Proceedings of the

20th Annual Conference on

Research in Undergraduate

Mathematics Education

Editors:

Aaron Weinberg

Chris Rasmussen

Je�rey Rabin

Megan Wawro

Stacy Brown

San Diego, California

February 23-25, 2017

Presented by

The Special Interest Group of the Mathematical Association of America

(SIGMAA) for Research in Undergraduate Mathematics Education

ii

Copyright @2017 left to authors. All rights reserved.

CITATION: In (Eds.) A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, and S. Brown, Proceedings of the

20th Annual Conference on Research in Undergraduate Mathematics Education, San Diego, California.

(ISSN 2474-9346).

iiii

Table of Contents

Best Paper Award

Leveraging real analysis to foster pedagogical practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Nicholas Wasserman, Keith Weber and William McGu�ey

Honorable Mention

Students’ epistemological frames and their interpretation of lectures in advanced mathematics . . . . . . 16

Victoria Krupnik, Keith Weber and Tim Fukawa-Connelly

Meritorious Citation

Set-based meanings for the truth of conditionals: A case study and a framework . . . . . . . . . . . . . . . . . . 31

Alec Hub and Paul Dawkins

Pre-journal Award

A new methodological approach for examining mathematical knowledge for teaching at the

undergraduate level: Utilizing task unfolding and cognitive demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Erica Miller

Conference Papers

Mathematical actions, mathematical objects, and mathematical induction . . . . . . . . . . . . . . . . . . . . . . . . 53

Rachel Arnold and Anderson Norton

Examining students’ procedural and conceptual understanding of eigenvectors and eigenvalues in

the context of inquiry-oriented instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Khalid Bouhjar, Christine Andrews-Larson, Muhammad Haidar and Michelle Zandieh

Students’ understanding of test statistics in hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Annie Childers, Darryl Chamberlain Jr., Aubrey Kemp, Leslie Meadows, Harrison Stalvey and

Draga Vidakovic

The role of visual reasoning in evaluating statements about real-valued functions: A comparison of

two advanced calculus students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Erika David, Kyeong Hah Roh and Morgan Sellers

Conceptual understanding of di�erential calculus: A comparative study . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Daria Gerasimova, Kathleen Matson, Robert Sachs and Margret Hjalmarson

Students ways of thinking in a traditional and inquiry-based linear algebra course . . . . . . . . . . . . . . . . . 116

Sarah Hough, Monica Mendoza and Elizabeth Thoren

Data cleaning in mathematics education research: The overlooked methodological step . . . . . . . . . . . . . 129

Aleata Hubbard

Mathematicians’ evaluations of the language of mathematical proof writing at the undergraduate

level in three di�erent pedagogical contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Kristen Lew and Juan Pablo Mejia-Ramos

xixi

Spatial training and calculus ability: Investigating impacts on student performance and cognitive style 156

Lindsay McCunn and Emily Cilli-Turner

Exploring student conceptions of binary operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Kathleen Melhuish and Joshua Fagan

Opportunities to learn from teaching: A case study of two graduate teaching assistants . . . . . . . . . . . . 181

Erica Miller, Nathan Wakefield and Yvonne Lai

Learning to notice and use student thinking in undergraduate mathematics courses . . . . . . . . . . . . . . . . 195

Anna Pascoe, Shari L. Stockero

Student conceptions of three-dimensional solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Stepan Paul and Monica Mendoza

Student mathematical connections in an introductory linear algebra course . . . . . . . . . . . . . . . . . . . . . . . 221

Spencer Payton

The e�ects of the epsilon-N relationship on convergence of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Zackery Reed

Hypophora: Why take the derivative? (no pause) Because it is the rate . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Kitty Roach

Students’ conceptions of mappings in abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Rachel Rupnow and Márcia M. F. Pinto

Emerging insights from the evolving framework of structural abstraction . . . . . . . . . . . . . . . . . . . . . . . . . 274

Thorsten Scheiner

A comparison of calculus, transition-to-proof, and advanced calculus student quantifications in

complex mathematical statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Morgan Sellers, Kyeong Hah Roh and Erika David

Exploring undergraduates’ experience of the transition to proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

John Smith, Mariana Levin, Younggon Bae, V. Satyam and Kevin Voogt

Knowledge about student understanding of eigentheory: Information gained from multiple choice

extended assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Kevin Watson, Megan Wawro, Sarah Kerrigan and Michelle Zandieh

Meta-representational competence with linear algebra in quantum mechanics . . . . . . . . . . . . . . . . . . . . . 326

Megan Wawro, Kevin Watson and Warren Christensen

Corequisite remediation and math pathways in Oklahoma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

Matthew Wilson and Michael Oehrtman

Using intuitive examples from women of color to reveal nuances about basis . . . . . . . . . . . . . . . . . . . . . . 346

Aditya Adiredja and Michelle Zandieh

Exploration of the factors that support learning: Web-based activity and testing systems in

community college algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

Shandy Hauk and Bryan Matlen

“Explanatory” talk in mathematics research papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Juan Pablo Mejia-Ramos and Matthew Inglis

xiixii

How limit can be embodied and arithmetized: A critique of Lako� and Núñez . . . . . . . . . . . . . . . . . . . . 383

Timothy Boester

Alice slowly develops self-e�cacy with writing proof frameworks, but her initial progress and sense

of self-e�cacy evaporates when she encounters unfamiliar concepts: However, it eventually returns . . 398

Annie Selden, John Selden and Ahmed Benkhalti

Questioning assumptions about the measurability of subdomains of mathematical knowledge for

teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Heather Howell, Yvonne Lai and Heejoo Suh

Those who teach the teachers: Knowledge growth in teaching for mathematics teacher educators . . . . 428

Shandy Hauk, Billy Jackson and Jenq-Jong Tsay

Raising reasoning through revision: A case study of an inquiry-based college geometry course . . . . . . . 440

Janessa Beach and Rebecca Dibbs

Developing student understanding: The case of proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

Darryl Chamberlain Jr. and Draga Vidakovic

Contributed Reports

Using women of color’s intuitive examples to reveal nuances about basis . . . . . . . . . . . . . . . . . . . . . . . . . 469

Aditya Adiredja and Michelle Zandieh

Comparing expert and learner mathematical language: A corpus linguistics approach . . . . . . . . . . . . . . 478

Lara Alcock, Matthew Inglis, Kristen Lew, Juan Pablo Mejia-Ramos, Paolo Rago and Chris Sangwin

Mathematics instruction leadership in undergraduate departments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Naneh Apkarian and Chris Rasmussen

Using learning trajectories to structure teacher preparation in statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 494

Anna Bargagliotti and Celia Anderson Rosseau

Interaction, activities, & feedback: A taxonomy of GTA professional development . . . . . . . . . . . . . . . . . 502

Daniel Bragdon, Jess Ellis and Jessica Gehrtz

Di�cult dialogs about degenerate cases: A proof script study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

Stacy Brown

Stages of development for the concept of inverse in abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

John Paul Cook and Rosaura Uscanga

Characterizing the nature of introduction to proof courses: A survey of R1 and R2 institutions

across the US . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

Erika David and Dov Zazkis

The role of visual reasoning in evaluating complex mathematical statements: A comparison of two

advanced calculus students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

Erika David, Kyeong Hah Roh, Morgan Sellers and Kody D’Amours

Features of explanatory proofs: An exploratory study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

Eyob Demeke

Undergraduate students’ holistic comprehension of a proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

Eyob Demeke

xiiixiii

Generalising univalence from single to multivariable settings: The case of Kyle . . . . . . . . . . . . . . . . . . . . 562

Allison Dorko

SCNI: A robust technique to investigate small-group learning at college . . . . . . . . . . . . . . . . . . . . . . . . . . 570

Fady El Chidiac

Infinitesimals-based registers for reasoning with definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

Rob Ely

Students’ attitudes toward listing and subsequent behavior solving counting problems . . . . . . . . . . . . . 588

Sarah Erickson

Tinker Bell’s pixie dust: Exploring the role of di�erentiation in emergent shape thinking . . . . . . . . . . . 596

Kristin Frank

Gender and discipline specific di�erences in mathematical self-e�cacy of incoming students at a

large public university . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Ulrike Genschel and Xuan Hien Nguyen

Exploring a pre-service teacher’s conceptions of area and area units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

Sayonita Ghosh Hajra and Betsy McNeal

Contextualizing symbols in word problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

Sayonita Ghosh Hajra and Victoria Kofman

Using oral presentations and cooperative discussions to facilitate learning statistics . . . . . . . . . . . . . . . . 629

Abeer Hasan and Sayonita Ghosh Hajra

Exploration of the factors that support earning: Web-based activity and testing systems in

community college algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

Shandy Hauk, Bryan Matlen and Larry Thomas

Supporting instructional change in mathematics: The role of online and in-person communities . . . . . 646

Charles Hayward and Sandra Laursen

Student proficiency with transformational geometry after a college proof-based geometry class . . . . . . 655

Meredith Hegg and Tim Fukawa-Connelly

Exploring experts’ covariational reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664

Natalie Hobson and Kevin Moore

How students interpret line and vector integral expressions: Domains, integrands, di�erentials, and

outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

Steven Jones and Omar Naranjo

Definite integrals versus indefinite integrals: How do students see them as the same or as di�erent? . . 682

Steven Jones and Cache Thompson

Using video in online work groups to support faculty collaboration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690

Karen Keene, Nicholas Fortune and William Hall

Undergraduate abstract algebra: Is teaching di�erent at ’teaching’ colleges? . . . . . . . . . . . . . . . . . . . . . . 698

Rachel Keller, Estrella Johnson, Valerie Peterson and Tim Fukawa Connelly

xivxiv

Mathematicians’ evaluations of the language of mathematical proof writing at the undergraduate

level in three di�erent pedagogical contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707

Kristen Lew and Juan Pablo Mejia-Ramos

An unexpected outcome: Students’ focus on order in the Multiplication Principle . . . . . . . . . . . . . . . . . 715

Elise Lockwood and Branwen Schaub

Computational thinking in and for undergraduate mathematics: Perspectives of a mathematician . . . 723

Miroslav Lovric and Ami Mamolo

“Explanatory” talk in mathematics research papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732

Juan Pablo Mejia-Ramos and Matthew Inglis

Managing tensions within a coordinated inquiry-based learning linear algebra course: The role of

worksheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

Vilma Mesa, Mollee Shultz and Ashley Jackson

Function sameness to “function” meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747

Alison Mirin

Students’ quantitative reasoning in a revenue maximization context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755

Thembinkosi Mkhatshwa and Helen Doerr

Preservice secondary teachers’ abilities to transfer from graphical to algebraic representations of

functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763

Kyunghee Moon

Preservice secondary teachers’ abilities to use representations and realistic tasks . . . . . . . . . . . . . . . . . . 771

Kyunghee Moon

Preservice elementary teachers’ understandings of greatest common factor versus least common

multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779

Kristin Noblet

Examining lecturer’s questioning in advanced proof-oriented mathematics classes . . . . . . . . . . . . . . . . . . 787

Teo Paoletti, Victoria Krupnik, Dimitri Papadopoulos, Joseph Olsen, Tim Fukawa-Connelly and

Keith Weber

Transitioning to proof with worked examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796

Dimitri Papadopoulos

Approaches to the derivative in Korean and the U.S. calculus classrooms . . . . . . . . . . . . . . . . . . . . . . . . . 804

Jungeun Park

Learning to notice and use student thinking in undergraduate mathematics courses . . . . . . . . . . . . . . . . 812

Anna Pascoe and Shari Stockero

Attention to detail: Norms for proof evaluation in a summer mathematics program . . . . . . . . . . . . . . . . 821

Cody Patterson and Xiaowen Cui

Student mathematical connections in an introductory linear algebra course . . . . . . . . . . . . . . . . . . . . . . . 829

Spencer Payton

Mathematicians’ collaborative silences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837

Matthew Petersen

xvxv

Graduate student instructors learning from peer observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846

Daniel Reinholz

McNuggets, bunnies, and remainders, oh my! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855

Nina Rocha and Jennifer Zakotnik

Students’ conceptions of mappings in abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863

Rachel Rupnow

A success factor model for calculus: The relative impact of and connections between factors a�ecting

student success in college calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871

Megan Ryals and Karen Keene

CxN: Investigating the creative proving process using neuroscience methods . . . . . . . . . . . . . . . . . . . . . . 879

Milos Savic, David Plaxco, Michael Wenger, Emily Cilli-Turner, Gail Tang, Houssein El Turkey

and Gulden Karakok

Emerging insights from the evolving framework of structural abstraction in knowing and learning

advanced mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886

Thorsten Scheiner and Má rcia Pinto

Physics students’ use of symbolic forms when constructing di�erential elements in multivariable

coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895

Benjamin Schermerhorn and John Thompson

A comparison of calculus, transition-to-proof, and advanced calculus student quantifications in

complex mathematical statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903

Morgan Sellers, Kyeong Hah Roh, Erika David and Kody D’Amours

Beyond the product structure for definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912

Courtney Simmons and Michael Oehrtman

Exploring undergraduates’ experience of the transition to proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 920

John Smith, Mariana Levin, Younggon Bae and V. Satyam

Principles for designing tasks that promote covariational reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928

Irma Stevens, Teo Paoletti, Kevin Moore, Biyao Liang and Hamilton Hardison

Common algebraic errors in calculus courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937

Sepideh Stewart and Stacy Reeder

Completeness and convergence: Interdependent development in the context of proving the

Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945

Stephen Strand

Angle measure, quantitative reasoning, and instructional coherence: The case of David . . . . . . . . . . . . . 954

Michael Tallman

Virtual manipulatives, vertical number lines, and Taylor series convergence: The case of Cody . . . . . . 963

Matthew Thomas and Jason Martin

Developing students’ reasoning about the derivative of complex-valued functions with the aid of

Geometer’s Sketchpad (GSP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972

Jonathan Troup

xvixvi

Calculus students’ reasoning about slope as a ratio-of-totals and its impact on their reasoning about

derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982

Jen Tyne

Students’ understanding of vectors and cross products: Results from a series of visualization tasks . . . 991

Monica VanDieren, Deborah Moore-Russo, Jill Wilsey and Paul Seeburger

Variations in precalculus through Calculus 2 courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001

Matthew Voigt, Chris Rasmussen and Naneh Apkarian

Using expectancy value theory to account for students’ mathematical justifications . . . . . . . . . . . . . . . . 1009

Keith Weber, Kristen Lew and Juan Pablo Mej́ı a-Ramos

Expert vs. novice reading of a calculus textbook: A case study comparison . . . . . . . . . . . . . . . . . . . . . . . 1018

Emilie Wiesner, Aaron Weinberg, John Barr and Kikki Upham

Mentor professional development for mathematics graduate student instructors . . . . . . . . . . . . . . . . . . . 1026

Sean Yee and Kimberly Rogers

Order of operations: A case of mathematical knowledge-in-use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035

Rina Zazkis

Theoretical Reports

Can/should students learn mathematics theory-building? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043

Hyman Bass

How limit can be embodied and arithmetized: A critique of Lako� and Núñez . . . . . . . . . . . . . . . . . . . . 1051

Timothy Boester

Those who teach the teachers: Knowledge growth in teaching for mathematics teacher educators . . . . 1060

Shandy Hauk, Billy Jackson and Jenq-Jong Tsay

Research in courses before calculus through the lens of social justice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068

Shandy Hauk, Allison Toney, April Brown and Katie Salguero

What should undergraduate mathematics majors understand about the nature of mathematical

knowledge? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080

Je�rey Pair

Making RUME for institutional change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089

Daniel Reinjolz

Inquiry as an access point to equity in the classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098

Gail Tang, Joussein El Turkey, Emily Cilli-Turner, Milos Savic, David Plaxco and Gulden Karakok

Leveraging real analysis to foster pedagogical practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107

Nicholas Wasserman, Keith Weber and William McGu�ey

Preliminary Reports

Characterizing mathematical digital literacy: A preliminary investigation . . . . . . . . . . . . . . . . . . . . . . . . . 1113

Todd Abel, Jeremy Brazas, Darryle Chamberlain Jr. and Aubrey Kemp

Blended processing: Mathematics in chemical kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119

Kinsey Bain, Adam Zabih, Alena Moon and Marcy Towns

xviixvii

Raising reasoning through revision: A case study of an inquiry-based college geometry course . . . . . . . 1127

Janessa Beach and Rebecca Dibbs

Exploring mathematics graduate teaching assistants’ developmental stages for teaching . . . . . . . . . . . . 1133

Mary Beisiegel

Flip vs. fold: What is so important about the rigidity of a motion? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140

Anna Marie Bergman, Dana Kirin and Ben Wallek

Examining students’ procedural and conceptual understanding of eigenvectors and eigenvalues in

the context of inquiry-oriented instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146

Khalid Bouhjar, Muhammad Haidar and Christine Andrews-Larson

Supporting students’ understanding of calculus concepts: Insights from middle-grades mathematics

education research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152

Steven Boyce and Kira Wyld

Signed quantities: Mathematics based majors struggle to make meaning . . . . . . . . . . . . . . . . . . . . . . . . . . 1158

Suzanne Brahmia and Andrew Boudreaux

Developing student understanding: The case of proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164

Darryl Chamberlain Jr. and Draga Vidakovic

Defining functions: Choices that a�ect student learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1170

Joshua Chesler

Students’ understanding of test statistics in hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176

Annie Childers, Darryl Chamberlain Jr., Leslie Meadows, Harrison stalvey, Draga Vidakovic and

Aubrey Kemp

Spatial training and calculus ability: Investigating impacts on student performance and cognitive

learning style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182

Emily Cilli-Turner and Lindsay McCunn

Applying variation theory to study modeling competencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189

Jennifer Czocher

Do students really know what a function is?: Applying APOS analysis to student small group

presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195

Tara Davis and Georgianna Martin

Teachers’ beliefs & knowledge of the everyday value of high school algebra & geometry: Is one more

useful than the other? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1201

Jennifer Dunham

Second semester calculus students and the contrapositive of the nth term test . . . . . . . . . . . . . . . . . . . . 1207

David Earls

Let’s talk about teaching: Investigating instructors’ social networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214

Kathleen Quardokus Fisher, Naneh Apkarian and Emily Walter

Instrumental genesis and generalization in multivariable calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219

Brian Fisher, Jason Samuels and Aaron Wangberg

xviiixviii

A continued exploration of self - inquiry in the context of proof and problem solving . . . . . . . . . . . . . . . 1225

Todd Grundmeier and Dylan Retsek

Undergraduates’ reasoning about integration of complex functions within three worlds of mathematics 1231

Brent Hancock

DNR-based professional development: Factors that a�ord or constrain implementation . . . . . . . . . . . . . 1238

Guershon Harel, Osvaldo Soto, and Brandon Olszewski

An exploration of students’ discourse using Sim2Bil within group work: A commognitive perspective 1243

Ninni Marie Hogstad and Olov Viirman

The role of mathematics faculty in the development of African American male mathematics majors . 1249

Christopher Jett

Abstraction and quantitative reasoning in construction of fractions as operators . . . . . . . . . . . . . . . . . . . 1256

Eun Jung

Students’ social adaptation to mathematical tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1261

Je�rey King

Instruction in precalculus and single-variable calculus: A bird’s eye view . . . . . . . . . . . . . . . . . . . . . . . . . 1267

Dana Kirin, Kristen Vroom, Sean Larsen and Naneh Apkarian

Students’ inclination to use visual images during problem solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273

Milé Krajcevski and Karen Keene

Students’ epistemological frames and their interpretation of lectures in advanced mathematics . . . . . . 1279

Victoria Krupnik, Keith Weber and Tim Fukawa-Connelly

Mathematical modeling and mathematical competencies: The case of biology students. . . . . . . . . . . . . . 1286

Yannis Liakos

A preliminary investigation of the reification of “choosing” in counting problems . . . . . . . . . . . . . . . . . . 1293

Elise Lockwood

Outcomes beyond success in a problem centered developmental mathematics class . . . . . . . . . . . . . . . . . 1299

Martha Makowski and Randi Congleton

Underrepresented students succeeding in math: The challenges and coping strategies of

mathematically talented post-secondary students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305

Martha Makowski

Student gesture use when explaining the second-derivative test and optimization . . . . . . . . . . . . . . . . . . 1311

Tim McCarty and Nicole Infante

Experts’ varied concept images of the symbol dx in integrals and di�erential equations . . . . . . . . . . . . . 1317

Tim McCarty and Vicki Sealey

Locating a realistic starting point for the guided reinvention of limit at infinity with community

college students prior to pre-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324

William McGu�ey

Exploring student conceptions of binary operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1330

Kathleen Melhuish

xixxix

The lead TA influence: Teaching practices focused on for an active learning classroom . . . . . . . . . . . . . 1336

Hayley Milbourne and Susan Nickerson

Professional development linking the concept of inverse in abstract algebra to function inverses in

the high school curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1342

Melissa Mills and Cara Brun

Textbook formations of independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1348

Adam Molnar and Steven Edalgo

Connecting secondary and tertiary mathematics: Abstract algebra and inverse . . . . . . . . . . . . . . . . . . . . 1354

Eileen Murray, Matthew Wright and Debasmita Basu

Self-assessment behaviors of undergraduate mathematics students: A preliminary report . . . . . . . . . . . 1360

Kedar Nepal, Kailash Ghimire, Ramjee Sharma and Manoj Thapa

Identification matters: E�ects of female peer role models di�er by gender between high and low

mathematically identified students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367

Susan Nickerson, Katie Bjorkman, Sei Jin Ko and David Marx

Considerations for explicit and reflective teaching of the roles of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373

Je�rey Pair and Sarah Bleiler-Baxter

Perturbing practice: The e�ects of virtual manipulatives as novel didactic objects on instruction . . . . 1380

Krysten Pampel

Comparing graph use in STEM textbooks and practitioner journals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386

Teo Paoletti, Madhavi Vishnubhotla, Zareen Rahman, Justin Seventko and Debasmita Basu

Student conceptions of three-dimensional solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393

Stepan Paul and Monica Mendoza

The e�ects of the epsilon-N relationship on convergence of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399

Zackery Reed

The e�ect of attending peer tutoring on course grades in calculus I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406

Brian Rickard and Melissa Mills

Mental constructions involved in di�erentiating a function to a function power . . . . . . . . . . . . . . . . . . . . 1412

Rachel Rupnow and Catherine Ulrich

Pedagogical practices for fostering mathematical creativity in proof-based courses: Three case studies 1418

Milos Savic, Houssein El Turkey, Gail Tang, Gulden Karakok, Emily Cilli-Turner, David Plaxco

and Mohamed Omar

The saga of Alice continues: Her progress with proof frameworks evaporates when she encounters

unfamiliar concepts, but eventually returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425

Annie Selden, John Selden and Ahmed Benkhalti

Studying pre-service teachers’ selection of representations in a technologically enhanced environment 1431

Robert Sigley, Muteb Alqahtani and Victoria Krupnik

Mathematicians’ interplay of the three worlds of the derivative and integral of complex-valued

functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436

Hortensia Soto and Michael Oehrtman

xxxx

Evaluation of graduate student professional development and instruction by mathematics

departments: Results from a national survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1442

Natasha Speer, Jess Ellis and Jessica Deshler

An explicit method for teaching generalization to pre-service teachers using computer programming . 1449

Cynthia Stenger, James Jerkins, Jessica Stovall and Janet Jenkins

The use of NCTM articles as reading assignments to motivate prospective elementary teacher

engagement in mathematics courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455

Krista Strand and Eva Thanheiser

Analysis of teachers’ conceptions of variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1462

Gabriel Tarr and April Strom

Implementation of pre and post class readings in calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469

Salam Turki and Houssein El Turkey

Opportunities to learn from teaching: A case study of two graduate teaching assistants . . . . . . . . . . . . 1475

Nathan Wakefield, Erica Miller and Yvonne Lai

Factors influencing instructor use of student ideas in the Multivariable Calculus Classroom . . . . . . . . . 1481

Aaron Wanbgerg, Tisha Jooks, Brian Fisher, Jason Samuels and Elizabeth Gire

Knowledge about student understanding of eigentheory: Information gained from multiple choice

extended assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487

Kevin Watson, Megan Wawro, Michelle Zandieh and Sarah Kerrigan

Meta-representational competence with linear algebra in quantum mechanics . . . . . . . . . . . . . . . . . . . . 1493

Megan Wawro, Kevin Watson and Warren Christensen

An investigation of the development of partitive meanings for division with fractions: What does it

mean to split something into 9/4 groups? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1499

Matthew Weber, Amie Pierone and April Strom

Corequisite remediation and math pathways in Oklahoma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505

Matthew Wilson and Michael Oehrtman

Instructor-generated concepts framework for elementary algebra in the college context . . . . . . . . . . . . . 1510

Claire Wladis

Career decision making strategies of calculus and developmental mathematics students . . . . . . . . . . . . . 1517

Xiangming Wu, Jessica Deshler, Edgar Fuller and Marcela Mera Trujillo

What constitutes a proof? Complementary voices of a mathematician and a mathematics educator

in a co-taught undergraduate course on mathematical proof and proving . . . . . . . . . . . . . . . . . . . . . . . . . 1523

Orit Zaslavsky and Jason Cooper

Poster Reports

Students’ thinking in an inquiry-based linear algebra course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1530

Sarah Hough, William Jacob, Monica Mendoza and Elizabeth Thoren

The influences of a teacher’s mathematical meaning of constant rate of change on his classroom

practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1532

Sinem Bas Ader and Marilyn Carlson

xxixxi

Supporting math emporium students’ learning through short instructional opportunities . . . . . . . . . . . 1534

Andrea Alt

Mathematics a�rmations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536

Geillan Aly

Developing and testing a knowledge scale around the nature of mathematical modeling . . . . . . . . . . . . 1538

Reuben Asempapa

Exploratory activities with dynamic geometry environment in axiomatic geometry . . . . . . . . . . . . . . . . 1540

Younggon Bae

STEM major mindset changes during their first undergraduate mathematics course . . . . . . . . . . . . . . . 1542

Laura Beene and Rebecca Dibbs

Building a cognitive model for symmetry: How well does an existing framework fit? . . . . . . . . . . . . . . . 1544

Anna Marie Bergman and Ben Wallek

Student understanding of the product layer of the integral in volume problems . . . . . . . . . . . . . . . . . . . . 1546

Krista Bresock and Vicki Sealey

Mathematics education as a research field: Reflections from ICME-13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548

Stacy Brown, Hortensia Soto and Spencer Bagley

Connecting reading comprehension research and college mathematics instruction . . . . . . . . . . . . . . . . . . 1550

Melanie Butler

Diagrams for the reasoning and proof of Amortization Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1551

Kuo-Liang Chang, Hazel McKenna and Thomas Mgonja

Technology use in the teaching and learning of an introductory statistics course: The case of Excel

and the ‘Knitr’ R-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553

Sher Chhetri

To factorize or not to factorize: Novice teachers’ struggles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555

Hyungmi Cho, Miyeong Na and Oh Nam Kwon

Upper-division physics student thinking regarding non-Cartesian coordinate systems . . . . . . . . . . . . . . . 1557

Warren Christensen, Brian Farlow, Marlene Vega and Michael Loverude

The mathematics attitudes and perceptions survey: New data and alignment with other recent findings1559

Warren Code and Wes Maciejewski

Online course component and student performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1562

Elizabeth DiScala and Yasmine Aki

JITAR online modules to improve math preparation of engineering students . . . . . . . . . . . . . . . . . . . . . . 1563

Alina Duca

A framework for characterizing a teacher’s decentering tendencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565

Ashley Duncan, Sinem Bas Alder and Marilyn Carlson

Some logical issues in RUME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567

Viviane Durand-Guerrier

xxiixxii

A comparison of faculty expectations and student perceptions of active engagement in a Calculus I

class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569

Belinda Edwards

Calculus students’ meanings for average rate of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1571

Wyatt Ehlke and Sayonita Ghosh Hajra

Empowered women In RUME: What have we been up to? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573

Jess Ellis, Stacy Muscgrave, Kathleen Melhuish, Eva Thanheiser and Megan Wawro

Designing and developing Likert items that capture mathematical problem solving . . . . . . . . . . . . . . . . 1575

James Epperson, Kathryn Rhoads and R. Cavender Campbell

Reducing abstraction in the Group Concept Inventory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577

Joshua Fagan and Kathleen Melhuish

Engaging in abstract algebra through game play: Group theory card game groups . . . . . . . . . . . . . . . . . 1579

Patrick Galarza

Research on concept-based instruction of calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1581

Xuefen Gao

Equity issues that (may) arise in active learning classrooms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583

Jessica Gehrtz, Richard Sampera and Jess Ellis

Conceptual understanding of di�erential calculus: A comparative study . . . . . . . . . . . . . . . . . . . . . . . . . . 1585

Daria Gerasimova, Kathleen Matson, Robert Sachs and Margret Hjalmarson

Variations of College Algebra instructors’ presentations of the mathematics: The case of solving

quadratic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593

Claire Gibbons

Investigating prospective teachers’ meanings of covariation before and after calculus coursework . . . . . 1595

Roser Giné

Improving undergraduate STEM education through adjunct mathematics instructor resources and

support (IUSE-AMIRS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597

Amir Golnabi, Eileen Murray Montclair and Zareen Rahman

Quantitative learning centers: What we know now and where we go from here . . . . . . . . . . . . . . . . . . . . 1599

Melissa Haire

Characterizing normative metacognitive activity during problem solving in undergraduate classroom

communities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1601

Emilie Hancock

Research in courses before calculus through the lens of social justice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603

Shandy Hauk, Allison Toney, April Brown and Katie Salguero

Exploring the contentspecific mathematical proving behavior of students in context: Opportunities

for extracting and giving mathematical meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605

Sarah Hough, William Jacob, Monica Mendoza and Alex Sacharuk

xxiiixxiii

Questioning assumptions about the measurability of subdomains of Mathematical Knowledge for

Teaching (MKT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607

Heather Howell, Yvonne Lai and Heejo Suh

Data cleaning in mathematics education research: The overlooked methodological step . . . . . . . . . . . . . 1609

Aleata Hubbard

Calculus instructor beliefs regarding student engagement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611

Carolyn James

Students’ mistakes and strategies when matching a function’s graph with its derivative . . . . . . . . . . . . 1613

Nathan Jewkes and Amy Dwiggins

Situational characteristics supporting instructional change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615

Estrella Johnson and Rachel Keller

Graphs display lengths, not locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617

Surani Joshua

Factors influencing instructor use of student ideas in the Multivariable Calculus classroom . . . . . . . . . 1619

Surani Joshua

All the math you need: An investigation into the curricular boundaries of mathematical literacy . . . . 1621

Gizem Karaali

Student beliefs about mathematics in an inquiry-based introduction to proof course . . . . . . . . . . . . . . . 1622

Shiv Karunakaran, Abigail Higgins and James Whitbread Jr.

Bringing evidenced-based practices to a large-scale precalculus class: Preliminary results . . . . . . . . . . . 1624

Karen Keene, Leslaw Skrzypek, Brooke Kott and Gregory Downing

Post-class reflections and calibration in introductory calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626

Taylor Kline and Rebecca Dibbs

Error detection in an introductory proofs course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1628

Tyler Kloefkorn, Jason Aubrey and Kyle Pounder

An Aative learning environment in introductory analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1630

Brynja Kohler and Patrick Seegmiller

Pre-service teachers’ use of informal language while solving a probabilistic problem . . . . . . . . . . . . . . . . 1631

Victoria Krupnik, Robert Sigley and Muteb Alqahtani

Student ways of framing di�erential equations tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633

George Kuster

Using evidence to understand and support an educational reform movement: The case of

inquiry-based learning (IBL) in college mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635

Sandra Laursen, Charles Hayward and Zachary Haberler

Educational point of Newton-Liebnitz formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636

Qun Lin and Rongrong Cao

Student reasoning with di�erentials and derivatives in upper-division physics . . . . . . . . . . . . . . . . . . . . . 1637

Michael Loverude

xxivxxiv

An alternate characterisation of developmental mathematics students . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1639

Wes Maciejewski and Cristina Tortora

Student attitudes, beliefs, and experiences related to counting problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1641

Samantha McGee, Sarah Erickson and Elise Lockwood

Algebra instruction at community colleges: An exploration of its relationship with student success . . . 1643

Vilma Mesa, Irene Duranczyk, Nidhi Kohli, April Strom, Laura Watkins and Angeliki Mali

Bridging the gaps between teachers’ and students’ perspectives of a culturally inclusive classroom . . . 1645

Thomas Mgonja and Juo-Liang Chang

Mathematical knowledge for teaching & cognitive demand: A comparative case study of precalculus

examples that involve procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647

Erica Miller

Coping with the derivative of an atypical representation of a common function . . . . . . . . . . . . . . . . . . . . 1649

Alison Mirin

Speaking with meaning about angle measure and the sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1651

Stacy Musgrave and Marilyn Carlson

Examining prospective teachers’ justifications of children’s temperature stories . . . . . . . . . . . . . . . . . . . 1653

Dana Olano�, Nicole Wessman-Enzinger and Jennifer Tobias

Classroom participation as an agent of socialization for identity shaping of preservice mathematics

teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655

Janet Omitoyin

Investigating student learning through team-based learning calculus instruction . . . . . . . . . . . . . . . . . . . 1657

Travis Peters, Elgin Johnston, Heather Bolles, Craig Ogilvie and Alexis Knaub

Analyzing focus groups of an experimental real analysis course: ULTRA . . . . . . . . . . . . . . . . . . . . . . . . . 1659

Ruby Quea and William McGu�ey

Extreme apprenticeship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1661

Johanna Rämö, Jokke Häsä and Juulia Lahdenperä

Hypophora: Why take the derivative? (no pause) because it is the rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663

Kitty Roach

Individual and group work with nonstandard problems in an ordinary di�erential equations course

for engineering students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665

Svitlana Rogovchenko, Yuriy Rogovchenko and Stephanie Tre�ert-Thomas

A topological approach to formal limits supported by technology: What concept images do students

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Leveraging research to support students’ quantitative and co-variational reasoning in an online

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Steven Silber

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Oyuki Aispuro, Jerica Bañares, Grant Dolmat, Todd CadwalladerOlsker and Sissi Li

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Emerging Insights from the Evolving Framework of Structural Abstraction

Thorsten Scheiner Márcia M. F. Pinto

University of Hamburg, Germany Federal University of Rio de Janeiro, Brazil

Only recently ‘abstraction from objects’ has attracted attention in the literature as a form of abstraction that has the potential to take account of the complexity of students’ knowing and learning processes compatible with their strategy of giving meaning. This paper draws attention to several emerging insights from the evolving framework of structural abstraction in students’ knowing and learning of the limit concept of a sequence. Particular ideas are accentuated that we need to understand from a theoretical point of view since they reveal a new way of understanding knowing and learning advanced mathematical concepts.

Keywords: Limit Concept; Mathematical Cognition; Sense-Making; Structural Abstraction

Introduction

Theoretical and empirical research shows the existence of differences in knowing and learning concerning different kinds of knowledge (diSessa, 2002). A general framework on abstraction cannot encompass the whole complexity of knowing and learning processes in mathematics. Rather, in investigating the nature, form, and emergence of knowledge pieces, various local learning theories may be developed, which will be quite specific to particular mathematical concepts, individuals, and their respective sense-making strategies. As a consequence, the complexity of knowing and learning processes in mathematics cannot be described or explained by only one framework. Instead, we acknowledge that comprehensive understanding of cognition and learning in mathematics draws on a variety of theoretical frameworks on abstraction.

The literature demonstrates significant theoretical and empirical advancement in understanding ‘abstraction-from-actions’ approaches, particularly the cognitive processes of forming a (structural) concept from an (operational) process (Dubinsky, 1991; Gray & Tall, 1994; Sfard, 1991). Abstraction-from-actions approaches take account of a certain sense-making strategy, namely what Pinto (1998) described as ‘extracting meaning’. However, only recently ‘abstraction from objects’ has attracted attention as a form of abstraction that provides a new way of seeing the complexity of knowing and learning processes compatible with students’ strategy of what Pinto (1998) described as ‘giving meaning’.

The purpose of this paper is to provide deeper meaning to a recently evolving framework of a particular kind of ‘abstraction from objects’: structural abstraction. The structural abstraction framework is evolving in the sense that the framework functions both as a tool for research and as an object of research (Scheiner & Pinto, 2016b). In more detail, we use the structural abstraction framework retrospectively as a lens through which we reinterpret a set of findings on students’ knowing and learning of the limit concept of a sequence. This reinterpretation is an active one in the sense that the framework serves as a tool to analyze a set of data, while the framework is also refined and extended since the reinterpretation produces deeper insights about the framework itself. Especially, these more profound insights are what we need to understand from a theoretical point of view since they have relevance for significant issues in knowing and learning advanced mathematical concepts. Such a dynamic view that is aligned with an interpretative approach seems to be promising in responding to questions that evolve while the object of consideration is still under investigation.

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We begin by providing an upshot of our synthesis of the literature on abstraction in knowing and learning mathematics. Our synthesis is to suggest an orientation toward the evolving framework of structural abstraction as an avenue to take account of an important area for consideration – that is, drawing attention to the complex knowing and learning processes compatible with students’ ‘giving meaning’ strategy. The structural abstraction framework constitutes the foundation of the second part of the paper providing emerging insights in knowing and learning the limit concept of a sequence. These insights offer theoretical advancement of the framework and deepen our understanding of knowing and learning advanced mathematics.

Mapping the Terrain of Research on Abstraction in Mathematics Education

Abstraction seems to have gained a bad reputation after been questioned by the situated cognition (or situated learning) paradigm, and, as a consequence, has almost disappeared from recent research discourse. This criticism rests primarily on traditional approaches on knowledge transfer through abstraction that led to an understanding of abstraction as a process of decontextualization and a confusion of abstraction with generalization. The recent contribution by Fuchs et al. (2003) shows that such classical approaches to abstraction still exist:

“To abstract a principle is to identify a generic quality or pattern across instances of the principle. In formulating an abstraction, an individual deletes details across exemplars, which are irrelevant to the abstract category […]. These abstractions […] avoid contextual specificity so they can be applied to other instances or across situations.” (p. 294)

Though various images of abstraction in the mathematics education literature can be identified (see Scheiner & Pinto, 2016a), several scholars argued against the image of abstraction as decontextualization. Van Oers (1998, 2001), for instance, argued that removing context must impoverish a concept rather than enrich it. Several other scholars have reconsidered and advanced our understanding of abstraction in ways that account for the situated nature of knowing and learning in mathematics. Noss and Hoyles (1996) introduced the notion of situated abstraction to describe “how learners construct mathematical ideas by drawing on the webbing of a particular setting which, in turn, shapes the way the ideas are expressed” (p. 122). Webbing in this sense means “the presence of a structure that learners can draw up and reconstruct for support – in ways that they can choose as appropriate for their struggle to construct meaning for some mathematics (Noss & Hoyles, 1996, p. 108). Hershkowitz, Schwarz, and Dreyfus (2001) introduced the notion of abstraction in context that they presented as “an activity of vertically reorganizing previously constructed mathematics into new mathematical structure” (p. 202). They identify abstraction in context with what Treffers (1987) described as ‘vertical mathematization’ and propose entire mathematical activity as the unity of analysis. These contributions demonstrate that research on abstraction in knowing and learning mathematics has made significant progress in taking account of the context-sensitivity of knowledge.

Several other contributions shape the territory in mathematics education research on abstraction. Mitchelmore and White (2004) indicate a distinction between abstraction in mathematics and abstraction in mathematics learning. They proposed to include a new meaning in the later, which seemed to be missing in the debate on the notion of abstraction, related to “formation of concepts by empirical abstraction from physical and social experience” (p. 329). Articulated to this understanding, Scheiner (2016) proposed a distinction between two forms of abstraction, namely abstraction from actions and

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abstraction from objects. This distinction has been further refined in Scheiner and Pinto (2014) arguing that the focus of attention of each form of abstraction takes place on physical objects (referring to the real world) or mental objects (referring to the thought world) (see Fig. 1).

abstraction on actions abstraction on objects

pseudo-empirical abstraction

empirical abstraction

reflective abstraction

mental

objects physical objects

structural abstraction

Fig. 1: A frame to capture various kinds of abstraction (reproduced from Scheiner & Pinto, 2014)

We consider this distinction as being productive in trying to capture some of the variety of images of abstraction in mathematics education (for details see Scheiner & Pinto, 2016a). It acknowledges Piaget’s (1977/2001) three kinds of abstraction, including pseudo-empirical abstraction, empirical abstraction, and reflective abstraction, that served as critical points of departure in thinking about abstraction in learning mathematics. Research on abstraction in mathematics has long moved beyond classifying and categorizing approaches in cognition and learning, based on similarities of the individuals constructs. For instance, Mitchelmore and White (2007), in going beyond Piaget’s empirical abstraction and in drawing on Skemp’s (1986) conception of abstraction, described abstraction in learning elementary mathematics concerning seeing the underlying structure rather than the superficial characteristics. Abstraction in learning advanced mathematics, however, is almost always defined in terms of encapsulation (or reification) of processes into objects, originating in Piaget’s (1977/2001) idea of reflective abstraction. Reflective abstraction is an abstraction from the subject's actions on objects, particularly from the coordination between these actions. The particular function of reflective abstraction is abstracting properties of an individual's action coordination. That is, reflective abstraction is a mechanism for the isolation of specific properties of a mathematical structure that allows the individual to construct new pieces of knowledge. Taking Piaget’s reflective abstraction as a point of departure, Dubinsky and his colleagues (Dubinsky, 1991; Cottrill et al., 1996; Arnon et al., 2014) developed the APOS theory describing the construction of concepts through the encapsulation of processes. Similar to encapsulation is reification – the central tenet of Sfard’s (1991) framework emphasizing the cognitive process of forming a (structural) concept from an (operational) process. In the same way, Gray and Tall (1994) described this issue as an overall progression from procedural thinking to proceptual thinking, whereas proceptual thinking means the ability to flexibly manipulate a mathematical symbol as both a process and a concept. Gray and Tall (1994) termed symbols that may be regarded as being a pivot between a process to compute or manipulate and a concept that may be thought of as a manipulable entity as procepts.

Scheiner (2016) revealed that the literature shows an unyielding bias toward abstraction from actions as the driving form of abstraction in knowing and learning advanced mathematics. This almost always exclusive view arises directly from the trajectory of our field’s history; originating in Piaget’s assumption that only reflective abstraction can be the source of any genuinely new construction of knowledge. While abstraction-on-actions

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approaches have served many purposes quite well, they cannot track detail of students’ knowing and learning processes compatible with the strategy of giving meaning. The recently evolving framework of structural abstraction has attracted attention as a promising tool to shed light into the complexity of students’ knowing and learning processes compatible with their strategy of ‘giving meaning’.

The Evolving Framework of Structural Abstraction

The distinction between abstraction from actions and abstraction from objects reflects Tall’s (2013) distinction between operational abstraction and structural abstraction. In contrast to Piaget (1977/2001) who dichotomized these two forms of abstraction, Tall (2013) argued that mathematical thinking emerges in these two forms: operational abstraction focusing on actions on objects and structural abstraction focusing on the properties (or structures) of objects. For instance, the development of geometry in the conceptual embodied world focuses mainly on structural thinking, the operational symbolic world blends both operational and structural thinking as new forms of number are introduced as extensional blends in algebra (see Tall, 2013). Obviously, mathematics education researchers used the term ‘structural’ in diverse ways, referring, for instance, to structural mathematics of axioms and definitions or to the properties of the structure of objects. In the following discussion, we departure from Scheiner’s (2016) understanding of structural abstraction as focusing on “the richness of the particular [that] is embodied not in the concept as such but rather in the objects that falling under the concept […]. This view gives primacy of meaningful, richly contextualized forms of (mathematical) structure over formal (mathematical) structure” (p. 175). Here we focus the attention to several core assumptions that orient the evolving framework of structural abstraction (see Scheiner, 2016; Scheiner & Pinto, 2016b): Concretizing through Contextualizing

Structural abstraction takes place on mental objects that, in Frege’s (1892a) sense, fall under a particular concept. These objects may be either concrete or abstract. Concreteness and abstractness, however, are not considered as properties of an object but rather as properties of an individual’s relatedness to an object in the sense of the richness of a person's representations, interactions, and connections with the object (Wilensky, 1991). From this point of view, rather than moving from the concrete to the abstract, individuals, in fact, begin their understanding of (advanced) mathematical concepts with the abstract (Davydov, 1972/1990). The ascending from the abstract to the concrete requires a concretizing process where the mathematical structure is particularized by looking at the object in relation with itself or with other objects that fall under the particular concept. The crucial aspect for concretizing is contextualizing, that is, setting the object(s) in different specific contexts. Different contexts may provide various senses (Frege, 1892b) of the concept of observation.

Complementizing through Recontextualizing

The central characteristic of the structural abstraction framework is that while, within the empiricist view, conceptual unity relies on the commonality of elements, it is the interrelatedness of diverse elements that creates unity within the approach of structural abstraction. The process of placing objects into different specific contexts allows specifying essential components. Structural abstraction, then, means attributing the particularized meaningful components of objects to the mathematical concept. Thus, the core of structural abstraction is complementarity rather than similarity. The meaning of advanced mathematical concepts is developed by complementizing diverse meaningful components of a variety of

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specific objects that have been contextualized and recontextualized in multiple situations. This perspective agrees with van Oers’ (1998) view on abstraction as related to recontextualization instead of decontextualization.

Complexifying through Complementizing

The structural abstraction framework takes the view that knowledge is a complex system of many kinds of knowledge elements and structures. Complementizing implies a process of restructuring the system of knowledge pieces. These knowledge pieces have been constructed through the above-mentioned process or are already constructed elements coming from other concept images, which are essential for the new concept construction. The cognitive function of structural abstraction is to facilitate the assembly of more complex and compressed knowledge structures. Taking this perspective, we construe structural abstraction as moving from simple to complex knowledge structures, a movement with the aim to build coherent and compressed knowledge structures. In Thurston (1990)’s words, when the latter is achieved we “can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process.” (p. 847). From the structural abstraction perspective, abstraction is acknowledged as a movement across levels of complexity (Scheiner and Pinto, 2014).

Emerging Insights from the Structural Abstraction Framework

In this section, we summarize emerging insights we gained so far by using the evolving framework of structural abstraction retrospectively as a lens through which findings on students’ (re-)construction of the limit concept of a sequence were reinterpreted. The study by Pinto (1998) provided the context in which she identified mathematics undergraduates’ sense-making strategies of formal mathematics. From a cross-sectional analysis of three pairs of students, two prototypical strategies of making sense could be identified, namely ‘extracting meaning’ and ‘giving meaning’:

“Extracting meaning involves working within the content, routinizing it, using it, and building its meaning as a formal construct. Giving meaning means taking one’s personal concept imagery as a starting point to build new knowledge.” (Pinto, 1998, pp. 298-299)

The literature on abstraction from actions provides several accounts of how students construct a mathematical concept compatible with their strategy of ‘extracting meaning’.

For instance, dynamic views of the limit of a sequence and the genetic decomposition of the limit concept of a sequence are intensively investigated by Arnon et al.’s (2014) APOS theory to respond to how students may construct the concept through the process of reflective abstraction. To mention a few recent investigations supported by the same theory, and compatible with the strategy of ‘extracting meaning’, Martinez-Planell, Gonzalez, DiCristina, and Acevedo (2012) focused on students’ understanding of series and investigated whether students saw series as a process without an end or as a sequence of partial sums, as stated by definition. They respond how students may construct the concept, by considering a distinction amongst their understandings of the concept of a sequence as a list of numbers or as a function defined in natural numbers (McDonald, Mathews, & Strobel, 2000), and concluded that even after formal training, students often think of sequences and series as an infinite, unending process.

However, there are almost no accounts of how students construct a concept compatible with their strategy of ‘giving meaning’, and the structural abstraction framework has shown to be enlightening with regard to this. Students who ‘give meaning’ seem to develop representations of the limit concept from their concept image and use them generically (see Yopp & Ely, 2016) for constructing and reconstructing the concept (see Pinto & Scheiner,

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2016). This means, such representations are not always generic in the sense of Mason and Pimm (1984) though they are used as if they were of that nature. Moreover, such representations may be productive in some, though not all contexts, in which they are needed. In spite of the striking differences in the knowledge constructions in each case study, that are made explicit by the nature of the representations construed and their use, the three case studies presented in Pinto (1998) on students’ strategy of giving meaning have in common a cohesion in their sense-making and in learning the formal mathematics concept (Pinto & Scheiner, 2016; Scheiner & Pinto, 2014). Pinto and Scheiner (2016) concluded that coherence amongst students’ sense-making and their (re-)construction of the formal content had been proven to be a central characteristic of those students who ‘give meaning’. This does not mean that the reconstructions a student made are configured in a “satisfactory reconstruction or accommodation” scenario (Vinner, 1991, p. 70); rather, that apart of the learning scenario, a student’s sense making is coherent with her or his learning of a mathematical concept.

It is important to note that the evolving framework of structural abstraction is problem driven, that is, addressing the need for bringing light into the complexity of students’ knowing and learning processes compatible with their strategy of ‘giving meaning’, rather than filling a theoretical gap just because it exists. The reinterpretation of empirical data on students’ strategies of giving meaning in the light of the theoretical framework of structural abstraction proved to be particularly fruitful – not only to provide deeper insights into the strategy of giving meaning but also as a way to deepen our understanding of the phenomenon of structural abstraction that revealed new theoretical developments (Pinto & Scheiner, 2016; Scheiner & Pinto, 2014). In the following sections, we highlight the main theoretical advancements.

The idea of complementizing meaningful components underlying the structural abstraction framework reflects the idea that whether an individual has ‘grasped’ the meaning of a mathematical concept is situated in specific contexts where the objects falling under the specific mathematical concept have been placed in. In the case studies revisited, these contexts include the formal mathematics one, where mathematical objects are presented as formal definitions and their properties are deducted through formal proofs. Such a diversity of situated or contextualized meanings supports Skemp’s (1986) viewpoint that “the subjective nature of understanding […] is not […] an all-or-nothing state” (p. 43). The reanalysis of the data indicates that the object of researchers’ observation should be directed to students’ partial constructions of the limit concept. These partial constructions may be specific and productive to particular contexts but may remain not fully connected and may be unproductive in other contexts (for instance, in making sense of the formal definition). The empirical data show that, in the case of the students who give meaning, several meaningful elements and relations in understanding the limit concept of a sequence are involved, although a few elements are missing (or distorted). However, some students are able to (re-)construct some meaningful components at need by making use of their partial constructions, while others are not able to do so.

Our reanalysis indicates that some students have developed resources that enable them to (re-)construct the limit concept of a sequence at need. Scheiner and Pinto (2014) focused on a case where a student developed a generic representation of the limit concept of a sequence that operates well in several, although not all, contexts and situations. This particular representation, however, allows the student to (re-)construct the limit concept in other contexts and situations. The reinterpretation of the data sheds light on the phenomenon that individuals may not ‘have’ all relevant, meaningful components, but, rather, they may have resources to generate some meaningful components and make sense of the context at need. In

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that sense, the ‘completeness’ of the complementizing process cannot ever be taken as absolute.

Several researchers suggested exposing learners to multiple contexts and situations. An important insight from using the structural abstraction framework retrospectively is that exposure to multiple contexts is at least important for particularizing meaningful components: various objects falling under a particular mathematical concept have to be set into different specific contexts in order to make visible the meaningful components or mathematical structure of these objects. In so doing, the objects may be ‘exemplified’ through a variety of representations, in which each representation has the same reference (the mathematical object); however, different representations may express different senses depending on the selected representation system (see Fig. 2).

limit as asymptote

ideaF

limit concept of a sequence

limit as cluster point

conception

...

conception...

ideaF

ideaF

...

...

ideaF

SEN

SEF

REF

EREN

CE

IDEA

FC

OM

PRES

SIO

N

Fig. 2: Reference, sense, and idea

The distinction between sense and reference has been specified by Frege (1892b) in his work Über Sinn und Bedeutung, indicating both the sense and the reference as semantic functions of an expression (a name, sign, or description). In short, the former is the way that an expression refers to an object, whereas the latter is the object to which the expression refers. According to Frege (1892b), to each representation correspondents a sense; the latter may be connected with an idea that can differ within individuals since people might associate different senses with a given representation. Though multiple contexts and situations are needed, a new context that does not provide a new sense will unlikely be productive for concept construction.

Research also indicates that students may have difficulties with the relationships between the sense and the reference as well as difficulties in maintaining the reference as the sense changes (Duval, 1995, 2006). Thus, one might assume that these difficulties may (at least partly) be overcome by providing students a particular resource (such as a generic

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representation of the mathematical concept) that serves as a guiding tool in complementizing the meaningful components indicated in the different senses. From this perspective, a ‘representation for’ is a tool that provides theoretical structure in constructing the meaning of the concept of observation. It necessarily reflects essential aspects of a mathematical concept but can have different manifestations (Van den Heuvel-Panhuizen, 2003). Concerning the learning of the limit concept of a sequence, the reinterpretation of the data indicates that a slightly modified version of a student’s representation (see Fig. 3) may support the complementizing process when the limit concept is recontextualized.

N

ԑ ԑ

Fig. 3: A generic representation for learning the limit concept of a sequence

Notice that this generic representation for learning the limit concept of a sequence takes account of several students’ common conceptions identified in the research literature, including those as (1) the limit is unreachable, (2) the limit has to be approached monotonically, and (3) the limit is a bound that cannot be crossed (see Cornu, 1991; Davis & Vinner, 1986; Przenioslo, 2004; Tall & Vinner, 1981; Williams, 1991).

The reanalysis of the empirical data gained from Pinto’s (1998) study has shown that students who gave meaning built a representation of the concept and, at the same time, used it as a representation for recognizing and building up knowledge – the reconstruction of the formal concept definition, for instance. The analysis shows that these students consistently used representations of mathematical objects to create pieces of knowledge; or, in other words, the representations were actively taken as representations for emerging new knowledge and making sense of the context and situation. This shift from constituting a representation of the limit concept to using this representation as a representation for (re-)constructing the limit concept in other contexts can be described in terms of shifting from a model of to a model for (Streefland, 1985) – a shift from an after-image of a piece of given reality to a pre-image for a piece of reality to be created. Adopting this view, we may indicate variations in knowledge structures related to the possible explanations that are considered. Models may involve acceptance of other hypothesis through deduction, causality or analogy.

This mental shift from ‘after-image’ to ‘pre-image’ indicates a degree of awareness of the meaningful components and the complexity of knowledge structure that allows the transition from a ‘representation of’ as a result of various representations expressing specific objects set in different contexts to a ‘representation for’ constructing and reconstructing the limit concept, if inter alia, in formal mathematical reasoning. We suggest that a generic representation, as presented in Fig. 3, may provide an instructional tool that supports raising the awareness of meaningful components in learning the limit concept of a sequence. In other

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words, such a generic representation may direct students’ perception of meaningful components although it does not enshrine mathematical knowledge.

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