Work in progress

- Working Draft: How do graduate students validate proofs?
Authors: S. Hauk, R. Cribari, A. B. Judd, A. M. Neumann, and J.-J. Tsay.

Current State:*Research report*under external review prior to submission.

In an article published in JRME, Selden and 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 by the Seldens: ``readings of and reflections on proofs to determine their correctness." Four PhD students who had each completed at least one year of graduate work in mathematics participated. Videotaped interviews centered on the participants' validation of four purported proofs. - Working Draft: Improving student performance and sense-making in Liberal Arts Mathematics
Authors: S. Hauk, R. A. Powers, A. B. Judd, J.-J. Tsay.

Current State:*Research report*in preparation.

Our goal is to contribute to investigation of the question: ``How do we increase student performance and sense-making in first-year 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 philosophies 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 they acclimate to college mathematics learning and teaching. (January 2004 Joint Meetings, abstract number 993-r1-1399). - Working Draft: The schema of multiplication for preservice elementary teachers.
Authors: J.-J. Tsay and S. Hauk

Current State:*Research report*in manuscript form.

Interviewees were presented seven simple numerical multiplication prompts. For each prompt the participant was asked to compute the product and to pose and visualize solving a problem, based on the prompt, that would be appropriate for working with schoolchildren. ParticipantsÕ computational work, drawings, explanations and justifications were analyzed using the multi-layered theoretical framework. Several participants had difficulty completing computations involving fractions and none of the twelve pre-service teachers could pose an effective and meaningful problem for prompts involving two fractions. Most problematic were the conceptions interviewees associated with properties they noticed: the property of negative number, of fraction, of multiplier, of negative number as multiplier, and of fraction as multiplier. Discussion of the twelve womenÕs understandings of multiplication is followed by implications for teacher preparation and suggestions for research and practice. In particular, problem-posing and visualization of problem-solving can serve as research tools to explore schematic mathematical understanding and can provoke prospective teachersÕ self-examination of their conceptual understanding of elementary mathematics topics. - Preliminary draft: A comparison of college student mathematical autobiographies: inner-city New York and suburban Southern California.
Authors: M. K. Davis and S. Hauk

Current State:*Research report*, data collected, in data analysis stage.

A follow-up to the first paper on 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 of the western tradition for the curriculum and teaching of school mathematics. - Preliminary draft: Does the evidence of authority prevail over the authority of evidence?
Authors: S. Hauk and M. K. Davis;

Current State:*Synthesis of action-research and practice*, data collection complete, preliminary data analysis complete, initial draft in preparation.

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 valued highly were teacher proclamation, answers from the text, and traditional mathematics classroom protocols. For some, the work of fellow students was a resource (hence cheating). Students for whom conjecture was difficult seemed to perceive themselves as the least credible resource. It appears that the more willing a student was to risk conjecture and the frustration of attempting to validate it, the more mathematically successful and mathematically self-aware the student became. - Preliminary draft: Case-studies of proof validation by
advanced mathematical thinkers: The case of Dr. Nichols.
Authors: S. Hauk, R. Cribari, R. Deon, & A. B. Judd

Current State:*Research report*, data collected, in data analysis stage.

Following the protocol used in earlier work [Selden & Selden, 2003], this is a case-study of an African-American male Ph.D. mathematician as he validates four purported proofs of a simple number-theoretic theorem. Interview prompts also included questions that delved into his epistemological views regarding proofs and proving. - Preliminary draft: Case-studies of proof validation by
advanced mathematical thinkers: The case of Ms. Silver.
Authors: S. Hauk, C. Dollard, A. Tisi, & A. F. Toney

Current State:*Research report*, data collected, in data analysis stage.

Following the protocol used in earlier work [Selden & Selden2003], this is a case-study of an African-American woman with a Master's degree in mathematics as she validates four purported proofs of a simple number-theoretic theorem. Interview prompts also included questions that delved into her epistemological views regarding proofs and proving.