Tuesday, July 30–Wednesday, July 31, 2002

Last updated July 24, 2002

- A printed book of abstracts will be available at the conference.
- Talks are listed in order of presentation.
- (*) denotes presenter(s) in cases of more than one author.

Tuesday morning | Tuesday late morning | Tuesday afternoon | |

Wednesday morning | Wednesday late morning | Wednesday afternoon | Plenary: Carlson |

Tuesday, 8:30-9:00 (A/B)

Brian Keller

Michigan State University

bkerller@math.msu.edu

Jered Wasburn-Moses*

Michigan State University

jeredwm@math.msu.edu

Dvora Peretz

Michigan State University

pere@math.msu.edu

**Improving Spatial Visualization Skills in Pre-Service Teachers**

The current study has three major purposes: to compare pre-service elementary teachers spatial visualization skills with those of the middle-school students studied in Ben-Haim, Lappan, and Houang (1985); to determine the effectiveness of a set of web-based instructional materials developed by the researchers at improving pre-service teachers spatial visualization skills; and to pilot techniques for using web-based assessment in research.

Pre-service teachers who had not taken a course in geometry (*n*=21)
scored better on average on the MGMP Spatial Visualization Test than the
middle-school students before treatment, but worse than those students
scored after treatment. Pre-service teachers currently enrolled in a
geometry course (*n*=196) outperformed the post-treatment
middle-school students. Additionally, those in the treatment group
(*n*=107) showed modestly greater gain than peers in the control
sections in spatial visualization skill. Techniques and results of
web-based assessment also will be discussed.

Tuesday, 8:30-9:00 (C)

Peter R. Atwood

Cornerstone University

patwood@cornerstone.edu

**Learning to Construct Proofs in a First Course on Mathematical Proof**

This study examined the conceptions of proof that undergraduate students
have upon entry to a transition course on mathematical proof, how they
develop skill in planning and reporting proofs, and obstacles they
encounter. The subjects were sophomores and juniors (*n*=16) in a
transition course at a fairly large university. Assessment of learning
was through quizzes and a final exam developed by the professor with
input from the researcher, augmented by case studies of six students.

The study confirmed obstacles previously identified in the literature: starting direct proofs and proofs by contrapositive, using definitions, and using universal and existential quantifiers. Additional obstacles detected were: choosing mathematical notation and representations, forming induction assumptions, and constructing proofs by contradiction.

This report concludes by suggesting avenues for further related research, and making recommendations for practice.

Tuesday, 9:05-9:35 (A/B)

Jeffrey S. Strickland

United States Military Academy

aj3948@usma.edu or jsstrick@earthlink.net

**Interactive learning using Maple 6®: An Alternative for Teaching
Linear Algebra**

Traditional courses in Linear Algebra are usually taught with lecture as the primary mode of instruction. This paper describes how a non-lecture mode of instruction was used in a mid-level course in Linear Algebra, with some measure of success.

The Research Questions were, "Does a non-lecture course in Linear Algebra engage students more in their own learning than a lecture course, and does this improve their conceptual understanding as measured by the Term End Exam (TEE)?"

The study incorporated a mixed case study and quasi-experimental,
posttest-only, comparison group design. Preliminary case study data
suggests that a non-lecture course does engage students more in their own
learning than a lecture course. The analysis of the quasi-experiment
employed a pooled *t*-test. The test value was 1.555 compared to a
critical value of 1.303. The experimental group's performance was better
than the comparison group on the TEE, at the .05 level of significance.

Tuesday, 9:05-9:35 (C)

April Judd*

University of Northern Colorado

judd8691@unco.edu

Shandy Hauk

University of Northern Colorado

hauk@fisher.unco.edu

David Tsay

University of Northern Colorado

tsay7018@blue.univnorthco.edu

**Case Study of a Ph.D. Mathematician Teaching College Algebra**

We offer a preliminary report aimed at providing a research foundation for practical efforts to improve teaching and learning in collegiate mathematics "service" courses. The primary participant in this case-study was a Ph.D. mathematician with twelve years college teaching experience. Every class meeting of each of his two college algebra courses for one semester were videotaped. Selected tapes were transcribed and analyzed through the lens of social cognitive theory. We describe the evolution of discourse between students and teacher and the relationships among curriculum intended, implemented, expected, and received. We also identify conflicts between intended and implemented curriculum in the course coordination processes, within the instructor himself, and between the instructor's implementation and students' expectations. Possible implications for teaching include: foci for training graduate students, suggestions for structuring of course coordination efforts to minimize curricular values conflicts, and potential action-research paradigms for instructors in similar circumstances.

Tuesday, 9:40-10:10 (A/B)

Ann Marie Murray

Hudson Valley Community College

murraann@hvcc.edu

**Notation, Notation, Notation: First-Time Calculus Students Discovering
the Product Rule**

This presentation is a report of the results on notation from a study that reported an analysis of students' understanding of the function concept and the notation used to describe it based on the observations of students as they attempted to discover the product rule.

The major findings of the qualitative study include: 1) students committed the common error identified by Leibniz as the first step in attempting to discover the product rule; 2) none of the students were able to clearly define the function concept; 3) students used examples and equations and a combination of incomplete ideas from multiple definitions to define the function concept; 4) the majority of students did not correctly interpret the notation used; 5) the behaviors were consistent with APOS theory.

Other findings of the study indicate that students who were successful in discovering the product rule were at an object level of understanding the function concept, had medium computational accuracy, remained connected throughout the interview, exhibited high mathematical confidence and were able to connect their verbal and notational work to the function concept.

Tuesday, 10:45-11:15 (A/B)

Shandy Hauk*

University of Northern Colorado

hauk@fisher.unco.edu

Jenq-Jong David Tsay*

University of Northern Colorado

tsay7018@unco.edu

RaKissa Dodgen Cribari, April Brown Judd, Andrew M. Neumann

University of Northern Colorado

dodg0922@unco.edu, judd8691@unco.edu, andyneumann@hotmail.com

**Validations of Proofs: How do Graduate Students Validate Proofs?**

We report on a pilot study of four mathematically trained graduate students as they read and reflected on, i.e. validated, four different purported proofs of a simple theorem. Unlike mid-level undergraduates in a comparable study, when moved to a conviction that some part of a purported proof was implausible these graduate students looked for counter-examples with persistence. The graduate students also considered large scale phenomena like logical structural details and stylistic clarity in determining the validity of a purported proof rather than relying primarily on an examination of the local, representational, details. We discuss the role of affective factors, in particular the interplay of forms of intuition with the validator's sense of conviction and consideration of the writer's intention. Finally, we address implications for the preparation of school and college teachers, who must validate and grade their students' work.

Tuesday, 11:20-11:50 (A/B)

Nell Rayburn*

Austin Peay State University

rayburnn@apsu.edu

Pam Crawford

Jacksonville University

pcrawfo@ju.edu

**An Analysis of College Students' Understandings of Scale**

This paper presents the results of the authors' study of the relationship between students' understanding of scale and their APOS function level. Sixty-four college students at levels ranging from developmental studies through senior level mathematics were given a questionnaire designed to assess their understanding of graphical representations of functions. Based on the questionnaire results, fifteen students were selected to participate in an audiotaped interview. The students were selected to give a cross-section by level and by performance on the questionnaire. The interview protocol was designed to give the authors further insight into the students' questionnaire answers and to provide information necessary to classify the students APOS level with regard to the function concept. Conclusions from this analysis will be presented as well as insights from an in-depth study of two of the interviews, chosen because of the richness of information in the interview responses.

Tuesday, 1:35-2:20 (A/B)

Chris Rasmussen*

Purdue University Calumet

raz@calumet.purdue.edu

Karen Whitehead*

Purdue University Calumet

Karen.Whitehead@valpo.edu

**Knowing Solutions with Rate in Differential Equations**

The purpose of this report is to frame and describe the increasingly complex ways in which students use and reason with rate in the context of first order differential equations. In particular, we elaborate on a particular type of knowing that Broudy (1977) called "knowing-with." He proposed that this form of knowing is different from two other types of knowing that are usually proposed: "knowing-that" and "knowing-how." We propose that knowing-with rate is an essential and not always recognized aspect of developing fluencies with important ideas and methods in differential equations and outline seven key ways of knowing-with rate that underpin students use of rate to predict and organize solution functions. These seven key ways of knowing-with rate are illustrated with examples drawn from the data corpus from two classroom teaching experiments in a university level differential equations course.

Tuesday, 2:25-3:10 (A/B)

Phil Clark

Arizona State University

phil.clark@asu.edu

Sean Larsen

Arizona State University

larsen@math.la.asu.edu

**Making Sense of the Unexpected: A Framework for Analyzing the Surprises
of Classroom Activity**

In this paper, we introduce a conceptual tool designed to help make sense of and react to the inevitable disparities between a teacher or researcher's hypothetical learning trajectory (Simon, 1995), and the actual interactive constitution of mathematical activities in the classroom. It is important for teachers and researchers alike to make sense of these disparities and the framework is intended for use by both of these groups. This conceptual framework emerged as we attempted to make sense of the data collected during the pilot stage of an investigation into learning elementary graph theory. It includes three phases: explanatory analysis, mathematical analysis, and pedagogical analysis. The three types of analyses that make up the framework contributed in meaningful ways to our understanding of the observed deviations from the HLT. We found this understanding to be invaluable as we considered potential instructional interventions.

Wednesday, 8:30-9:00 (A/B)

Todd Grundmeier*

University of New Hampshire

tag2@cisunix.unh.edu

**Problem Posing in a Mathematics Content Class for Pre-Service
Teachers**

This talk is based on the presenter's dissertation research incorporating problem posing into a mathematics content class for pre-service elementary and middle school teachers. This presentation will focus on addressing the following questions,

- How do participants' problem posing abilities change over the course of a semester in which they are engaged in problem posing activities?
- As they are experiencing mathematics from a problem posing perspective what changes occur in pre-service teachers' beliefs about the teaching and learning of mathematics?

Wednesday, 8:30-9:00 (C)

Corinne Schaeffer

SUNY at Buffalo

schaeffe@velocity.net

**Mathematics Majors and Their Understanding of Functions**

The purpose of this research is to investigate the understanding of functions formed by three high achieving students with significant exposure to undergraduate mathematics. Of particular interest is determining the extent to which these advanced students are capable of working flexibly on moderately non-routine tasks involving functions, and identifying the strengths and weaknesses in their understanding of functions and related concepts. A brief overview of the project will be presented along with a few of the tasks used in the interview sessions. Preliminary findings indicate that these high performing students demonstrate varying levels of flexibility. For example, one student has great difficulty making progress with tasks where the algebraic representation for the function is unavailable, while for another student this is not a barrier. Additionally, in cases where solutions are obtained, the ability to justify the solution path varies indicating the formation of a limited depth of understanding.

Wednesday, 9:05-9:35 (A/B)

Elsa Medina*

Cal Poly State University-San Luis Obispo

emedina@calpoly.edu

Beth Chance

Cal Poly State University-San Luis Obispo

bchance@calpoly.edu

**Merging Pre-Calculus and Statistics Content for the Preparation of
Pre-Service Elementary School Teachers**

In this talk, we will discuss a new two-quarter sequence of classes we piloted which integrate college algebra and data analysis concepts for future elementary school teachers. The concepts of functions, modeling, problem solving, and the use of technology to make connections between different representations were emphasized. The goals of these courses are to: (a) integrate algebra and statistics content designed to prepare students for their pre-professional courses and for their future in teaching, (b) emphasize problem solving and hands-on learning, and incorporate appropriate computer technology, (c) be designed cooperatively by mathematics and statistics faculty and be team taught by a mathematics educator and a statistics educator. We will discuss the activities we designed to integrate the two content areas and how the students reacted to them as well as the team teaching experience, the two textbooks, and the technology.

Wednesday, 9:40-10:10 (A/B)

Jennifer Earles Szydlik*

University of Wisconsin Oshkosh

szydlik@uwosh.edu

Stephen D. Szydlik*

University of Wisconsin Oshkosh

szydliks@uwosh.edu

**Exploring Changes in Elementary Education Majors' Mathematical Beliefs
Using a Model of the Classroom as a Culture **

In our presentation, we will describe the culture of a mathematics classroom for preservice elementary teachers that is designed to establish sociomathematical norms (Yackel and Cobb, 1995) that foster autonomy. Data for this work include classroom videotape, and student survey responses and transcribed interviews from both the beginning and end of the course. The data provides both a description of the classroom culture and elementary education students' mathematical beliefs, and they reveal that student beliefs became more consistent with autonomous behavior during the course. Interviewed students attributed this change to specific social and sociomathematical norms including aspects of small group work, work on significant problems with underlying structures, a broadening in the acceptable methods of solving problems, the focus on explanation and argument, and the norm that the mathematics was generated by the students and not the instructor.

Wednesday, 9:40-10:10 (C)

Cindy Stenger

University of North Alabama

cstenger@mnu.edu

**A Characterization of Differentiating Mathematical Thinking Skills and
Views in Immature Undergraduates Before and After a Cooperative Research
Project**

Mathematics educators believe that thinking, reasoning, and problem solving should be central goals of mathematics education. Educators have described ways to implement these goals through cooperative learning, projects, group work, and technology implementation. Yet, we still do not have a workable knowledge of how undergraduates' view mathematics, what mathematical thinking skills they possess, and how to improve or begin to develop mathematical thinking in undergraduates. In my research I have identified mature and immature mathematical thinkers and the skills and views that provide the greatest differentiation between the two groups. For example, it was found that immature students did not understand the importance of mathematics to them personally or the commitment, personal responsibility, and perseverance involved in doing mathematics. The current research focuses on three students characterized as immature and provides an in-depth look at the differentiating skills and views portrayed by these students before and after cooperative research.

Wednesday, 11:20-11:50 (A/B)

Jennifer Olszewski*

Purdue University Calumet

olszewj@axp.calumet.purdue.edu

Kevin Dost*

Purdue University Calumet

dostman027@aol.com

Chris Rasmussen*

Purdue University Calumet

raz@calumet.purdue.edu

**Looking for a Circle and Finding an Apple: The Role of Tools and Bodily
Activity in Mathematical Learning **

As part of a larger study investigating students learning dynamical systems, this report examines how one university student developed powerful ways of knowing acceleration with a tool called the water wheel. Our goal in this analysis is to better understand the use of tools and bodily activity in mathematical learning and how they suggest alternative characteristics to knowing. In particular, we develop the idea of "knowing-with" as it relates to the use of tools and bodily activity. In so doing, we contribute to a line of research that seeks to develop constructs and perspectives for inferring the quality of students' mathematical experiences.

Wednesday, 1:00-1:45 (A/B)

Eric Hsu

San Francisco State University

erichsu@math.sfsu.edu

**Switching Contexts: Undergraduates and Experts Reason about Calculus**

In this talk, we describe our findings from a study on the conceptual development of undergraduate calculus students. We gathered data from interviews with students enrolled in a first year calculus course at a major southwestern public university. We analyze student understanding by tracing the connectedness of different conceptual contexts, and their strategic use of switching between contexts. We use a combination of qualitative analysis of student discourse and graphical tools from network analysis. We also compare the performance on a similar task by graduate students and PhDs.

Wednesday, 2:40-3:25 (A/B)

Gilda de La Rocque Palis *

Puc-Rio, Brazil

gilda@mat.puc-rio.br

Lynne Ipiña *

University of Wyoming

ipina@uwyo.edu

**Building and Interpreting Function Graphs. Analyzing Students´
Mental Constructions **

This research, based on the APOS (Action-Process-Object-Schema) theoretical perspective, addresses the following questions: How can we describe the mental constructions that a student might make in order to develop his understanding of: 1. the concept of a Cartesian graph of a real function of one variable, given by an algebraic rule. 2. the concept of a real function defined by a vertical line test satisfying curve drawn in the Cartesian plane. During the presentation of this work we shall describe our theoretical analysis of the learning of these concepts articulating it with empirical data we have collected.

Wednesday, 4:00-4:30 (A/B)

Karen Marrongelle*

Portland State University

marrongelle@mth.pdx.edu

Michael Keynes*

Purdue University Calumet

keynes@calumet.purdue.edu

Chris Rasmussen*

Purdue University Calumet

raz@calumet.purdue.edu

**Pedagogical Tools: A Conceptual Resource for Teaching**

The purpose of this research is to identify and describe different types of instructional moves that build on students' ideas and further their mathematical development, based on analysis of classroom videorecordings from three different instructors in differential equations. We use the term pedagogical tool to mean something that an instructor intentionally uses in the whole class setting to build on students' ideas for the purpose of furthering their mathematical understandings. Analysis of the data corpus suggests two different types of pedagogical tools used by these instructors: (1) presentation of alternatives that may or may not have been considered by students, and (2) unconventional or informal notational records-of student reasoning. These two pedagogical tools will be discussed and illustrated with examples from the videotaped data corpus.

Wednesday, 4:40-5:40

Marilyn Carlson

Arizona State University

marilyn.carlson@asu.edu

**A Calculus Project: The Role of Theory in Research and Curriculum
Development**

An overview of the project will be followed by a discussion of the theoretical perspectives that have guided investigations of students' developing understandings of the major conceptual strands of first semester calculus. The role of theoretical perspectives in this research project will be elaborated by providing specific illustrations of how the theory has guided the project design and implementation, and how how the results have informed the refinement of the theoretical perspectives. In addition, I will discuss the role that theory has played in guiding the development of curricular modules for function, covariation, limit, and accumulation.

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