Last updated October 20, 2003
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Plenary Talk: Steven Williams
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Panel-Simon, King, & Speer Plenary Talk- Celia Hoyles |
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Panel-Confrey & Sloan Plenary Talk-Ricardo Nemirovsky |
Thursday, October 23, 7:30 –
8:30
Plenary talk
Brigham Young
University
Theories
of Advanced Mathematical Thinking: What they Look Like, What they Do For Us,
and What they Do To Us
Karl Jaspers claimed, "There is no escape from philosophy," pointing
out that even in rejecting philosophy we practice a kind of philosophy. In a
similar way, we cannot escape theory in our work studying the teaching and
learning of mathematics. The question is not whether theory guides our work,
but rather, to what extent is it explicit and helpful. Using examples of
theories developed to look at undergraduate mathematics education, I will
explore how theories both illuminate and hide the phenomena we are interested
in, and how what is implicit in our theories colors what we can know. I will
also explore why, as Kurt Lewin suggested, "there is nothing so practical
as a good theory," and how we can make our theories work for us.
Friday, October 24
8:00 – 8:30, Session #1 (S-CT)
Salon
A
Ann Ryu
University of California, Berkeley
Nicole Gillespie
University of California, Berkeley
Suzanna Loper
University of California, Berkeley
sloper@uclink.berkeley.edu
Nathaniel Brown
University of California, Berkeley
Knowing
as an activity, part 1: Producing explanatory accounts and negotiating
standards of accountability
This talk is the first of two linked sessions in which we examine the cognitive
activity of mathematics and science students in a range of diverse settings in
order to illustrate that knowing is a complex, contingent, situated activity.
By rephrasing the question ("What is known?") in terms of activity
("How is cognitive activity accomplished? What form does it take?")
we can better address issues of process important for instructional and
curricular design, and examine knowing in both "clean" and
"messy" settings (i.e. interviews vs. classrooms). By closely
attending to the detail available in video records, we illustrate that knowing
depends on the interactional setting and the cultural-historical context, which
includes social relations, material artifacts and representational forms. This
session focuses on the cognitive activities of producing explanatory accounts
and negotiating standards of accountability. The following session focuses on
the cognitive activities of argumentation and the negotiation of technical
terms.
Salon B
Natasa
Sirotic*
Simon Fraser University
nsirotic@telus.net
Rina Zazkis
Simon Fraser University
zazkis@sfu.ca
Irrational numbers: Dimensions of knowledge
This report is part (B) of a larger research that investigates preservice
secondary teachers' understanding of irrationality of numbers. In particular,
we focus here on the ideas of density and non-denumerability of irrationals, as
well as their "place" in the set of reals with respect to rationals.
The basic assumption of the conceptual framework that we adopt (following
Tirosh. et al study of rational numbers) is that learners' mathematical
knowledge is embedded in a set of connections among algorithmic, intuitive and
formal dimensions of knowledge. The results suggest that very few prospective
teachers are aware of the striking abundance of irrationals. There was a great
reliance on decimal representation and a tendency to view the infinite decimal
expansion of a number as a process rather than as an existing entity. We find
that underdeveloped intuitions are often related to weak formal knowledge and a
lack of algorithmic experience. In our presentation, we shall discuss some
implications for teaching practices that could address this lacuna.
Salon
C
Sean
Larsen
Arizona State University
Supporting the Guided Reinvention of the
Concepts of Group and Group Isomorphism: An Evolving Local Instruction Theory
In this talk I will describe a local instruction theory that emerged over the course of a sequence of three teaching experiments in elementary group theory. Each of the experimental "classrooms" consisted of two university students and the teacher/researcher. These teaching experiments were guided by the instructional design theory of Realistic Mathematics Education (RME). The goal was to promote the guided reinvention of the concepts of group and group isomorphism. The local instruction theory consists of a sequence of instructional activities and a justification for the sequence. The concepts of group and group isomorphism are seen as first emerging as models of students’ informal mathematical activity and then evolving into models for more formal mathematical reasoning. This process was driven by the students’ participation in classroom mathematical practices. The individual students’ mathematical development is seen as reflexively related to their participation in these mathematical practices.
Salon
D
Margret Hjalmarson*
Purdue University
Heidi Diefes-Dux
Purdue University
Richard Lesh
Purdue University
Data Analysis in Context by First-year Engineering Students
First-year engineering students completed a small group problem solving activity in statistics using either MATLABâ or Microsoft Excelâ. The response was a procedure for ranking airlines based on the likelihood of arriving on time. Upon analysis of 60 responses, four cycles of response were identified to classify products based on the types of statistics, clarity of the response and the consideration of the context of the problem. Groups with responses in the first cycle only computed an average for each airline. Groups in later cycles computed other statistics such as standard deviation or generated graphs (histograms, cumulative distribution plots). A critical difference between cycles was the ability to identify relevant statistical measures and explain the procedure clearly so a third party could use it. This preliminary study raises questions about the use of technology, the application of statistics and group interactions in undergraduate engineering courses.
Friday, October 24
8:35 – 9:05, Session #2 (S-CT)
Salon A
Nathaniel Brown
University of California, Berkeley
njsb@uclink.berkeley.edu
Suzanna Loper
University of California, Berkeley
sloper@uclink.berkeley.edu
Nicole Gillespie
University of California, Berkeley
ngillesp@uclink.berkeley.edu
Ann Ryu
University of California, Berkeley
Knowing as an activity, part 2: Argumentation and
the negotiation of technical terms
This talk is the second of two linked sessions in which we examine the
cognitive activity of mathematics and science students in a range of diverse
settings in order to illustrate that knowing is a complex, contingent, situated
activity. By rephrasing the question ("What is known?") in terms of
activity ("How is cognitive activity accomplished? What form does it
take?") we can better address issues of process important for
instructional and curricular design, and examine knowing in both
"clean" and "messy" settings (i.e. interviews vs.
classrooms). By closely attending to the detail available in video records, we
illustrate that knowing depends on the interactional setting and the
cultural-historical context, which includes social relations, material
artifacts and representational forms. The first session focused on the
cognitive activities of producing explanatory accounts and negotiating
standards of accountability. This session focuses on the cognitive activities
of argumentation and the negotiation of technical terms.
Salon B
Karen Allen *
Purdue University Calumet
karen.whitehead@valpo.edu
Chris Rasmussen
Purdue University Calumet
raz@calumet.purdue.edu
Undergraduate
students’ mental operations in systems of differential equations (SDES)
As part of a semester long classroom teaching experiment in a first course in
differential equations at the university level, we conducted task-based
interviews with six students immediately after their study of first order
differential equations to obtain baseline data on their conceptual resources
for learning systems of differential equations (SDEs). Interpretative analysis
of the interview data generated three themes pertaining to reasoning for
students as they worked on tasks dealing with SDEs. First, students used their
conceptions of rate developed in first order differential equations as well
before as a reasoning tool. Second, students used quantification as a mental
operation. Quantification creates an image (not a specific number) for a new
quantity by mental operating on other quantities. Third, students enacted what
we call a function-variable cheme in their efforts to reason about linear
systems of differential equations.
Salon C
Stephen R. Campbell
Simon Fraser University
An Overview of Kant's Theory of Mathematical Cognition and Its
Implications for Research in Mathematics Education
Over two centuries ago Immanuel Kant developed a comprehensive, systematic, and definitive constructivist theory of cognition. The fundamental component of this influential thinker's cognitive theory concerns the nature of mathematical cognition. Overall, Kant's theory of cognition has been notoriously controversial among mathematicians and philosophers alike, especially in its claim that all knowledge, and in particular mathematical knowledge, is grounded in human experience-constructed in accord with rational principles constituting sensory intuition and conceptual understanding. In emphasizing the interdependence of intuition and understanding, Kant's theory attempts to reconcile traditional empiricist and rationalist views on the development of cognition as being solely either a "bottom-up" or "top-down" phenomenon. This presentation will focus on the historical context and contemporary relevance of Kant's theory of mathematical cognition-thereby providing some insight into why Piaget claimed to be "profoundly Kantian." The talk will conclude with discussion on ways in which Kant's theory informs contemporary research in mathematics education.
Salon D
Gabriela Buendía *
Universidad Autónoma de Hidalgo
buendiag@hotmail.com
Francisco Cordero
Cinvestav-IPN
fcordero@mail.cinvestav.mx
Periodicity in a social practice framework
This research presents a study of the periodical aspect of functions. The main
result is a social-epistemology of periodicity whose elements are extracted
from the activity the individual realizes while constructing mathematical knowledge. This kind
of epistemology has allowed us to design a situation which gives evidence of
the relation between periodicity and the act of predicting. Our focus is
on the relation between social practices, as intentional human activities, and
construction of mathematical knowledge through meaning reconstruction. We
propose a framework in mathematics education in which there is a systemic
interaction between the social, didactic, epistemological, and cognitive knowledge
dimensions. The result of this conjunction has been called socioepistemological
approach and its nature sets the possibility to study social mechanisms
of construction of the mathematical knowledge.
Note: This research is funded under a grant from the CONACYT project: Construcción social del conocimiento matemático avanzado. Estudios sobre la reproducibilidad y la obsolescencia de situaciones didácticas: de la investigación a la realidad del aula (CLAVE: 41740-S).
Salon E
Georgia Tolias *
DePaul University
David Jabon
DePaul University
Quantitative Literacy Among College Students: Investigating the
Concept of Percent
A deep
understanding of percents is of practical utility to any numerically literate
person Unfortunately, too many college students don't possess an adequate
framework within which to interpret quantitative information in
contemporary contexts that makes explicit use of percents. This study
investigates students' conceptions of percents using quantitative and
qualitative research methods. An analysis of student errors on written
assignments is presented in conjunction with a theory-based explanation of the
students' conceptual understanding on questions during clinical interviews. The
combined results serve as a rationale for curricular changes in a
university-level quantitative reasoning course. The results of this study
indicate that students have at best a mechanical understanding of the
relationship between percents, fractions, and decimals which is preliminary to
knowing how to deal with percents in more sophisticated problems
situations. Problems involving successive percent change are particularly
difficult for students-precluding them from a deep understanding of exponential
phenomenon.
Salon A
Chris Rasmussen*
Purdue University Calumet
raz@calumet.purdue.edu,
Jennifer Olszewski
Purdue University Calumet
olszeskij1@calumet.purdue.edu
Kevin Dost
Purdue University Calumet
dostk1@calumet.purdue.edu
Do Graphs of Solution Functions Ever Touch? A Case Study of
Students Reasoning about the Uniqueness of Solutions to First Order Ordinary
Differential Equations
The purpose of this report is to characterize the processes by which students
can come to understand the uniqueness of solution functions to differential
equations where formal mathematical understandings emerge from informal ways of
reasoning. We describe the processes by which this occurred in terms of
three complementary themes: refining asymptotic intuition, shifts in
perspective about rate of change, and the interplay between empirical evidence
and logical necessity. The first theme characterizes how students refine their
initial intuitive theory that non-equilibrium solutions always approach
equilibrium solutions symptotically. The second theme describes a process by
which students shift from reasoning about rate as an adjective to reasoning
about the rate as a function with its own adjectival properties. The third
theme details students' arguments as they navigate between empirical
observations (based on real-world situations or on observable graphs and/or
vectors) and justifications that such and such has to be the case.
Salon B
Michael Oehrtman
Arizona State University
oehrtman@math.asu.edu
Students' Reasoning
about Limit Concepts: Common Metaphors and Foundations for the Development of
Rigor and Formalism
Eight major metaphorical contexts emerged from a study of the instrumental
structure of college calculus students' spontaneous reasoning about limit
concepts. Their reasoning within five of these contexts met the tudy's criteria
for support of implicative reasoning, commitment to the context of the
metaphor, evolution in understanding of the mathematical structure, and
evolution in understanding of the metaphorical ontext. Labeled
"strong" metaphors, these involved reasoning about limits in terms of
a collapse in dimension, approximation, closeness in a spatial domain, a
physical size limit beyond which nothing smaller exists, and the treatment of
infinity as a number. Aspects of students' reasoning with strong metaphors
often exhibited traces of language from instructional sources and could be
followed through time in the students' own evolving understanding. Students'
reasoning within the other three contexts, involving motion, zooming, and
arbitrary smallness, exhibited none of the criteria listed, and were thus
labeled "weak" metaphors.
Salon A
Maria L. Blanton*
University of Massachusetts Dartmouth, USA
mblanton@umassd.edu
Despina A. Stylianou*
City College, The City University of New York, USA
dstylianou@ccny.cuny.edu
M. Manuela David
Universidade Federal de Minas Gerais, Brazil
Manuela@fae.ufmg.br
Instructional Scaffolding and the Zone of
Proximal Development: An Examination of Whole-Class Discourse and Student
Learning in Mathematical
Proof
This presentation focuses on the role of instructional scaffolding in the development of undergraduate students' ability to read and write mathematical proof. We extend our previous work, which built a framework for understanding whole class discourse on proof, by delineating the constructs constituting this framework, interpreting the framework using classroom data on whole-class proof constructions, and identifying how the constructs accessed students' zones of proximal development. The question on which we are currently focusing our attention is how we can know that the teacher, through the four forms of scaffolding, accessed students' ZPD. Our current coding and analysis suggest a tentative hypothesis: Students' proposal of new ideas and their subsequent elaboration and justification of these ideas in a way that furthered the >construction of a proof indicates their development within the ZPD. We will present part of the data and the coding that support this hypothesis.
Salon B
Sally Jacobs
Scottsdale Community College
sally.jacobs@sccmail.maricopa.edu
Advanced Placement BC Calculus Students' Understanding of Variable
The purpose of this exploratory study was to describe the general notions about “variable” that Advanced Placement BC calculus students hold, as well as their conceptions about variable in the context of function, limit and derivative. The study's theoretical perspective draws on the works of Freudenthal, Schoenfeld, and Janvier. The findings indicate that these calculus students think about variable in qualitatively different ways, depending on whether they have a calculational versus conceptual orientation to the mathematical task at hand. When their orientation is calculational, they are less likely to exhibit a view of variable as varying. They also tend to conceptualize variable differently depending on the context. In the context of the symbolic limit expression, they regard the variable x occurring in “x --> a” differently from the way they regard x occurring in “f(x).” In the context of derivative, they exhibit weak conceptions of variable as actually varying.
Friday late morning, October
24
11:15 – 11:45, Session #5
(PR)
Salon A
Tami Martin*
Illinois State University
Roger Day*
Illinois State University
Using Performance-Based Assessment to Assess Teachers' Pedagogical Content Knowledge
We will discuss ways in which performance assessment may be used to assess pedagogical content knowledge in novice teachers. We developed a performance assessment instrument that identifies and describes teacher attributes related to the content knowledge, pedagogical knowledge, and pedagogical content knowledge of preservice teachers. For several years, we have used this criterion-referenced instrument to assess student teacher performance. Because the instrument has also been used for self-assessment, it has become a vehicle for reflection as well as for assessment. For our presentation, we will describe the instrument and its attributes. We will share our perspectives on its use as a tool for assessing pedagogical content knowledge and we will provide specific examples of the assessment of preservice teachers' pedagogical content knowledge based on the instrument. We will solicit participants' reactions to and suggestions for the instrument and its use as well as its viability as a research tool.
Salon B
Kadian M. Howell
University of Maryland
kmhowell@wam.umd.edu
The Impact of
Academic-Centered Peer Interactions on First-Year Success and Retention in
Mathematics, Science, and Engineering
The presenter will give an overview of a developing research project intending
to study the impact of course related and non-course related academic-centered
peer interactions on the academic success and retention of first-year
undergraduates in mathematics, science, and engineering (MSE) programs.
The purpose of this research is two-fold. Firstly, it will inform
mathematicians and mathematics educators about promoting and utilizing various
peer interactions to support the teaching and learning of mathematics.
Secondly, it will provide information to university decision-makers about
facilitating academic enhancement opportunities among peers in MSE programs
that will promote diversity and success of students entering the math and
science pipeline.
Salon C
Eva
Thanheiser
evat@sunstroke.sdsu.edu
Preservice Elementary Teachers’
Conceptions of Multidigit Whole Numbers
Researchers have shown that American elementary school teachers and preservice elementary school teachers (PSTs) are proficient in executing algorithms but not in explaining them conceptually. Extending that work is this examination of what PSTs do and do not understand about the underlying structure of multidigit whole numbers in our base-ten numeration system and how that understanding relates to algorithms. A framework of conceptual structures PSTs hold for multidigit whole numbers is introduced. Implications for instruction and teacher education are discussed.
Salon D
Serkan Hekimoglu
The University of Georgia
shekimog@coe.uga.edu
College Students'
Perceptions of Calculus Teaching and Learning
A 55-item questionnaire were distributed to 510 college
students in calculus classes to investigate their views on mathematics
including their perception about using technology, calculus learning and
teaching. The statistical analysis revealed students' views on calculus, using
technology, and calculus learning and teaching differs significantly across
students who were taught by mathematics professors, graduate teaching
assistants, and instructors. Students' views in technology-integrated calculus
classes were different from the ones in traditional calculus classes. The
overall comparison of students in first semester calculus class with the ones
who were in second semester calculus class revealed students' views on calculus,
using technology, and calculus learning and teaching became more static as they
moved to upper-level mathematics classes. Students are not satisfied with their
calculus education and they have difficulty seeing connections between their
academic area and what they learned in calculus classes.
Salon E
David Kung
St. Mary's College of Maryland
dtkung@smcm.edu
Pedagogical Content
Knowledge of Former Emerging Scholars Teachers
Former Calculus teaching assistants were interviewed about their knowledge
surrounding the concept of the limit. Interviews centered around two
teaching tasks, and sought to understand their Pedagogical Content Knowledge
(PCK) in this area and how it was developed. All subjects showed an
awareness of multiple student strategies, but varied widely in articulating the
typical misconceptions documented in the literature. For all of the
subjects, experiences in Emerging Scholars programs played a vital role in the
development of their PCK. This is preliminary work, and discussion will
focus on the methods, the teaching tasks, and possible directions for future
research.
Friday, October 24
1:15
– 3:15
Martin Simon
Penn State University
Karen King
Michigan State University
Natasha Speer
Michigan State University
Research on the Mathematical Preparation of Teachers
The topic of this session will be the mathematical preparation of teachers (K-16) and how educational research (K-16) can inform work in this area. In this session, panelists will pose and frame questions, audience members will have an opportunity to contribute to discussions, and the panelists will respond to audience-generated ideas. In particular, the session will focus broadly on two questions: What do we know about the mathematical preparation of teachers and why is this such an important issue? How has the research community examined these issues and how might we learn more?
Friday, October 24
3:45 – 4:15, Session #6 (PR)
Salon A
Mark Burtch
Arizona State University
mark.burtch@sccmail.maricopa.edu
The Role of
Conjecturing in the Development of Social Norms in a Differential Equations
Course
This preliminary report will present an ongoing research project designed to
study the roles that conjecturing can play in the development of social norms
and mathematical understanding in an undergraduate differential equations
course. The research project also seeks to characterize conjecturing as a
mathematical activity and to describe possible curricular and pedagogical
strategies that make conjecturing effective. Several examples of
conjecturing that were observed in a semester long classroom teaching
experiment will be explored to illustrate the current state of the project.
Salon B
Jessica Knapp
Arizona State University
Knapp@mathpost.asu.edu
The Role of Examples in Student Proof Schemes
Empirical examples play an important role in convincing College Algebra
students of algebraic facts, even when they are capable of generalized
reasoning. When students are asked to convince themselves about the truth of a
statement, they use examples in several ways. While some of the example usage data
from my study fit within the hierarchy described by Balacheff (1988); there are
other reasons students spend time looking at and considering examples,
particularly for the purposes of authority and reality verification.
Students who use an example authoritatively place value on "concrete"
justification. They tend to believe a proof could contain a mistake, but
an example "shows it works." Some students exhibited the belief that
even after a symbolic proof is provided, there is still some doubt as to its
validity prior to making numeric substitutions, computing the result, and
observing its truth for a specific case, i.e. reality verification.
Salon C
Enrique Galindo*
Indiana University-Bloomington
egalindo@indiana.edu
Patricia Salinas
Instituto Tecnologico y de Estudios Superiores de Monterrey-Monterrey
Mexico
npsalinas@itesm.mx
Angeles Dominguez*
Instituto Tecnologico y de Estudios Superiores de Monterrey-Monterrey
Mexico
angeles.dominguez@itesm.mx
Undergraduate
students'
understanding of calculus in a technology-supported environment
This ongoing study seeks to document the nature and extent of students' understanding
of calculus ideas after taking an introductory college calculus course
supported by technology. Twenty-nine undergraduate students participated
in the study. The class used a new textbook, inspired by non-standard analysis.
A set of computer-supported tasks was designed for several units in the course.
Three main pieces of software were used in the class: the software MathWorlds,
the software Graphing Calculator, and a spreadsheet. The integration of the
different representations made accessible by each piece of software was seen as
a medium for making meaning, and for communicating information and meaning. A
first analysis of the data will be available for this presentation. The
findings will help identify important characteristics of computer-supported
tasks so that they are both good tools to help reveal students’ understandings,
and effective ways to help students build their understandings of calculus
ideas.
Salon D
Jamie Sutherland
University of Wisconsin - Madison
sutherla@math.wisc.edu
Alternative Values
in Mathematics Placement Assessment
This study provides an evaluation of a university mathematics department's
program for placing entering freshmen into first year classes with a focus on
the values communicated by the program. As a result, the study found a
lack of nontraditional values such as diversity in the make-up of courses and
students' individual differences reflected in the program. This talk will
focus on the piloting of a supplemental practice that was developed in order to
fill this need. I will describe the practice and provide initial results
concerning the effect of the practice on students' potential for success as
well as their resulting view of the department's values.
Salon E
Raven Wallace* CANCELED
Michigan State University
ravenmw@msu.edu
Andreas Stylianides*
University of Michigan
Helen Siedel
University of Michigan
An analysis of mathematics textbooks for
undergraduate elementary education majors
Friday, October 24
4:20 – 5:00, Session #7
(L-CT)
Salon A
Andrew Izsák
The University of Georgia
izsak@coe.uga.edu
Joseph F. Wagner
Xavier University
Coordination classes as a lens for understanding the development
and generalization of mathematical modeling knowledge
Izsák (2000), Wagner (2003), and others have argued that theoretical accounts couched in terms of process and object understandings, or in terms of general thinking processes, do not provide researchers with tools for analyzing the moment-by-moment difficulties and accomplishments that students experience as they solve modeling problems. We apply the notion of coordination classes (diSessa & Sherin, 1998) to understand at a fine-grained level how students coordinate their understandings of mathematics and of problem situations to model and construct solutions to problems of elementary probability and algebra. In particular, we demonstrate how coordination classes can afford precise descriptions of students' emerging understandings of problem situations and students' ways of constructing solutions to problems. The examples suggest that coordination classes could be used in future research to gain insight into students learning to solve still other classes of modeling problems.
Salon B
Ju,
Mi-Kyung
Ewha
Kwon, Oh Nam*
Seoul
Perspective
Mode Change in Mathematical Narrative: Social Transformation of Views of
Mathematics in a University Differential Equations Class
As part of a 2-year developmental research project of RME(Realistic Mathematics
Education)-based and reform-oriented differential equations course in a Korean
university, this research has investigated students’ views of mathematics in order to understand the social nature
of mathematics education. The result of classroom discourse analysis has shown
that the students’ ways of talking
about mathematics changed significantly. One of discourse patterns identified
is the switch from “the
third-person perspective”
to “the first-person
perspective”. This change
in the mode of talking about mathematics is interpreted as an indication of the
changes in students’ views
towards mathematics. In the process of the transformation, it turns out that
the lecturer played a crucial role in the sense that she presented the cultural
way of talking about mathematics. As
a practitioner of mathematics, she has been socialized in the specific cultural
ways of talking about mathematics legitimized by the mathematics community.
Friday, October 24, 5:05 –
6:05
Celia Hoyles
University of London
In this presentation I report some
results from a longitudinal study of mathematical reasoning among a large
sample (n=1512) of high-attaining students in England, from age 13 1/2 years to
age 15 1/2 years. I will describe how we were able to identify and characterize
clear progress in reasoning in response to standard items in algebra and
in geometry. However, on less standard items, that for example required an
explanation, we found at best modest progress along with some regression.
Reasons for these results will be discussed.
Saturday, October 25
8:00 – 8:30, Session #8 (S-CT)
Salon A
Susan Gray*
University of New England
sgray@une.edu
Cary Moskovitz
Duke University
E-mail:cmosk@duke.edu
College Students'
Interpretations of Descriptive Statistical Concepts Represented in
Histograms
The interpretation of histograms is a fundamental goal of statistics
instruction. We designed three final exam questions to assess how well
this goal was being achieved in our statistics courses. Students were asked to
identify histograms with similar means and standard deviations, and a histogram
with a skewed distribution. Of 159 students, less than half answered all
three questions correctly. Students had the most difficulty with the skewed
distribution, often relating skewness to a shift in the position of the
distribution along the x-axis rather than to an asymmetrical shape.
Errors in identifying similar means and standard deviations were most
frequently the result of confusing the two measures. That is, students
identified histograms with similar means as having the same standard deviations
and vice versa. In addition to discussing these results, other histogram
interpretation errors are documented , and instructional implications are
considered.
Salon B
Jered Wasburn-Moses
Michigan State University
jeredwm@math.msu.edu
Student approaches to proof-writing as problem-solving in introductory real
analysis
The purpose of this pilot study is to begin to develop a framework describing
students' proof-writing and problem-solving efforts in introductory real
analysis, with particular attention to the students' perspectives on their
practices. This study explored the following questions: (1) What strategies did
a student in an introductory real analysis class employ in trying to answer
exam questions, and what meaning did she give to those activities? and (2) What
role did scratch work play in her problem-solving and proof-writing processes?
A variety of data was collected, including student and instructor interviews,
document analysis, and classroom observation. Based on these data, a
preliminary framework was developed. The majority of responses on the exam were
informed directly by memory, either of the "surface structure" or
"deep structure" of the problem; limited heuristics were employed on
other problems. Scratch work generally served an organizational role, but
occasionally served as a "parallel workspace."
Salon C
David Benitez
Universidad Autonoma Coahuila
Luis Moreno-Armella*
Cinvestav, Mexico D.F.
Mathematical Arguments and their formalization
within a Dynamic World
Recent mathematics curriculum reforms have pointed out the relevance of using
technology in the teaching and learning of mathematics. Working within a
Dynamic World of Geometry, enhances the students' capacity to explore
"geometric objects", and to develop mathematical experiments with
those objects and provides the students with adequate tools for conjecturing
and justifying mathematical statements. We present empirical work done with
students (18-20 years olds) related with these activities of conjecturing and
proving. The results we will present are inaccessible by means of paper and
pencil technology. Our study is part of an investigation of the ecology of
mathematical propositions and the natural way to find their proper context of
generalization. The idea that the dynamic environment provides conceptual tools
that enable students to go beyond visual arguments and come closer to genuine
proofs, is one that we believe worth to work with in didactical environments.
Salon D
Christine Stewart
Procedural Change in
Mathematics: Tales of Adoption and Resistance
* No abstract submitted *
Salon E
Sunday
A. Ajose
East
Carolina University
Using Multiple Representations To Solve Problems:
The Case of an Old Nemesis
Does
the use of multiple representations improve the likelihood of success in
solving mathematical problems? This question was explored using the famous
"students and professors" problem which states: Write an
equation using the variables S and P to represent the statement:
"There are six times as many students as professors at this
university" Use S for the number of students and P for the number of
professors. Four random samples of college students employed
different representations in their efforts to solve the problem. One group
simply read the problem and wrote the corresponding equation. Another restated
the problem before solving it. A third group drew pictures of the problem
before solving it. The fourth group restated the problem and drew pictures before
writing the required equation. However, a chi-squared test detected no
difference in the proportions of people in the four groups who succeeded in
solving the problem!
Saturday, October 25
8:35 – 9:05, Session #9 (S-CT)
Salon A
Marguerite George
Arizona State University
george@mathpost.la.asu.edu
A New View of Teacher Revoicing: Consequences
for the College Mathematics Classroom
Revoicing refers to when one person repeats, recasts, summarizes, rephrases or
translates what another says. Researchers have shown that teacher revoicing is
used, for example, to introduce terminology, emphasize presupposed information,
and create alignments and oppositions within an argument. Using the teacher
modeling process developed by the Teacher Model Group at UC Berkeley, this
qualitative study may be the first to view teacher revoicing in the college
mathematics classroom. The teachers in this study exemplify interesting uses of
teacher revoicing not previously discussed in the literature: to keep a
"level playing field", to allow an error, and to foster autonomy.
These ways of using teacher revoicing along with previous uses help to foster
and sustain discussion in the mathematics classroom.
Salon B
Ernesto Colunga*
Instituto Tecnologico y de Estudios Superiores de Monterrey-Monterrey
Mexico
jcolunga@itesm.mx
Enrique Galindo*
Indiana University-Bloomington
egalindo@indiana.edu
The
Cognitive Development of Students from 9th Grade to College in the Learning of
Linear and Quadratic Functions
The cognitive development of students from 9th grade to college in the learning
of linear and quadratic functions was examined in terms of an operational-structural
model of learning. Toward this aim, Sfard’s three-phase model of conceptual
development was adapted to help understand students' mathematical concept
formation. This study hypothesized that students of upper academic levels, who
have been working longer with the function concept, move more in the evolution
of the process-object continuum toward the reification stage, than students at
lower academic levels. The 150 participants were students enrolled in
10th-grade, and in the first, fourth, and last semesters of the school of
engineering. For both the linear and quadratic functions, significant
differences were found regarding the student’s stages of conception between the
high school students in their tenth grade, and the undergraduate students in
the school of Engineering. No significant differences were found among the
undergraduate students.
Salon C
Gary Hagerty
Black Hills State University
GaryHagerty@bhsu.edu
Using Web Based Technology to enhance College Algebra
During the fall of 2002, half of our College Algebra students used web-based
software to replace traditional textbook assignments. The remaining
students worked traditional assignments. This study included 251
students, eight sections and four full time Professors. Each professor
taught two sections; one with technology and one without. Tests used in
this study reflected the traditional tests previously used. The study
considered the following factors: ACT scores, traditional/non-traditional
students, student's opinion of their math skills, student's opinion of
computers, and whether or not they used web-based technology. In
evaluating the results, the only factor that had a statistically significant
difference was whether or not the students used web-based software.
Students using technology performed statistically better at a 0.001
p-level. This session will discuss the procedures of the study, the
results and some of the directions we are looking at moving based on these
results
Salon D
Kate Riley
California Polytechnic State University
Prospective secondary mathematics teachers’ conceptions of proof and
refutations
A research study was conducted to investigate prospective secondary mathematics
teachers' conceptions of proof and refutation as they neared completion of
their preparation programs. The researcher administered the questionnaire to 23
prospective secondary mathematics teachers. Examining prospective teachers'
ability to complete mathematical proofs show that only 57% of the participants
were able to write a valid proof of a proof common to the high school geometry
curriculum. Only 39% were able to write a valid proof about even integers and
only 39% were able to recognize and refute a false conjecture. Results suggest
that the vision of the MAA (1998) and the NCTM (2000) recommendations for
teaching reasoning and proof to all students K - 12, and in all mathematics
content areas, may not be attainable by all prospective teachers. The results
imply that some participants have not retained their knowledge of proof or
their knowledge of proof is insufficient.
Salon E
Maria Trigueros*
ITAM
Bernadette Baker*
Drake University
bernadette.baker@drake.edu
Clare Hemenway
University of Wisconsin Marathon
chemenwa@uwc.edu
Calculators and Writing
in Learning Transformations in College Algebra
One typical pre-calculus approach introduces students to transformations of
basic functions to help them develop a better understanding of functions. There
is no research focusing on how or if this type of course achieves its goal. The
present study addresses this issue as well as the difficulties students face
when working with the concept of transformations of functions. This research
attempts to explain, in terms of APOS (Action, Process, Object, Schema) theory,
the difficulties that students exhibited in one particular course and to gain
insights into why many students were not as successful as expected.
Through the analysis of detailed interviews with 24 students, this study
describes students' conceptions of transformation, and shows that this is a
difficult concept that involves many subtleties that must be taken into account
in designing instruction.
Saturday, October 25
9:10 – 9:50, Session #10
(L-CT)
Salon A
*Michael McDonald
Occidental College
mickey@oxy.edu
*Kirk Weller
University of North Texas
wellerk@unt.edu
Anne Brown
Indiana University South Bend
abrown@iusb.edu
A genetic
decomposition of infinite iterative processes and their encapsulation into
objects
We report on the development of college students' conceptions of infinite
iterative processes. Two sets of interviews were conducted with students
attempting to solve a set theory problem involving an infinite union of power
sets. We used APOS Theory to begin to describe the types of mental
constructions that students appear to make as they come to understand the
concept of infinite iterative processes. A process conception of infinite
iteration develops as the individual becomes able to coordinate multiple
instantiations of finite iterative process. The individual needs to conceive of
the infinite process as being complete, and see it as a totality. As he or she
attempts to construct an action of evaluation on the process, the encapsulation
of the process into an object, called its transcendent object, may occur.
This object, the state at infinity, is understood to be related to, but beyond
the objects produced by the process.
Salon B
Neil Portnoy*
Stony Brook University
nportnoy@math.sunysb.edu
Thomas Mattman*
California State University, Chico
tmattman@csuchico.edu
Undergraduates
study knot invariants as functions: What understandings are revealed?
Twenty-one students in a college geometry course engaged in a two-week
curriculum module in knot theory. The module focused on classification of knots
using various knot invariants including the Jones polynomial. A knot invariant
is a function from the set of all knots to some other set. The curriculum made
explicit connections between the term “invariant” and the mathematical
construct of function. This study indicates a wide variation among students in
the depth of understanding of functions and in their flexibility to move from
their experience with real functions to a context in which function provides a
framework for the study of advanced mathematics. A substantial portion of the
population did not have the ability to see a knot invariant as a function or to
differentiate between a relation and a function in the context of knot theory,
whereas a small number of students were more successful in the transition.
Salon A
Stacy A. Brown
University of Illinois, Chicago - Institute for Mathematics and Science
Education
stbrown@uic.edu
The Evolution of
Students'
Understandings of Mathematical Induction
In this talk, I will present results from a series of teaching experiments in
which I examined how students' ways of understanding and ways of thinking
change as they explore, discuss, and resolve proof-by-mathematical-induction-appropriate
problem situations. The design of these experiments was informed by a
theoretical perspective that is a synthesis of two complementary theories:
Brousseau's Theory of Didactical Situations and Harel's theory of intellectual
need (Necessity Principal). I will provide an account of how a cohort of
students' proof schemes and ways of understanding progressed through three
stages: pre-transformational, restrictive transformation, and transformational.
I will also report on the various didactical and epistemological obstacles the
students encountered at each stage. The results of the study indicate that the
students’ conceptions of what constitutes a convincing argument changed in
response to a series of shifts in the students’ understandings of generality.
Salon B
Stephen J. Hegedus
University of Massachusetts Dartmouth
shegedus@umassd.edu
Improving
understanding of core algebra and calculus ideas in a connected SimCalc
classroom
The paper reports some of the major findings of a three-year research study of
using SimCalc Mathworlds in undergraduate classrooms (NSF Grant # REC-0087771).
We aimed to study the profound potential of combining the representational
innovations of the computational medium with the new connectivity affordances
of increasingly robust and inexpensive hand-held devices in wireless networks
linked to larger computers. We present results from a 45-item pre-post test
statistical analysis, which outline significant increases in students¹
understanding of core algebra and calculus ideas such as rate, variable,
parametric variation and the fundamental theorem of Calculus. The effect of the
intervention in terms of gain relative to prior knowledge is also presented utilizing
Hake¹s Statistic. We triangulate these results with qualitative thematic
portfolios of the classroom concentrating on participation and engagement as
fundamental social structures evident in connected classrooms that give rise to
such strong learning gains.
Salon A
Katrina
Piatek-Jimenez
University of Arizona
jmnz@math.arizona.edu
Struggles and Strategies of Undergraduate Students when Writing Proofs
The transition from computational mathematics to theoretical mathematics tends
to be difficult for many undergraduate mathematics students. In this
study, through task-based interviews, I took a qualitative look at some of the
struggles students had when trying to write mathematical proofs and what
strategies they found to be successful. I found that the students in my
study had great difficulties dealing with the notion of infinity.
Mathematical notation was also a great difficulty for many students, especially
the idea of keeping certain notation arbitrary within proofs. These
students also demonstrated difficulties with understanding the structure of
mathematical statements and with deviating from the structure of direct
proofs. Symbolic logic proved to be a useful tool for many of these
students. Symbolic logic not only aided many students in constructing
valid proofs, but also was successfully used by students for the process of
validating proofs.
Salon B
LeeAnna Rettke
Arizona State University
Students Completing Proofs in Groups: Successes and Struggles
Constructing proofs can be considered a problem solving activity. This work provides an analysis of a proof writing activity by a group of junior level university geometry students, using Carlson & Bloom’s problem solving framework (in review). Within this framework I have identified the group’s expectation for members to share their resources and to take part in checking what has been put forward for group consideration. In addition, I have classified two types of difficulties that students face within the problem solving process: logic-related difficulties and content-related difficulties. It appears that the degree to which the individual’s problems are resolved depends on the group’s awareness of the problem itself and the group’s attempt to address the problem. Sometimes this requires any number of members to convince the other members of their conjecture which distinguishes the problem solving process for the group from that of the individual.
Salon C
Serkan Hekimoglu
The University of Georgia
shekimog@coe.uga.edu
What do client
disciplines want?
In this study, various departmental
disciplines' expectations from collegiate level mathematics courses are
investigated. Although each client discipline's mathematics requirements for
their students vary in a wide range, faculty members' expectations from
mathematics classes remained the same. Their expectations can be summarized in
the following categories: developing skills to work cooperatively to solve
problems, being able to transfer and connect their mathematical knowledge with
their academic discipline classes, and being comfortable with using technology
to solve problems. They all expressed the desire to see students who can think
critically and operate flexibly in their use of mathematical knowledge. They
all expressed their concern that students did not have a well-developed
conceptual understanding of mathematical concepts, even though they were able
to master algebraic skills. They also expressed their willingness and desire to
take part in curriculum development for undergraduate mathematics classes.
Salon D
Lara Alcock
lalcock@rci.rutgers.edu
Rutgers University, USA
Teaching
mathematical reasoning
The work presented here addresses the question of what mathematicians know
about students' mathematical reasoning, and is based on interviews with five
mathematicians experienced in teaching an introductory course in proof-based
mathematics. The results reported will be that: 1) there is a split in the student
population between "strong" students and those who write statements
that are "obviously wrong" and fail to "think critically";
2) a common pedagogical response to this is to introduce "rules" in
the form of proof templates and/or a restricted vocabulary; 3) mathematicians
do not evaluate proofs according to these rules, but according to checks
of the implications of any statement. I will discuss the meaning of
"thinking critically" and "obviously wrong", and explore
the nature of these checks. I will also discuss future research, and
suggest that it may be more effective to teach the systematic generation of
example objects than the "rules of reasoning".
Saturday,
October 25
1:15 – 3:15
Jere Confrey
Washington University in St. Louis
jconfrey@wustl.edu
Barry Sloan
National Science Foundation
Saturday,
October 25
3:45 – 4:15, Session #13 (PR)
Salon A
Cynthia O. Anhalt
University of Arizona
Robin A. Ward
University of Arizona
Mathematical
Representation and Academic Task: An Investigation of Prospective Elementary
Teachers’ Planning for Mathematics Instruction
A study was carried out involving thirty-one K-8 teacher candidates enrolled in an elementary mathematics methods course to investigate their planned uses of mathematical representation and academic task when planning for instruction in mathematics. The teacher candidates submitted lesson plans at three intervals during a semester-long methods course which were coded based on the planned uses of mathematical representations and academic task. Analysis of the data yielded interesting trends in the uses of mathematical representations and academic tasks. Recommendations highlighting the potential benefits of incorporating the knowledge base on mathematical representations and academic task into a mathematics methods course are offered and a closing discussion on the development of these teacher candidates’ pedagogical content knowledge through their choices of mathematical representations and academic task follows.
Salon B
Michael Grasse & Tami
Martin CANCELED
Tracing Students' Informal Understanding of Rate of Change in a
Physical, Multi-Representational Context
Salon C
Trey Cox
Arizona State University
A Preliminary Look at a Professional
Learning Community for Secondary Precalculus Teachers
This talk reports on an ongoing pilot study on the formation of a professional learning community (PLC) to promote change in the teachers’ mathematical views and classroom practices. The preliminary results regarding teachers’ views of the nature of mathematics show most participants have an integrated view of the nature of mathematics meaning that they fell on a continuum somewhere between an applied and pure view of mathematics.Other findings show their views about the nature of mathematics education consist of a preference for social constructivist teaching methods. However in actuality as seen in classroom observations and subsequent interviews, I have found that most do not teach in a constructivist fashion because they didn’t believe they have adequate time to do so.
Salon D
Jean J. McGehee
UCA
Professional
Development and Curriculum Alignment Impacts Student Achievement and Preservice
Courses
When the ultimate goal of a professional development
project is student learning and achievement, the results are a comprehensive
project that not only changes teacher practice and knowledge, but also affects
the way teacher educators organize preservice courses. The documented
results of this project will show the impact on both inservice and preservice
teachers.
Saturday, October 25
4:20 – 5:00, Session #14
(L-CT)
Salon A
Michelle J. Zandieh*
Arizona State University
Denise Nunley
Arizona State University
Sean Larsen
Arizona State University
Mathematizing Notions of Symmetry and
Congruence Using Transformations
This report examines the interplay between university students' intuitive and perceptual reasoning with symmetry and congruence and more mathematically structured views of these constructs through the use of isometric transformations. We find that students intuitively perceive symmetries of a figure and then struggle to make explicit these assumptions and mathematize them adequately for use in further reasoning. The data analyzed for this report comes from a larger research project studying the mathematical reasoning of students in a university geometry course. This data was collected during a classroom teaching experiment conducted in undergraduate geometry class using Henderson's (2001) Experiencing Geometry in Euclidean, Spherical and Hyperbolic Spaces. Three examples will be considered in which students struggle to mathematize their notions of symmetry or transformation in order to create a definition (e.g. angle congruence) or a proof (e.g. isosceles triangle theorem).
Salon B
Rina Zazkis *
Simon Fraser University
zazkis@sfu.ca
Natasa Sirotic
Simon Fraser University
nsirotic@telus.net
On irrational numbers and representations
This report is part (A) of a larger research that investigates preservice
secondary teachers' understanding of irrationality of numbers. Specifically, we
focus here on how irrational numbers can be (or cannot be) represented and how
different representations influence participants' responses with respect to
irrationality. As a theoretical perspective we use the distinction between
transparent and opaque representations, that is, representations that
"show" some features of numbers while "hide" other
features. The results suggest that often participants do not rely on given
transparent representation (i.e. 53/83) in determining whether a given number
is rational or irrational. Further, the results indicate participants' tendency
to rely on a calculator, preference towards decimal over the common fraction
representation, confusion between irrationality and infinite decimal
representation, regardless of the structure of this representation. As a
general recommendation for teaching practice we suggest a tighter emphasis on
representations and conclusions that can be derived from considering them.
Saturday, October 25
5:05 – 6:05
Ricardo Nemirovsky
TERC Inc
Thought, Language, and the
Use of Mathematical Notations
In this presentation we will review ideas about the relationship between thought and language and their relevance to how the use of mathematical notations relates to the stream of thoughts of the symbol-users. We will take as a starting point the work of Vygotsky in his “Thought and Language” book, outline some of his insights, explore the ideas of Edward Sapir as a contribution that surmounts some limitations in Vygotsky’s approach, discuss the notion of “translation” applied to the issue of combining multiple mathematical representations, and illustrate several points with a video episode that took place in a high school mathematics classroom; during this episode students derived a quadratic equation for a graphically defined function.
Sunday, October 26
8:00 – 8:40, Session #15
(L-CT)
Salon A
Keith
Weber
Rutgers University
Formal and intuitive
understandings of isomorphic groups and their roles in formal mathematical
reasoning
Salon B
Nathalie Sinclair*
Michigan State University
Peter Liljedahl
Simon Fraser University
Pre-service teachers
re-learning rational numbers with the colour calculator
In undergraduate
mathematics courses, pre-service elementary school teachers are often faced
with the task of re-learning some of the concepts they themselves struggled
with in their own schooling. They are expected to develop a profound
mathematical background in a subject that makes many anxious and, for most,
poses serious conceptual difficulties. It seems misguided to ask these
pre-service elementary teachers to learn the concepts they will have to
teach--and which they themselves had difficulty learning--in the same way they
have previously encountered them. In our research, we investigate the notion of
re-learning aspects of rational numbers. Using a computer-based microworld,
called the Colour Calculator, we report on an experiment aimed at investigating
(1) what the participants learned and re-learned about rational numbers, as
well as how the properties and relationships they learned interact with their
prior knowledge and (2) how the participants' attitudinal orientations towards
rational numbers were affected by the Colour Calculator.
Draga Vidakovic
Georgia State University
Discourse created by students’ homework presentations
This
study is an investigation of classroom discourse created during students’
presentation of homework assignments. It focuses on questions: (i) What kind of
discourse dominates students’ homework presentations? and (ii) In what ways do
the students value such discourse? The setting for the study was a modern
geometry course designed primarily for high school mathematics teachers. Results indicate that a
univocal-dialogic discourse was dominant in homework presentations. Students
valued homework presentations as an activity that helped them build their
conceptual understanding of the material. With the interchange of ideas and
opinions during the presentation, students developed better presentation
skills, built reflective thinking, decentered from their own thinking and
perspectives, developed logical and critical thinking, learned how to accept
criticism and, very important for them, enjoyed the process of learning.
Lara
Alcock*
Rutgers University, USA
lalcock@rci.rutgers.edu
Adrian Simpson
University of Warwick, UK
A.P.Simpson@warwick.ac.uk
Visual reasoning in
real analysis
This presentation is based on a qualitative investigation that followed
eighteen students through two pedagogically different first courses in real
analysis. This investigation found that "visual" students were
similar in that they focused on mathematical constructs as objects, were quick
to draw general conclusions about sets of mathematical objects and had a strong
sense of conviction in their own assertions. However, they differed in
regard to their understanding of what objects belong to key mathematical sets,
and whether they showed an ability and inclination to use formal definitions to
construct general arguments about these. These differences were found to
be related to student expectations about their own role as learners of
mathematics. We explore this relationship, and suggest that visual images
can provide a strong basis for understanding real analysis, but only if the
student is motivated to make considerable effort to link these with the formal
representations.
Sunday, October 26
9:30 – 10:00, Session #17
(S-CT)
Salon A
Angeles
Dominguez
Instituto Tecnologico y de Estudios Superiores de Monterrey-Monterrey
angeles.dominguez@itesm.mx
The purpose of this study was to develop a better understanding of college students' conception of variable. This study used Küchemann’s (1978) and Philipp’s (1992) categorization of variables: labels, constants, parameters, unknowns, generalized numbers, varying quantities, and abstract symbols. It was found that students recognized and used, with little difficulty, variables as unknowns and as generalized numbers. As expected, the more advanced the mathematics course, the more sophistication there was to the students’ solutions, and the less difficulty they had using variables. Also, students in the more advanced mathematics course talked more about their reasoning and were more articulate in their explanations for their solutions. As a result of categorizing the problems used for the interviews, four roles of variable emerged: unknown, generalized numbers, varying quantities, and abstract symbols. The definition of these four roles of variables considers how the variables are used and represented
Salon B
Peter Liljedahl
Simon Fraser University
Nathalie Sinclair
Michigan State University
Computer Microworlds: Thickening Students’
Mathematical Experiences
Much of mathematics is based on definitions, and much of learning mathematics is based on the acquisition, retention, recollection, and application of these definitions in a variety of situations. Despite their general applicability, definitions are often "thin". They are compact and decontextualized – streamlined for easy consumption, and application. Meanwhile, the "thick" contexts from which they are born, and in which they are to be applied, are richly varied and complex. In this study, we examine how a computer-based microworld called Number Worlds was used to create a contextually rich environment for a group of 90 preservice elementary school teachers in the area of elementary number theory. From their interactions with this environment the participants very clearly developed "thicker" understandings of number theory concepts such as primes, factors, and multiples. Gone were the "thin" definitions: replacing them were richly varied discussions of patterns, numbers of factors, distribution of factors, and randomness.
Salon C
Mario Sánchez Aguilar*
Cinvestav-IPN
Departamento de Matemática Educativa
Rosa María Farfán Márquez
Cinvestav-IPN
Departamento de Matemática Educativa
Communication of
mathematical concepts on distance mathematical education
In our research we
make an adaptation of a theorical framewok proposed by Bosch & Chevallard
(1999). This theorical framework considers to the ostensive objects like
representations of concepts and mathematical ideas, but considering its
institutional dimension like constituent ingredients of the tasks, techniques,
technologies and theories. We are studying the ostensive objets used by
students during interaction processes. These are distance mathematics students,
and they use communication mediated by computer. In particular we are interested in studying the
characteristics of these ostensive objects and analyzing them during the
processes of interaction of the students.During our presentation we will show
some of the results.
Note: This research is part of a
research program funded by the Consejo
Nacional de Ciencia y Tecnología of México: Construcción Social del
Conocimiento Matemático Avanzado. Estudios sobre la reproducibilidad y la
obsolescencia de situaciones didácticas: De la investigación al aula. Clave
U41740-S
Salon D
Francisco Cordero
Centro de Investigación y Estudios Avanzados del IPN
Reconstructing meaning of the asymptotic aspects in a socio-epistemological approach
Our specific problem of research concerns the asymptotic functions that are taught in the post secondary level. It consists of students not being able to reconstruct meanings of the asymptotic aspects because there is no frame of reference which helps them achieve this. The research contributes with indicators for such frame, through a situation design of the asymptotic aspects. This was supported by the socio-epistemological approach, which assumes that when dealing with didactic phenomena of mathematics the construction of such is social. Hence, the meanings of knowledge are reconstructed through the institutional experience, where the human activity or the social practices are the generators of such reconstruction. Like this, the socio-epistemology of the asymptotic aspects deals with two moments in order to achieve the reconstruction of meanings: a) the asymptote when it questions the shape of the tendency and b) the asymptote when it questions the ratio of the tendency. Besides, it forced to treat graphics, not as representations, but as arguments that allow the reconstruction of meaning.
Note: This research is funding under a grant from the CONACYT about project: Construcción social del conocimiento matemático avanzado. Estudios sobre la reproducibilidad y la obsolescencia de situaciones didácticas: de la investigación a la realidad del aula (CLAVE: 41740-S).
Salon E
Patricia E Balderas-Cañas
Universidad Nacional Autónoma de México
Obstacles,
media and activities in mathematics teaching at engineering school
A case study on a test basis with undergraduate and graduate engineering students let me realize two cognitive obstacles associated with two cognitive conflicts (NCTM, 1992). The first, related with conditional probability of a future event, obstructed the comprehension of probability trees to solve word problems. The second, an undetermined conceptualization instead of a variable one, when students used Excel work sheet models of linear programming problems, prevented them to identify variable restrictions on decisions. All of the involved students were enrolled on my statistics and applied linear algebra classes. From previous ideas, I would point out two recommendations: Math teachers should use portfolios and projects in order to “reflect the mathematics that student should know and be able to do (NCTM, 2000)” and try to reduce cognitive conflict arising. Both tools let us strength concept learning and problem solving strategies acquisition, and promote conversation as a learning media.
Sunday, October
26
10:30 – 11:00, Session #18
(PR)
Salon A
Kwon, Oh Nam *
Seoul National
Ju, Mi-Kyung
Ewha
Cho, Kyoung-Hee
Ewha
Students’ Conceptual Understanding of Differential Equations in a Reformed
Course
The purpose of this research is to analyze students’ conceptual understandings of differential equations in order to examine the effect of the RME-DE in comparison to the TRAD-DE approach. Data were collected from the two Differential Equations courses at a Korean university during the 2002 fall semester. The data analysis reveals a significant difference in the strategies that the students used to answer the conceptual questions. The RME-DE students demonstrated a more comprehensive understanding of differential equations than the TRAD-DE students. The results indicate that the RME-DE students achieved a better overall understanding of differential equation concepts than their TRAD-DE counterparts. In particular, the RME-DE students developed much stronger capacities of graphical representations in differential equations concepts than TRAD-DE students.
Salon B
Ana
González-Ríos *
University of Puerto Rico at Mayaguez
ana@math.uprm.edu
Gladys DiCristina-Yumet *
University of Puerto Rico at Mayaguez
gladysd@math.uprm.edu
Rafael Martínez-Planell *
University of Puerto Rico at Mayaguez
rafael@math.uprm.edu
Infinite Series: A
Preliminary Genetic Decomposition
This presentation will report on an ongoing research project that examines
college students' cognitive constructions of the concept of infinite series
using APOS Theory. A very brief description of the research paradigm and the
theoretical perspective used, will be given, together with comments on related
literature. The evolution of a preliminary genetic decomposition and the design
and implementation of the assessment techniques used to collect data will be
described. The analysis of the data will be presented with examples from
interviews to 20 students. The analysis of the data was guided by the
preliminary genetic decomposition and at the same time the data will serve to
support it. A brief discussion of how the results could suggest modifications
to the preliminary genetic decomposition will follow. The talk will end by
considering the steps that will be taken to further the study.
Salon C
Nicole
Engelke
Arizona State University
Engelke@mathpost.asu.edu
Related Rates: The
Difficulties Students Encounter
The purpose of this study was to gain a better understanding of the process
involved in working related rate problems, including knowledge of the obstacles
that first semester calculus students commonly encounter when completing these
problem types. It is my hypothesis that a critical component of solving related
rates problems is being able to diagram and visualize the situation together
with understanding variable, function and derivative. Although most
students were able to label their diagrams with the constants given in the
problem, they did not distinguish between quantities that vary in time from
those that remain constant. By not constructing a mental model that
attends to the changing quantities, students focused on applying a set of
learned procedures, such as find a formula and differentiate, rather than
conceptual knowledge. This resulted in frustration and incorrect
responses. Activities that may facilitate improved understanding of these
problems are suggested.
Salon D
Jason
K. Belnap
University of Arizona
belnap@math.arizona.edu
Putting TAs into Context: Understanding
the Graduate Mathematics Teaching Assistant
Graduate
mathematics teaching assistants (MTAs) have become an integral part of
undergraduate mathematics instruction at universities. Many graduate teaching
assistants (TAs) teach their own courses, yet training for TAs tends to be
short, minimal, and having limited impact on MTA teaching practices. Why?
We do not know; little has been done to uncover the complexity of the MTA
experience. This year-long, qualitative, multi-case study addresses the
"why" question, by identifying and elements of the MTA experience
influencing MTA development and teaching practices. With data collection
over and analysis underway, several sources of influence have already emerged,
such as supervisors, peers, course structure, departmental policy, etc;
analysis is currently underway to identify how each impacts MTA teaching
practices. In this talk, I will present the background of the study, some
of the data, and then give time for audience feedback/discussion regarding some
data being analyzed.
Sunday, October 26
11:05 – 11:35, Session #19
(S-CT)
Salon A
Jennifer
Christian Smith*
The University of Texas at Austin
jenn.smith@mail.utexas.edu
Catherine Stacy*
The University of Texas at Austin
cstacy@mail.utexas.edu
The role of ownership in the construction and writing of proofs in an inquiry-based number theory course
During an exploratory study of undergraduates enrolled in traditional and inquiry-based sections of a number theory course, we found that students in the inquiry-based sections appeared to take ownership of the process of constructing and writing proofs, while the students in the traditional sections appeared to view proofs almost as exercises to be completed as quickly and effortlessly as possible. For example, when presented with a statement to prove, students in the inquiry-based sections typically spent several minutes choosing an appropriate proof strategy, while students in the traditional sections tended immediately to try several strategies without considering each very closely. There were also differences between each group's use of symbolic representations of mathematical statements and the forms of the written proofs produced. These preliminary results suggest that inquiry-based teaching strategies may encourage students to take ownership of mathematical ideas and processes in ways that students in traditional courses may not.
Salon B
James
Vicich
Scottsdale Community College
james.vicich@sccmail.maricopa.edu
Teaching
Mathematical Problem Solving to Developmental Undergraduates: The Hidden Agenda
This study investigated the problem-solving behaviors of seven undergraduate
developmental algebra students enrolled in a one-semester course at a two-year
college. The primary assessment was analysis of three forty-five-minute,
videotaped sessions that recorded students' mathematical problem-solving
performance on four different tasks, over twelve weeks. Each taped
session included a follow-up interview that allowed the researcher to probe
students' perceptions, beliefs, and meanings. Analysis of problem-solving
behaviors included the categories of resource knowledge, engagement, planning,
monitoring/control, heuristic strategies, verification, beliefs, and
mathematical habits. A three-tiered taxonomy of student problem-solving
performance emerged that included the categories of sufficient, limited, and
obstructed. Students in the sufficient category consistently performed better in
the categories of resource knowledge, control, and verification. Students
whose problem-solving performance was classified as obstructed often had
difficulty accessing resource knowledge and demonstrated ineffective or no
monitoring abilities. Students' performance in the limited category fell
between sufficient and obstructed.
Salon C
Irene Bloom
Arizona State University
Horizontal and Vertical Mathematization of Extended Analyses Tasks
This study investigated the mathematical development of prospective high school mathematics teachers, and involved analyses of student solutions to “extended analysis” tasks for evidence of mathematization and mathematical abstraction. The results of this study revealed that these tasks were effective in igniting powerful mathematical thinking in preservice secondary mathematics teachers. Examination of student data on the final project revealed that all the students were able to use the techniques learned during the semester to successfully and meaningfully extend their problem in a way that can be described as vertical mathematization. Some extensions evidenced higher levels of abstraction than others, but all were able to explore some aspect of the mathematics beyond the problem setting. Even students who were less successful with their final project appeared to emerge from the experience with an appreciation for the value and role these tasks had on their mathematical development.
Salon D
Jenifer Bohart
Arizona State University
jen@mathpost.asu.edu
Strategies in Statistics: Choosing the Forest or the Trees
The results of clinical interviews indicate that although subjects know how to
use the strategies learned in their statistics classes, they have trouble
deciding when to use them. On any given problem, subjects would either (a) view
the sample as a whole (seeing the "forest"), or (b) concentrate on
the individual observations or trials that made up that sample (seeing the
"trees"). Focusing on individual elements of the sample, subjects
would recall various "Stats Class Strategies" such as
independence. This approach was successful on problems comparing the
likelihood of two samples of the same size. When comparing small and
large samples, however, students using this strategy would ignore the sample
size altogether. In contrast, subjects choosing to view the sample
as a whole did not consider the relationship between individual elements of the
sample. This intuitive approach was often influenced by the
representativeness heuristic.
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This document was compiled by Draga
Vidakovic.
Last modified October
6, 2003