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«The Mathematics Educator 2013 Vol. 22, No. 2, 11–30 The Association between Teachers’ Beliefs, Enacted Practices, and Student Learning in ...»

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The Mathematics Educator

2013 Vol. 22, No. 2, 11–30

The Association between Teachers’ Beliefs,

Enacted Practices, and Student Learning in

Mathematics

Drew Polly, Jennifer R. McGee, Chuang Wang, Richard

G. Lambert, David K. Pugalee, and Sarah Johnson

Mathematics educators continue to explore ways to improve student

learning. Of particular interest are the relationships between teachers’

instructional practices, their beliefs towards mathematics teaching, and

student learning outcomes. While some studies have found empirical links between teachers’ enactment of specific instructional practices and gains in student learning, there is no conclusive connection between beliefs, instructional practices, and gains in student learning outcomes.

This study examines a few critical relationships between: teachers’ beliefs and instructional practices, teachers’ beliefs and student learning outcomes, and teachers’ instructional practices and student learning outcomes. Data from 35 teachers and 494 elementary school students indicated significant relationships between teacher beliefs and practices but not between teacher beliefs or instructional practice when related to student achievement in mathematics measured by curriculum-based tests.

Implications for the design of professional development and for further research related to mathematics teachers’ beliefs, their instructional practice and their student learning outcomes are also shared.

Dr. Drew Polly is an Associate Professor in the Department of Reading and Elementary Education at UNC Charlotte. His research interests include supporting the enactment of learner-centered instruction and standards-based mathematics pedagogies in elementary school classrooms.

Dr. Jennifer R. McGee is an Assistant Professor in the Department of Curriculum and Instruction at Appalachian State University. Her research interests include program evaluation, and validating instruments to measure the self-efficacy of classroom teachers.

Dr. Chuang Wang is an Associate Professor in Educational Leadership and a Research Associate in the Center for Educational Measurement and Evaluation (CEME) at UNC Charlotte. His research interests include the evaluation of professional development programs.

Dr. Richard G. Lambert is a Professor in Educational Leadership and the Director of the Center for Educational Measurement and Evaluation (CEME) at UNC Charlotte. His research interests include the evaluation of large-scale professional development and teacher evaluation programs.

Dr. David K. Pugalee is a Professor in Middle, Secondary, and K-12 Education and the Director of the Center for Science, Technology, Engineering, and Mathematics (STEM) Education at UNC Charlotte. His research interests include examining ways to integrate writing and litearcy into the mathematics curriculum.

Sarah Johnson is a Doctoral Candidate in the Curriculum and Instruction program at UNC Charlotte. Her research interests include examining how to best support secondary teachers' use of reform-based pedagogies.

Polly, McGee, Wang, Lambert, Pugalee, and Johnson

–  –  –

Improving Student Learning in Mathematics Mathematics educators and policy makers continue to examine how to best increase student learning outcomes in mathematics (Braswell, Daane, & Grigg, 2003; Gonzalez et al., 2004; Stigler & Hiebert, 1999; United States Department of Education [USDE], 2008; Wu, 2009). Despite mixed results, researchers have found empirical links between specific instructional practices and student learning outcomes (Carpenter, Fennema, Franke, Levi, & Empson, 2000; USDE, 2008; Wenglinsky, 1999). These instructional practices reflect a student-centered view on teaching mathematics, in which students engage in mathematically rich tasks and are supported by classroom teachers who pose questions and modify instruction based on students’ mathematical thinking (Carpenter, Fennema, & Franke, 1996; National Council for Teachers of Mathematics, 2000).

In recent years, critics to this student-centered approach to teaching mathematics have emerged, citing a need to focus more on basic facts and mathematical algorithms (Marshall, 2006). The recently published report from the United States National Math Panel (USDE, 2008) found that evidence suggesting one specific approach being more effective than others was inconclusive. Some studies (e.g., Fennema, Carpenter, Franke, Levi, Jacobs, & Empson,1996; Gonzalez et al., 2004; Polly, 2008) have found empirical links between student-centered approaches to teaching mathematics and statistically significant gains in student learning outcomes, but there is still a gap in the literature regarding the interplay between teachers’ instructional practices and student learning.

Teachers’ Beliefs in Mathematics

Teachers’ beliefs towards mathematics and their impressions of effective mathematics teaching have been associated with teachers’ enacted instructional practices (Fennema et al., 1996), their use of curricula (Remillard, 2005; Stein & Kim, 2008), and their willingness to enact student-centered pedagogies (Heck, Banilower, Weiss, & Rosenberg, 2008; McGee, Wang, & Polly, Beliefs and Student Learning 2013; Remillard & Bryans, 2004). Discussion of the composition of teachers’ mathematical beliefs has gone on for decades. Ernest (1991) argued that a mathematics teacher’s belief system has three parts; the teacher’s ideas of mathematics as a subject for study, the teacher’s idea of the nature of mathematics teaching, and the teacher’s idea of the learning of mathematics. Askew, Brown, Rhodes, Johnson and William (1997) characterized the orientations of teachers towards each of these components as transmission (T), discovery (D) or connectionist (C). Swan (2006) posited that an individual teacher’s conception of mathematics teaching and learning might combine elements of each of them, even where they appear to be contradictory.





Swan (2006) explained Askew et al.’s (1997) categories in detail. Transmission-oriented teachers believe that mathematics is a set of factual information that must be conveyed or presented to students, and typically enact didactic, teacher-centered methods.

Discovery-oriented teachers view mathematics as a set of knowledge best learned through student-guided exploration, and frequently tend to focus on designing effective classroom experiences that are appropriately sequenced. Lastly, connectionist-oriented teachers view mathematics as an intertwined set of concepts, and they rely heavily on experiences to help students learn about the connections between mathematical topics.

The enactment of student-centered and standards-based pedagogies requires teachers to embrace both a discovery and a connectionist stance (Swan, 2007). Teachers are charged with the role of designing learning environments and facilitating students’ exploration of concepts through a variety of hands-on activities and games (Mokros, 2003). Following these activities, teachers guide students’ discussions of the activities and help them make explicit the mathematical concepts that were embedded in the tasks. In order for the implementation of standards-based instruction to be effective, teachers must facilitate both the activities and the discussion of the mathematics. Although discovery and connectionist dispositions are related to the philosophical underpinning of standards-based mathematics curricula, the transmission view is contradictory. Transmissionoriented teachers relate best to traditional curricula in which information is presented and followed by substantial practice Polly, McGee, Wang, Lambert, Pugalee, and Johnson opportunities. Therefore, it is reasonable to expect that teachers who embrace standards-based mathematics curricula are oriented towards either the discovery or connectionist views.

Teachers’ Mathematics Instruction Mathematics education researchers have classified teachers’ instruction in numerous ways. Qualitative studies (e.g., Cohen, 2005; Henningsen & Stein, 1997; Peterson, 1990; Schifter & Fosnot, 1993), typically using case study or ethnographic methodologies, provide intensive and longitudinal data about teachers’ instruction. Some studies (e.g., Fennema et al., 1996;

Hufferd-Ackles, Fuson, & Sherin, 2004; Polly & Hannafin, 2011;

Schifter & Simon, 1992) have embraced a multi-methods approach, in which qualitative observation data are quantified using a rubric or scale. These reports then provide numerical data for teachers’ instruction as well as descriptions of their enacted pedagogies. Lastly, survey studies (e.g., Heck, Banilower, Weiss, & Rosenberg, 2008) collect self-reported data from teachers on their instructional practices. These survey studies sometimes are done in isolation, or coupled with classroom observations to increase the reliability of the self-reported data.

Researchers have attempted to classify teachers’ instructional practices, such as teacher or student centered, in a variety of ways (Garet, Porter, Desimone, Birman, & Yoon, 2001; Heck et al., 2008; Swan, 2006; Tarr, Reys, Reys, Chavez, Shih, & Osterlind, 2008). These researchers have observed that teachers’ instructional practices may vary based on the concept they are teaching, and the types of curricula resources utilized. Further, although teachers’ practices may shift slightly, their self-report of their instructional practices typically aligns to observed instructional practices (Desimone, Porter, Garet, Yoon, & Birman, 2002; Swan, 2006).

Gaps in the Research

Student-centered mathematics instruction and beliefs that are standards-based (discovery and connectionist) have potential to lead to greater student learning outcomes than those pedagogies and beliefs that are more teacher-centered (Fennema et al., 1996;

Wenglinsky, 1999). However, there is a lack of empirical studies Beliefs and Student Learning that link both teachers’ beliefs about mathematics teaching and learning and their instructional practices to student learning outcomes. In this study we aim to examine the links between (a) teachers’ beliefs and student learning outcomes, (b) teachers’ instructional practices and student learning outcomes, and (c) teachers’ beliefs, instructional practices, and student learning outcomes.

This study was guided by the following research questions:

1. How are teachers’ beliefs regarding mathematics teaching and learning associated with their teaching practices in mathematics?

2. Are there significant differences between grade levels and school districts with respect to student gains in mathematics achievement following the intervention?

3. How are teachers’ beliefs regarding mathematics teaching and learning associated with their students’ learning of mathematics?

4. How are teachers’ beliefs regarding mathematics teaching and learning associated with their students’ learning of mathematics?

–  –  –

Participants Participants in this study included 53 elementary school teachers (grades K though 5) that were involved in a mathematics professional development program focused on standards-based instruction. All teachers were certified to teach elementary school and taught in two school districts near a large city in the southeastern United States. They were identified as teacher-leaders from their respective schools as a requirement to participate in the professional development. Thirty-two teachers were from a large urban school district and the remaining 21 teachers were from a neighboring suburban school district. Thirty-seven percent (n =

20) hold only a bachelor’s degree, 30% (n = 16) hold a master’s degree, and one teacher holds a bachelor’s degree and certification specific to their content area. The rest did not report their highest degree held. Eighty-seven percent (n = 46) identified as Caucasian while 13% (n = 7) identified as African American.

Polly, McGee, Wang, Lambert, Pugalee, and Johnson Participants also included 688 students who were in the participating teachers’ classrooms. Gender and ethnicity were reported by teachers for their aggregate classrooms. Fifty percent (n = 344) of the students were female and 50% (n = 344) were male. Thirty-nine percent (n = 268) of the students were Caucasian, 34% (n = 234) were African American, 20% (n = 138) were Hispanic, 4% (n = 28) were Asian, and 3% (n = 21) were identified by their teachers as “Other.” Fourteen percent (n = 96) were identified as Limited English Proficient (LEP) and 10% (n =

69) were identified as having Individualized Education Plans (IEP).

Instruments Teacher’s beliefs The teachers’ beliefs questionnaire (Appendix A) was developed by Swan (2007) to examine teachers’ espoused beliefs about mathematics, mathematics teaching, and mathematical learning. For each of those three dimensions, teachers report the percentage to which their views align to each of the transmission, discovery, and connectionist views. Participants were instructed that the sum of the three percentages in each section should total 100.

Swan (2007) noted a clear distinction between the transmission orientation and the remaining two orientations but not a very clear distinction between the discovery and connectionist orientations. Further, discovery and connectionist categories both aligned with standards-based orientations to teaching and learning of mathematics (McGee et al., in press).

Therefore, we coded teachers into two categories: transmission and discovery/connectionist. Teachers were coded as discovery/connectionist if they indicated at least 50% in either discovery or connectionist category. Due to the alignment of both the discovery and connectionist categories with standards-based orientations to mathematics teaching, the data on teachers’ beliefs were analyzed as a dichotomous variable; “1” represented teachers with transmission views toward teaching mathematics and “0” stood for teachers with a discovery/connectionist views toward teaching mathematics.

Beliefs and Student Learning

Teachers’ practices



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