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RUME XVI CONFERENCE SCHEDULE |
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THURSDAY February 21, 2013 |
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OPENING SESSION 12:00 12:30 PM |
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Session 1 Preliminary Reports 12:40 1:10 PM |
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Wait a
MinuteIs That Enough to Make a Difference? Daniel Reinholz and Mary Pilgrim
Colorado G Abstract: The one-minute paper (Stead, 2005) is a
technique for facilitating communication between students and the
teacher and promoting reflection. In this paper we focus on the types
of questions students ask and how they may be related to success. We
present preliminary results from an introductory university-level
calculus course, indicating that the nature of questions asked by more
successful and less successful students are different, suggesting that
the types of reflections that students engage in may have a significant
impact on the efficacy of such an intervention. |
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Rethinking Business
Calculus in the Era of Spreadsheets Mike May Colorado H Abstract: The author is Writing and electronic
book to support the teaching of calculus to business students with
the assumption that they will use a spreadsheet as their main
computational engine. With the change in technology, it is appropriate
to rethink the content of the course as a different technology makes
different tasks accessible. This study looks at what the content of the
course should be. It compares the official learning objects of the
course, with de facto learning objectives obtained by analyzing final
exams from 20 sections of the course, and with the results of a survey
of the faculty of the business school, the client discipline. It is
intended that this preliminary study establishes a baseline that can be
used to evaluate the effectiveness of the new approach to the course. |
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Expert Performance on
Routine and Novel Integral Application
(Volume) Problems Krista Toth and Vicki Sealey Abstract: Past research has shown that students
struggle when applying the definite integral concept, and these
difficulties stem from incomplete understanding of the integrals
underlying structure. This study aims to provide insight into the
construction of effective mental structures for integrals by examining
experts solutions to volume problems. Seven mathematics faculty
members from a large, public university solved three calculus-level
volume problems (two routine, one novel) in videotaped interview
sessions. Preliminary analysis shows that the experts have a rich
understanding of definite integrals, and the few instances of errors
seemed to be a result of inattention as opposed to a deficit in
understanding. Their problem-solving process was highly structured and
detailed. The experts visual representations varied from sparse and
static to fully 3-dimensional and dynamic. We hope to use this and past
student data to construct a framework for analyzing student
understanding of integral volume problems. |
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How Pre-Service Teachers
in Content Courses Revise Their
Mathematical Communication Nina White Matchless Abstract: Math content courses aim to develop
mathematical reasoning and communication skills in future teachers.
Instructors often assign problems requiring in-depth written
explanations to develop these skills. However, when a students
conception is incorrect, does written feedback from the instructor
create the cognitive dissonance necessary to effect realignment of the
students understanding? These conceptions may be mathematical (what
is a fraction?) or meta-mathematical (what constitutes a
justification?). Assigning problem revisions theoretically creates
space for cognitive dissonance by having students rethink their
solutions. I investigate a revision assignment in a course for future
teachers to understand the nature of students revisions and the
possible impetuses for these revisions. In particular, I find
preliminary evidence that students revisions demonstrate changes in
their language, mathematics, and use of examples and representations.
Further, students adoption of new representations in their solutions
are largely due to observing peers presentations rather than to
instructor feedback. |
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Opportunity to Learn from
Mathematics Lectures Emilie Wiesner, Tim Fukawa-Connelly, and Aaron
Weinberg Denver Ballroom 1-4 Abstract: Many mathematics students experience
proof-based classes primarily through lectures, although there is
little research describing what students actually learn from such
classroom experiences. Here we outline a framework, drawing on the idea
of the implied observer, to describe lecture content; and apply the
framework to a portion of a lecture in an abstract algebra class.
Student notes and interviews are used to investigate the implications
of this description on students' opportunities to learn from
proof-based lectures. Our preliminary findings detail the behaviors,
codes, and competencies that an algebra lecture requires. We then
compare those with how students behave in response to the same lecture
with respect to sense-making and note-taking, and thereby how they
approach opportunities to learn. |
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Assessment of Students
Understanding of Related Rates
Problems Costanza Piccolo and Warren Code Abstract: This study started with a thorough
analysis of student work on problems involving related rates of change
in a first-year differential calculus course at a large,
research-focused university. In two sections of the course, students'
written solutions to geometric related rates problems were coded and
analyzed, and students' learning was tracked throughout the term. Three
months after the end of term, "think-aloud'' interviews were conducted
with some of the students who completed the course. The interviews and
some of the written assessments were structured based on the
classification of key steps in solving related rates proposed by Martin
(2000). Our preliminary findings revealed a widespread, persistent use
of algorithmic procedures to generate a solution, observed in both the
treatment of the physical and geometric problem, and the approach to
the differentiation, and raised the question of whether traditional
exam questions are a true measure of students' understanding of related
rates. |
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Session 2 Contributed Reports 1:20 1:50 PM |
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Switcher and Persister
Experiences in Calculus I Jessica Ellis, Chris Rasmussen, and Kristin Duncan Gold Coin Abstract: Previous reports show that not only are
too few students pursuing Science, Technology, Engineering, or
Mathematics (STEM) fields, but also many who originally intend to
pursue these fields leave after their experiences in introductory STEM
courses. Based on data gathered in a national survey, we will present
an analysis of 5381 STEM intending students enrolled in introductory
Calculus in Fall 2010, 12.5% of whom switched out of a STEM trajectory
after their experience in Calculus I. When asked why these students no
longer intended to continue taking Calculus (an indicator of continuing
their pursuit of a STEM major), 31.4% cited their negative experience
in Calculus I as a contributing factor. We analyze student and their
instructor survey responses on various aspects of their classroom
experience in Calculus I to better understand what aspects of this
experience contributed to their persistence. |
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Critiquing the Reasoning
of Others: Devils Advocate and Peer Interpretations
as Instructional Interventions Aviva Halani, Owen Davis, and Kyeong Hah Roh
Abstract: This study investigated the ways in which
college mathematics teachers might encourage the development of student
reasoning through critiquing activities. In particular, we focused on
identifying situations in which the instructional interventions were
implemented to encourage the critiquing of arguments and in which
students explained anothers reasoning. Data for the study come from
two teaching experiments one from the domain of combinatorics and the
other from real analysis. Through open coding of the data, Devils
Advocate and Peer Interpretations emerged as effective interventions
for the creation of sources of perturbation for the students and for
assisting in the resolution of a state of disequilibrium. These two
interventions differ in design and in the type of reasoning students
evaluate, but they both provoke students to further develop their
reasoning, and therefore their understanding. We discuss the
implications of these interventions for both research and teaching
practice. |
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Entity versus Process
Conceptions of Error Bounds in Students Reinvention of
Limit Definitions Robert Raish, Michael Oehrtman, Jason Martin,
Brian Fisher, and Craig Swinyard Denver Ballroom 1-4 Abstract: We report results from a guided
reinvention of the definition of sequence convergence conducted in
three second-semester calculus classes. This report contributes to the
growing body of research on how students come to understand and reason
with formal limit definitions, focusing on the emergence of students
understanding of the epsilon quantity, conceived in terms of error
bounds. Using Sfard's framework of the condensation of processes to
entities, we mapped the possible conceptual trajectories followed by
the students in the study. In this report, we detail our map, these
trajectories and students reasoning about other aspects of the formal
definition, and the influence of reasoning about approximations and
error analyses in students progression. |
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Perspectives that Some Mathematicians Bring to University Course Materials Intended for Prospective Elementary Teachers Elham Kazemi and Yvonne Lai
Abstract: Many elementary teachers receive their certification in undergraduate contexts, where they are taught mathematics content courses by mathematics faculty. However, there is a disconnect between courses typically taught in mathematics department, such as service courses for engineering or advanced mathematics courses, and mathematics content courses for teachers often both in the instructors' experience with the material as well as in the way the courses are taught. In this paper, we report on four mathematicians' reviews of one set of materials for content courses for prospective elementary teachers. We report on perspectives these mathematicians brought to the materials regarding mathematics, mathematical knowledge for teaching, and teaching. We report on analysis of these perspectives for what may be visible and invisible about mathematical knowledge for teaching and the work of teaching. |
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Session 3 Preliminary Reports 2:00 2:30 PM |
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Not All Informal
Representations are Created Equal Kristen Lew, Juan Pablo Mejia-Ramos, and Keith
Weber Denver Ballroom 1-4 Abstract: Some mathematics educators and mathematicians have suggested that students should base their proofs on informal reasoning (Garuti et al. 1998). However, the ways in which students implement informal representations are not well understood. In this study, we investigate informal representations made by undergraduates during proof construction. Their use of informal representations will be compared to mathematicians use of informal representations as described in Alcock (2004) and Samkoff et al. (2012). Further, an analysis of different types of informal representations will investigate the necessity to treat these different representations more carefully in the future. |
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Fostering Students
Understanding of the Connection Between Function and Derivative: A
Dynamic Geometry Approach Dov Zazkis Abstract: Students difficulties with relating the
graphs of functions to the graphs of their derivatives have been well
documented in the literature. Here I present a Geometers Sketchpad
based applet, which was used as part of a technologically enriched
Calculus I course. Individual interviews with students conducted after
this in-class activity show evidence of varied and powerful student
problem solving strategies that emerged after participation in the
activity. |
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Determining the Structure
of Student Study Groups Gillian Galle Abstract: Although students are expected to spend
time outside the classroom furthering their understanding of the
material, there has been little verification of what students actually
do when they study. This project observed undergraduate students
studying together outside of the classroom setting in order to
determine what study groups formed and what structures described these
groups. Elements of social network analysis were employed to identify
the groups that students formed. Transcripts of the study sessions were
coded and frequency counts were established for each type of student
interaction in order to characterize the roles students assumed while
studying. This paper discusses the process of identifying the study
groups and sets the groundwork for sharing the student roles. One main
finding of this work is that the presence of a student recognized as an
authority or facilitator of the group impacts the type of conversations that occur in the group setting. |
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Assessing Pre-Service
Teachers Conceptual Understanding of Mathematics Using Praxis
II Data Revathi Narasimhan Abstract: We summarize the preliminary results of a study of conceptual understanding of mathematics by pre-service secondary school math teachers. Our research involves the statistical analysis of data from an actual mathematics Praxis II licensure exam, which was administered nationwide. Through a quantitative, item by item analysis, using a classification of these test items by conceptual difficulty, we obtain insight into the conceptual issues that pre-service teachers have great difficulty with. Our preliminary results show a significant gap between computational and abstract mathematical processes. This in turn, affects the ability of pre-service teachers to be fluent in the domains of both subject and pedagogical content knowledge. |
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A Case Study on a Diverse
College Algebra Classroom:
Analyzing Pedagogical Strategies to
Enhance Students Mathematics
Self-Efficacy Michael Furuto and Derron Coles Gold Coin Abstract: Shifting demographics show America rapidly
diversifying, yet research indicates that an alarming number of diverse
students continue to struggle to meet learning outcomes of collegiate
mathematics curriculum. Consequently, recruitment and retention of
diverse students in STEM majors is a pervasive issue. Using a
sociocultural perspective, this study examined the effect of two
pedagogical strategies (traditional instruction and cooperative
learning) in a diverse College Algebra course on enhancing students
mathematics self-efficacy. Particular attention was paid to
investigating the role student discourse and interaction play in
facilitating learning, improving conceptual understanding, and
empowering students to engage in future self-initiated communal
learning. The goal is to develop an effective classroom model that
cultivates advancement in content knowledge and enculturation into the
STEM community, culminating in a higher retention rate of diverse
students in STEM. Preliminary data analysis suggests that a hybrid
model encompassing both traditional instruction and cooperative
learning successfully enhances students self-efficacy. |
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A Multidimensional
Analysis of Instructional Practices Melissa Mills Abstract: This study is an investigation of the questions that are asked by four faculty members who were teaching advanced mathematics. Each question was analyzed along three dimensions: the expected response type of the question, the Blooms Taxonomy level, and the context of the question within the mathematics content. |
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COFFEE BREAK 2:40 3:10 PM |
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Session 4 Contributed Reports 3:20 3:50 PM |
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Pat Thompson, Neil Hatfield, Cameron Byerly and
Marilyn Carlson Abstract: High school mathematics teachers must have coherent systems of mathematical meanings to teach mathematical ideas well. One hundred five teachers were given a battery of items to discern meanings they held in with respect to quantities, variables, functions, and structure. This paper reports findings on a sample of items that, by themselves, should alert college mathematics professors that foundational understandings they assume students have in advanced mathematics courses likely are commonly missing. |
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Using Cognitive Science
with Active Learning in a Large Lecture College Algebra Course David Miller and Matthew Schraeder Abstract: At a research university near the east coast, researchers have restructured a College Algebra course by formatting the course into two large lectures a week, an active recitation size laboratory class once a week, and an extra day devoted to active group work called Supplemental Practice (SP). SP was added as an extra day of class where the SP leader has students work in groups on a worksheet of examples and problems, based off of worked-example research, that were covered in the previous weeks class material. Two sections of the course were randomly chosen to be the experimental group and the other section was the control group. The experimental group was given the SP worksheets and the control group was given a question-and-answer session. The experimental group significantly outperformed the control on a variety of components in the course, particularly when the number of SP days was analyzed. |
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On the Plus Side: A
Cognitive Model of Summation Notation Steve Strand and Sean Larsen
Abstract: This paper provides a framework for analyzing and explaining successes and failures when working with summation notation. Cognitively, the task of interpreting a given summation-notation expression differs significantly from the task of expressing a long-hand sum using summation notation. As such, we offer separate cognitive models that 1) outline the mental steps necessary to carry out each of these types of tasks and 2) provide a framework for explaining why certain types of errors are made. |
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Preservice Elementary
Teachers Understanding of Greatest
Common Factor Story Problems Kristen Noblet Abstract: Little is known about preservice elementary teachers mathematical knowledge for teaching number theory concepts, like greatest common factor or GCF. As part of a larger case study investigating preservice elementary teachers understanding of topics in number theory, both content knowledge and pedagogical content knowledge (Shulman, 1986), a theoretical model for how preservice elementary teachers understand GCF story problems was developed. An emergent perspective (Cobb & Yackel, 1996) was used to collect and analyze data in the form of field notes, student coursework, and responses to task-based one-on-one interviews. The model resulted from six participants responses to three sets of interview tasks where participants discussed concrete, visual, and story problem representations of GCF. In addition to discussing the model and relevant empirical evidence, I suggest language with which to discuss GCF representations. |
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Session 5 Preliminary Reports 4:00 4:30 PM |
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Crossing Community
Boundaries: Collaboration Between Mathematicians and
Mathematics Educators Sarah Bleiler Abstract: Effective mathematics teachers are able to make connections between mathematical content and pedagogy in their professional practice. One of the most readily prescribed approaches for facilitating teachers ability to make such connections is through the development of collaborations between mathematicians and mathematics educators in venues related to teacher professional development. Most prior research related to collaborative endeavors between these two groups has focused on the products, rather than the process, of collaboration. In this preliminary research report, I present the results of an interpretative phenomenological case study that investigated the team-teaching experiences of a mathematician and a mathematics educator within the context of an undergraduate mathematics teacher preparation program. I present extracts from interviews that highlight the instructors perceptions related to crossing the boundaries of their professional communities of practice, and engage participants in discussion about relevant boundary crossing in their own institutional contexts. |
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Gaya Jayakody Abstract: This study looks at the interplay between
the concept image and concept definition when students are given a task
that requires direct application of the definition of continuity of a
function at a point. Data was collected from 37 first year university
students. It was found that different students apply the definition to
different levels, which varied from formal deductions (based on the
application of the definition) to intuitive responses (based on rather
loose and incomplete notions in their concept image). |
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An Examination of Proving
Using a Problem Solving Framework Milos Savic
Abstract: A link between proving and problem solving has been well established in the literature (Furinghetti & Morselli, 2009; Weber, 2005). In this paper, I discuss similarities and differences between proving and problem solving by using the Multidimensional Problem-Solving Framework created by Carlson and Bloom (2005) on Livescribe pen data from a study of proving (Author, 2012). I focus on two participants proving processes: Dr. G, a topologist, and L, a mathematics graduate student. Many similarities were revealed by using the Carlson and Bloom framework, but also some differences distinguish the proving process from the problem-solving process. In addition, there were noticeable differences between the proving of the mathematician and the graduate student. This study may influence a proving-process framework that can encompass both the problem-solving aspect of proving and the differences found. |
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Talking Mathematics: An
Abstract Algebra Professors Teaching
Diaries Sepideh Stewart, John Paul Cook, Ralf Schmidt,
and Ameya Pitale Abstract: The world of a mathematician, with all its creativity and precision is fascinating to most people. This study is an account of collaboration between mathematicians and mathematics educators. In order to examine a mathematicians daily activities, we have primarily employed Schoenfelds goal-orientated decision making theory to identify his Resources, Orientations and Goals (ROGs) in teaching an abstract algebra class. Our preliminary results report on a healthy and positive atmosphere where all involved freely express their views on mathematics and pedagogy. |
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Scaling Up Reinvention:
Developing a Framework for Instructor
Roles in the Classroom Jungeun Park, Jason Martin, and Michael Oehrtman Abstract: Studies have shown that students have difficulty with the concept of limit, especially when reasoning about formal limit definitions. We conducted a five-day teaching experiment (TE) in a second semester calculus classroom in which students were asked to reinvent a formal sequence convergence definition. Author 3 (2011) detailed how pairs of students reinvented sequence convergence definitions but did not attempt the same instructional heuristic in the classroom. Our analysis focused on the instructor prompts and the TE students' subsequent group discussion through their use of key words and visuals in revising their definition. An interview with the instructor was conducted to investigate his intention of using specific prompts and his thinking about the TE group's choice of words and visuals. In our preliminary analysis, we found that the roles of the instructor were extended beyond those roles previously reported as roles for facilitators with pairs of students. |
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An Analysis of First
Semester Calculus Students Use of
Verbal and Written Language When
Describing the Intermediate Value
Theorem Vicki Sealey and Jessica Deshler Abstract: This preliminary report describes the
second stage of data collection and analysis in a larger study that
examines students written and verbal language when studying basic
theorems in a first-semester calculus course. We examine students
difficulties with understanding and using mathematical language and
notation in both formal written work and informal verbal descriptions.
Not surprisingly, the students in our study rarely use formal
mathematical language without being prompted to do so. One surprising
result was that while many students do understand the mathematical
notation in the theorems, and can illustrate this graphically when
prompted, they still do not use this notation when providing their own
written (or verbal) description of a theorem. Preliminary results
suggest that our biggest obstacle as teachers is not in getting our
students to understand the notation, but instead lies in convincing our
students of the power that comes from this notation in describing a
concept, thus encouraging our students to use this notation in their
own written work. |
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Session 6 Contributed Reports 4:40 5:10 PM |
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Students Conceptions of
Mathematics as a Discipline George Kuster Abstract: Researchers have found that students
beliefs about mathematics impact the way in which they learn and
approach mathematics in general. The purpose of this study is to
categorize college students various conceptions concerning mathematics
as a discipline. Results from this study were used to create a preliminary
framework for categorizing student conceptions. The results of this study indicate that the
conceptions are numerous and range greatly in complexity. The results also suggest the need for
further study to qualify the various student conceptions and the roles they play in students'
understanding of and approach to performing mathematics. |
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Using Disciplinary
Practices to Organize Instruction of
Mathematics Courses for Prospective Teachers Yvonne Lai Abstract: One challenge of teaching content courses for prospective teachers is organizing instruction in ways that represent the discipline with integrity while serving the needs of future teachersfor example, choosing math problems that provide a logical development of a topic while also addressing mathematical knowledge for teaching. This paper examines the work entailed in structuring in-class work in mathematics courses for teachers. It argues that practices of teaching that are mathematicalsuch as representing ideas, grounding reasoning in mathematical observations available to the class, using definitions, or using mathematical languagecan be used to negotiate mathematical and pedagogical aims, and therefore can be used to organize instruction of mathematical knowledge for teaching while simultaneously developing a disciplinary understanding. |
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On the Emergence of
Mathematical Objects: The Case of e^(az) Ricardo Nemirovsky and Hortensia Soto-Johnson Abstract: In this report we propose an alternate account of mathematical reification as compared to Sfards (1991) description, which is characterized as an instantaneous quantum leap, a mental process, and a static structure. Our perspective is based on two in-service teachers exploration of the function , using Geometers Sketchpad. Using microethnographic analysis techniques we found that the long road to beginning to reify the function entailed interplay between body-generated motion and object self-motion, kinesthetic continuity between different sides of the same thing, cultural and emotional background of life with things-to-be, and categorical intuitions. Our results suggest that perceptuomotor activities involving technology may serve as an instrument in facilitating reification of abstract mathematical objects such as complex-valued functions. |
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Commonly Identified
Students Misconceptions About Vectors
and Vector Operations Aina Appova and Tetyana Berezovski Abstract: In this report we present the commonly identified error patterns and students misconceptions about vectors, vector operations, orthogonality, and linear combinations. Twenty three freshmen students participated in this study. The participants were non-mathematics majors pursuing liberal arts degrees. The main research question was: What misconceptions about vector algebra were still prevalent after the students completed a freshmen-level linear algebra course? We used qualitative data in the form of artifacts and students work samples to identify, classify, and describe students mathematical errors. Seventy four percent of students in this study were unable to correctly solve a task involving vectors and vector operations. Two types of errors were commonly identified across the sample: a lack of students understanding about vector operations and projections, and a lack of understanding (or distinction) between vectors and scalars. Final results and conclusions include research suggestions and practitioner-based implications for teaching linear algebra in high school and college. |
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POSTER SESSION & RECEPTION 5:30 6:30 PM |
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Thats Nice but is it
worth sharing Daniel Reinholz Abstract: "Group-worthy" problems (cf. Featherstone et al., 2011) are nontrivial, often have multiple solution paths and require multiple competencies; in short, they provide opportunities for students to engage in meaningful groupwork. In contrast, many standard tasks degenerate into one student teaching the other, because the tasks do not have proper affordances to support collaborative learning. I present on "peer-worthy" problems, a set of problems that are useful for pairs of students to work on in settings other than full-blown collaborative groupwork. Peer-worthy problems should satisfy a number of the following criteria; they: (1) are nontrivial, (2) have multiple solution paths, (3) require students to generate examples, and (4) involve explanation. In this poster I contrast student interactions around two problems - one peer-worthy, the other a standard task. |
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The Flipped Classroom
Model for College Algebra: Effects on
Student Achievement Jerry Overmyer The past few years have seen a substantial
rise in the use and interest in a teaching and learning paradigm most
commonly known as the flipped classroom. It is called the flipped class
model because the whole classroom/homework paradigm is "flipped". In
its simplest terms, what used to be classwork (the lecture) is done at
home via teacher-created videos and what used to be homework (assigned
problems) is now done in class. This quantitative research compares 5
sections (n = 144) of college algebra using the
flipped classroom methods with 6 sections (n = 181) of
traditional college algebra and its effect on student achievement as
measured through a common final exam. The data will be analyzed using
ANOVA with interactions of the intervention measured with gender and
ACT mathematics scores. |
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An Annotation Tool
Designed to Interface with Webwork: Interpreting Students
Written Work Nicole Engelke, Gulden Karakok, and Aaron Wangberg Abstract: We present how we are using tablets with an open-source online homework system to collect students written work to calculus problems. The new whiteboard feature captures all student written work in real time. An annotation tool has also been incorporated into the system. Through this tool, we are examining how students solve chain rule problems and what actions they take to correct their mistakes. At this poster, we will allow users to try out the annotation tool and provide results of how we have used it to date. |
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Bethany Fowler, Kristin Frank,
Hyunkyoung Yoon and Marilyn Carlson Abstract: This poster illustrates and describes student thinking when responding to applied problems to relate two quantities that cannot be directly related by a single formula. Students who understood the meaning of the directive to define one quantity in terms of another, and who also conceptualized variables as representing varying values that a quantity can assume were successful in constructing a meaningful formula to relate the values of two quantities that cannot be directly related by a single formula. Students who failed to construct meaningful formulas during their solution process either held the view that a variable is an unknown value to be solved for or could not meaningfully interpret the directive of the problem statement. |
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Stefanie Livers Abstract: This program evaluation study examines the
impact of providing professional development on coaching strategies and
mathematics pedagogy to university supervisors on their supervision
practice and teacher candidates instructional practice and beliefs
about mathematics. The mixed-methods study was designed to answer the
following two questions: What are the effects of training university
supervisors in mathematics pedagogy and coaching strategies on their
supervision practices of elementary teacher candidates? What are the
effects of training university supervisors in mathematics education and
coaching strategies on elementary teacher candidates instructional
practices and their beliefs about mathematics? The qualitative data consisted of supervisors background experience, observations, and interviews. Quantitative data included teacher candidates performance on the Reformed Observation Teaching Protocol (RTOP) and belief scores from the Mathematics Beliefs Instrument (MBI). Analysis of the data revealed that university supervisors support changed as a result of the professional development, thus changing the beliefs of teacher candidates. |
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PLENARY SESSION 6:30 8:30 PM
The Role and Use of
Examples in Learning to Prove Dr. Eric Knuth University of Wisconsin, Madison Abstract: Proof is central to
mathematical practice, yet a perennial concern in mathematics education
is that students of all ages struggle to understand the nature of
evidence and justification in mathematics. Mathematics education
scholars have suggested that overreliance on examples to justify the
truth of statements is an underlying reason for students difficulties
learning to prove. As such, example-based reasoning has typically been
viewed as a stumbling block to overcome. My colleagues and I, however,
view example-based reasoning as an important object of study and posit
that examples play both a foundational and essential role in the
development, exploration, and understanding of conjectures, as well as
in subsequent attempts to develop proofs of those conjectures. In this
talk, I will discuss research related to the role and use of examples
in learning to prove as well as implications for teaching mathematics
at both the secondary and tertiary levels. |
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FRIDAY February
22,
2013 |
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Session 7 Preliminary Reports 8:30 9:00 AM |
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Comparing a
Flipped Instructional Model in an undergraduate Calculus III Course Nicholas Wasserman, Scott Norris
and Thomas Carr Colorado C Abstract: In this small comparative study, we explore the impact of flipping the instructional delivery of content in an undergraduate Calculus III course. Two instructors collaborated to determine daily content and lecture notes; one instructor altered the instructional delivery of the content (not the content itself), utilizing videos to communicate procedural course content to students out-of-class, with time in-class spent on conceptual activities and homework problems. With similar numbers (n=41 and n=40) and types of students in each class, student performance on tests for both classes will be compared to determine any significant differences in achievement related to flipping the instructional delivery of content. |
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Development and Analysis
of a Basic Proof Skills Test Sandra Merchant and Andrew Rechnitzer Abstract: We have developed a short (16 question) basic skills test for use in our institution's transition-to-proof course that assesses basic skills required to succeed in such a course. Using this test in our core introductory proof course, we have found that students are generally deficient in a number of skills assumed by instructors. In addition, using this test as a pre/post-test we have found that in this course students are learning some concepts well, but that learning gains on other concepts are much below desired levels. Finally, administration of the test to students in a higher level course has allowed us to assess retention of these skills. At this preliminary stage these skills appear to be retained into higher-level proof courses, but more data collection is needed, as well as a more extensive instrument to assess proof skills, rather than simply basic logic and comprehension. |
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A Microgenetic Study of
One Students Sense Making About the Temporal Order of Delta
and Epsilon Aditya Adiredja Abstract: The formal definition of a limit, or the epsilon delta definition is a critical topic in calculus for mathematics majors development and the first chance for students to engage with formal mathematics. This report is a microgenetic study of one student understanding of the formal definition focusing on a particularly important relationship between epsilon and delta. diSessas Knowledge in Pieces and Knowledge Analysis provide frameworks to explore in detail the structure of students prior knowledge and their role in learning the topic. The study documents the progression of the students claims about the dependence between delta and epsilon and explores relevant knowledge resources. |
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An Investigation of
Pre-Service Secondary Mathematics
Teachers Development and
Participation in Argumentation Lisa Rice Abstract: This study investigates how two professors and pre-service secondary mathematics teachers engage in argumentation and proof in two courses. One course under investigation is a geometry course; the second is a methods of teaching mathematics course. The research also studies the how professors and pre-service teachers construct arguments and proofs. Examining the classroom discourse to understand how it may impact argumentation practices is another aspect of the research. Case study and grounded theory approaches are used to guide the data collection and analysis. Some data collected include interviews with the two professors and pre-service teachers and observations of the two courses and the pre-service teachers classrooms during their student teaching. Data analysis so far indicates the geometry professor engages students in argumentation and proof in multiple ways. |
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Emergent Modeling and
Riemann Sum Kritika Chhetri and Jason Martin Abstract: This research focuses on mental challenges that students face and how they resolve these challenges while transiting from intuitive reasoning to constructing a more formal mathematical structure of Riemann sum while modeling real life contexts. A pair of Calculus I students who had just received instruction on definite integral defined using Riemann sums and illustrated as area under the curve participated in multiple interview sessions. They were given contextual problems related to Riemann sums but were not informed of this relationship. Our intent was to observe students transitioning from model of to model for reasoning while modeling these problem situations. Results indicate that students conceived of five major conceptions during their first task and their reasoning from the first task that became a model for reasoning about their next task. In this paper we detail those conceptions and their reasoning that became model for reasoning on the second task. |
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Students Way of Thinking
About Derivative and its
Correlation to Their Ways of Solving
Applied Problems Shahram Firouzian Abstract: Previous researchers have examined students understanding of derivative and their difficulties in solving applied problems and/or their difficulties in applying the basic knowledge of derivative in different contexts. There has not been much research approaching students ways of thinking about derivative through the lens of applied questions. In this research, first I categorized the students way of thinking about the basic concept of derivative by running a survey of questions addressing the different ways of thinking about derivative based on the existing research works. While analyzing these surveys, I used grounded theory and added more ways of thinking about derivative. I specially noticed very incomplete ways of thinking about derivative as described below. Since my goal was looking at the students ways of thinking about derivative through the lenses of applied questions, I also piloted my applied questions survey with 51 multivariable calculus students. I noticed a lot of students struggling with defining variables (the initial translation as described below) and if they could define the variable, a lot of them struggled on applying their ways of thinking about derivative into solving the applied problem. These difficulties are great venues to study their ways of thinking about derivative using their struggle in the applied questions. This is a summary of my initial works on this ongoing research, the goal of which is to shed new insights into students solving of applied problems. |
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Session 8 Contributed Reports 9:10 9:40 AM |
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Preparing Students for
Calculus April Brown Judd and Terry Crites Abstract: This quantitative study compared the implementation of a problem-based curriculum in precalculus and a modular-style implementation of traditional curriculum in precalculus to the historical instructional methods at a western Tier 2 public university. The goal of the study was to determine if either alternative approach improved student performance in precalculus, improved student efficacy around learning mathematics and better prepared students for success in a calculus sequence. The study used quantitative data collection and analysis. Results indicate students who experienced the problem-based curriculum should be better prepared to learn calculus but mixed results in terms of retention and success in calculus. |
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Partial
Unpacking and Indirect Proofs: A Study
of Students' Productive
Use of the Symbolic Proof Scheme Stacy Brown Abstract: This paper examines mathematics majors' evaluations of indirect proofs and of the compound statements used in such forms of proofs. Responses to survey items with a cohort of 23 students and six 1-hour clinical interviews, indicate that the students who could successfully evaluate indirect arguments and who could successfully recognize logically equivalent statements, tended to use partially unpacked (Selden & Selden, 1995) versions of the statement and the proofs and, in so doing, demonstrated a productive use of the symbolic proof scheme. Whereas, both successful and unsuccessful students tended to use proof frameworks (Selden & Selden, 1995). Moreover, successful students' approaches are suggestive of activities, which are rarely found in introductory proof texts, yet may benefit novice proof writers. |
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Megan Wawro and David Plaxco
Abstract: The purpose of this study is to
investigate students concept images of span and linear (in)dependence
and to utilize the mathematical activities of defining, example
generating, problem solving, proving, and relating to provide insight
into these concept images. The data under consideration are portions of
individual interviews with linear algebra students. Grounded analysis
revealed a wide range of student conceptions about the span and/or
linear (in)dependence. The authors organized these conceptions into
four categories: travel, geometric, vector algebraic, and matrix
algebraic. To further illuminate participants conceptions of span and
linear (in)dependence, the authors developed a framework to classify
the participants engagement into five types of mathematical activity:
defining, proving, relating, example generation, and problem solving.
This framework could prove useful as a means of providing finer-grained
analyses of students conceptions and the potential value and/or
limitations of such conceptions in certain contexts. |
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Coherence from Calculus to
Differential Equations Jennifer Czocher, Jenna Tague, and Greg Baker Abstract: Despite recent research efforts to make calculus more coherent with other fields, instructors still express dissatisfaction in the mathematical preparation of their students. Even further, we suggest that there are coherence issues within the field of mathematics. In this paper, we expose and examine an epistemological mismatch between how calculus is expected to be known in calculus and how calculus is expected to be used in differential equations. |
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COFFE BREAK 9:50 10:20 AM |
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Session 9 Contributed Reports 10:30 11:00 AM |
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Odd Dialogues on Odd and
Even Functions Dov Zazkis
Abstract: A group of prospective mathematics teachers was asked to imagine a conversation with a student centered on a particular proof regarding the derivative of even functions and produce a script of this imagined dialogue. These scripts provided insights into the script-writers mathematical knowledge, as well as insights into what they perceive as potential difficulties for their students, and by extension difficulties they may have had themselves when learning the concepts. The paper focuses on the script-writers understandings of derivative and of even/odd functions. |
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Secondary Teachers
Development of Quantitative Reasoning David Glassmeyer, Michael Oehrtman, and Jodie
Novak Abstract: This study was designed to document the development of teachers ways of thinking about quantitative reasoning, one of the standards for mathematical practice in the Common Core State Standards. Using a models and modeling perspective, the authors designed a model-eliciting activity (MEA) that was implemented in a graduate mathematics education course focusing on quantitative reasoning. Teachers were asked to create a quantitative reasoning task for their students, which they subsequently revised three times in the course after receiving instructor, peer, and student feedback. The MEA documented the development of the teachers models of quantitative reasoning, and the findings of this study detail one group of three teachers development over the course. Findings include an overall model of teachers development that is both generalizable and sharable for other researchers and teacher educators. |
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Students Reconciling
Notions of One-to-One Across Two Contexts Michelle Zandieh, Jessica Ellis, and Chris
Rasmussen Abstract: This research is part of a larger study. In previous work we created a framework for analyzing student understanding that incorporates five clusters of metaphorical expressions as well as properties and computations that students spoke about when discussing function or linear transformation. In this paper we apply this framework to the setting of students reconciling their understandings of one-to-one in the context of precalculus-type functions with their understandings of one-to-one in the context of linear algebra. Ideally we would like students to be able to recognize a similar structure for one-to-one in each context, and thereby to strengthen their overall understanding of the notion of one-to-one. This proposal provides four vignettes that we found illustrative of the way students reasoned about one-to-one within and across the two contexts. More broadly we find the case of one-to-one as prototypical of the struggles students have in seeing similarities across contexts. |
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On Mathematics Majors
Success and Failure at Transforming
Informal Arguments into Formal Proofs Bo Zhen, Juan Pablo Meija-Ramos, and Keith Weber Abstract: In this paper, we examine 26 instances in which mathematics majors attempted to write a proof based on an informal explanation. In each of these instances, we represent students informal explanations using Toulmins (1958) scheme, we use Stylianides (2007) conception of proof to identify what one would need to accomplish to transform the informal explanation into a proof. We then compare this to the actions that the participant took in attempting to make this transformation. The results of our study are categories of actions that led students to successfully construct valid proofs and actions that may have hindered proof construction. |
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Session 10 Contributed Reports 11:10 11:40 AM |
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Mathematicians
Example-Related
Activity
When
Proving
Conjectures Elise Lockwood, Amy B. Ellis and
Eric Knuth Abstract: Examples play a critical role in mathematical practice, particularly in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, the role that examples play in the process of exploring and proving conjectures has not received the same attention. In this paper, results are presented from interviews conducted with six mathematicians. In these interviews, the mathematicians explored and attempted to prove several mathematical conjectures and also reflected on their use of examples in their own mathematical practice. Their responses served to refine a framework for example-related activity and shed light on the ways that examples arise in mathematicians work. Illustrative excerpts from the interviews are shared, and five themes that emerged from the interviews are presented. Educational implications of the results are also discussed. |
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Transfer of Critical
Thinking Disposition from Mathematics to
Statistics Hyung Kim and Tim Fukawa-Connelly Abstract: In this study we draw on the constructs of
eagerness, flexibility and willingness to characterize the necessary
disposition for critical thinking that is required in learning
statistics in addition to specific content knowledge (Enis, 1989). We
investigated the challenges that students who are highly successful in
mathematics might have in doing statistics and found that while a
student might have an inquisitive disposition and good proficiency with
the foundational mathematical concepts such as functions and function
transformations, that same student might struggle in statistics. Even
concepts that are seemingly related to their mathematical counterparts
such as what is a variable when considering a population and sample may
cause problems as the question is distinct enough from the mathematical
sense. We suggest that such students may experience greater than usual
affective problems in a statistics class and may, therefore, give up
easier and earlier than students who were less successful
mathematically. |
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The Emergence of Algebraic Structure: Students Come to
Understand Zero-Divisors John Paul Cook Abstract: Little is known about how students learn the basic ideas of ring theory. While the literature addressing student learning of group theory is certainly relevant, the concept of zero-divisor in particular is one for which group theory has no analog. In order to better understand how students come to understand zero-divisors, this talk will present results from a study that investigated how students can capitalize on their intuitive notions of solving equations to reinvent the definitions of ring, integral domain, and field. In particular, the emergence and progressive formalization of the concept of zero-divisor at various stages of the reinvention process will be detailed and discussed. |
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Performance and
Persistence Among Undergraduate Mathematics
Majors Joe Champion and Ann Wheeler Abstract: There is little mixed methods research into the patterns of course taking, performance, and persistence among mathematics majors, in general, and among secondary mathematics majors, in particular. Drawing from a sample of 42,825 mathematics enrollment records at two universities over a six-year period, this study presents quantitative summaries of mathematics majors' performance and persistence in undergraduate mathematics courses alongside qualitative themes from interviews of nine secondary mathematics majors at one of the universities. Implications include potential strategies for mathematics programs and faculty to support the success of mathematics majors in undergraduate mathematics coursework, with special emphasis on prospective secondary mathematics teachers. |
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LUNCH 11:50 12:50 AM |
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Session 11 Theoretical Reports 1:00 1:30 PM |
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Daniel Reinholz Abstract: Perceived mathematical authority plays an important role in how students engage in mathematical interactions, and ultimately how they learn mathematics. This paper elaborates the concept of mathematical authority (Engle, 2011) by introducing two concepts: scope and relationality. This elaborated view is applied to a number of peer-interactions in a specialized peer-assessment context. In this context, self-perceived authority influenced the way feedback was framed (as either questions or assertions).
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Todd Cadwalladerolsker, David Miller, and Kelly Hartmann Abstract: This theoretical paper describes model
analysis and our adaptation of this method to the study of proof
schemes in a transition to proof course. Model analysis accounts for
the fact that students may hold more than one idea or conception at a
time, and may use different ideas and concepts in response to different
situations. Model analysis is uniquely suited to study students' proof
schemes, as students often hold multiple, sometimes conflicting proof
schemes, which they may use at different times. Model analysis treats
each students complete set of responses as a data point, rather than
treating each individual response as a separate data point. Thus, model
analysis can capture information on the self-consistency of a students
responses. Data was collected in a Transition to Proof course and
analyzed using both traditional descriptive statistics and model
analysis. We find that model analysis offers significant insights not
offered by traditional analysis. |
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Implications of Realistic
Mathematics Education for Analyzing
Student Learning Estrella Johnson Abstract: The primary goal of this work is to
articulate a theoretical foundation based on Realistic Mathematics
Education (RME) that can support the analysis of student learning, both
individual and collective, by documenting changes in local activity. To
do so, I will build on previous work on the analytic implications of
the Emergent Perspective, such as Rasmussen and Stephans (2008)
analytic approach to documenting the establishment of classroom
mathematical practices. The Emergent Perspective is broadly consistent
with RME, but the existing analytic methods related to the Emergent
Perspective fail to draw on the theoretical constructs provided by RME.
For instance, current analytic methods fail to draw on the RME Emergent
Models heuristic to inform the analysis of the development of
mathematical practices related to models of/for student mathematical
activity. Here I will be explicitly considering the roles that RME
constructs could play in analytic processes consistent with the
Emergent Perspective. |
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Jason Mark Dolor Abstract: In the past decade, educators and statisticians have made new suggestions for teaching undergraduate statistics, in light of these new recommendations it is important to (re)evaluate how individuals come to understand statistical concepts and how such research should impact curricular efforts. One concept that plays a major role in introductory statistics is hypothesis testing and the computation of the test statistic to draw conclusions in a hypothesis test. This proposal presents a theoretical approach through the development of a hypothetical learning trajectory of hypothesis testing by utilizing sampling distributions as the building block of coming to understand statistical inference. In addition, this proposal presents a way this hypothetical learning trajectory may support the development of research-based curricula that foster an understanding of the test statistic and its role in hypothesis testing. |
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Session 12 Preliminary Reports 1:40 2:10 PM |
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Computational Thinking in
Linear Algebra Spencer Bagley and Jeff Rabin Abstract: In this work, we examine students' ways of
thinking when presented with a novel linear algebra problem. We have
hypothesized that in order to succeed in linear algebra, students must
employ and coordinate three modes of thinking, which we call
computational, abstract, and geometric. This study examines the
solution strategies that undergraduate honors linear algebra students
employ to solve the problem, the variety of productive and reflective
ways in which the computational mode of thinking is used, and the ways
in which they coordinate the computational mode of thinking with other
modes. |
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Pre-service
Secondary Mathematics Teachers Statistical Preparation:
Interpreting
the
News Joshua Chesler Abstract: Undergraduate mathematics programs must prepare teachers for the challenges of teaching statistical thinking as advocated in standards documents and statistics education literature. This preliminary report presents initial results from a study of pre-service secondary mathematics teachers at the end of their undergraduate educations. Although nearly all had completed a required upper-division statistics course, most were challenged by two tasks which required a critical analysis of the use of statistics in newspaper articles. Some patterns emerged in the incorrect answers, including a tendency to focus on potential sampling issues which were not relevant to the tasks. The session will explore the nature and sources of these difficulties with statistical thinking and statistical communication and it will explore the implications for undergraduate mathematics and statistics teacher preparation. |
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The Effects of Formative
Assessment on Students Zone of Proximal Development in
Introductory Calculus Rebecca Dibbs and Michael Oehrtman Abstract: One of the challenges of teaching introductory calculus is the large variance in student backgrounds. Formative assessment can be used to target which students need help, but little is known about why formative assessment is effective with adult learners. The purpose of this qualitative study was to investigate which functions of formative assessment help instructors to provide the scaffolding needed to help students in an introductory calculus course progress through their Zones of Proximal Development during the weekly group labs. By providing students a low-stakes opportunity to demonstrate their current understanding, students were able to evaluate their progress and ask further questions after the activity was completed; this information was used to plan the discussion in the next class period. This discussion provided the scaffolding students needed to progress through the activities as well as providing peripheral participation opportunities for students who would not ordinarily ask questions during class. |
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Verifying Trigonometric
Identities: Proof and Students
Perceptions of Equality Benjamin Wescoatt Abstract: This preliminary study explores how students perceptions of the equality of trigonometric expressions evolve during the process of verifying trigonometric identities (VTI). If students already view the purported equality as being true, VTI may not offer much in the way of learning experiences for students. Using a semiotic perspective to analyze student work, this study attempts to describe the evolution of students perceptions of identities upon application of VTI, focusing on the components of the students VTI process that contribute to the evolution. Initial analyses of interviews conducted while students proved identities indicate that students are not fully convinced that the identities are initially true. However, successful VTI, signaled, for example, by the use of an idiosyncratic equality construction, endows equality on not only the purported identity but on ancillary equality statements generated as part of the VTI process. |
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Student Responses to Team
Based Learning in Tertiary
Mathematics Courses Judy Paterson, Louise Sheryn, and Jamie Sneddon
Abstract: Starting in 2009 we have implemented a Team Based Learning (TBL) model of delivery in two mathematics courses and one mathematics education course involving a total of 295 students. Qualitative data from evaluations, observations and interviews is used to begin to answer four questions raised by the Seldens (2001) regarding teaching mathematics at tertiary levels. Our analysis indicates that students say that TBL creates an environment in which they are active, have productive arguments and discussions and benefit from immediate feedback. There is scant evidence of any group being disadvantaged by this model of delivery. |
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Session 13 Contributed Reports 2:20 2:50 PM |
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Venn
Diagrams as Visual Representations of Additive and Multiplicative
Reasoning
in
Counting
Problems Aviva Halani
Abstract: This case study explored how a student could use Venn diagrams to explain his reasoning while solving counting problems. An undergraduate with no formal experience with combinatorics participated in nine teaching sessions during which he was encouraged to explain his reasoning using visual representations. Open coding was used to identify the representations he used and the ways of thinking in which he engaged. Venn diagrams were introduced as part of an alternate solution written by a prior student. Following this introduction, the student in this study often chose to use Venn diagrams to explain his reasoning and stated that he was envisioning them. They were a powerful model for him as they helped him visualize the sets of elements he was counting and to recognize over counting. Though they were originally introduced to express additive reasoning, he also used them to represent his multiplicative reasoning. |
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Stephenie Anderson Dyben,
Hortensia Soto-Johnson and Gulden Karakok Abstract:
This study explores
in-service high school mathematics teachers conception of various
forms of a complex number and the ways that they transition between
different representations (algebraic and geometric) of these forms.
Data were collected from three high school mathematics teachers via a
ninety- minute interview after they completed professional development
on complex numbers. Results indicate that these teachers do not
necessarily objectify exponential form of complex numbers and only
conceptualized it at the operational level. On the other hand, two
teachers were very comfortable with Cartesian form and showed
process/object duality by translating between different representations
of this form. It appeared that our participants ability to develop a
dual conception of complex numbers was bound by their conceptualization
of the various forms, which in turn was hindered by their
representations of each form. |
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on the Design and
Implementation of An Undergraduate Course on Mathematical Orit Zaslavsky, Pooneh Sabouri, and Michael Thoms Abstract: The goal of our study was to characterize
the processes and to identify the ways in which different kinds of
expertise (mathematics vs. mathematics education) unfolded in the
planning and teaching of an undergraduate course on Mathematical Proof
and Proving (MPP), which was co-taught by a professor of mathematics
and a professor of mathematics education. The content of the course
consisted of topics that were supposed to be familiar to the students,
i.e., high school level algebra, geometry, and basic number theory. In
particular, we looked for instances that would help understand how each
expertise contributed to the course and complemented the other. The
findings indicate that by co-teaching and constantly reflecting on
their thinking and teaching, the instructors became aware of the added
value of working together and the unique contribution each one had. |
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Noelle Conforti Preszler Abstract: At small liberal arts colleges, a single calculus sequence must successfully accommodate students from various majors, such as mathematics, biology, chemistry, and economics. This qualitative case study considers mathematics professors' perspectives about the required nature of calculus in various disciplines, attempts to identify how calculus instructors teach with the aim of preparing students to apply calculus knowledge in their future coursework, and how the disciplinary focus of their students affects professors' design and teaching of calculus courses. Framed using aspects of teaching and learning shown to promote transfer of knowledge, results suggest that the professors teach for understanding and allow in-class processing time, but could improve their emphasis on applying calculus in non-mathematics disciplines. This study contributes to the growing body of undergraduate mathematics education research intended to document undergraduate teaching practices. |
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COFFE BREAK 2:50 3:20 PM |
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Session 14 Preliminary Reports 3:30 4:00 PM |
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Chanakya Wijeratne Abstract: Previous studies have shown that the
normative solutions of the Pin-Pong Ball Conundrum and the Pin-Pong
Ball Variation are difficult to understand even for learners with
advanced mathematical background such as doctoral students in
mathematics. This study examines whether this difficulty is due to the
way they are set in everyday life experiences. Some variations of the
Pin-Pong Ball Conundrum and the Pin-Pong Ball Variation and their
abstract versions set in the set theoretic language without any
reference to everyday life experiences were given to a doctoral student
in mathematics. Data collected suggest that the abstract versions can
help learners see beyond the metaphorical language of the paradoxes.
The main contribution of this study is revealing the possible negative
effect of the metaphorical language of the paradoxes of infinity on the
understanding of the learner. |
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Todd Grundmeier, Dylan Retsek and
Dara Stepanek Abstract: Self-inquiry is the process of posing
questions to oneself while solving a problem. The self-inquiry of
thirteen undergraduate mathematics students was explored via structured
interviews requiring the solution of both mathematical and
non-mathematical problems. The students were asked to verbalize any
thought or question that arose while they attempted to solve a
mathematical problem and its nonmathematical logical equivalent. The
thirteen students were volunteers who had each taken at least four
upper division proof-based mathematics courses. Using transcripts of
the interviews, a coding scheme for questions posed was developed and
all questions were coded. While data analysis of the posed questions is
ongoing, initial analysis suggests that the good mathematics students
focus more questions on legitimizing their work and fewer questions on
specification of the problem-solving task. Additionally, the
self-inquiry of fast problem solvers mirrored that of the strong
students with even less focus on specification questions. |
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Kelly Bubp Abstract: Despite the importance of intuitive and
analytical reasoning in proof tasks, students have various difficulties
with both types of reasoning. Such difficulties may be attributed to
insufficient intuition, logical reasoning skills, or concept images.
However, dual-process theory asserts that intuition can form faulty
representations of tasks based on systematic errors before analytical
reasoning can respond. Thus, students' difficulties could be attributed
to systematic intuitive errors rather than inadequate intuitive or
analytical reasoning. In this study, I conducted task-based interviews
with four undergraduate and one graduate mathematics major in which
they completed prove-or-disprove tasks. In this paper, I discuss the
systematic intuitive errors committed by these students on a
monotonicity task. These errors led all five students to believe
incorrectly that the statement in the task was true. Furthermore, each
student engaged in correct mathematical reasoning guided by their
incorrect intuitive representations. |
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A Coding Scheme for
Analyzing Graphical Reasoning on Second
Semester Calculus Tasks Rebecca Schmitz Abstract: As a first step in studying students spatial reasoning ability, preference, and their impact on performance in second semester calculus, I ran a pilot study to develop interview tasks and a coding scheme for analyzing the interviews. Four videotaped interviews were conducted with each of the five participants and the video was coded for graphical reasoning. I will discuss my coding scheme and share some preliminary results. I hypothesize that the coding scheme may help identify a students preference and ability for spatial reasoning. |
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Mathematicians Tool Use
in Proof Construction Melissa Goss, Jeffrey King, and Michael Oehrtman Abstract: The goals on teaching proof are to help
students develop an understanding of proof that is consistent with that
shared and practiced by mathematicians of today. This study sought to
describe the tools and reasoning techniques used by mathematicians to
construct and write proofs. Task-based clinical interviews were
conducted with 3 research mathematicians in varying research fields.
The tasks were upper-undergraduate and lower-graduate level proofs from
linear algebra, basic analysis, and abstract algebra. Data were coded
based on a framework constructed from Deweys theory of inquiry and the
characterizations of conceptual insight and technical handle.
Preliminary results indicate the task of discovering a conceptual
insight that can potentially lead to a proof can be problematic, and
there are distinct moments in the construction process when the problem
changes from why should this be true? to how can I prove that? |
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Initial Undergraduate
Student Understanding of Statistical
Symbols Samuel Cook and Tim Fukawa-Connelly Abstract: In this study we use the tradition of semiotics to motivate an exploration of the knowledge of, and facility with, the symbol system of statistics that students bring to university. We collected a sample of incoming mathematics majors in their first semester of study, prior to taking any statistics coursework and engaged each in a task-based interview using a think-aloud protocol with questions designed to assess their fluency with basic concepts and symbols of statistics. Our findings include that students find symbols arbitrary and difficult to associate with the concepts. Second, that generally, no matter the amount of statistics that students took in high school, including Advanced Placement courses, they generally have relatively little recall of topics. Most can calculate the mean, median and mode, but they generally remember little beyond that. Finally, students have difficulty connecting practices or procedures to meaning. |
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Session 15 Preliminary Reports 4:10 4:40 PM |
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Proof Structure in the Context of Inquiry Based Learning Alyssa Eubank, Shawn Garrity and
Todd Grundmeier
Abstract: Data was collected from the final exams of 68 students in three sections of an introductory proofs course taught from an inquiry-based perspective. Inquiry-based learning (IBL) gives authority to students and allows them to present to their peers, rather than the instructor being the focus of the class and the authority on proof. This data was analyzed with a focus on proof structure. The selected final exam problems included concepts that were introduced prior to the course and others that were new to students. This research utilizes an adaptation of Toulmins method for argumentation analysis. Our goal was to compare the proof structures generated by these students to previous research that also applied some form of Toulmins scheme to mathematical proof. There was significant variety of proof structures, which could be a result of the IBL atmosphere. |
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Analyzing Calculus Concept Inventory Gains in Introductory Calculus Matthew Thomas and Guadalupe Lozano
Abstract: Research in science education, particularly physics education, indicates that students in Interactively-Engaged classrooms are more successful on tests of basic conceptual knowledge. Despite this, undergraduate mathematics courses are dominated by lectures in which students take a passive role. Given the value of such tests in assessing students' conceptual knowledge, the method for measuring such change is largely unexplored. In our study, students were given one such inventory, the Calculus Concept Inventory, in introductory Calculus classes as a pretest and posttest. We address issues of how gains might be measured on this instrument using two techniques, and the implications of using each of these measures. |
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Jennifer Czocher Abstract: Mathematical modeling perspectives continue to become viable lenses for examining students' mathematical thinking in novel contexts. Thus, it is vital to re-visit past foundational work in problem solving in order to connect new ideas and interpretations with accepted knowledge. The objective of this paper is to examine results from a well-known problem setting (The Cell Problem) to explore alternative interpretations of students' mathematical work. |
Kathleen Melhuish Abstract: This report considers student proof construction in small groups within an inquiry-orientated abstract algebra classroom. During an initial analysis, several cases emerged where students used familiar knowledge from another mathematical domain to provide informal intuition. I will report on two episodes in order to illustrate how this intuition could potentially aid or hinder the construction of a valid proof. |
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Warren Christensen
Abstract: An initial investigation into students
understanding of Eigen theory using semi-structured interviews was
conducted with students at the end of a first-semester course in
quantum mechanics. Many physics faculty would expect students to have
mastery of basic matrix multiplication after a course in Linear
Algebra, and especially so after fairly extensive use of matrices in
quantum mechanics in the context of Ising model spin problems. Using a
previously published interview protocol by Henderson et al, student
reasoning patterns were investigated to probe to what extent there
reasoning patterns were similar to those identified among Linear
Algebra students. Reasoning patterns appeared quite consistent with
previous work; that is, students used superficial algebraic
cancellation, and demonstrated difficulty interpreting their result
even when they arrived at a correct solution. The interview protocol
was modified slightly to probe whether or not students felt the tasks
they were engaging in were mathematical or physics-related. Additional
questions were added at the end of the protocol about how these
concepts were used in their quantum mechanics course. Students were
somewhat successful relating them to Hamiltonians and energy
eigenvalues, but couldnt articulate the type of physical situations
where they might be useful. |
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Erin Glover and Jennifer Noll
Abstract: This preliminary report addresses the need
for research that explores how technology changes the way students
think about statistics and the ways technology can be used to enable
students to construct models to solve statistical problems. This study
focuses on student challenges interpreting a single trial of a
statistical experiment and setting up TinkerplotsTM simulations.
Sixteen students in a lower division introductory statistics course
worked on a task involving the "One Son Policy", a situation in which
families continue to have children until they have a boy. Students
interpretations of what a single trial represented in the One Son
activity fell into four categories, three of which were completing the
task. Further, students had difficulty with using the technology to set
up and interpret a simulation to address the question. These results
suggests that the process of setting up a computer simulation to answer
a statistical question is quite complex. |
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Session 16 Contributed Reports 4:50 5:20 PM |
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Students Axiomatizing in a Classroom Setting Mark Yannotta
Abstract: The purpose of this paper is to examine descriptive axiomatizing as a classroom mathematical activity. More specifically, if given the opportunity, how do students select axioms and how might their intellectual needs influence these decisions? These two case studies of axiomatizing address these questions and elaborate on how students engage in this practice within a classroom setting. The results of this research suggest that while students may at first be resistant to axiomatizing, this mathematical activity also affords them opportunities to create meaning for new mathematical content and for the axiomatic method itself. |
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Understanding Abstract Algebra Concepts Anna Titova Abstract: This study discusses various theoretical perspectives on abstract concept formation. Students reasoning about abstract objects is described based on proposition that abstraction is a shift from abstract to concrete. Existing literature suggested a theoretical framework for the study. The framework describes process of abstraction through its elements: assembling, theoretical generalization into abstract entity, and articulation. The elements of the theoretical framework are identified from students interpretations of and manipulations with elementary abstract algebra concepts including the concepts of binary operation, identity and inverse element, group, subgroup. To accomplish this, students participating in the abstract algebra class were observed during one semester. Analysis of interviews and written artifacts revealed different aspects of students reasoning about abstract objects. Discussion of the analysis allowed formulating characteristics of processes of abstraction and generalization. The study offers theoretical assumptions on students reasoning about abstract objects. The assumptions, therefore, provide implications for instructions and future research. |
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Kevin Moore, Teo Paoletti, Jackie Gammaro, and
Stacy Musgrave
Abstract: An extensive body of research exists on
students function concept in the context of graphing in the Cartesian
coordinate system (CCS). In contrast, research on student thinking in
the context of the polar coordinate system (PCS) is sparse. In this
report, we discuss the findings of a teaching experiment that sought to
characterize two undergraduate students thinking when graphing in the
PCS. As the study progressed, the students capacity to engage in
covariational reasoning emerged as critical for their ability to graph
relationships in the PCS. Additionally, such reasoning enabled the
students to understand graphs in the CCS and PCS as representative of
the same relationship despite differences in appearance. Collectively,
our findings illustrate the importance of covariational reasoning for
conceiving graphs as relationships between quantities values and that
graphing in the PCS might create one opportunity to promote such
reasoning when combined with graphing in the CCS. |
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Using Metaphors to Support Students Ability to Reason about Logic Paul Christian Dawkins and Kyeong Hah Roh Abstract: In this paper, we describe an inquiry-oriented method of using metaphors to support students development of conventional logical reasoning in advanced mathematics. Our model of instruction was developed to describe commonalities observed in the practice of two inquiry-oriented real analysis instructors. We present the model via a general thought experiment and one representative case study of a students metaphorical reasoning. Part of the success of the instructional method relates to its ability to help students reason about, assess, and communicate about the logical structure of mathematical activity. In the case presented, this entailed a students shift from using properties to describe examples to using examples to relate various properties. The metaphor thus imbued key example sequences with meta-theoretical significance. We introduce the term wedge to describe such examples that distinguish oft-conflated properties. We also present our analytical criteria for empirically verifying the specific influence of the metaphorical aspect of instruction. |
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PLENARY SESSION 5:30 6:30 PM How to Support Students in Constructing More
Formal Mathematics Dr. Koeno Gravemeijer
Eindhoven University of Technology Abstract: Formal mathematical
knowledge is hard to transmit to students. For it typically concerns
knowledge on a higher level of understanding than the students possess.
Instead of trying to transmit mathematical knowledge, we may aim at
helping students to construct new mathematical knowledge by building on
what they already know. This presents mathematics educators with the
difficult task of building on the informal knowledge of the students
while working towards the conventional formal mathematics one is aiming
for. I will discuss the theory of realistic mathematics education (RME)
that offers guidelines on how to reconcile informal, experientially
real starting points with conventional, abstract, mathematical
endpoints. Key here are processes of generalizing and formalizing.
There are, however, various pitfalls that threaten the realization of
such processes in regular classrooms. |
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DINNER ON YOUR OWN |
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SATURDAY February 23, 2013 |
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Session 17 Preliminary Reports 8:30 9:00 AM |
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Providing Opportunities
for College-Level Calculus Students
to Engage in Theoretical
Thinking Dalia Challita and Nadia Hardy Abstract: Previous research has reported an absence of a theoretical thinking component in college-level Calculus courses; moreover, valid arguments can be made for or against the necessity and feasibility of incorporating such a component. Our belief is, however, that students who wish to engage in theoretical thinking should be given the chance to do so in such a course. The current report presents a preliminary analysis of a study we conducted in a Calculus class in which we presented students with tasks, in the form of quizzes, intended to provoke a type of behavior that is indicative of theoretical thinking. Using Sierpinska et al.s (2002) model as a basis for theoretical thinking we show that students were indeed engaged in theoretical thinking through these tasks. Our preliminary analysis of the results suggests that despite constraints often faced by instructors of such courses, incorporating such a component is indeed feasible. |
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Students Knowledge
Resources About the Temporal Order of
Delta and Epsilon Aditya Adiredja and Kendrice James Abstract: The formal definition of a limit, or the epsilon delta definition is a critical topic in calculus for mathematics majors development and the first chance for students to engage with formal mathematics. Research has documented that the formal definition is a roadblock for most students but has de-emphasized the productive role of their prior knowledge and sense making processes. This study investigates the range of knowledge resources included in calculus students prior knowledge about the relationship between delta and epsilon within the definition. diSessas Knowledge in Pieces provides a framework to explore in detail the structure of students prior knowledge and their role in learning the topic. |
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Two
Students' Interpretation of Rate of Change in Space Eric Weber Abstract: This paper describes a model of the understandings of two first-semester calculus students, Brian and Neil, as they participated in a teaching experiment focused on exploring ways of thinking about rate of change of two-variable functions. I describe the students construction of directional derivative as they attempted to generalize their understanding of one-variable rate of change functions, and characterize the importance of quantitative and covariational reasoning in this generalization. |
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Conceptualizing Vectors
in College Geometry: A New Framework
for Analysis of Student
Approaches and Difficulties Oh Hoon Kwon Abstract: This article documents a new way of conceptualizing vectors in college geometry. The complexity and subtlety of the construct of vectors highlight the need for a new framework that permits a layered view of the construct of vectors. The framework comprises three layers of progressive refinements: a layer that describes a global distinction between physical vectors and mathematical vectors, a layer that recounts the difference between the representational perspective and the cognitive perspective, and a layer that identifies ontological and epistemological obstacles in terms of transitions towards abstraction. Data was gathered from four empirical studies with ninety-eight total students to find evidence of the three major transition points in the new framework: physical to mathematical coming from the first layer, geometric to symbolic and analytic to synthetic from the second layer, and the prevalence of the analytic approach over the synthetic approach while developing abstraction enlightened by the third layer. |
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Jump Math Approach to
Teaching Foundations Mathematics in
2-Year College Shows Consistent
Gains in Randomized Field Trial Taras Gula and Carolyn Hoessler Abstract: Many first year college students struggle with foundational mathematics skills even after one semester of mathematics. JUMP math, a systematized program of teaching mathematics, claims that its approach, though initially designed for K-8, can strengthen skills at the foundations college math level as well. Students in sixteen sections of Foundations Mathematics at a college in Canada were randomly assigned to be taught with either the JUMP math approach or a typical teaching approach. Students were measure before and after on their competence (Wechsler test of Numerical Operations) and attitudes (Mathematics Attitudes Inventory) to identify any improvements. Results showed that students in JUMP classes had modest, but consistently higher improvements in competence when compared to students in non-JUMP classes, even after controlling for potential confounding variables, while improvements in Math Attitudes showed no differences. |
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Identifying Change in
Secondary Mathematics Teachers Melissa Goss, Robert Powers, and Shandy Hauk Abstract: Like several other research groups, we
have been investigating multiple measures for capturing change in
middle and high school teachers mathematical pedagogical content
knowledge (PCK). This article reports on results among 14 teachers (of
16 enrolled) who have completed a virtual masters program in
mathematics education. The degree program seeks to develop content
proficiency, cultural competence, and pedagogical expertise for
teaching mathematics. Analysis included pre- and post-program data from
classroom observations and written PCK assessment. Results indicate
significant changes in curricular content knowledge on the observation
instrument and significant changes in discourse knowledge on both the
observation instrument and the written assessment. Additional path
analysis suggests teacher discourse knowledge as measured by the
written assessments is significantly related to discourse knowledge as
measured by the post-program observation. |
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Session 18 Contributed Reports 9:10 9:40 AM |
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A Dialogic Method of
Presenting Proofs: Focus on Fermats
Little Theorem Boris Koichu and Rina Zazkis Abstract: Twelve participants were asked to de |
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Understanding
of
Commutativity
and
Associativity Steven Boyce Abstract: The purpose of this study is to
investigate a relationship between mathematical content knowledge and
pedagogical knowledge of content and students (Hill, Ball, &
Shilling, 2008), in the context of algebra. As participants in a paired
teaching experiment, mathematics education doctoral students revealed
their understandings of commutativity and associativity (cf. Larsen,
2010). Although the participants knowledge of childrens initial
understandings of algebra and familiarity with mathematics education
literature influenced their own mathematics reasoning, the difficulties
they encountered were similar to those of undergraduates without such
pedagogical content knowledge. |
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Abstract: Combinatorial enumeration has a variety of
important applications, but there is much evidence indicating that
students struggle with solving counting problems. In this paper, the
use of the problem-solving heuristic of solving smaller, similar
problems is tied to students facility with sets of outcomes. Drawing
upon student data from clinical interviews in which post-secondary
students solved counting problems, evidence is given for how numerical
reduction of parameters can allow for a more concrete grasp of
outcomes. The case is made that the strategy is particularly useful
within the area of combinatorics, and avenues for further research are
discussed.
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Cameron Byerley and Neil Hatfield Abstract: In this study, seventeen math education
majors completed a test on fractions and quotient. From this group, one
above-average calculus student was selected to participate in a
six-lesson teaching experiment. The major question investigated was
what constrains and affords the development of the productive meanings
for division and fractions articulated by Thompson and Saldanha
(2003)? The students thinking was described using Steffe and Olives
(2010) models of fractional knowledge. The report focuses on the
students part-whole meaning for fractions and her difficulty
assimilating instruction on partitive meanings for quotient. Her
part-whole meaning for fractions led to the resilient belief that any
partition of a length of size m must result in m, unit size pieces. It
was non-trivial to develop the basic meanings underlying the concept of
rate of change, even with a future math teacher who passed calculus. |
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COFFEE BREAK 9:50 10:20 AM |
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Session 19 Preliminary Reports 10:30 11:00 AM |
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Difficulties
in
Using
Variables
A
Tertiary
Transition
Study Ileana Borja-Tecuatl, Asuman Okta
and Mara Trigueros Abstract: This article describes the results obtained from a diagnostic instrument to establish the difficulties in understanding and using variables that engineering students have at the moment of their entrance to a public Mexican university, that does not examine the candidates prior to admittance. Once the difficulties were established, a 1-year treatment based on the 3 Uses of Variables Model was applied to foster a rich conception of variable in the students. The purpose of this treatment was on one hand to enrich the students concept of variable, to make it possible that they consider variables as dynamic objects that not only represent unknowns, but also describe relations between the objects they represent, and that may vary their usage along one same problem. On the other hand, we wanted to set the basis to study how a poor/rich conception of variable interferes with understanding the solution of a linear equations system. |
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Gestures: A Window to
Mental Model Creation Nancy Garcia and Nicole Engelke
Abstract: Gestures are profoundly integrated into communication. This study focuses on the impact that gestures have in a mathematical setting, specifically in an undergraduate calculus workshop. We identify two types of gesture dynamic and static and note a strong correlation between these movements and diagrams produced. Gesture is a primary means for students to communicate their ideas to each other, giving them a quick way to share thoughts of relative motion, relationships, size, shape, and other characteristics of the problem. Dynamic and static gestures are part of the students thinking, affecting how they view the problem, sway group thinking, and the construction of their diagrams. |
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Development of Students
Ways of Thinking in Vector Calculus Eric Weber and Allison Dorko Abstract: In this talk, we describe the development of the ways of thinking of 25 vector calculus students over the course of one term. In particular, we characterize the generalizations that students made within and across interviews. We focus on the construction of the semi-structured pre and post interviews, trace the construction of explanatory constructs about student thinking that emerged from those interviews, and describe how those constructs fit within the broader literature on student thinking in advanced calculus. We conclude by exploring implications for future research and practical applications for educators. |
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Characteristics of
Successful Programs in College Calculus:
Pilot Case Study Sean Larsen, Estrella Johnson and Steve Strand Abstract: The CSPCC (Characteristics of Successful Programs in College Calculus) project is a large empirical study investigating mainstream Calculus 1 to identify the factors that contribute to success, to understand how these factors are leveraged within highly successful programs. Phase 1 of CSPCC entailed large-scale surveys of a stratified random sample of college Calculus 1 classes across the United States. Phase 2 involves explanatory case study research into programs that are successful in leveraging the factors identified in Phase 1. Here we report preliminary findings from a pilot case study that was conducted at a private liberal arts university. We briefly describe the battery of interviews conducted at the pilot site and discuss some of the themes that have emerged from our initial analyses of the interview data. |
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Cooperative Learning and
Traversing the Continuum of Proof
Expertise Martha Byrne Abstract: This paper describes preliminary results of a study aimed at examining the effects of working in cooperative groups on acquisition and development of proof skills. Particular attention will be paid to the varying tendencies of students to switch proof methods (direct, induction, contradiction, etc) based on their level of proof expertise. |
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Students Sense-Making in
Mathematics Lectures Aaron Weinberg, Tim Fukawa-Connelly, and Emilie
Wiesner Abstract: Many mathematics students experience
proof-based classes primarily through lectures, although there is
little research describing what students actually learn from such
classroom experiences. Here we outline a framework, drawing on the idea
of the implied observer, to describe lecture content; and apply the
framework to a portion of a lecture in an abstract algebra class.
Student notes and interviews are used to investigate the implications
of this description on students' opportunities to learn from
proof-based lectures. Our preliminary findings detail the behaviors,
codes, and competencies that an algebra lecture requires. We then
compare those with how students behave in response to the same lecture
with respect to sense-making and note-taking, and thereby how they
approach opportunities to learn. |
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Session 20 Contributed Reports 11:10 11:40 AM |
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Preservice Teachers
Mathematical Knowledge for Teaching and Concepts of Teaching
Effectiveness: Are They Related? Jathan Austin Colorado D Abstract: Mathematical knowledge for teaching (MKT) is essential for effective teaching of elementary mathematics. Given the importance of MKT, MKT and conceptions of teaching effectiveness should not develop independently. The purpose of this study was to examine whether and how K-8 pre-service teachers MKT and personal mathematics teacher efficacy beliefs are related. Results indicated overconfidence in teaching ability was prevalent, with the majority of participants exhibiting a strong sense of personal mathematics teacher efficacy but low levels of MKT. Pre-service teachers with high levels of MKT, however, reported a more accurate assessment of their teaching effectiveness. Results also indicated that examining pre-service teachers self-evaluations of MKT is helpful for understanding pre-service teachers personal mathematics teacher efficacy beliefs. Moreover, the results of this study point to the inadequacies of existing measures of teacher efficacy beliefs that do not parse out differences in efficacy beliefs according to a number of contextual factors. |
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Students Emerging
Understandings of the Polar Coordinate
System Teo Paoletti, Kevin Moore, Jackie Gammaro, and
Stacy Musgrave Abstract: The Polar Coordinate System (PCS) arises in a multitude of contexts in undergraduate mathematics. Yet, there is a limited body of research investigating students understandings of the PCS. In this report, we discuss findings from a teaching experiment concerned with exploring four pre-service teachers developing understandings of the PCS. We illustrate the role students meanings for angle measure played while constructing the PCS. Specifically, students with a stronger understanding of radian angle measure more fluently constructed the PCS than their counterparts. Also, we found that various aspects of the students understandings of the Cartesian coordinate system (CCS) became problematic as they transitioned to the PCS. For instance, mathematical differences between the polar pole and Cartesian origin presented the students difficulties. Collectively, our findings highlight important understandings that can support or prevent students from developing a robust conception of the PCS. |
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Understanding Mathematical
Conjecturing Jason Belnap and Amy Parrott Abstract: In this study, we open up discussions
regarding one of the unexplored aspects of mathematical sophistication,
the inductive work of conjecturing. We consider the following
questions: What does conjecturing entail? How do the conjectures of
experts and novices differ? What characteristics, behaviors, practices,
and viewpoints distinguish novice from expert conjecturers? and What
activities enable individuals to make conjectures? To answer these
questions, we conducted a qualitative research study of eight
participants at various levels of mathematical maturity. Answers to our
research questions will begin to provide an understanding about what
helps students develop the ability to make mathematical conjectures and
what characteristics of tasks and topics may effectively elicit such
behaviors, informing curriculum development, assessment, and
instruction. |
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Building Knowledge for Teaching Rates of Change: Three Cases of Physics Graduate Students Natasha Speer and Brian Frank Abstract: Over the past two decades education researchers have demonstrated that various types of knowledge, including pedagogical content knowledge, influence teachers' instructional practices and their students learning opportunities. Findings suggest that by engaging in the work of teaching, teachers acquire knowledge of how students think, but we have not yet captured this learning as it occurs. We examined whether novice instructors can develop such knowledge via the activities of attending to student work and we identified mechanisms by which such knowledge development occurs. Data come from interviews with physics graduate teaching assistants as they examined and discussed students' written work on problems involving rates of change. During those discussions, some instructors appear to develop new knowledgeeither about students thinking or about the contentand others did not. We compare and contrast three cases representing a range of outcomes and identify factors that enabled some instructors to build new knowledge. |
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LUNCH 11:50 12:50 AM |
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PLENARY SESSION 1:00 2:00 PM How Visualization Techniques Shape the Landscape of
Mathematics and Science Dr. Loretta Jones
University of Northern Colorado Abstract: Dynamic visualizations of
scientific and mathematical processes can be powerful learning tools.
However, their use has not always enhanced learning and, in fact, has
sometimes been found to mislead learners. Collaborations among the
research communities of cognitive science, education, mathematics, and
physical and life sciences are helping science and mathematics
education researchers to study the instructional uses of visualization
techniques. These collaborations reveal how students perceive and
interpret various kinds of visualizations and animations and show how
to develop design principles for creating and using effective
instructional visualizations. This presentation will examine research
paradigms developed for investigating these areas. |
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COFFEE BREAK 2:10 2:40 PM |
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Session 21 Theoretical Reports 2:50 3:20 PM |
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Does/Should
Theory Building Have a Place in the Mathematics Curriculum? Abstract: Mathematicians distinguish two modes of their practice problem solving and theory building. While problem solving has a robust presence in the mathematics curriculum, it is less clear whether theory building does, or should, have such a place. I will report on a curricular design to support a kind of simulation of mathematical theory building. It is based on the notion of a common structure problem set (CSPS). This is a small set of mathematical problems with a two-part assignment: I. Solve the problems; and II. Find and articulate a mathematical structure common to all of them. Some examples will be presented and analyzed. Relations of this construct to earlier ideas in the literature will be presented, in particular to the notion of isomorphic problems, and cognitive transfer. Designing effective instructional enactments of a CSPS is still very much in an experimental stage, and feedback about this would be welcome. |
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Paul Christian Dawkins Abstract: Sjogren (2010) suggested that formal proof could be understood as an explication (Carnap, 1950) of informal proof. Explication describes the supplanting of an intuitive or unscientific concept by a scientific or formal concept. I clarify and extend Sjogrens claim by applying Carnaps criteria for explication (similarity, exactness, and fruitfulness) to definitions, theorems, axioms, and proofs. I synthesize a range of proof-oriented research constructs into one overarching framework for representing and analyzing students proving activity. I also explain how the analytical framework is useful for understanding student difficulties by outlining some results from an undergraduate, neutral axiomatic geometry course. I argue that mathematical contexts like geometry in which students have strong spatial and experiential intuitions may require successful semantic style reasoning. This demands that students construct rich ties between different representation systems (verbal, symbolic, logical, imagistic) justifying explication as a reasonable analytical lens for this and similar proof-oriented courses. |
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The Action, Process,
Object, and Schema Theory of Sampling Neil Hatfield Abstract: This paper puts forth a new theoretical perspective for students understanding of sampling. The Action, Process, Object, and Schema Theory for Sampling serves as a potential bridge between Saldanhas and Thompsons Multiplicative Conception of Sampling and APOS Theory. This theoretical perspective provides one potential way to describe the development of a students conception of sampling. Additionally this perspective differs from most other perspectives in that it does not focus on the sample size the student uses or the sampling method, but rather how the student understands sampling in terms of a sampling distribution. |
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Illustrating a Theory of
Pedagogical Content Knowledge for Secondary and
Post-Secondary Mathematics Instruction Shandy Hauk, Allison Toney, Billy Jackson, Reshmi
Nair, and Jengq-Jong Tsay Abstract: The accepted framing of pedagogical content knowledge (PCK) as mathematical knowledge for teaching has centered on the question: What mathematical reasoning, insight, understanding, and skills are required for a person to teach mathematics? Many have worked to address this question, particularly among K-8 teachers. What about teachers with broader mathematics knowledge (e.g., from algebra to proof-based understandings of topics in advanced mathematics)? There is a need for examples and theory in the context of teachers with greater mathematical preparation and older students with varied and complex experiences in learning mathematics. This theory development piece offers background and examples for an extended theory of PCK as the interplay among conceptually-rich mathematical understandings, experience of teaching, and multiple culturally-mediated classroom interactions. |
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Session 22 Contributed Reports 3:30 4:00 PM |
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Undergraduate
Students
Models
of
Curve
Fitting Shweta Gupta Abstract: The Models and Modeling Perspectives (MMP)
has evolved out of research that began 26 years ago. MMP research uses
Model Eliciting Activities (MEAs) to elicit students models of
mathematical concepts. In this study MMP were used as conceptual
framework to investigate the nature of undergraduate students models
of curve fitting. Participants of this study were prospective
mathematics teachers enrolled in an undergraduate mathematics problem
solving course. Videotapes of the MEA session, class observation notes,
and anecdotes from class discussions served as the sources of data for
this study. Iterative videotape analyses as described in Lesh and
Lehrer (2003) were used to analyze the videotapes of the participants
working on the MEA. Results of this study discuss the nature of
students models of the concept of curve fitting and add to the
introductory undergraduate statistics education research by
investigating the learning of the topic curve fitting. |
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On the Sensitivity of
Problem Phrasing Exploring the
Reliance of Student Responses on
Particular Representations of Infinite
Series Danielle Champney Abstract: This study will demonstrate the ways in which students ideas about convergence of infinite series are deeply connected to the particular representation of the mathematical content, in ways that are often conflicting and self-contradictory. Specifically, this study explores the different limiting processes that students attend to when presented with five different phrasings of a particular mathematical task - (1/2)^n - and the ways in which each phrasing of the task brings to light different ideas that were not evident or salient in the other phrasings of the same task. This research suggests that when attempting to gain a more robust understanding of the ways that students extend the ideas of calculus in this case, limit one must take care to attend to not only students reasoning and explanation, but also the implications of the representations chosen to probe students conceptions, as these representations may mask or alter student responses. |
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Session 23 Preliminary Reports 4:10 4:40 PM |
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Guided Reflections on
Mathematical Tasks: Fostering MKT in College Geometry Josh Bargiband, Sarah Bell and
Tetyana Berezovski Abstract: This study is a part of ongoing research
on development of Mathematical Knowledge for Teaching (MKT) in
mathematical content courses. Reflective practice represents a central
theme in teacher education. The purpose of this reported study was to
understand the role of guided reflections on mathematical tasks in a
college geometry course. We were also interested in understanding how
guided reflections on mathematical tasks would effect teachers
development of MKT. Our research data consist of participants
reflections, teaching scenarios, and pre-post test results. In this
study we developed a workable framework for data analysis. Audience
discussion will address questions related to the proposed analysis
framework and development of MKT in college mathematics courses. |
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Effects of Collaborative
Revision on Beliefs about Proof
Function and Validation Skills Emily Cilli-Turner Abstract: Although there is much research showing
that proof serves more than just a verification function in
mathematics, there is little research documenting which functions of
proof undergraduate students understand. Additionally, research
suggests that students have difficulty in determining the validity of a
given proof. This study examines the effects of a teaching intervention
called collaborative revision on student beliefs regarding proof and on
student proof validation skills. Student assessment data was collected
and interviews were conducted with students in the treatment course and
in a comparison course. At the end of this study, we will produce a
categorization of the proof functions that students appreciate, as well
as a determination of the value of the teaching intervention on
students abilities to correctly classify proofs as valid or invalid. |
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Assessing Student
Presentations in an Inquiry-Based Learning
Course Christina Eubanks-Turner Abstract: Inquiry-Based Learning (IBL) is an instruction method that puts the student as the focal point of the learning experience. An integral part of many IBL proof-based courses is presentation of proofs given by students. In this report, I introduce an assessment rubric used to evaluate student presentations of theoretical exercises from a presentation script. The W.I.P.E. rubric is based on several different assessment models, which emphasize proof writing and comprehension. The rubric has been created to evaluate in-class presentations in undergraduate Abstract Algebra courses for math and math education majors, which offer graduate credit. |
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The Role of Time in a
Related Rate Scenario Katherine Czeranko Abstract: Students who graduate with an engineering or science degree using applied mathematics are expected to synthesize concepts from calculus to solve problems. First semester calculus students attempting to understand the derivative as a rate of change encounter difficulties. Specifically, the challenges arise while making the decision to apply an average rate of change or an instantaneous rate of change (Zandieh, 2000) to the problem. This paper discusses how students view the derivative in an applied mathematical setting and investigates how the concept of time and other related quantities contribute to the development of a solution. |
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Calculus Students
Understanding of Volume Allison Dorko and Natasha Speer Abstract: Understanding the concept of volume is important to the learning/understanding of various topics in undergraduate mathematics. Researchers have documented difficulties that elementary school students have in understanding volume, but we know little about college students understanding of this topic. This study investigated calculus students volume understanding. Clinical interview transcripts and written responses to volume problems were analyzed. Findings include: (1) some calculus students find surface area when directed to find volume, believing that the surface area computation accounts for the objects three-dimensional space; and (2) some calculus students find volume using reasoning and formulae that contain surface area and volume elements. Comparisons with research on elementary school students thinking and implications for the teaching and learning of volume-dependent topics from calculus are also presented. |
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BREAK 5:00 6:00 PM |
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AWARDS BANQUET & PLENARY SESSION 6:00 9:00 PM PLENARY TALK: Benefits and Limitations of Complimenting Qualitative
Research with Quantitative Studies: Lessons Learned from Five Studies on
Proof Reading Dr.
Keith
Weber
Rutgers University Abstract: Most mathematics educators agree that research in our discipline is most effective when it combines theoretical, qualitative, and quantitative approaches. Yet the research presented at the annual RUME conferences is dominated by qualitative studies, with research reports on quantitative studies being relatively rare. Until recently, this trend was present in my research projects as well, which nearly exclusively employed qualitative studies. Over the last few years, I have begun complimenting this qualitative work with quantitative studies. The purpose of this presentation is to describe what I have learned through this process in the context of five quantitative studies on the reading of mathematical proof. The findings represent my views of how quantitative studies can compliment qualitative research, including how such studies can be conducted and the benefits and limitations in engaging in this type of research. I conclude that quantitative research is a useful but underused methodology in the RUME community. |