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Krichgraber, J. Hillel, M. Niss, & A. Schoenfeld (Eds.), The teaching and learning of mathematics at university level: An ICMI Study (pp. 273-280). Dordrecht: Kluwer.

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344 15TH Annual Conference on Research in Undergraduate Mathematics Education



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Capstone courses have been recommended as a way to connect the mathematics pre-service secondary mathematics teachers learn in college to the school mathematics they will teach in their own classrooms. Yet little is known about the status of these courses across the U.S, in whether they are offered, the topics that are covered, the curriculum used, and the pedagogical approach, among other aspects of the course. We will present findings from a 2011 survey of U.S. colleges and universities that investigated the status of such capstone courses at these institutions. Discussion will be centered around the importance and future of such courses in teacher preparation programs.

Keywords: capstone course, teacher preparation, secondary, mathematics Research Issue Hodgson (2001) recognized that pre-service secondary school teachers “have no explicit occasion for making connections with the mathematical topics for which they will be responsible in school, nor of looking at those topics from an advanced point of view” (p.

509). Such an experience is important as these future teachers need a “deep conceptual understanding of the school mathematics content which falls under their responsibility” (Hodgson, 2001, p. 512), and this should occur before their entry into their profession.

Addressing this same concern, the Conference Board of the Mathematical Sciences (CBMS) recommended that pre-service high school teachers complete “a 6-hour capstone course connecting their college mathematics courses with high school mathematics” (2001, p. 8).

Since that time, there have been a handful of reports on implementations of individual courses that fit this description (e.g., Hill & Senk, 2004; Loe & Rezak, 2006; Shoaf, 2000;

Van Voorst, 2004). However, the status of the mathematics capstone course in the United States is largely unknown; there has thus far been no systematic study of the extent or characteristics of its varied implementations. The goals of this research study are to uncover the status of capstone courses across the United States, to understand what is offered to preservice high school mathematics teachers, and to investigate whether CBMS recommendations are being followed by programs that prepare future high school mathematics teachers.

Methodology In 2011, we conducted a survey of universities that may offer an upper-level capstone course either in the mathematics department or in the college of education for mathematics majors pursuing secondary certification. From the 1,713 institutions listed by the Carnegie Foundation for the Advancement of Teaching (Carnegie Classifications, 2011) we 15TH Annual Conference on Research in Undergraduate Mathematics Education 345 selected a stratified random sample of 200 institutions, weighted appropriately for each of nine classification groups (e.g., PhD granting institutions with high research activity). For the purposes of the survey, we defined a capstone as a course taken at the conclusion of a program of study for pre-service secondary mathematics teachers that places a primary focus on providing at least one of the following: (1) bridges between upper-level mathematics courses, (2) connections to high school mathematics, (3) additional exposure to mathematics content in which students may be deficient, or (4) experiences with communicating with, through, and about mathematics (Loe & Rezac, 2006).

The survey investigated the prevalence and nature of courses fitting this description. In particular, it included questions about course logistics such as the department, title, duration, textbook(s), technology, and other resources used in the course.

The survey also included questions relating to the nature of the course; specifically, data were collected about the description of the course in the universities’ course catalogs, the course goals, the instructional style, and the content. To provide a more complete picture of the current state of capstone courses, data were also collected about instructors’ backgrounds and their levels of academic freedom. Data collection was completed in November 2011.

Questions to be considered by the audience The discussion portion of the presentation will be framed by an initial presentation of the general findings of our study. Specifically, we will share findings about commonalities and differences of capstone courses across the various types of institutions. Then, we will pose

two questions to the audience for discussion:

1. What are your experiences with capstone courses in relation to the national landscape, and what more would you like to learn about capstone courses, instructors, and students? During future phases of this project, we will be soliciting institutions for a follow-up interview that will collect data to help us look more deeply at the methods used and nature of content taught in capstone courses.

The discussion will provide direction and context for the next phase of the study.

2. What resources or collaborations have the potential to support institutions wanting to offer new capstone courses or to improve the existing capstone experience? The limited research on and discussion about capstone courses are cause for concern that institutions are building courses from the ground up without a sense of how their efforts fit with others. This discussion may lead to a sharing of ideas about capstone resources and, potentially, the formation of networks of support or collaboration.

The presenters will document the discussions and will share session notes (via email) with attendees and interested parties.

Implications for the preparation of pre-service secondary mathematics teachers Ten years after the recommendation for capstone courses by the CBMS, mathematics education researchers continue to emphasize the need for pre-service mathematics teacher training programs to make connections between university-level mathematics, teaching methods, and high school content (e.g., Artzt et al., 2011). This preliminary report will help uncover the extent to which this need is being addressed. Furthermore, the results of the survey may offer direction to mathematics departments wishing to create or to improve capstone courses. The discussion portion of the session will guide future phases of 346 15TH Annual Conference on Research in Undergraduate Mathematics Education our research agenda and will potentially foster impactful collaborations. Capstone courses offer great promise for enhancing pre-service teacher training; the research presented in this session, and the discussion it provokes, will provide insight into the popularity of this relatively new course, the variety of implementations, and the future of the capstone course.


Artzt, A., Sultan, A., Curcio, F. R., & Gurl, T. (2011). A capstone mathematics course for prospective secondary mathematics teachers. Journal of Mathematics Teacher Education, Online First, June 14, 2011.

Carnegie Classifications. (2011). Carnegie Foundation for the Advancement of Teaching. Retrieved from http://classifications.carnegiefoundation.org/resources/ Conference Board of the Mathematical Sciences (CBMS). (2001). The mathematical education of teachers. Providence, RI: American Mathematical Society.

Hill, R., & Senk, S. (2004). A capstone course for prospective high school mathematics teachers. Mathematicians and Education Reform Newsletter, 16(2), 8–11.

Hodgson, B. (2001). The mathematical education of school teachers: Role and responsibilities of university mathematicians. In D.A. Holton (Ed.) The teaching and learning of mathematics at the university level: An ICMI study (pp. 501 – 518). Boston, MA: Kluwer Academic Publishers.

Loe, M., & Rezak, H. (2006). Creating and implementing a capstone course for future

secondary mathematics teachers. The Work of Mathematics Teacher Educators:

Continuing the Conversation, 3, 45-62.

Shoaf, M. M. (2000). A capstone course for pre-service secondary mathematics teachers. International Journal of Mathematical Education in Science and Technology, 31(1), 151–160.

Van Voorst, C. (2004). Capstone mathematics course for teachers. Issues in the Undergraduate Mathematics Preparation of School Teachers, 11.

15TH Annual Conference on Research in Undergraduate Mathematics Education 347 Title Student Understanding in the Concept of Limit in Calculus: How Student Responses Vary Depending on Question Format and Type of Representation Author Rob Blaisdell, University of Maine Abstract Research indicates that calculus students have difficulties with limit. However, underlying reasons for those difficulties and possible influences of question format have not been examined in detail. Since limit is foundational to calculus it would help the mathematics education community to know not only the difficulties students have, but also how questions used to assess knowledge affect responses. Data for this study came from surveys administered to 111 first semester calculus students. Survey questions focused on limit in multiple representations including graphs, mathematical notation and definitions. Questions were multiple choice and free response. Student difficulties documented in previous research were evident in this population.

Findings also indicated that difficulties students exhibited in one question were sometimes different then the difficulties those same students exhibited when asked about the same idea in a different representation. Students in general had less difficulty with graphical representations than mathematical notation or definition questions.

Keywords Undergraduate students’ thinking Multiple representations Survey question design Introduction Knowledge of how students understand mathematical topics can help inform and improve instruction. Because the limit concept is a foundational concept in calculus it would help the mathematics education community to know not only the difficulties that students have, but also how the questions used to assess their knowledge affect their responses. While research into student ideas and thinking has flourished, research into how students interact with the questions they are given is lacking. This study extends existing work on student thinking about limits by examining how students respond to questions given in different formats. In particular, students were asked questions that involved mathematical notation/symbols and ones that were based on graphs to investigate whether students demonstrated different levels of success on the differently formatted questions.

Other researchers have found that question format can significantly influence student responses.

Some of this research has been performed in the context of attitudinal surveys (Tanur, 1992;

Schuman & Presser, 1981) and the areas of confirmation bias (Nickerson, 1998), answer confidence (Koriat, Lichtenstein & Fischhoff, 1980) and response elicitation (Garthwaite, Kadane & O’Hagan, 2005). However this issue of links between question format and what data on student thinking is generated has not been examined for student thinking about limits.

Knowing whether students perform differently on questions in different formats could be of 348 15TH Annual Conference on Research in Undergraduate Mathematics Education importance to researchers examining student thinking of limit as well as instructors who use written tasks to assess student learning. Knowledge about student thinking from this study will be used in future research on college mathematics instructors' knowledge of student thinking about the concept of limit in calculus.

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