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With this project we seek to fill a gap in the knowledge that exists about how mathematics faculty members new to teaching with inquiry-based learning [IBL] methods learn to use these approaches. Specifically, we seek to produce accounts of the process of learning to teach using IBL from faculty who are new to the method and contrasting that process with faculty who consider themselves advanced users of IBL. At the undergraduate level, inquiry-based learning in mathematics finds its roots in views of R. L. Moore of the University of Texas. Moore believed students should build their own understanding and work through the course material individually. As peer collaboration and group work have come to be valued (National Council of Teachers of Mathematics [NCTM], 2000), Moore’s insistence on individual work has fallen out of favor. Instructors have adapted Moore’s values with time, and now, dubbed IBL, the method refers to a spectrum of instructional styles that allow students to work in small groups, consult outside resources, or pose and seek answers to their own questions. Present in all IBL classrooms, however, is an emphasis on student presentations and active student participation with very limited lecturing (Coppin, Mahavier, May, & Parker, 2009). A growing body of research has been documenting the positive impact of IBL methods on students’ gains in the cognitive and social domains (Hassi, 2009; Laursen & Hassi, 2009, 2010; Laursen, Hassi, & Crane, 2009; Laursen, Hassi, Crane, & Hunter, 2010). In particular, Laursen and colleagues (Laursen, et al., 2010) report that students in IBL courses report higher cognitive gains than students in non-IBL courses, in terms of mathematical thinking, understanding of concepts, and application of mathematical knowledge. Students in IBL courses, and in particular future teachers, reported higher cognitive gains about teaching. Given that lecturing is a predominant model of instruction in college mathematics classrooms (Lutzer, Rodi, Kirkman, & Maxwell,

2007) we asked: How do faculty learn to teach with these new methods? What kinds of concerns 15TH Annual Conference on Research in Undergraduate Mathematics Education 507 do they have? And how do resources such as experience, other colleagues, books, or conferences and workshops help them in developing a better sense of what it means to use IBL methods in teaching mathematics?

The literature on teacher learning to teach mathematics is extensive at the K-12 level, but is more limited in the post-secondary level. Investigations at the post-secondary level suggest a developmental path in the process of learning to teach (Nardi, Jaworksi, & Hegedus, 2005;

Nyquist & Sprague, 1998). Nyquist and Sprague (1998) suggest that teaching assistants’ concerns, discourse, and relationships with students and colleagues progress through a series of stages. Initially, teaching assistants focus on themselves (“Will my students like me?”) and on their own survival; next, they worry about managing discussions or handling classroom participation; and in later stages, they start to focus on students’ understanding and learning outcomes. These shifts from concerns about the self, to concerns about managing teaching, and finally to concerns about students’ learning and understanding, determine a path that we might expect as instructors teach with a new method. Nardi and her colleagues (2005) worked with tutors at the University of Oxford over an 8-week period doing individual interviews in which they were prompted to reflect on aspects of their teaching. The researchers identified four stages of pedagogical awareness—naïve and dismissive, intuitive and questioning, reflective and analytic, and confident and articulate (p. 293)—which, they propose, reveal a spectrum of awareness about students’ difficulties, strategies to overcome those difficulties, and selfreflection about teaching practices. Because they claim that instructor awareness can feed into other teaching formats (p. 293), we could anticipate comparable stages of awareness as teachers face a new instructional method for the first time. Other accounts of teaching with inquiryoriented curriculum (Marrongelle & Rasmussen, 2008; Speer & Hald, 2008; Stephan & Rasmussen, 2002) point at specific dilemmas that instructors face, in particular navigating the need to stay away for lecturing and moving toward more discussion-based classes. This literature is informative and allows us to think that there might be common concerns faculty have when they start teaching using IBL methods, and that these concerns may change and evolve as faculty teach other IBL courses.

Methods There are two primary sources of data collected over a one-year period: on-line teaching logs filled every other week and three interviews with faculty, at the beginning of the year, half way through, and the end of the year. In the pilot phase of the study, we worked with four instructors, all new to the method, having been through one week-long workshop the previous summer.

The on-line teaching logs request information on time spent on various types teaching activities (homework review, lecturing, large-group discussion, small group work, student presentations, assessment, class preparation, mathematical content, and pacing); challenges faced and concerns about these activities, solutions found to resolve these challenges, and resources used. The initial interview seeks to get baseline information about their understanding of IBL, what are necessary and sufficient conditions for a successful IBL course, and their anticipated learning goals for the students. The intermediate interview seeks to get information on the students, the curriculum, the instruction, and their assessment practices; in addition we explicitly ask instructors to tell us what they have learned about themselves, the students, teaching, and mathematics through teaching with IBL. The interview also asks for information on specific entries in the logs. The 508 15TH Annual Conference on Research in Undergraduate Mathematics Education final interview asks a combination of questions from the initial interview (e.g., their understanding of IBL) and the intermediate interview (e.g., students, curriculum, assessment).

The log data have been analyzed by finding themes across all the comments (N=36) submitted by the four instructors over a one-year period, attending first to the type of teaching activity. The themes were then used to code across the comments and refined into five categories: Student preparation, motivation, and engagement; Coverage; Rigor; Difficulty of the material; and Student Learning. We are currently analyzing the interview data.

Findings The instructors most frequently reported concerns about Student Preparation, Motivation, and Engagement with the material (14/36). For example they mentioned that the students would come to class with incomplete homework or with no evidence of having worked on some of the assignments (e.g., “The only challenge was in the most recent class when none of the students had a proof of Euler's Theorem.” Instructor 4). Instructors were concerned that student motivation waned towards the end of the term, presumably due to other commitments the students had (e.g., “My students are starting to feel the end of the semester, and they all seem quite worn down. I'm worried that their lack of enthusiasm will have a detrimental effect on their ability to keep being productive in the class.” Instructor 1) or that they appeared, at times, to be less engaged than they should be (e.g., “well overall, it is good, but I guess before spring break, their mind were somewhere else.” Instructor 3). Coverage, (8/36) was a concern shared by all instructors. As the method relies on students’ discovering the material, this theme is not unexpected, of course, and the instructors tended to compare time with their experience with non-IBL courses (e.g., “We didn't get to the division algorithm until day 5, and usually this is covered by day 2 when I'm lecturing!” Instructor 1). Departments were mentioned as a source of the pressure to cover the material (e.g., “pressure from the department to reach a level of content (namely reach the fundamental theorem of calculus), at this point it seems impossible unless I switch to a lecture format.” Instructor 3). But the pressure also came from the time that it takes to go through the discovery process (e.g., “I designed this course for prospective secondary math

teachers to end with the proof of the three impossible constructions of Euclidean geometry:

doubling the cube, squaring the circle and trisecting the angle. Everything was set up to get us there; it ties into the course we've taken in math history, and the 2-quarter sequence in geometry.

And, we aren't going to make it. It's a disappointment to me.” Instructor 2). Rigor and Difficulty of the material were each mentioned with the same frequency (6/36). Rigor referred to instructors’ dissatisfaction that the students were not learning to be careful in writing proofs (e.g., “Students were getting a little too informal in class, particularly when it came to giving proofs by induction. I struggled with how to get them to write out formal proofs by induction.” Instructor 1). Instructors also mentioned the difficulty of the content or assignments as a challenge (“The material that we are currently covering is a notch or two up in difficulty from what we have been doing all semester.” Instructor 4), which tied to students’ waning interest in some cases, led to disengaged classes. Finally, the two comments that we classified as Student Learning referred to the areas of assessment. Instructor 2 showed concern that in spite of designing a test that was quite similar to the homework assigned, students’ scores were around 78% with 2 students failing. This instructor adds: “I was disappointed to see scores as low as they were when the students weren't asked to do anything that was significantly new.” Instructor 4 showed concern about finding ways to assess students’ knowledge using other means beyond homework and presentations. We took these comments as referring to student earning because 15TH Annual Conference on Research in Undergraduate Mathematics Education 509 they appear to indicate worry about the measures we use (through assessments) of what students know.

Discussion It is interesting that these instructors voice concerns that are focused on whether students like them or the method but more about managing instruction: keeping students engaged, ensuring that they are prepared for class, regulating the difficulty of the material and the rigor of students’ productions, and handling pressures to cover material. It is less evident that the instructors worry about students’ learning. Although not definitive, this analysis gives us information about what types of concerns to expect from the larger sample. Up to now we have collected 131 teaching logs from a new sample of 28 instructors and we are in the process of analyzing these to identify trends over time and trends by instructors' experience.

Questions for the audience

1. What types of questions could be added to the logs so that student learning can become more visible? 2. We propose a developmental path but other possible interpretations are viable. What could be other frameworks that could be used to analyze these data?

References Coppin, C. A., Mahavier, W. T., May, E. L., & Parker, G. E. (2009). The Moore Method: A pathway to learner-centered instruction. Washington, DC: The Mathematical Association of America.

Hassi, M.-L. (2009). Empowering undergraduate students through mathematical thinking and learning. Paper presented at the 15th International Conference of Adults Learning Mathematics, Lancaster, PA.

Laursen, S., & Hassi, M.-L. (2009, January 5-8). Inquiring about inquiry: Progress on research and evaluation studies of inquiry based learning in undergraduate mathematics at four campuses. Paper presented at the Joint Mathematics Meetings of the MAA and AMS, Washington, DC.

Laursen, S., & Hassi, M.-L. (2010). Benefits of inquiry based learning for undergraduate college mathematics students. Paper presented at the Annual meeting of the American Educational Research Association, Denver, CO.

Laursen, S., Hassi, M.-L., & Crane, R. (2009, July 16-18). First findings from evaluation studies of the IBL Mathematics Projects. Paper presented at the 12th Annual Legacy of R. L.

Moore conference, Austin, TX.

Laursen, S., Hassi, M.-L., Crane, R., & Hunter, A.-B. (2010). Student outcomes from Inquiry Based Learning in mathematics: A mixed method study. Paper presented at the Joint Meeting of the Mathematical Association of America and the American Mathematical Society, New Orleans, LA.

Lutzer, D. J., Rodi, S. B., Kirkman, E. E., & Maxwell, J. W. (2007). Statistical abstract of undergraduate programs in the mathematical sciences in the United States: Fall 2005 CBMS Survey. Washington, DC: American Mathematical Society.

510 15TH Annual Conference on Research in Undergraduate Mathematics Education Marrongelle, K., & Rasmussen, C. L. (2008). Meeting new teaching challenges: Teaching strategies that mediate between all lecture and all student discovery. In M. Carlson & C.

L. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 167-177). Washington, DC: Mathematical Association of America.

Nardi, E., Jaworksi, B., & Hegedus, S. (2005). A spectrum of pedagogical awareness for undergraduate mathematics: From 'tricks' to 'techniques'. Journal for Research in Mathematics Education, 36(4), 384-316.

National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Nyquist, J. D., & Sprague, J. (1998). Thinking developmentally about TAs. In M. Marincovich, J. Protsko & F. Stout (Eds.), The professional development of graduate teaching assistants (pp. 61-88). Bolton, MA: Anker.

Speer, N., & Hald, O. (2008). How do mathematicians learn to teach? Implications from research on teachers and teaching for graduate student professional development. In M. Carlson & C. L. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 305-317). Washington, DC: Mathematical Association of America.

Stephan, M., & Rasmussen, C. L. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21, 459-490.

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