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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.

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.

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.

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.

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.

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.

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.