Racial Consciousness and Faculty Behavior in STEM Classrooms

Presenters: Nicole M. Joseph, University of Denver; Joi Spencer, University of San Diego; Kate R. Johnson, Brigham Young University; and Richard Kitchen, University of Denver
Location: AMTE Conference, Irvine, California
Abstract/Description:
Exploring racial consciousness’ influence on faculty behavior, White and faculty of color share narratives that reveal how they hold one another, and themselves, accountable for racial equity in mathematics.

Facing Resistance in the Preparation of Critical Mathematics Teachers

Presenters: Kate R. Johnson, Brigham Young University and Alisa Belliston, University of Wisconsin-Madison
Location: AMTE Conference, Irvine, California
Abstract/Description:
When preparing critical mathematics teachers, mathematics teacher educators may face resistance. We highlight two cases to illustrate the natures of possible resistance and provide tools for illuminating the invisible beliefs and assumptions that disrupt opportunities to learn about critical pedagogies.

Attributes of Student Mathematical Thinking that is Worth Building on in Whole-Class Discussion

Presenters: Laura R. Van Zoest, Western Michigan University; Shari L. Stockero, Michigan Technological University-Houghton; Napthalin A. Atanga, Western Michigan University; Blake E. Peterson and Keith R. Leatham, Brigham Young University; and Mary A. Ochieng, Western Michigan University
Location: PMENA Conference, East Lansing, Michigan
Abstract/Description:
This study investigated the attributes of 297 instances of student mathematical thinking during whole-class interactions that were identified as having the potential to foster learners’ understanding of important mathematical ideas (MOSTs). Attributes included the form of the thinking (e.g., question vs. declarative statement), whether the thinking was based on earlier work or generated in-the-moment, the accuracy of the thinking, and the type of the thinking (e.g., sense making). Findings both illuminate the complexity of identifying student thinking work building on during whole-class discussion and provide insight into important attributes of MOSTs that teachers can use to better recognize them.

Uncovering Teachers’ Goals, Orientations, and Resources Related to the Practice of Using Student Thinking

Presenters: Shari L. Stockero, Michigan Technological University-Houghton; Laura R. Van Zoest, Western Michigan University; Annick Rougee, University of Michigan; Elizabeth H. Fraser, Western Michigan University; Keith R. Leatham and Blake E. Peterson, Brigham Young University
Location: PMENA Conference, East Lansing, Michigan
Abstract/Description:
Improving teachers’ practice of using student mathematical thinking requires an understanding of why teachers respond to student thinking as they do; that is, an understanding of the goals, orientations and resources (Schoenfeld, 2011) that underlie their enactment of this practice. we describe a scenario-based interview tool developed to prompt teachers to discuss their decisions and rationales related to using student thinking. We examine cases of two individual teachers to illustrate how the tool contributes to (1) inferring individual teachers’ goals, orientations and resources and (2) differentiating among teachers’ uses of student thinking.

Intellectually Engaging Problems: The Heart of a Good Lesson

Presenters: Blake E. Peterson, Brigham Young University
Location: NCTM Conference, Boston, Massachusetts
Abstract/Description:
A common characteristic of good lessons worldwide is that students are intellectually engaged in solving and reasoning through rich mathematical problems. I will share several problems that I have seen during observations in Japan and have subsequently used in the U.S. I will also discuss some features I have found common among these rich problems.

Preliminary Steps Toward Developing a Theory of Productive Use of Student Mathematical Thinking

Presenters: Laura R. Van Zoest, Western Michigan University; Shari L. Stockero, Michigan Technological University-Houghton; Blake E. Peterson and Keith R. Leatham, Brigham Young University; Napthalin Atanga, Western Michigan University; Lindsay Merrill, Brigham Young University, and Mary Ochieng, Western Michigan University
Location: NCTM Conference, Boston, Massachusetts
Abstract/Description:
Presentations consider (1) the nature of student thinking (ST) available to teachers during instruction, (2) teachers’ perceptions of productive use of ST, and (3) teachers’ abilities to recognize and respond to ST. The work will be discussed in the broader context of developing a theory of productive use of ST.

Shifting Opportunities to Teach and Learn in Common Core “Aligned” Textbooks: Implications for Depth and Equity

Presenters: Dawn Teuscher, Brigham Young University
Location: NCSM Conference, Boston, Massachusetts
Abstract/Description:
Analyses of new middle grades textbooks across Ratio and Proportion and Geometry domains of the Common Core will be reported. Data will be shared related to mathematical content, types of representations, and comparisons. We will discuss how access to mathematics based on curriculum use poses a potential equity gap in implementing the Common Core.

8 by 8, Connecting Teaching Practices and Student Mathematical Practices

Presenters: Scott Hendrickson, Brigham Young University and Dawn Barson, Alpine School District
Location: NCSM Conference, Boston, Massachusetts
Abstract/Description:
The Common Core State Standards describes eight Mathematical Practice Standards for students’ engagement in mathematical work. NCTM introduced eight Mathematics Teaching Practices in Principles to Actions. How are these sets of practices related? Using video vignettes we will examine how effective teaching elicits authentic mathematical work.

Adding Explanatory Power to Descriptive Power: Combining Zandieh’s Derivative Framework with Analogical Reasoning

Presenters: Steven Jones and Kevin Watson, Brigham Young University
Location: RUME Conference, Pittsburgh, Pennsylvania
Abstract/Description:
The derivative is an important foundational concept in calculus that has applications in many fields of study. Existing frameworks for student understanding of the derivative are largely descriptive in nature, and there is little by way of theoretical frameworks that can explain or predict student difficulties in working with the derivative concept. In this paper we combine Zandieh’s framework for understanding the derivative with “analogical reasoning” from psychology into the “merged derivative-analog framework.” This framework allows us to take the useful descriptive capabilities of Zandieh’s framework and add a layer of explanatory power for student difficulties in applying the derivative to novel situations.

Promoting Students’ Construction and Activation of the Multiplicatively-Based Summation Conception of the Definite Integral

Presenters: Steven Jones, Brigham Young University
Location: RUME Conference, Pittsburgh, Pennsylvania
Abstract/Description:
Prior research has shown how the multiplicatively-based summation conception (MBS) is important for making sense of definite integral expressions in science contexts. This study attempts to accomplish two goals. First, it describes introductory lessons on integration from two veteran calculus teachers as a way to possibly explain why so few students draw on the MBS conception when making sense of definite integrals. Second, it reports the results from a design experiment intended on promoting not only the construction of the MBS conception, but its priming for activation when students see and interpret definite integrals expressions.

Students’ Understanding of Concavity and Inflection Points in Real-World Contexts: Graphical, Symbolic, Verbal, and Physical Representations

Presenters: Steven Jones and Michael Gundlach, Brigham Young University
Location: RUME Conference, Pittsburgh, Pennsylvania
Abstract/Description:
Little research has been conducted into student understanding of concavity and inflection points. Much of what we know comes incidentally from studies looking at the calculus activity of sketching the graphs of functions. However, since concavity and inflection points can be useful in conveying information in disciplines like science, engineering, technology, and economics, it seems important to study how students understand these two concepts in these contexts. This study attempts to provide insight into this area.

Students’ Generalizations of Single-Variable Conceptions of the Definite Integral to Multivariate Conceptions

Presenters: Steven Jones, Brigham Young University; Allison Dorko and Eric Weber, Oregon State University
Location: RUME Conference, Pittsburgh, Pennsylvania
Abstract/Description:
Prior research has documented several conceptualizations students have regarding the definite integral, though the conceptualizations are largely based off of single-variable integral expressions. No research to date has documented how students’ understanding of integration becomes generalized for multivariate contexts. This paper describes six conceptualizations of multivariate definite integrals and how they connect to students’ prior conceptions of single-variable definite integrals.

How Do Japanese Teachers Critically Analyse a Lesson During Lesson Study?

Presenters: Doug Corey, Brigham Young University and Hiroyuki Ninomiya, Saitama University
Location: AMTE Conference, Orlando, Florida
Abstract/Description:
We analyzed video of three Japanese lesson study sessions connected to elementary or middle school math lessons. We use the discussion to better understand what Japanese teachers view as most important in a lesson and the frame which they use to view a lesson. We discuss how some ideas used by the Japanese could potentially be useful for US teachers and US professional developers.

Transformational Geometry in New Middle Grades Textbooks: What do Teachers Need to Know?

Presenters: Dawn Teuscher, Brigham Young University
Location: AMTE Conference, Orlando, Florida
Abstract/Description:
PSTs curricular reasoning is necessary to analyze curriculum and make decisions about planning, implementation, and reflecting. This session will provide participants an opportunity to examine textbooks and participate in a curriculum analysis activity that we have used with our PSTs.

Engaging Preservice Teachers’ in Probing Student Thinking Through the Video-based Model Seeing, Trying, Reflecting (STiR)

Presenters: J. Matt Switzer, Texas Christian University and Dawn Teuscher, Brigham Young University
Location: AMTE Conference, Orlando, Florida
Abstract/Description:
We will share the iterative video-based See it, Try it, and Reflect on it (STiR) model of making practice studyable was implemented in methods courses at two universities. We share our findings that the model promotes preservice teachers’ learning as they probe student thinking.

Seeing Through Your Student’s Eyes

Presenters: Blake E. Peterson, Brigham Young University
Location: AMTE Conference, Orlando, Florida
Abstract/Description:
Anticipating student mathematical thinking is broadly discussed as a valuable teaching practice. Specifically, it is emphasized as part of the lesson study process and is the first of the five practices discussed by Smith and Stein (2011). Learning to anticipate student thinking requires teachers to see mathematics through their students’ eyes. In my own teaching as well as in my work with preservice teachers, I have come to value seeing mathematics through students’ eyes as well as to recognize the challenges in doing so. In this talk, I will share some interesting ways students see mathematics and discuss the pedagogical benefits of looking at mathematics through their eyes.

Defining and Developing Teaching Practices Related to Responding to Students’ Mathematical Thinking

Presenters: Corey Webel, University of Missouri; William Deleeuw, University of Missouri; Susan Empson, University of Texas at Austin; Victoria Jacobs, University of North Carolina at Greensboro; Tonia Land, Drake University, Keith R. Leatham and Blake E. Peterson, Brigham Young University; Shari L. Stockero, Michigan Technological University-Houghton; and Laura Van Zoest, Western Michigan University
Location: AMTE Conference, Orlando, Florida
Abstract/Description:
This session builds on research on professional noticing of students’ mathematical thinking by unpacking different ways of conceptualizing the teaching practice of responding to student thinking. Four projects focused on defining and developing this practice will be presented and discussed.

Does Common Core Teaching Lead to Improved Student Learning?

Presenters: Johanna Barmore, Harvard; David Blazar, Harvard, Charalambos Y. Charalambous, University of Cyprus; Doug Corey, Brigham Young University; Heather C. Hill, Harvard; Andrea Humez, Boston College; and Erica Litke, Harvard
Location: International Conference on Education, Honolulu, Hawaii
Abstract/Description:
Policy-makers in the U.S. have asked teachers both to implement Common Core Standards and improve student achievement. While many assume that these goals work in concert, research suggests that links between teaching quality and student outcomes may be more tenuous. We explore whether implementation of new Common Core-aligned achievement tests might strengthen these relationships, focusing on a test considered a model for these assessments and an observational instrument aligned with the Common Core.