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Measuring mathematical knowledge for teaching: a longitudinal study using two measures

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Abstract

This longitudinal study examines growth in teacher knowledge as measured by two popular assessments—Learning Mathematics for Teaching (LMT) and Diagnostic Teacher Assessments in Mathematics and Science (DTAMS). Using data collected from 24 teachers, we compare the extent to which each assessment captured teacher learning during a K-8 mathematics content/pedagogy hybrid course and a general mathematics content course. Teachers made large gains on both measures, but the LMT better captured gains made during the hybrid course, whereas DTAMS better detected gains during the mathematics course. Patterns in teacher performance suggest substantive differences between specialized mathematical knowledge for teaching and everyday mathematics knowledge. The theoretical and practical implications of these findings are discussed.

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Notes

  1. The areas of “horizon knowledge” and “knowledge of curriculum” were more tentatively placed in this diagram when published by Ball et al. (2008). These two areas are beyond the scope of this paper.

  2. In our own attempt to classify such items, we estimated that roughly two-thirds of the items focus specifically on specialized content knowledge.

  3. Although the authors of this study were not involved with LMT or DTAMS development, they have attended LMT training and have consulted with the LMT and DTAMS developers, as needed, over the course of this study.

  4. Video available at http://www.learner.org/series/modules/express/pages/ccmathmod_20.html.

  5. The LMT developers originally used a convenience sample to pilot the forms, and the mean for that sample was made to correspond with “0” when establishing the scale. Hence, a score below 0 on the LMT scale is not necessarily “below average” among the general population of US teachers. Indeed, Hill (2010) recently found that the performance of a nationally representative sample of elementary teachers was slightly lower than that of the original pilot sample on identical items.

  6. In an effort to reduce the amount of testing while still maintaining regular assessments, we administered LMT alone during two points in the program. We chose the LMT over DTAMS on those occasions, because the elementary LMT measures had more established reliability and the items appeared to more closely align with program goals.

  7. We were initially concerned about our lack of ability to closely monitor various factors outside the program that could affect teachers’ knowledge during the intervening year. Teachers reported attending 34 h of mathematics-related professional development activities each year, in addition to our program. The focus of these activities varied by teacher and school, but rarely (if ever) involved specific mathematics content knowledge. Some examples include a workshop on integrating mathematics and reading and a Response-To-Intervention initiative. Moreover, teachers took additional courses during this time, but those courses focused on science as well as educational philosophy, psychology, and equity. Given the similarities in teachers’ LMT scores on December 2008 and January 2010, we are less concerned about these external factors increasing teachers’ relevant knowledge during that time. It is possible that teachers gained in some areas that are better captured by DTAMS than by LMT, which would artificially boost DTAMS gains in the second course. However, given the math content focus of DTAMS, it is unlikely that this was a major factor in this study.

  8. Algebra received specific focus in both courses, but was also interwoven throughout each course. For example, in the hybrid course, teachers were encouraged to use algebraic methods (e.g., using variables to generalize patterns) to solve problems in various curricular areas. In the content course, the ALEKS system was used throughout the semester to foster teachers’ algebra skills.

  9. Given that this study involves a sample that is not randomly chosen from a particular population, significance levels are reported here to indicate the strength of patterns identified, as opposed to indicating that such patterns would hold in a general population. However, given the relatively small sample involved, the significance tests can be viewed as rather conservative indicators of associations that are noteworthy and could hold in broader populations.

  10. We used two-tailed tests as a more conservative approach to testing statistical significance. Given the small sample size and the consequent danger of Type I errors, we take note of p-values between .05 and .10 and consider them “marginally significant.” It is also worth noting that the math content gain on DTAMS during the hybrid course is significant with a one-tailed t test at p = 0.04.

  11. The gain is significant with a one-tailed t test at p = 0.028.

  12. We analyzed the items on which teachers decreased in performance, but we could detect no consistent pattern that would illuminate reasons for the decrease.

  13. Several middle school teachers’ who did note a connection with their teaching said they would use course ideas for extra enrichment or extension activities, as opposed to enhancing their everyday curriculum and instruction.

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Acknowledgments

This project was funded by No Child Left Behind, Title II, Part B, US Department of Education. The authors are grateful for the support of Dr. Barbara Hug, principal investigator of the Illinois Mathematics and Science Partnership Grant that provided professional development to the teachers in this study. The authors are also deeply indebted to the teachers who so generously gave of their time to participate in this research. The authors are solely responsible for the content of this article.

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  1. Rice University, 6100 Main MS-172, Houston, TX, 77005-1827, USA

    Yasemin Copur-Gencturk

  2. University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, USA

    Sarah T. Lubienski

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  1. Yasemin Copur-Gencturk
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Correspondence to Yasemin Copur-Gencturk.

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Copur-Gencturk, Y., Lubienski, S.T. Measuring mathematical knowledge for teaching: a longitudinal study using two measures. J Math Teacher Educ 16, 211–236 (2013). https://doi.org/10.1007/s10857-012-9233-0

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