PROJECT OVERVIEW: Purpose: The IMAP study was based upon the assumption that prospective elementary school teachers (PSTs ) content knowledge and beliefs will be enhanced if they are provided with opportunities to learn about children's mathematical thinking while they are learning mathematics. By providing PSTs opportunities to develop more nuanced beliefs about mathematics, teaching, and learning early in their undergraduate experiences, we might launch them on a different growth trajectory that may orient them toward learning mathematics from a relational or meaning-making, rather than an instrumental, perspective (Skemp, 1978). In addition to developing the mathematics early field experience in which preservice teachers engaged, researchers developed a web-based computerized beliefs survey and a paper-and-pencil content instrument; they created a published set of 25 selected video clips, a searchable set of 216 video clips, and a set of video case studies to be used in mathematics content or mathematics education courses.
Intervention: We developed a Children's Mathematical Thinking Experience (CMTE), and we implemented it using two approaches. The CMTE-L PSTs worked with children at a local elementary school in half the CMTE meetings, six occasions, forming pairs of PSTs who together interviewed and tutored individual children (see Ambrose, 2004, and Philipp, Thanheiser, & Clement, 2002, for more extensive descriptions of the CMTE-L). CMTE-V students engaged in many of the same group activities as the PSTs in the CMTE-L, including viewing video developed for the CMTE, analyzing problem types, and anticipating children s solution strategies. In addition, they viewed video of interviews conducted in the CMTE-L, but they did not work directly with children.
We also developed a Mathematical Observation and Reflection Experience (MORE). PSTs assigned to the MOREs made 14 weekly visits to local elementary schools and each week submitted a 1 2 page reflection paper about the visit. The PSTs also wrote mid- and end-of-semester reflections about the experience.
Setting: Preservice teachers who participated in the large-scale study in Fall 2001 attended a large state university. Preservice teachers from the same university participated in pilot studies for the project in Fall 2000, Spring 2001, and Fall 2002. Children interviewed and observed by project participants were in ethnically and socioeconomically diverse schools.
Research Design: The students who participated in this study were enrolled in a first mathematics course designed for PSTs. Each student was assigned to one of four treatment groups or to the control group. Those in two of the treatment groups were provided with opportunities to study children s thinking in the Children s Mathematical Thinking Experiences (CMTE) treatments. Participants in the other two treatment groups observed mathematics classes at elementary schools in Mathematical Observation and Reflection Experiences (MORE) treatments. The CMTE treatments were college courses, albeit experimental and optional, for which the students received college credit, whereas the MORE treatments were not courses.
Prospective elementary school teachers (N = 159) enrolled in a mathematics course were randomly assigned to (a) concurrently learn about children's mathematical thinking by watching children on video (CMTE-V) or working directly with children (CMTE-L), (b) concurrently visit elementary school classrooms of conveniently chosen (MORE-C) or specially selected teachers (MORE-S, or (c) a control group. We employed a modified random assignment, constrained by the students personal class and work schedules and the times scheduled for school visits; most students were available for at least two treatments. PSTs in the control group were also enrolled in the mathematics course and completed the instruments administered to all PSTs in the study but participated in no additional fieldwork.
We designed, for pre and post assessment for the semester-long study, a web-based computer instrument to measure seven beliefs about mathematics and the way students learn mathematics and a content instrument measuring understanding of the mathematical content of interest in this study. We designed rubrics on which to base beliefs and content scores and statistically analyzed the change scores in both areas. We invited graduate students external to the project (several from other universities) to meet at our research site to code our assessments. Our project researchers trained the coders but had no input on final codes assigned by the coders. The responses were blinded so that neither coders nor trainers could determine in which treatment the respondents were enrolled or whether the responses came from pre- or post-assessments. Participants beliefs changes scores on each of the seven beliefs were analyzed using a polychotomous log-linear odds ratio using the ordered logit procedure in the STATA software package (Long & Freese, 2001). An ordered logit procedure generates a log-linear regression and corresponding goodness-of-fit statistic to test the prediction that change scores vary by group assignment.
Findings: Those who studied children's mathematical thinking while learning mathematics developed more sophisticated beliefs about mathematics, teaching, and learning and improved their mathematical content knowledge more than those who did not. Furthermore, beliefs of those who observed in typical classrooms underwent less change than the beliefs of those in the other groups, including those in the control group. Of the 18 significant differences in beliefs-scores change in this study resulting with use of the Holm's procedure, 10 involved students in the MORE-C, 7 involved students in the control group, and 1 involved students in the MORE-S, with the students in these groups showing significantly less increase on beliefs scores than students in one of the two CMTE treatments. Although no pairwise content comparisons among groups were significant, the means for those who studied children's thinking were higher than for those who did not. The pattern data supported the content advantage for those who studied children s thinking. We pooled the two groups according to whether they focused on children s mathematical thinking, and results of a t-test showed that these differences were significant at the .05 level. The effect size was 0.26 (Cohen s d = 0.2645), signifying that the mean change on the content test for PSTs in the CMTEs was about one-fourth standard deviation higher than the mean change for PSTs in the other groups.
The results of this project have been used to develop curriculum resources that integrate content knowledge and pedagogy effectively in the preparation of elementary school mathematics teachers. These resources include, in addition to the content and belief surveys, software created for use in content courses, a published set of 25 selected video clips, a searchable set of 216 video clips, and a set of video case studies to be used in mathematics content or mathematics education courses.
PROJECT PUBLICATIONS: Ambrose, R. (in press). Initiating change in prospective elementary school teachers' orientations to mathematics teaching by building on beliefs. Journal of Mathematics Teacher Education.
Ambrose, R., Clement, L., Philipp, R. A., & Chauvot, J. (2004). Assessing prospective elementary school teachers' beliefs about mathematics and mathematics learning: Rationale and development of a constructed-response-format beliefs survey. School Science and Mathematics, 104(2) 56 69.
Ambrose, R., Philipp, R., Chauvot, J., & Clement, L. (2003). A web-based survey to assess prospective elementary school teachers' beliefs about mathematics and mathematics learning: an alternative to Likert scales. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 2003 joint meeting of PME and PMENA (Vol. 2, pp. 33 39). Honolulu, HI: CRDG, College of Education, University of Hawaii.
Clement, L., Chauvot, J., Philipp, R., & Ambrose, R. (2003). A method for developing rubrics for research purposes. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 2003 joint meeting of PME and PMENA (Vol. 2, pp. 221 227). Honolulu, HI: CRDG, College of Education, University of Hawaii.
Clement, L., Philipp, R. A., & Thanheiser, E. (2002). Placing children between prospective elementary school teachers and mathematics: The mathematical early field experience. In D. S. Mewborn, P. Sztajin, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the twenty-fourth annual meeting: North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 1097 1104). Columbus, OH: Clearinghouse for Science, Mathematics, and Environmental Education.
Integrating Mathematics and Pedagogy. (2004). IMAP Content-Survey manual. San Diego, CA: Center for Research in Mathematics and Science Education, San Diego State University. http://www.sci.sdsu.edu/CRMSE/IMAP/pubs.html
Integrating Mathematics and Pedagogy. (2003). IMAP Web-Based Beliefs Survey. San Diego, CA: Center for Research in Mathematics and Science Education, San Diego State University. http://www.sci.sdsu.edu/CRMSE/IMAP/survey/
Integrating Mathematics and Pedagogy. (2004). IMAP Web-Based Beliefs-Survey manual. San Diego, CA: Center for Research in Mathematics and Science Education, San Diego State University. http://www.sci.sdsu.edu/CRMSE/IMAP/pubs.html
Philipp, R. A., Ambrose, R., Lamb, L. C., Sowder, J. T., Schappelle, B. P., Sowder, L., & Thanheiser, E. (accepted for publication). Effects of early field experiences on the mathematical content knowledge and beliefs of prospective elementary school teachers: An experimental study. Journal for Research in Mathematics Education.
Philipp, R. A., & Sowder, J. T. (2002). Using eye-tracking technology to determine the best use of video with prospective and practicing teachers. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 233 240). Norwich, England: University of East Anglia.
Philipp, R. A., Thanheiser, E., Clement, L. (2002). The role of a children's mathematical thinking experience in the preparation of prospective elementary school teachers. International Journal of Educational Research, 37, 195-210.
Presentations
Ambrose, R. (2002, April). They all hate math: Getting beyond our stereotypes of prospective elementary school teachers. Paper presented at the annual meeting of the American Educational Research Association, New Orleans.
Ambrose, R. C., & Vincent, C. (2003, April). Blending mathematics learning with an early field experience: What do prospective elementary school teachers learn? Paper presented at the annual meeting of the American Educational Research Association. Chicago.