Presentations

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.

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.

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."

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.