Current Research Activities [*]
Links are (mostly) to PDF file format versions of papers/proposals.
Funded Proposal FIPSE 2006: Video cases for novice college mathematics instructor development (880K PDF)
Principal Investigator (PI):
Shandy Hauk (University of Northern Colorado)
Co-PIs: Natasha Speer (Michigan State University), Dave Kung (St. Mary's College, MD), Jenq-Jong Tsay (University of Texas, Pan American)
Senior Personnel: Nikita Patterson (Georgia State), Angelo Segalla (California State University, Long Beach), Cindy Kronauge (External Evaluator, Prime Solutions, CO)
Abstract
The research and development team is a group of specialists in mathematics education and
mathematics who firmly believe the time has come for graduate mathematics programs to
prepare TAs to teach well and that professional development through video cases is an excellent, sustainable, method.
Our goal for the 3 year grant period (2007-2010) is the creation of a book-DVD package containing at least 10 video cases and
supporting textual, video, and audio materials on college mathematics teaching. The volume is planned for the Conference Board for the Mathematical Sciences (CBMS) Issues in Mathematics Education series; the same publisher that brought us Friedberg et alia's (2001) textual case materials for college mathematics instruction.
Paper: Multiplication schema for signed number:
Case study of three prospective teachers (PDF)
Authors: Jenq Jong Tsay and
Shandy Hauk
Current State: Mathematical Sciences and Mathematics Education, 1(1), 33-37 (2006). [Click here for issue Table of Contents]
Abstract
This study investigated the pedagogical content knowledge that a college learner
who is a prospective teacher might construct for teaching two-factor multiplication. In
particular in this report, we attended to learners' cognitive structures for signed number
multiplication, described in terms of actions, processes, objects, and schema. In closing,
we suggest problem-posing, visualization of problem solving, and identifying the
isomorphic relationship between computation and visualization as tools for improving
both future research and the college mathematics preparation of teachers.
Paper: Mathematical autobiography among college learners in the United States (PDF)
Author:
Shandy Hauk
Current State: Adults Learning Mathematics International Journal, 1(1), 36-56 (2005).
Abstract
This study examines the K-12 mathematical experiences of U.S. university
students via an expressive writing assignment: a mathematical autobiography.
The mathematical autobiographies of 67 college students, out of over 300
enrolled in 16 sections of a college liberal arts mathematics course, were
analyzed deeply using constant-comparative methods. Four categories of
experience connected to aspects of mathematical self-regulation emerged as
significant to the student-authors: locus of control for mathematics knowledge
and learning, self-evaluation of mathematical ability, emotionally-charged
epistemological views of mathematics, and mathematical decision-making habits.
Interviews of 18 of the 67 students provided support and clarification of
the analysis. An argument, grounded in existing research, for increased
mathematical self-regulation as a result of completing the mathematics
autobiography is made. Finally, connections are drawn between learning
and psychological theories to support the conjecture that the assignment
may be as useful to novice university teachers as it is to their students.
Paper: Negotiating reform: Implementing Process Standards in culturally responsive
professional development. (uncorrected printer's proof, PDF file)
Authors: Jeff Farmer, Shandy Hauk, Andrew Newmann.
Current State: High School Journal, 88(4), 59-71 (2005).
Abstract.
The paper presents the guiding ideas behind our
culturally responsive approach to teacher professional development and an
overview of how those tenets inform, tacitly and directly, our efforts to
realize the promise of the National Council of Teachers of Mathematics' five
Process Standards. A review of the primary obstacles teachers face in implementing
these standards in their own teaching and learning is followed by a description of
the design elements in a university-based professional development program.
Our goal is to provide an example of the foundations upon which an evolving
approach to professional development planning has grown. We discuss research
on what constitutes effective teacher professional development while noting
the paucity of programs that embrace recognized needs. We do not give a prescription
for effective teacher development. Instead, we speak as teacher-educators
about the necessary philosophical and self-evaluative underpinnings to
effective culturally responsive professional development and to our approach to creating an
environment where it is safe to leave the isolation of forced autonomy
and be reflective about community, mathematical activity, and intellectual engagement.
Paper: Student perceptions of the web-based homework program WeBWorK
in moderate enrollment college algebra courses
Authors: Shandy Hauk & Angelo Segalla;
Current State: Journal of Computers in Mathematics and Science Teaching, 24(3), 229-253 (2005).
Abstract. Twelve of 19 college algebra classes used WeBWorK and 7 used traditional paper and pencil homework (PPH). Given the quantitative result that no significant difference in performance between WeBWorK and PPH classes was found, a qualitative analysis of 358 student and instructor surveys revealed three primary categories of student perceptions related to WeBWorK: views about its usefulness, intentionality in engaging with mathematics, and challenges to student beliefs about mathematics. Student and instructor comments are reported within the context of self-regulated learning theory. Results support the conjecture that WeBWorK is at least as effective as traditionally graded paper and pencil homework for students learning college algebra.
Paper: Why can't calculus students
access their knowledge to solve non-routine problems?
Authors: Annie Selden, John Selden, Shandy Hauk, and Alice Mason;
Current State: In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.),
Research in Collegiate Mathematics Education. IV (pp. 128-153),
Providence, RI: American Mathematical Society (2000)
ABSTRACT. In two previous studies we investigated the non-routine
problem solving abilities of students just finishing their first year
of a traditionally taught calculus sequence. This paper reports on a
similar study, using the same non-routine first-year differential
calculus problems, with students who had completed one and one-half
years of traditional calculus and were in the midst of an
ordinary differential equations course. More than half of these students were
unable to solve even one problem and more than a third made no substantial
progress toward any solution. A routine test of associated algebra and
calculus skills indicated that many of the students were familiar with
the key calculus concepts for solving these non-routine problems;
nonetheless, students often used sophisticated algebraic methods rather
than calculus in approaching the non-routine problems. We suggest a
possible explanation. These students may have had too few tentative
solution starts in their problem situation images to help prime recall
of the associated factual knowledge. We also discuss the importance
of this for teaching.
Draft Manuscript: Fostering college students' autonomy
in written mathematical justification
Authors: Shandy Hauk and
Matt Isom;
Current State: Submitted.
Abstract
The focus of this study is the influence of regular structured writing about
college algebra topics (acronym PSOLVE) on locus of control, flexibility of
articulation, and accuracy of student work on mildly non-routine problems.
We analyze and compare the problem solutions offered by college algebra
students who had written about mathematical content using PSOLVE with the
work of those who had not. The writing assignments provided students a
framework for expressing their thoughts about mathematical actions, processes,
structures, and language. Augmenting traditional college algebra curriculum
with the PSOLVE writing assignments produced a noticeable increase in students'
ability to communicate their ideas. Because the students can clearly discuss
what they think they know, PSOLVE assignments may help an instructor gain
insight into students' primitive knowledge of mathematical concepts. We also
discuss potential benefits of the PSOLVE augmentation for the development of
college mathematics teaching research and practice.
Working Draft of Paper:
The schema of multiplication in preservice elementary teachers
Authors: Jenq-Jong Tsay & Shandy Hauk
Current State: Submitted.
Abstract
The study was a clinical interview exploration among prospective elementary
teachers of two factor multiplication. Taking action-process-object-schema
(APOS) theory as a foundation, the efforts of three pre-service teachers to
compare their solution processes for context-free numeric prompts like 4 x 3
with the solution of associated story problems were analyzed. The interview protocol,
aimed at stimulating pedagogical content knowledge efforts, required participants
to pose and solve the associated word problems themselves. Results presented are
the analyses of the three pre-service teachers' understandings of integer and
positive rational number multiplication, their apparent understandings about
the isomorphism between decontextualized and contextualized problem-solving,
and evidence of aspects of pedagogical content knowledge development.
The presentation of results is followed by a discussion of participants'
understandings, their pedagogical content knowledge, and suggestions
regarding the practice of problem-posing in the mathematical preparation
of prospective elementary teachers.
Manuscript:
A comparison
of web-based and paper and pencil homework on student performance in college algebra
Authors: Shandy Hauk, Robert A. Powers, Alan Safer, Angelo Segalla;
Current State: Submitted.
Abstract. The study investigated differences in mathematics achievement between college students using web-based homework (WBH) and those doing traditional homework. Twelve of 19 college algebra classes used the WBH software WeBWorK and 7 used traditional paper and pencil homework (PPH). A test of algebra skills was administered pre- and post-course. Quantitative analyses revealed no significant differences in performance by ethnicity or instructor between the two homework treatments even when analysis controlled for previous mathematics achievement. However, women in some WeBWorK classes had statistically significantly higher score gains than women in PPH sections. Results support the conjecture that WeBWorK is at least as effective as traditionally graded paper and pencil homework for students learning college algebra.
Manuscript: Diary of a graduate teaching assistant (GTA).
Authors: S. Hauk, R. Cribari, M. Chamberlin, A. Brown Judd, R. Deon, A. Tisi, & H. Khakhail;
Current State: Under review.
Abstract.
Selected entries are presented from a mathematics graduate
teaching assistant (GTA)Ís journal across three years.
The excerpts are followed by an explanation of both their
source and purpose along with a discussion of current
recommendations and resources for GTA development.
Working Draft of Paper:
Improving student performance and sense-making in Liberal Arts Mathematics
Authors: Shandy Hauk, Robert A. Powers, April Judd, Jenq-Jong Tsay;
Current State: in preparation.
Our goal is to contribute to investigation of the question:
"How do we increase student performance and sense-making in college
mathematics while building college teaching effectiveness?"
More specifically, the liberal arts mathematics (LAM) research project
underway is focused on ways to foster the development, in tandem, of
flexibility of thought (cognition) and sense-making drive (affect)
among undergraduates and the Graduate Teaching Assistants (GTAs) who teach them.
We examined college student conceptions of mathematical problem-solving
and mathematical task efficacy using quantitative and qualitative
research methods. Analysis of 553 valid surveys and of problem-task
based interviews of eight students, six months after their LAM course,
suggests that the greatest impacts on student views were connected to
LAM instructors' teaching philosophy and number of years teaching experience.
Together, survey and interview results:
(1) point to directions for the preparation of GTAs before and during teaching,
(2) demonstrate the benefits to undergraduates of directly addressing
aspects of acculturative stress when teaching, and
(3) throw light on the cognition and affect interaction among students
as the acclimate to college mathematics learning and teaching.
(January 2004 Joint Meetings, abstract number 993-r1-1399).
Manuscript:
Preservice elementary teachers' understanding of logical inference
Authors: Shandy Hauk, Harel Barzilai, Homer Austin, April Judd, Jenq-Jong Tsay;
Current State: in preparation.
Particularly challenging for prospective teachers is recognizing and accommodating their own difficulties with logical reasoning in both natural language and abstract mathematical contexts. This paper reports on the contextualization and logical reasoning efforts of five students as they responded to interview prompts involving nonsense, natural, and mathematical representations of conditional statements. The students came from sections of Mathematics for Elementary Teachers which had been involved in a survey using the same types of logic prompts. For example, the following are three, logically equivalent, items from the three (nonsense, natural, and mathematical language) sections of the survey:
X and Y are statements.
A3. Suppose that X implies Y. Suppose X is false. Is Y false? (Yes) (No) (Not necessarily)
Paper: Does the evidence of authority prevail over the authority of evidence?
In this study of nine sections of college algebra, students did research in either mathematics history or the philosophy of mathematics and gave oral presentations in small groups. This curricular extension was accompanied by a concerted effort to change the classroom culture to one in which traditional appeals to authority were not highly valued and well-argued conjecture was expected. The data considered are student interviews, essays, and performance on a common final exam. Among the resources students value highly are teacher proclamation, answers from the text, and traditional mathematics classroom protocols. For some, the work of fellow students is a resource (hence cheating). Students for whom conjecture is difficult seem to perceive themselves as the least credible resource. It appears that the more willing a student is to risk conjecture and the frustration of attempting to validate it, the more mathematically successful and mathematically self-aware the student becomes.
In an article to be published in JRME, Selden&Selden (2003) conducted an exploratory study on how undergraduate students validate proofs. To extend their research, we have chosen to look at how mathematically trained graduate students in a Ph.D. program validate proofs. "Validation" here is meant in the sense defined in Selden&Selden (2003): "readings of and reflections on proofs to determine their correctness." Four graduate students who had each completed at least one year of graduate work in mathematics and were in the Ph.D. program at a western Carnegie-Doctoral I university participated in this study. Data collection was videotaped interviews centering on the participants' process of validating proofs as well as interviewer's notes and the participants' work written on the paper provided during the interview. All interviews were transcribed.
The study's Research Questions: * What criteria/strategies do graduate students use in determining whether an argument proves a theorem? * What role does affect play in proof validation for graduate students? * Do graduate students use examples during the validation process? If so how does their use compare to the way undergraduates have used them in previous research?
The study is qualitative and grounded theoretical in nature so the theoretical
perspective emerged from the data. The palette of affective, cognitive, and
behavioral aspects involved in proof validation is different (though not
distinct) from that involved in proof construction. Our theoretical perspective
hinges on the fine points of conviction.
Emerging from
the interview data were indicators of kinds of conviction, i.e., the
interviewee's degree of belief or disbelief in the truth of a statement or
validity of an argument. Of course, an individual's sense of conviction is not
always absolute. Someone who has written a proof and believes it correct may
still need to do a few examples to provide the explanation (Hanna, 1989&1990) that
allows him to be certain (convinced) that the theorem is true (Balacheff, 1987;
Fischbein&Kedem, 1982).
We use the terms possible, plausible, and probable (in ascending order) to indicate levels of conviction between absolutely convinced something is false and absolutely convinced something is true.
Conviction Continuum Conviction that a statement or |---------|----------|----------|----------| theorem is: False Possible Plausible Probable True
Paper: Case Study of a Ph.D. Mathematician Teaching College Algebra
The case study to be presented is part of a larger research program. The primary participant in the study is a Ph.D. mathematician with twelve years college teaching experience (six years part-time while a graduate student, six years full-time after the Ph.D.). Every class meeting of each of his two college algebra courses for one semester were videotaped. These classes were two among 54 sections taught that semester at a large, western, Research I university. The data for this case study are the videotapes, the transcriptions of some of them, student written work, and interviews with the instructor. Interview data from students were not included in this study but will be considered in future work. The research is qualitative and grounded theoretical (Strauss&Corbin, 1998). The research questions in this first case study are:
The social cognitive aspects of the conflicts between educational factors fell into the three categories proposed by Bandura's (1986) social cognitive theory: personal, behavioral, and environmental. For example, the larger environment established by the departmental course coordinators for college algebra was fraught with sudden changes in policy. Emergent from the data were a collection of strategy changes (by the instructor and by the students) in negotiation of social norms for self-regulation within the classroom milieu. Also clear from the semesters' videotapes were the evolution of what Yackel and Cobb (1996) described (in a school setting) as socio-mathematical norms; we look at these through the lens of social cognitive theory, in particular, the sub-area of self-efficacy theory (Bandura, 1997).
This case study is aimed at providing a research foundation for practical efforts - such as might build on the recent work of Friedberg et al. (2001). Among the implications for teaching to be discussed are: possible foci for training of graduate students before they enter full-time college teaching; suggestions for structuring of course coordination efforts to minimize curricular values conflicts; potential action-research paradigms for instructors in similar circumstance to the present case-study within their respective departments.
Paper: Working title - A comparison of college student mathematical autobiographies: inner-city New York and suburban Southern California;
A follow up to the first paper on evidences of mathematical self-awareness in
college student mathematical autobiography. The two populations considered are
vastly different on several scales. However, preliminary analysis of student
essays indicates there are certain affective and meta-affective aspects of
mathematical self-awareness which are independent of learning history and may
depend primarily on the nature and western tradition of teaching in school
mathematics.
References
N. Balacheff (1987). "Processus de preuve et situations de validation [Proof processes and validation situations]," Educational Studies in Mathematics, 18, 147-176.