Abstract
This chapter reports the results of a cross-national study designed to examine the mathematics knowledge and the mathematical pedagogical content knowledge attained by prospective primary teachers at the end of their formal preparation and before they begin to teach. The study used survey methods to collect data from nationally representative samples of pre-service university-based teacher education programs and their future teachers in Botswana, Chile, Chinese Taipei, Germany, Malaysia, the Philippines, Poland, Russia, Singapore, Spain, Switzerland, Thailand, and the United States. Descriptive and multivariate analyses show that future teachers’ individual characteristics, such as levels of achievement in previous schooling, programs’ selection policies, and opportunities to learn the content and the pedagogy of the mathematics school curriculum, were associated with higher levels of knowledge and dispositions toward teaching and learning mathematics. Results support teacher education policies directed at (a) raising the level of subject knowledge required for program selection and graduation and (b) increasing the level of complexity and cognitive demand of the opportunities to learn mathematics and mathematics pedagogy offered to future primary mathematics teachers.
TEDS-M and the study contained in this chapter were supported by funding provided by a grant from the National Science Foundation Award No. REC – 0514431 (M.T. Tatto, PI). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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Notes
- 1.
‘Note that in the U.S. ‘primary’ usually refers to grade K-3, while ‘elementary’ is used for grades K-5 or K-6. In this chapter the term elementary is used as was used in TEDS-M (see Tatto et al., 2012 pp. 29–32 for specific definitions within countries as to what grades are included as primary or secondary).’
- 2.
One of the unique contributions of TEDS-M was arriving at common definitions of terms to support measurement across the participating countries. One such term is “program type,” used to refer to the variety of programs that prepare future primary teachers across the participating countries. Program type refers to clusters of programs that share similar purposes and structural features—such as the credential earned, the range of school grade levels for which teachers are prepared, and the degree of subject-matter specialization for which future in teachers are prepared—that correspond to distinct pathways to becoming qualified to teach (Tatto et al., 2012, pp. 27–28).
- 3.
The final composition of the OTL indicators was done based on the logical organization of courses as judged by experts, after repeated piloting of the questions ultimately used to develop them. The Comparative Fit Index, or CFI, was used to test the degree to which the indicators were internally consistent; the CFI depends in large part on the average size of the correlations in the data, where an acceptable model is indicated by a CFI larger than .93, but .85 is acceptable (see Bollen, 1989). Another approximation is the Tucker Lewis index (TLI), which is relatively independent of sample size (Marsh, Balla, McDonald, 1988), where values over .90 or.95 are considered acceptable (e.g., Hu & Bentler, 1999). The Root Mean Square Error of Approximation, or RMSEA, is another test of model fit; good models are considered to have a RMSEA of .05 or less, while models whose RMSEA is .1 or more have a poor fit. Fit Indices provided evidence that the groupings that formed the opportunity to learn indicators make sense (i.e., tertiary level mathematics CFI .911, TLI .954, RMSEA .044, school-level mathematics CFI .97, TLI .973, RMSEA .057). The reliabilities for the OTL scales were unweighted and were estimated using jMetrik 2.1 (Meyer, 2011). The reliability estimates were based on the congeneric measurement model, which allows each item to load on the common factor at different levels, and allows item error variances to vary freely (each item can be measured with a different level of precision). This was considered by the TEDS-M study research team to be the most flexible measurement model and the most appropriate for measures with few items. The reliabilities for the opportunity to do class reading on research on mathematics teaching and learning for the primary sample is .85; for the opportunity to learn in a coherent program, the reliability is .96.
- 4.
The reliabilities for the beliefs scales were unweighted and were estimated using jMetrik 2.1 (Meyer, 2011). The reliability estimates were based on the congeneric measurement model, which allows each item to load on the common factor at different levels, and allows item error variances to vary freely (each item can be measured with a different level of precision) as described in footnote 3 for the OTL scales. For the international sample, the reliability for the beliefs scale “mathematics as a set of rules and procedures” for future primary teachers is .94 and .92 for “learning mathematics through active involvement.”
- 5.
For the international sample, the reliabilities for the mathematics content knowledge and the mathematics pedagogical content knowledge scores were .83 and .76, respectively. Reliabilities tend to be high if there is a great deal of variation in the sample relative to the size of the standard error. The reliability will be low if one of the following occurs: there is a small standard deviation in the sample, or there is a large standard error (e.g., the test was too easy for a particular sample; this was the case for Chinese Taipei and Singapore).
- 6.
Several TEDS-M items were provided by other studies, including Study of Instructional Improvement (SII) Learning Mathematics for Teaching/Consortium for Policy Research in Education (CPRE), University of Michigan, School of Education, Ann Arbor, MI, supported by NSF grants REC-9979873, REC- 0207649, EHR-0233456 & EHR 0335411. Developing Subject Matter Knowledge in Math Middle School Teachers (P-TEDS), Michigan State University, supported by NSF Grant REC-0231886. Knowing Mathematics for Teacher Algebra (KAT), Michigan State University, supported by NSF Grant REC-0337595 (TEDS-M received 2006 publication copyright for those items).
- 7.
Hierarchical Linear Modeling or HLM (Raudenbush & Bryk, 2002; Raudenbush, Bryk, & Congdon, 2004) is a statistical method that helps compute regressions at multiple levels, estimating a regression within each program and combining them to see if there is a common regression across programs within a given country. If regression slopes vary across programs, it is possible to examine program-level characteristics that may explain such variation, and to explore the program factors that may show a relationship with future teachers’ outcomes. The analysis was done using a two-level HLM model in which future teachers were nested within their teacher education programs within countries. The descriptive statistics for the institutions and future teachers are in Appendix Table 8.4.
- 8.
Regression analysis (ordinary least squares or OLS) is a method that helps to explore the relationship between a dependent variable, in this case mathematics knowledge for teaching (defined as MCK and MPCK), and one or more explanatory variables, in this case individual and program variables. The descriptive statistics for the institutions and future teachers are in Appendix Tables 8.5 and 8.6.
- 9.
The definition of primary grades is “the first stage of basic education which starts normally between the ages of 5–7,” according to UNESCO’s International Standard Classification of Education (UNESCO-UIS, 2006). Since the age of children enrolled in primary education varies, as does the grade span for which teachers are prepared to teach, countries were asked to define the grade range. For TEDS-M, the grade span for which teachers are prepared to teach in the different countries is included in Table 8.1 and ranges from Grades 1 to 10 for generalists and Grades 1 to 12 for mathematics specialists.
- 10.
Anchor points are dependent on the number of items included in a particular measure. In the assessment 2/3 of the items measured MCK while 1/3 measured MPCK, therefore for MPCK only one anchor point was defined at the proficient level.
- 11.
According to the Common Core State Standards, “To help students meet the standards, educators will need to pursue, with equal intensity, three aspects of rigor in the major work of each grade: conceptual understanding, procedural skills and fluency, and application. Conceptual understanding: The standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures. Procedural skills and fluency: The standards call for speed and accuracy in calculation. Students must practice core functions, such as single-digit multiplication, in order to have access to more complex concepts and procedures. Fluency must be addressed in the classroom or through supporting materials, as some students might require more practice than others. Application: The standards call for students to use math in situations that require mathematical knowledge. Correctly applying mathematical knowledge depends on students having a solid conceptual understanding and procedural fluency” (CCSS, 2016).
- 12.
The ICC reports the percent of variance between programs, where 100-ICC (100 minus the ICC) is the percent of variance within programs. This is always reported simply as the ICC for percent of variance between groups. The % of variance explained in the last two rows then simply states how much variance each model explains. In Table 8.2, in Poland, 59% of the variance in MCK performance is between programs, 41% is within programs. The model including all variables explains 5% of the variance within programs (i.e., student characteristics do not explain much of the variance in their performance within program); that is, the student characteristics explain 5% of the 41% variance within programs. The model also explains 85% of the variance between programs; that is, the program characteristics explain 85% of the 59% variance between programs.
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Tatto, M.T. (2018). The Mathematical Education of Primary Teachers. In: Tatto, M., Rodriguez, M., Smith, W., Reckase, M., Bankov, K. (eds) Exploring the Mathematical Education of Teachers Using TEDS-M Data. Springer, Cham. https://doi.org/10.1007/978-3-319-92144-0_8
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