Background/The Problem Research suggests that student beliefs about the nature of mathematics tend to be very different from those of mathematicians. While mathematics professionals broadly view mathematics as a coherent and logical discipline, novice students often see the subject as a body of facts and procedures that are at best loosely connected. As Schoenfeld (1992) writes, this latter perspective "trivializes mathematics [and] a curriculum based on mastering a corpus of mathematical facts and procedures is severely impoverished - in much the same way that an English curriculum would be considered impoverished if it focused largely, if not exclusively, on issues of grammar." Much research has been done on the mathematical beliefs of prospective teachers, but there is scarce literature on the views of lower-level collegiate mathematics students. It is the goal of this SoTL research to investigate the mathematical views of students in one lower level course. Research Questions: What do students enrolled in a problem-based mathematics course believe about the nature of mathematics? Do students who take this class change their attitudes and beliefs about the nature of mathematics and/or about their ability to do mathematics? If so, how? PBIS Courses at UW-Oshkosh: All students at UW-Oshkosh are required to take at least one mathematics course. While some students fulfill this requirement as part of their major, most other students end up taking a Problem-Based Inquiry Seminar (PBIS) course. The PBIS courses are small, interactive, student-centered classes emphasizing active student problem-solving. UW-Oshkosh offers three different PBIS courses. The current focus is on PBIS-188: Introduction to Modern Mathematics and its Applications. Topics include the mathematics of voting, political power, apportionment of representatives, and graph theory, offered in a problem-solving context. Content is relevant, but a primary goal is to influence student views of what mathematics is. Course Goal: Help students to better understanding what mathematics is and how we actually go about doing mathematics. Specific objectives: Helping students to understand how mathematics can be used to solve meaningful problems within the realms of political science, networks, and efficient routing. Cultivating student expertise in critical thinking, abstract reasoning, problem solving and creativity. Deepening student understanding of what it means to "think mathematically," and helping students to become more effective at it. Understanding what constitutes a sound mathematical argument, and strengthening the ability to make such an argument. Developing skills associated with the scientific method, including rational inquiry, data collection, analysis, theory formulation, and hypothesis testing. Continuing to develop effective written and oral communication skills. Note that the last five objectives, in particular, involve developing intellectual resources that form the heart of a liberal arts education. Course Philosophy: Students should behave as real mathematicians do: Collect data in order to see patterns. Make conjectures. Model physical situations. Examine simpler cases first. Reduce problems to ones already solved Verify conjectures using data. Classify based on structure. Use precise language. Make rigorous arguments Approach: Class is problem-based, with a mixture of individual, small group and large group activities. Making arguments about why patterns and mathematical situations make sense is emphasized, while lecture and algorithmic thinking is de-emphasized. The links below give examples of typical course problems that students work on in PBIS-188. Summer 2007 Handshakes.pdf Smalltown.pdf painttown.pdf summer.pdf power.pdf

Project Summary According to Hutchings (2000), the subject of my SoTL research is a "What is?" question. Specifically, I would like to know what PBIS students believe about the nature of mathematics. When I refer "beliefs about the nature of mathematics," I need to be very precise. I am talking about neither attitudes ("I like/don't like mathematics") nor self-efficacy ("I feel confident balancing my checkbook"). The question I am trying to answer is what students believe about mathematics as a discipline: Is it a coherent subject? Is it a static field of already known ideas? Is it useful? Is it logical? Anecdotally, I see tremendous growth in some PBIS students over the course of a semester. They become better at creating arguments, at using definitions carefully, and at making and testing conjectures. I would like to find a way measure student mathematical growth of this type to see if, in fact, students do "grow" in this course. Ongoing

Study Methodology Administer a survey instrument to PBIS Students at the beginning of the semester. Score each question from -2 to 2 reflecting agreement level of agreement with mathematicians. This will provide each student with an aggregate score from -22 to 22. Interview a sample of students to look for common themes as well as reinforcement of survey data. At the end of the course, administer the same survey and interview students again. This data will help us to better understand whether the course has any impact on student beliefs. Administer the survey to professional mathematicians to help establish validity. In fall 2007 piloted a survey designed to assess student beliefs about the nature of mathematics (see the file "beliefs07.pdf" in the link box below). I administered the instrument on the first day of class in two sections of my PBIS class in September, and repeated the assessment on the last day of the semester. This yielded sample pre and post data from about 40 students. The survey generally allows for "Likert"-type responses measuring level of agreement with statements about mathematics. Student beliefs about mathematics are evaluated based on their level of agreement with the broader mathematical community. I intend to administer a "final" version of the survey to my PBIS students in the fall semester. As in the pilot survey, I will administer the instrument at both the beginning and end of the semester as a means of measuring "growth" in students conception of mathematics. Interview data will be used to help validate the survey data. I also intend to administer the survey to a sample of students other than the PBIS students to test the reliability of the instrument. Fall 2007 beliefs07.pdf

Annotated List of Helpful Resources & References Mathematicians writing on mathematics: Davis, P.  and D. Hersch, 1995.  The mathematical experience. Birkhauser Press Hardy, G.H.1940. A mathematicians apology.  Cambridge University Press. Courant, R. and Herbert Robbins, 1996. What is mathematics? 2nd Edition. Oxford University Press. Math Ed/SoTL Resources: Schoenfeld, Alan, 1992. "Learning to think mathematically:  problem solving, metacognition, and sense making in mathematics."  in Handbook of research on teaching and learning, Grows D., Ed.  NCTM. Swan, Malcolm, 2006.  "Designing and using research instruments to describe the beliefs and practices of mathematics teachers." Research in Education 75: 58-70. Hutchings, P. (2000). "Introduction to Opening Lines; Carnegie Foundation for the Advancement of Teaching." Helpful Web Resources: Bruce Cooperstein's Problem Solving Course Portfolio Ongoing

Preliminary Results (Spring 2008) 43 PBIS student responses in Fall 2007 pilot survey. Aggregate mean for initial survey of PBIS students at the start of the semester was 3.70. This reflects very weak agreement with mathematics community. Possible score ranged from -22 to 22, but actual scores ranged from -5 to 14. Initial mean for 13 sophomore-level mathematics majors was 7.85. Mean for 3 professional mathematicians was 14.

Aggregate mean from survey at the end of the semester was 5.19 (scores ranged from -3 to 12). The positive change in the mean (toward more agreement with mathematicians) is statistically significant at the .05 level. On some individual questions, students showed fairly dramatic positive change over the semester (especially Questions 1, 2, 5). On others, the change was negligible, including very small negative movement on some items (Questions 6, 10). More complete results from the pilot survey can be found by clicking on the link below. Pilot Survey Results.pdf

Ongoing Issues Survey instrument still needs work. Professional mathematicians did not score quite as highly as I would have expected. Although having all mathematicians in agreement is likely impossible, we would like a higher mean for mathematicians. Informal discussions with mathematicians indicate that some questions in particular may be problematic (see #10). Change in aggregate mean is statistically significant, but is it really significant? Does the score change reflect a meaningful change in student beliefs? Hopefully, interviews will help to clarify this issue.

Spring 2008 and ongoing

Career Relevance & Impact Ongoing