Framework

Alternative routes to teaching certification in the United States have grown in number and account for an ever-increasing portion of the new teachers in American public schools. The majority of these routes are early entry programs, in which new teachers begin teaching before satisfying the requirements for full licensure. Policy arguments supporting alternative routes presume that new teachers can develop the knowledge, skills, and dispositions needed for teaching competently while on the job. In this study, I focus on teaching mathematics in the domain of multiplicative reasoning (for example, including fractions, ratios, and proportions) because this content is often difficult for students and teachers and, although multiplicative reasoning is a primary concern in the middle grade curriculum, it also plays a significant role in elementary and high school mathematics.

Recent work on teacher knowledge builds on Shulman's (1986) idea of pedagogical content knowledge to define mathematical knowledge for teaching (MKT) as the mathematics entailed in the work of teaching (e.g., Ball, Thames, & Phelps, 2008). In the last decade, success in identifying and measuring MKT has enabled researchers to establish empirical relationships between MKT and student achievement (e.g., Hill, Rowan, & Ball, 2005; Baumert et al., 2010). In two separate national samples of middle and elementary teachers, Hill (2007, 2010) found teachers' grade level and years of teaching experience to be positively correlated with their MKT, and called for more research describing the content knowledge of alternatively certified teachers.

Moreover, in the study of elementary teachers, Hill (2010) found a disappointingly low correlation (0.25) between MKT and teachers' self-concept of mathematics.  Hill observed this result is concerning because program evaluations employ teachers' reports of learning, but in addition, the finding calls into question the expected relationship between motivation beliefs and knowledge (e.g., Newton, 2009). Bandura (1977) proposed self-efficacy beliefs, a judgment of one's ability to succeed in some activity, as the root of motivation, governing effort and perseverance.  Teaching self-efficacy beliefs, in which teaching is the activity in question, have been successfully measured and related empirically with a variety of student outcomes including achievement (see Tschannen-Moran, M. & Woolfolk-Hoy, A., 2001, for a review).

Methods

This quantitative study is descriptive; I seek to answer the following research questions: How are teaching self-efficacy beliefs related to teachers' content knowledge for teaching in the domain of multiplicative reasoning? How are the MKT and teaching self-efficacy beliefs of early entry teachers different than those of traditionally certified teachers? Regression analysis of cross-sectional data is used to answer these questions; I do not presume to address questions of causality.

Data

The measure of content knowledge uses 26 items developed for the Measures of Effective Teaching Project. Because self-efficacy beliefs are task and situation specific, I developed a measure of teaching self-efficacy beliefs tailored to the domain of multiplicative reasoning by adapting existing measures for science teaching self-efficacy (Enochs & Riggs, 1990; Roberts & Henson, 2000).

The sample is derived from an ongoing census of Texas mathematics teachers currently being conducted by researchers at Michigan State University (see http://usteds.msu.edu/project_overview.asp). Teachers who complete the initial survey are invited to take an additional survey. Data from the second survey are reported here. Because of the timeframe for data collection, the sample reported in this study is partial.  To date, 122 of the approximately 700 teachers invited to take the second survey have agreed to take it, but only 35 answered more than 25% of the survey questions. Of those 35 teachers, 33 answered all or all but 1 of the survey questions.  Over the next few months, surveys will be sent to almost all of the approximately 70,000 mathematics teachers in Texas.

Results

The low n of 35 in the partial sample reported here is insufficient to do the necessary psychometric reliability and validity work (including IRT score estimation and factor analyses) and leads to inflated standard errors so fewer estimates are statistically significant.  However, even this small (but possibly biased) sample yields some interesting preliminary results under classical test theory.

Table 1. Correlations among predictors of MKT.

			Teaching Experience	Highest Grade Taught	GTES	MTSE
MKT			.19	.45 **	.13	.41 *
Teaching Experience				.26	.06	.39 *
Highest Grade Taught					.15	.56
GTES						.01 ***
* p < .05	** p < .01	*** p < .001

Correlations among predictors of MKT (Table 1) suggest that domain-specific self-efficacy beliefs are more correlated with MKT than scores on the General Teaching Efficacy Scale (the GTES, see Hoy & Woolfolk,1993) and that grade level is also moderately correlated with MKT. The sample further suggests that teachers at different grade levels have different scores for MKT in the multiplicative reasoning domain (Figure 1), but that early entry teachers were not found to have significantly different MKT than traditional route teachers (Figure 2).

Figure 1. Significant (p = .012) differences exist among the mean MKT scores of teachers with experience at different grade levels.

Figure 2. Differences between the mean MKT scores of early entry and traditional route teachers are not significant (p = .29).

A series of regression equations (Table 2) suggests that teaching self-efficacy beliefs account for 16% of the variance in MKT scores, and by adding teachers' grade level to the model, the variance explained rises to 24%. Only 14% of variance in MKT was accounted for in a regression model for MKT with 14 predictors (Hill, 2010). The regression coefficients in this model are not significant. However, a simulation increasing the sample size to 128 but preserving the observed distribution suggests that with a larger number of observations the parameter estimates for mathematics teaching self-efficacy and grade level would be significant.

Table 2. Regression models of MKT.

		Model 1 (n = 32)		Model 2 (n = 32)	Model 3 (simulated n = 128 )
Intercept		.01 (.17)		.00 (.17)	.00 (.08)
MTSE		.38 * (.17)		.19 (.21)	.19 * (.10)
GTES		.13 (.17)		.09 (.17)	.09 (.08)
Highest Grade Taught				.33 (.20)	.33 *** (.10)
R2		.16		.24	.24
* p < .05	** p < .01		*** p < .001

Implications

Clearly, these results are very tentative and preliminary.  Once the full sample is obtained, and if models similar to those presented here fit the full data set, these results would suggest that early entry teachers are not significantly more knowledgeable than their traditional route counterparts, contrary to policy arguments that favor alternative certification routes because they ostensibly increase the number of mathematically knowledgeable teachers in schools. Moreover, the domain-specific measure of teaching self-efficacy appears to be orthogonal to a general measure of teaching self-efficacy, and together these beliefs explain a large portion of the variance in teachers' domain-specific MKT. If the relationship between self-efficacy beliefs and content knowledge holds up in the larger data set, future research might explore the role of self-efficacy beliefs in teacher development designed to increase content knowledge (for example, in early entry certification routes).

References

Ball, D., Thames, M., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.

Bandura, A. (1977). Self-efficacy:  Toward a unifying theory of behavioral change. Psychological Review, 84, 191-215.

Baumert, J., Kunter, M., Blum, W., Brunner, M. Voss, T., Jordan, A., et al. (2010). Teachers' mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47, 133–180.

Eddy, Colleen M. (2006) The affects of a middle grades teacher education program on preservice teachers choice of teaching strategies and mathematical understanding. Ed.D. dissertation, Baylor University, United States -- Texas. Retrieved October 20, 2008, from Dissertations & Theses: A&I database. (Publication No. AAT 3195295).

Enochs, L. G., & Riggs, I. M. (1990). Further development of an elementary science teaching efficacy belief instrument: A preservice elementary scale. School Science and Mathematics, 90(8), 695-706.

Hill, H., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42, 371–406.

Hill, H.C. (2007). Mathematical knowledge of middle school teachers: Implications for the No Child Left Behind Policy initiative. Educational Evaluation and Policy Analysis (29), 95-114.

Hill, H. C. (2010). The nature and predictors of elementary teachers' mathematical knowledge for teaching. Journal for Research in Mathematics Education, 41, 513–545.

Hoy, W. K., & Woolfolk, A. E. (1993). Teachers' sense of efficacy and the organizational health of schools. Elementary School Journal, 93, 335–372.

Netwon, K. J. (2009). Instructional practices related to prospective elementary school teachers' motivation for fractions. Journal of Mathematics Teacher Education, 12, 89– 109.

Roberts, J.K., & Henson, R.K. (2000). Self-efficacy teaching and knowledge instrument for science teachers (SETAKIST): A proposal for a new efficacy instrument. Paper presented at the Annual Meeting of the Mid-South Educational Research Association.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

Tschannen-Moran, M. & Woolfolk Hoy, A. (2001). Teacher efficacy: Capturing an elusive concept. Teaching and Teacher Education 17, 783-805.