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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.

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.