Resolving the chicken-or-egg causality dilemma: The longitudinal interplay of teacher knowledge and teacher beliefs

Teaching and Teacher Education 37 (2014) 130e139

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Resolving the chicken-or-egg causality dilemma: The longitudinal interplay of teacher knowledge and teacher beliefs Sigrid Blömeke a, *, Nils Buchholtz b, Ute Suhl a, Gabriele Kaiser b a b

Humboldt University of Berlin, Department of Education, D-10099 Berlin, Germany University of Hamburg, Department of Education, D-20146 Hamburg, Germany

h i g h l i g h t s
Student teachers’ early level of PCK determines their later knowledge levels.
Early beliefs of student teachers determine their later beliefs.
Student teachers’ PCK causally influences their later beliefs.
The higher student teachers’ PCK is, the more constructivist are their beliefs.
Differential beliefs do not lead to differential PCK.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 6 April 2013 Received in revised form 10 October 2013 Accepted 14 October 2013

To examine the longitudinal relation between knowledge and beliefs and to determine cause and effect, 183 mathematics teachers were assessed three times during their first years of teacher education on their mathematics pedagogical content knowledge (MPCK) and their beliefs about teaching and learning. The data revealed that prior MPCK predicted later achievement. Prior beliefs also determined later ones. In addition, MPCK affected later beliefs: Higher MPCK at the first measurement resulted in more constructivist beliefs at later time points. By contrast, beliefs did not predict later MPCK. If constructivist teacher beliefs are to be fostered, teacher education should strengthen MPCK. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Causality Chicken-or-egg dilemma Longitudinal study Autoregressive path model Cross-lagged effect Teacher learning Teacher knowledge Teacher beliefs

Identifying causes and effects and examining longitudinal relations are two of the most difficult tasks in teacher research around the world. Teacher characteristics are so intertwined that it is difficult to disentangle causes and effects in cross-sectional studies. Thus, we are not fully able to resolve this fundamental research problem of teacher learning in this paper, either, but we designed a prospective studydusing lower secondary mathematics teachers in Germany as an exampledto address this research gap

* Corresponding author. Tel.: þ49 30 2093 1911; fax: þ49 30 2093 1828. E-mail addresses: [email protected] (S. Blömeke), nils. [email protected] (N. Buchholtz), [email protected] (U. Suhl), [email protected] (G. Kaiser). 0742-051X/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tate.2013.10.007

with respect to the question “Which comes first during teacher education: teacher knowledge or teacher beliefs”?1 Answers to this question will help the international audience to understand the development that occurs during teacher education. As is true in many countries, mathematics teacher education in Germany is characterized by high drop-out rates (between 30 and 40% in the first years at a university; Dieter, Brugger, Schnelle, & Törner, 2008). Given the high national and international demand for mathematics teachers (KMK, 2011), such high drop-out rates are critical. Therefore, there is a need for scientific investigation into

1 We are very grateful for the reviewers’ comments. They helped to improve the paper substantially.

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how teacher education can be improved. This includes, among other approaches, an analysis of the development of teacher knowledge and teacher beliefs in a longitudinal design. 1. Theoretical framework 1.1. Teacher knowledge and teacher beliefs Teacher knowledge can be subdivided into different facets that have been frequently discussed in the literature (Baumert & Kunter, 2006; Shulman, 1985). According to current research, two subjectrelated and one generic facet of teacher knowledge can be distinguished: content knowledge (CK), pedagogical content knowledge (PCK; including curricular knowledge), and general pedagogical knowledge (GPK). With respect to lower secondary mathematics teachers in Germany, Baumert et al. (2010) provided evidence that it is mainly teachers’ PCK that is related to student achievement in mathematics. Therefore, we focused on this facet of teacher knowledge in our study. Shulman (1986) defines PCK as subject-specific knowledge that is relevant for teachers with regard to “the ways of representing and formulating the subject that makes it comprehensible to others” (p. 9). On the one hand, PCK includes a pedagogical perspective on teaching and learning: “Pedagogical content knowledge also includes an understanding of what makes the learning of specific topics easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning of the most frequently taught topics and lessons. If those preconceptions are misconceptions, which they so often are, teachers need knowledge of the strategies most likely to be fruitful in recognizing the understanding of learners [.]” (Shulman, 1986, p. 9). It is this notion of “what counts as professional knowledge from the perspective of improving outcomes for diverse learners” (Timperley & Alton-Lee, 2008) that drove much of our conceptualization. Besides such pedagogical knowledge for dealing with diversity, PCK includes curricular knowledge, which covers the selection and arrangement of the material that is to be taught and learned (Shulman, 1987). Bromme (1995, 1997) points to the connection between Shulman’s (1987) pedagogical perspective and curricular knowledge: “In order to find appropriate forms of presentation of the content, to determine the arrangement of topics and to weight which topics are treated more intensively, subject-specific pedagogical knowledge is necessary” (Bromme, 1997, p. 197). With respect to mathematics pedagogical content knowledge (MPCK) specifically, this theoretical framework means that MPCK includes knowledge about how to present fundamental mathematical concepts to K-12 students, some of whom may have learning difficulties. Before instruction in the classroom can begin, the mathematics content and concepts must be selected appropriately, prepared with respect to the learners’ preconditions, and connected to a range of different teaching strategies (Krauthausen & Scherer, 2007; Vollrath, 2001). Knowledge about the ways in which students learn is part of such an MPCK conceptualization as well. Mathematics teachers should be able to use the language of mathematics appropriately in order to communicate mathematical ideas, ask questions of varying complexity, identify common misconceptions, provide feedback, and react with appropriate intervention strategies. MPCK also includes information about how to deal with the consequences in future lessons if a key topic in the curriculum was removed or taught in a different context. Beliefs can generally be defined as “understandings, premises or propositions about the world that are felt to be true” (Richardson, 1996, p. 103). Beliefs are, thus, not a well-defined construct (Pajares, 1992). Clear distinctions from terms such as attitudes,

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perceptions, or conceptions are rare. Rodd (1997) points out that beliefs rely on evaluative and affective components. At the same time, the distinction between beliefs and knowledgedfor PCK in particulardis more an analytical tool than that it can strictly be kept up (Furinghetti & Pehkonen, 2002). Several efforts have been made to categorize the beliefs of mathematics teachers (Op ’t Eynde, De Corte, & Verschaffel, 2002). If the beliefs facets that are selected include both the content being taught and the professional task that needs to be mastered, evidence suggests that there would be a link between these facets and K-12 student achievement (Bromme, 2005). The link between beliefs and student achievement exists via teacher knowledge. Beliefs are a crucial aspect of teachers’ perceptions of teaching situations and thus influence their choice of teaching methods (Leder, Pekhonen, & Törner, 2002). Beliefs also influence which parts of their knowledge teachers choose to draw from in class (Leinhardt & Greeno, 1986). Staub and Stern (2002) provided evidence that, in particular, constructivist beliefs about the teaching and learning of mathematics are significantly related to K-12 student achievement in mathematics: Teachers who tend to hold more constructivist beliefs tend to have students who are better able to solve complex problems. Therefore, we focused on this facet of teacher beliefs in our study.

1.2. The development of knowledge and beliefs during teacher education There are very few longitudinal studies that have examined the development of teacher knowledge during teacher education with standardized tests. Whereas much research exists on the professional development of practicing teachers (see e.g., Clarke & Hollingsworth, 2002), most studies on teacher growth during their training have been case studies (Kagan, 1992); have used selfreported data (e.g., Grossman & Richert, 1988), which carry the risk of being biased by differences in the future teachers’ educational aspirations (Blömeke, 2014); or have used nonstandardized distal indicators such as coursework or grades, which carry the risk of being biased by differences in institutional curriculum standards (e.g., Turner, 2008). The international comparative study “Mathematics Teachers in the 21st Century (MT21)” was one of the few studies that tested future mathematics teachers’ MPCK in a standardized way. As indicated by mean differences between student cohorts in different years of teacher education as well as their standard deviations, the results of this study suggested that teachers’ knowledge grows significantly during teacher education and that, at the same time, its variance increases (Schmidt, Blömeke, & Tatto, 2011). However, the MT21 study had methodological limitations because it was based on only cross-sectional data from different student-teacher cohortsdbeginning students, mid-program students, and students in their final year of teacher educationdso that the relations between the cohorts in terms of the future teachers’ rank ordering could not be analyzed. Longitudinal data on the development of knowledge that allows for such a relational approach has been collected on K-12 students. From these studies, we know that prior knowledge significantly determines later achievement (Simmons, 1995). An explanation for this phenomenon is that higher prior knowledge facilitates the acquisition of new knowledge, for example, by supporting the integration of new information into existing schemata, the modification of knowledge structures, or the compilation and chunking of knowledge (Anderson & Lebière, 1998). Thus, one objective of our study was to be the first to use longitudinal data to examine whether the MPCK of future mathematics teachers grows during

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teacher education and whether the knowledge assessed at the beginning of their training predicts their later achievement. Whether teacher beliefs develop during teacher education is less clear. Using cross-sectional data, the MT21 study pointed toward significant changes. In Germany, the country examined in the present paper, future mathematics teachers had on average more constructivist and less transmission-oriented beliefs about the teaching and learning of mathematics at the end of their training than at the beginning (Schmidt et al., 2011). However, other studies have found that beliefs are resistant to change such that future teachers’ mean belief scores did not differ significantly across different measurement occasions (Kane, Sandretto, & Heath, 2002; Pajares, 1992). As an explanation, these authors had hypothesized that the future teachers’ prior school experience functions as an “apprenticeship of observation” (Lortie, 1975). This is why Richardson (1996) refers to teacher education as a “weak intervention” (with respect to mathematics teachers, see Ball, Lubienski, & Mewborn, 2001). With the occurrence of either stability or change, results have suggested a significant longitudinal relation of teacher beliefs such that prior beliefs predict later ones. Thus, a second objective of our study was to use prospective data to examine the mean differences in and the longitudinal relations between future teachers’ beliefs about teaching and learning. 1.3. The relation between teacher knowledge and teacher beliefs The community of educational researchers has long argued that a significant relation exists between teacher knowledge and teacher beliefs. However, the literature has not been able to identify the cause and the effect, meaning that studies have yet to determine the direction of this relation: Does the MPCK of future teachers determine their beliefs about the teaching and learning of mathematics, or do their beliefs determine their MPCK? Reasonable arguments can be provided for both directions: Differential teacher knowledge may be the consequence of differential teacher beliefs because these beliefs may function as filters. Perhaps only the parts of student teachers’ knowledge that match their beliefs are integrated into their cognitive schemas (Kane et al., 2002; Lortie, 1975; Wideen, Mayer-Smith, & Moon, 1998; Zeichner, 1986). However, an alternative hypothesis could be that pre-existing differences in teacher knowledge promote the differential development of beliefs because the cognitive processes may lead to differential patterns of perception, and a certain level of knowledge may be needed before future teachers are able to value constructivist approaches. Reciprocal effects are possible as well, meaning that both constructs may function as both cause and consequence. Thus, the history of teacher research has been fraught with controversy. Whereas longitudinal relations cannot be examined in a methodologically sound way in cohort or case studies, it is possible to model the concurrent relation between teacher knowledge and teacher beliefs. In fact, most studies discussed above have pointed toward significant relations. In particular, the MT21 study revealed positive correlations between MPCK and constructivist beliefs of future lower secondary mathematics teachers in Germany at the beginning, during, and at the end of teacher education (Schmidt et al., 2011). Blömeke, Suhl, and Döhrmann (2012) confirmed this relation based on a representative sample of future primary teachers at the end of their training that had taken part in the “Teacher Education and Development Study in Mathematics (TEDS-M)” (Tatto et al., 2008). With respect to currently employed German mathematics teachers, Kunter et al. (2007) as well as Kunter and Klusmann (2010) provided similar evidence.

A major open question concerns which facet causes which effect. Correlations are a necessary precondition for determining cause and effect, but the direction of the relation cannot be inferred. Thus, the cause-and-effect relation between teacher knowledge and teacher beliefs is still a matter of international debate. It is addressed in this paper. 1.4. Research questions and significance of the study In this study, we aimed to disentangle the causes and the effects of teacher knowledge and teacher beliefs by following 183 mathematics teachers from the beginning of their teacher education as the first measurement occasion through their second and third year with additional two measurements. Complex and controversial issues often require sophisticated methodologies. This study relied on scaling the knowledge data based on Item Response Theory (IRT; Fischer & Molenaar, 1995) and subsequently applying autoregressive path analyses with cross-lagged effects (Bollen & Curran, 2006). In this case, resolving the “chicken-egg” dilemma required that we had knowledge of information about at least four parameters: (1) the future teachers’ MPCK0 at the beginning of their teacher education (i.e., t0); (2) the teachers’ beliefs about the teaching and learning of mathematics TBLM0 at t0; (3) the teachers’ MPCK0þ1 at t0þ1, namely, after spending some time in teacher education; and (4) the teachers’ beliefs TBLM0þ1 at t0þ1. If differential teacher knowledge causes differential teacher beliefs, one would expect a significant relation between MPCK0 and TBLM0þ1, but no relation between TBLM0 and MPCK0þ1. Alternatively, if differences in beliefs lead to differences in teacher knowledge, one would expect a significant relation between TBLM0 and MPCK0þ1 but no relation between MPCK0 and TBLM0þ1. Finally, if reciprocal effects exist, one would expect both a significant relation between MPCK0 and TBLM0þ1 and a significant relation between TBLM0 and MPCK0þ1. The stability of these effects can be tested across our three measurement occasions. With respect to the growth of MPCK across measurement occasions, we hypothesized that the mean MPCK level assessed at later time points would significantly exceed the prior levels. With respect to the longitudinal relation in terms of the future teachers’ rank ordering between the three time points, we hypothesized that prior knowledge would significantly predict later achievements. With respect to change in the future teachers’ beliefs across measurement occasions, we hypothesized that the mean constructivist view on teaching and learning would be significantly higher at later time points than at prior ones. With respect to the longitudinal relation in terms of the future teachers’ rank ordering between the three time points, we hypothesized that prior beliefs would significantly predict later ones. It may also be that long-term effects exist in such a way that teacher characteristics at the beginning of teacher education influence their characteristics not only 1 year later but also 2 years later (i.e., a second-order relation; we would expect a significant relation between MPCK0 and MPCK0þ2 in this case or between TBLM0 and TBLM0þ2). To the best of our knowledge, this is the first work on the longitudinal relation between teacher knowledge and teacher beliefs that has relied on standardized knowledge tests. The longitudinal interplay of MPCK and beliefs about the teaching and learning of mathematics plays a crucial role in the development of mathematics teachers because they are both related to

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teaching quality and K-12 student achievement in mathematics. Examining which early student teacher characteristics significantly contribute to later ones can provide important information about early markers of future teachers’ subsequent development and may thus provide starting points for special support during teacher education. 2. Method

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a psychological perspective on mathematics pedagogy, which refers to MPCK aspects influenced by psychological processes such as diagnosing mathematical thinking or misconceptions and designing interventions or assessments for the mathematics classroom;
a perspective related to teaching methods, which refers to MPCK aspects of instructional arrangements specific for the teaching of mathematics such as questions about curricula or educational standards.

2.1. Context of the study and participants Mathematics teacher education for secondary schools in Germany was used as an example to examine our research questions. Germany offers a teacher education program that starts with a university-based program totaling up to 5 years. A practical training phase based at a school and at state teacher training institutions then follows for between 1 and 2 years. A high-school exit exam, the so-called “Abitur,” is required for admission to a university. Future secondary teachers major in two subjects. In mathematics, their program includes courses on analysis and linear algebra in the introductory phase and continues with advanced university mathematics. Courses on mathematics pedagogy and general pedagogy have to be attended too. During the second practical training phase, the university graduates teach part-time in schools and simultaneously attend courses in general pedagogy and mathematics pedagogy. In order to capture the development of future mathematics teachers and to examine the concurrent and longitudinal relations between knowledge and beliefs from the beginning of the program, the study was designed as a longitudinal study. A sample of 183 German future secondary school teachers was tested on their MPCK as well as surveyed on their beliefs about the teaching and learning of mathematics on three measurement occasions during their first (2008), second (2009), and third years (2010) of mathematics teacher education. The students were recruited via colleagues from the field of mathematics education. Five German universities from Northrhine-Westphalia and Hesse took part in the study. The assessment of the students lasted for 90 min, and they took part voluntarily. As the implementation of the study depended on situational conditions such as the size of classes and access to the future teachers, the sample must be characterized as a nonrepresentative convenience sample. To account for panel attrition, we restricted our analyses to those 183 first-year students who participated in at least two of the three measurement occasions. Most of these had left high school in 2007. Their grade point average in the Abitur was 2.2 (SD ¼ .6). About half of the sample was male, and about 30% had a nonGerman family background. The future teachers were tested during regular university classes in mathematics pedagogy. Due to the structure of teacher education, they had only limited experience with teaching. 2.2. Instruments 2.2.1. Mathematics pedagogical content knowledge The MPCK test was conceptualized based on the theoretical framework presented above and on Shulman’s definition of PCK in particular. To operationalize this construct with respect to mathematics, different research perspectives of mathematics pedagogy were taken into account (Bigalke, 1974):
a content-related perspective on mathematics pedagogy, which refers to MPCK aspects influenced by mathematical content such as mathematical concepts, formalization, or typical student errors;

The validity of these three perspectives was confirmed through expert reviews. The key topics of the items used to capture them with our test were:
analyzing basic mathematical concepts underlying student solutions and student errors, identifying learning approaches to mathematical content;
analyzing the usefulness of problems given to students as a starting point for learning processes, diagnosing the adequacy of student solutions;
providing basic mathematical experiences, supporting mathematical thinking, dealing with heterogeneity;
developing assessments of mathematics achievements, applying methods of performance evaluation;
diagnosing misconceptions and dyscalculia, developing interventions, creating individual learning plans;
developing teaching arrangements such as genetic learning or concept learning;
knowing educational standards for teaching mathematics, mathematics curricula, textbooks. A substantial proportion of the items dealt with illustrative mathematical arguments that were supposed to support a deeper understanding of mathematics by students (Kirsch, 1987). With this conceptualization, MPCK is aligned with a constructivist view on how people can best learn mathematics. Learners are perceived as individuals who create their mathematical understanding by themselves. They incorporate new mathematical content into their prior knowledge through a reflective process and are thus the subjects of their own learning processes. With respect to teaching, a constructivist approach means that mathematics teachers orient their strategies toward student activities. Students are supposed to develop their own learning experiences largely independently of each other and the teacher. The teacher’s authority as an expert is not downgraded by such a concept, but his or her role is modified. The teacher acts as a selector and provider of learning opportunities (Toh, Ho, Chew, & Riley, 2003). His or her task is specifically “the transformation of content knowledge . in ways that allow learners to construct knowledge during classroom practice. Teachers derive PCK from their understandings of content, their own teaching practice, and their own schooling experience” (Hausfather, 2002, p. 64). As part of a constructivist approach, teachers take the environmental context of learning into account. Their understanding of their students includes students’ abilities and learning strategies, developmental levels, attitudes, motivation, and prior conceptions (Hausfather, 2002). The alignment of the items and the overall test with core aspects of the German standards for mathematics teacher education and typical tasks of mathematics teachers in secondary schools (DMV, GDM, & MNU, 2008; KMK, 2008) was confirmed through expert reviews. In addition to using this substantive heuristic to lead the development of items and tests, a distinction between three types of cognitive demands was used as a heuristic to enrich item development and to ensure that the cognitive processes typically

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Table 1 Cognitive processes covered by the MPCK test. Remembering

Applying

Evaluating

Knowledge about - content-specific learning approaches and paradigmatic examples - levels of conceptual rigor and formalization - mathematics education as general education - facets of mathematical competences (e.g., modeling, problemsolving) - heterogeneity in mathematics education, assessment, and diagnostics - dyscalculia and mathematical giftedness - educational standards, curricula, and textbooks

Dealing with - basic concepts and specialized learning approaches in the preparation of teaching materials - adequate opening of lessons - methods of performance assessment in mathematics education - educational standards, curricula, and textbooks for teaching mathematics when developing learning objectives; - application of individual diagnostic procedures

Explaining - the diagnosis of mathematical problems of students and possible student solutions - the interconnection between mathematics education and its corresponding mathematical background - misconceptions of students - subject-specific interventions by teachers - standards in mathematics education, curricula, and comparative studies - goals, methods, and limits of assessment in mathematics education - methods for dealing with heterogeneity - learning processes by gifted student and specific learning requirements

performed in a mathematics classroom were covered by our test: remembering MPCK, applying MPCK, and evaluating classroom solutions and strategies (Anderson & Krathwohl, 2001). Examples of these processes are provided in Table 1. The cognitive processes and the substantive distinctions between content-related, psychological, and teaching-method-related perspectives on mathematics pedagogy together formed a matrix-type blueprint for the item development during which each cell had to be represented by several items. A balanced incomplete block design with two test forms was used with 47 items at the first, 33 items at the second, and 34 items at the third measurement occasion. Each time, 31 items served as anchor items (i.e., they were administered at all three measurement occasions). Figs. 1 and 2 present two item examples that represent the three substantive perspectives of MPCK and the three cognitive processes. The first example was also part of TEDS-M. The solution of A) requires the identification of the misconception that multiplication always yields a greater result and division always a smaller result (applying). On the basis of this diagnosis, B) expects an appropriate visual representation that can be used to explain why .2  6 ¼ 1.2 such as a drawing that shows the addition of six units of size .2 (e.g., boxes in which two out of 10 subdivisions are colored). Fig. 2 represents another item example. This problem requires a sufficient mathematical understanding of the difference quotient and the derivative as the local rate of change (remembering). Furthermore, the interpretations developed by the students must be evaluated. The correct solutions are: A) both correct, B) none correct, and C) one correct. The data were scaled using a three-dimensional IRT model with one dimension for each measurement occasion (see Fig. 3). In a first calibration step, the item parameters were estimated, thereby

fixing the item parameters for each anchor item to be equal across measurement occasions. During calibration, items that were omitted or not reached (due to time constraints) were treated as missing values (Rost, 2004; Hartig & Kühnbach, 2006; Te Marvelde, Glas, Van Landeghem, & Van Damme, 2006). In a second step, the person parameters were estimated. Here, items that were not reached were still treated as missing values, whereas items that were omitted were treated as incorrect responses. Weighted likelihood estimates (WLE; Warm, 1989) were used as person parameters and linearly transformed on a scale with a mean of 100 test points and a standard deviation of 20 at the third measurement occasion. The higher the scale score of a person, the more test items he or she had solved correctly, thus indicating a broader knowledge of mathematics pedagogy. The scaling was implemented with the software package Conquest (Adams, Wilson, & Wang, 1997). The reliability of the scale was good at each measurement occasion (t1: .80; t2: .77; t3: .83). 2.2.2. Surveying constructivist beliefs about teaching and learning mathematics In addition to testing the future teachers’ MPCK, we also surveyed their beliefs about the teaching and learning of mathematics. Teacher beliefs refer to affective ideas about the character and nature of teaching and learning processes, the identities and roles of students and teachers, as well as the institutional and social context of education (Reusser, Pauli, & Elmer, 2011). A precise distinction between knowledge and beliefs is not possible. Whereas some authors regard beliefs and knowledge as overlapping constructs (Woolfolk Hoy, Davis, & Pape, 2006), others assign beliefs to knowledge or understand knowledge as a function of beliefs (Philipps, 2007). In contrast to knowledge as a cognitive structure, beliefs are more strongly influenced by emotions, values, and personal experiences in our conceptualization. This affective component distinguishes beliefs from knowledge and thus enabled us to apply an analytical though still heuristic separation of these constructs (Entwistle & Peterson, 2004; Thompson, 1992). The future teachers surveyed in our study were asked about their beliefs about the teaching and learning of mathematics. They had to indicate their level of agreement with various statements on a 6point Likert scale ranging from 1 (strongly disagree) to 6 (strongly agree). Following Staub and Stern (2002) and other well-established conceptualizations (e.g., Peterson, Fennema, Carpenter, & Loef, 1989), we distinguish between a “transmission view” and a “constructivist view” as different classes of beliefs about the teaching and learning of mathematics. As the constructivist scale is closely related to our conceptualization of MPCK, we used it in our study. The constructivist perspective is characterized by studentcentered teaching and learning.2 The learning process is regarded as an active self-directed construction process that is supported by the provision of a suitable learning environment and the guidance of the instructor. The five-item scaledvalidated multiple times by expert reviews and confirmatory factor analysesdwas also used in studies such as MT21 or TEDS-M:
“In mathematics, it is not only important to find the right solution, but also to understand why the solution is correct.”
“Teachers should give students the opportunity to find their own ways to solve a problem”

2 By contrast, the transmission-oriented perspective assumes that students acquire knowledge through their teacher’s presentations of content. It includes, for example, the following statements: “Students learn mathematics best by attentively following the explanations of the teacher" or "When students deal with mathematical tasks, the result is more important than the approach."

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Jeremy notices that when he enters 0.2 × 6 into a calculator his answer is smaller than 6, and when he enters 6 ÷ 0.2 he gets a number greater than 6. He is puzzled by this, and asks his teacher for a new calculator! A)

What is Jeremy’s most likely misconception?

B)

Draw a visual representation that the teacher could use to model 0.2 × 6 to help Jeremy understand WHY the answer is what it is? Fig. 1. Item example from our MPCK test (Source: Tatto et al., 2012).


“Students may well find solutions to mathematical tasks without the help of the teacher”
“It helps students if they discuss certain different ways to an approach”
“Teachers should encourage students to find their own solutions to mathematical tasks, even if they are not efficient” A confirmatory factor analysis indicated an adequate fit of the scale to the data from our sample too. The scale’s reliability (Cronbach’s a) was between .63 and .68 at the three measurement occasions and was thus deemed acceptable. The future teachers’ responses were averaged across all five statements. High values indicate that teachers regarded mathematics learning as an active process in which K-12 students conduct their own inquiries. 2.3. Data analysis To examine our research questions with regard to the development of MPCK and constructivist beliefs over time, we used socalled autoregressive path models (Bollen & Curran, 2006; Jöreskog & Sörbom, 1979). In this kind of model, the score of the MPCK

scale at time t0þ1 is predicted on the basis of the MPCK score at t0 (i.e., “autoregressive” or regressed on itself), and the MPCK score at t0þ2 is predicted on the basis of the MPCK score at t0þ1 (this is the second autoregressive component of the model). The same procedure was applied to beliefs. To model the longitudinal dependencies between MPCK and TBLM, we added so-called cross-lagged effects; for example, we hypothesized that the MPCK score at t0þ1 depended not only on the MPCK score at t0 but also on the TBLM score at t0 (i.e., “lagged” for an effect across time and “cross” because this is an effect on another variable). Vice versa, the longitudinal dependencies were hypothesized with respect to the effects of MPCK at t0 on TBLM at t0þ1 and with respect to the later relations. Finally, following current practices in research, we hypothesized time-specific correlations between knowledge and beliefs (i.e., MPCK was assumed to be correlated with TBLM at each measurement occasion). Fig. 4 shows the baseline model that was the result of this reasoning. It is an autoregressive path model that includes all cross-lagged effects and time-specific correlations. Because only direct effects within each measurement occasion or between occasions were allowed, this type of model is also called a first-order

Fig. 2. Another item example from our MPCK test.

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autoregressive path model. Hypothesizing effects from an earlier time point on a much later one (e.g., from measurement occasion 1 on measurement occasion 3) would transform the model into a second-order path model. All models were estimated with manifest variable indicators to avoid unnecessary complexity, given the sample size in particular, and using maximum-likelihood procedures. We combined this path-analysis approach with a modelselection procedure to explore the relations anddgiven the sample sizedto achieve the most parsimonious model. We used a series of path models to test different hypotheses about the structural relation between teacher knowledge and teacher beliefs, beginning with the above-mentioned first-order autoregressive model. As we assumed that t0 might influence not only t1 but also t2, we introduced a second-order relation between measurement occasions 1 and 3, followed by a step-by-step elimination of nonsignificant relations. The fit of these different path models to our data was evaluated using comparative and global fit indices (Fan, Thompson, & Wang, 1999; Hu & Bentler, 1999). For the comparative fit index (CFI), estimates > .95 indicate a very good fit, and CFI estimates > .90 indicate a good model fit. For the global fit index Root Mean Square Error of Approximation (RMSEA), estimates < .05 indicate a very good fit, and RMSEA estimates < .08 indicate a good model fit. All path analyses were implemented in MPlus, a software package specifically designed for statistical modeling techniques of this kind (Muthén & Muthén, 2008).

This means that the rank ordering of the future teachers remained much the same with respect to these characteristics. By contrast, the time-specific correlations between teacher knowledge and teacher beliefs (underlined) varied from moderate to nonsignificant. The same variation between moderate and nonsignificant relations was found for the longitudinal interplay of knowledge and beliefs (neither bold nor underlined in rows 4e6 of Table 2). When we compared the level of knowledge across the three measurement occasions (row 7, columns 1e3; Table 2), the data revealed that MPCK did not grow significantly between the first and second measurement occasions, but it grew by more than one third of a standard deviation between the second and third measurement occasions of mathematics teacher education. The same development applied to the future teachers’ beliefs (row 7, columns 4e6). They did not change significantly between t0 and t0þ1, but they changed by more than one third of a standard deviation between t0þ1 and t0þ2. The model fit statistics (Table 3) indicated that the first-order baseline model (see Fig. 4), which included all possible crosslagged effects and time-specific correlations as well as the direct effects of MPCK or TBLM from one measurement occasion to the respective next one, did not fit the data very well. Once we introduced the second-order effects of MPCK and TBLM from time point one to time point three, the fit improved substantially. This secondorder baseline model, which also claimed cross-lagged effects from MPCK to beliefs and at the same time cross-lagged effects from beliefs to MPCK, fit the data well. The model that was applied to test the hypothesis that differential MPCK causes differential beliefs but not vice versa (i.e., eliminating the cross-lagged effects from beliefs to MPCK) did not worsen the model fit significantly compared to the second-order baseline model. By contrast, the test of the model that represented the alternative, that is, that differential beliefs cause differential MPCK but not vice versa (i.e., eliminating the cross-lagged effects from MPCK to beliefs), resulted in a worse fit to the data. Thus, we can state that neither the hypothesis that there would be reciprocal effects nor the hypothesis that beliefs would affect knowledge was supported by our data. The final parsimonious model in which the nonsignificant paths were eliminated is displayed in Fig. 5. With respect to both MPCK and TBLM, the direct effects were particularly large during the first year of teacher education. However, taking into account the finding that the teacher characteristics at the third time point were affected not only by the second time point but also by the first one, it seemed as though the autoregressive relation remained stable across time (see also the sizes of the correlations in Table 2).

3. Results

4. Discussion

Descriptive statistics (the mean, standard error, standard deviation, as well as the minimum and maximum) and the manifest correlations (Pearson’s r) of all variables are reported in Table 2. The data revealed that the three measurement occasions of both MPCK and TBLM (indicated in bold) were strongly related to each another at about the same magnitude across all measurement occasions.

The unclear causal relation between teacher knowledge and teacher beliefs has led to many heated debates about how to model the chicken-and-egg dilemma. Results from cross-sectional studies have pointed to a significant positive relation, which was supported with respect to MPCK and TBLM by our study for two measurement occasions too (see the time-specific correlations at t0 and t1 in Table 2). Yet, identifying causality and consequences is important if teacher education intends to set appropriate foci. Our prospective study provides the first evidence for this issue. First, the data revealed that subsequent teacher knowledge and teacher beliefs are strongly caused by the respective prior traits. As hypothesized, this applies to MPCK and TBLM. In both cases, not only was the third measurement occasion related to the second one and the second one to the first one, but the third measurement occasion was also significantly predicted by the knowledge or beliefs held by the future mathematics teachers when they entered

T1

T2

T3

Items

Measurement occasion

Fig. 3. Three-dimensional IRT model with the three measurement occasions as latent variables.

Fig. 4. Possible pathways linking teacher knowledge and teacher beliefs (first-order baseline model).

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137

Table 2 Descriptive statistics and correlations for MPCK and TBLM.

1 MPCK0 2 MPCK0þ1 3 MPCK0þ2 4 TBLM0 5 TBLM0þ1 6 TBLM0þ2 Mean SE SD Minemax

1

2

3

4

5

6

.55** .58** .15* .27** .15y 91.61 1.19 17.6 48.0e144.2

.64** ns .22** ns 93.75 1.24 19.0 48.0e187.0

.16* .22** ns 100.0 1.48 20.0 60.2e163.0

.48** .44** 4.93 .03 .53 3.33e6.00

.41** 4.92 .04 .60 2.33e6.00

5.15 .03 .44 4.00e6.00

Note. MPCK ¼ Mathematics pedagogical content knowledge; TBLM ¼ Future teacher beliefs about the teaching and learning of mathematics; SE ¼ standard error; SD ¼ Standard deviation. yp < .10. *p < .05. **p < .01. ***p < .001.

teacher education. Thus, a future teacher’s entering qualifications may predetermine what can be achieved in mathematics teacher education. These results are in line with current research findings in cognitive psychology and research on K-12 student achievement (Anderson & Lebière, 1998; Simmons, 1995). Higher prior knowledge facilitates the acquisition of new knowledge, for example, by supporting the integration of new information into existing schemata, the modification of knowledge structures, or the compilation and chunking of knowledge. Second, our main research question was about the cross-lagged effects of MPCK and TBLM. The results of the path models are clear: The future teachers’ beliefs about the teaching and learning of mathematics do not determine their later knowledge about how students learn mathematics; how to select mathematics content appropriately; how to prepare it appropriately with respect to the students, simplify it, and connect it to teaching strategies; how to evaluate consequences for future lessons if a key topic in the curriculum were to be removed or taught in a different context; or how to ask questions of varying complexity, identify common misconceptions, provide feedback, and react with appropriate intervention strategies. By contrast, the level of such knowledge is significantly linked to the teachers’ later beliefs about the teaching and learning of mathematics. The higher a future teachers’ initial level of MPCK, the more strongly the teachers regard later mathematics learning as an active process in which K-12 students conduct their own inquiries and develop approaches to problem solving. This result does not imply that beliefs cannot function as filters, but it stands in contrast to the often-made follow-up inference that future teachers’ knowledge acquisition depends on their beliefs (Kane et al., 2002; Lortie, 1975; Pajares, 1992; Richardson, 1996). Our data do not support this second inference, although we have to be careful about making conclusions that are too far-reaching in

Table 3 Results from the autoregressive path models.

c2 (df) First-order baseline model Second-order baseline model Second-order without CR-L beliefs effects Final model (Fig. 5)

c2/df CFI

26.6 (4)*** 6.7

RMSEA RMSEA 90% CI RMSEA < .05

.89 .18

.11; .24

***

1.1 (2) ns

.6

1.00 .00

.00; .12

ns

2.2 (4) ns

.6

1.00 .00

.00; .08

ns

3.9 (7) ns

.6

1.00 .00

.00; .05

ns

Note. Baseline models ¼ including all possible cross-lagged (CR-L) effects; c2 ¼ chisquare value; df ¼ degrees of freedom; CFI ¼ comparative fit index; RMSEA ¼ root mean square error of approximation; CI ¼ confidence interval. ***p < .001.

this respect given the broad nature of MPCK that we are examining. In any case, it seems that a certain level of MPCK is needed before it is possible for future teachers to value constructivist approaches. Third, our data indicate significant growth in MPCK during teacher education and significant change in the future teachers’ constructivist beliefs. Both developments happen between the second and third measurement occasions, most likelydalthough this has to remain an open question at this point of timeddue to opportunities to learn. Before we draw our conclusions, we would like to point out the methodological limitations of our study. One limitation is the sensitivity of our knowledge test and the beliefs survey. The changes in MPCK and TBLM during the first 3 years were limited to about one third of a standard deviation. This is a substantial change, but we may have captured only a small proportion of professional growth. Due to the organization of mathematics teacher education in Germany (and in many other countries), most of the mathematics pedagogy classes are taken toward the end of teacher education. Thus, it may be that more and other effects would show up in the following years or once these future teachers take jobs as teachers. Another caveat of our approach is implied by the autoregressive nature of our model. An event that follows another could be caused by a third variable (the post hoc ergo propter hoc fallacy). Due to the sample size, we were not able to exclude potentially confounding biases. This limitation applies to characteristics such as gender or grade-point average. More research is therefore needed in this respect. Also because of the sample size, we used manifest variables that included measurement error to examine growth. Such a practice could lead to an underestimation of the autoregressive parameters. The study was conducted in Germany. Results from relational analyses can often be generalized across countries, at least in western countries such as Australia, England, Norway, Spain, or the U.S. if not beyond (see, e.g., Blömeke, Suhl, Kaiser, & Döhrmann, 2012). However, replication studies would strengthen the outcomes of our study with respect to an international audience.

Fig. 5. Parsimonious multivariate second-order autoregressive path model.

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Longitudinal studies are urgently needed anyway; thus, we strongly recommend applying such a design more often in mathematics teacher education research in other countries. 4.1. Conclusions Given the close relation between teacher knowledge, teacher beliefs, and K-12 student achievement, it is important to ascertain whether and how mathematics teacher education contributes to the development of teacher knowledge and teacher beliefs. As evidence exists for the value that is added by teachers holding a constructivist view on mathematics instruction (Staub & Stern, 2002), teacher education should try to strengthen such a view. Student achievement can this way obviously be strengthened with respect to complex problem solving abilities without losing strength in more routine-oriented algorithmic skills. The results of our study indicate that improving the knowledge base of mathematics teachers may be a promising way to achieve this objective. If replication studies confirm that prior MPCK has a causal effect on constructivist beliefs, it will be important to enhance future mathematics teachers’ opportunities to learn mathematics pedagogy and support them in developing their pedagogical content knowledge. Which types of opportunities for learning are best in this respect remains an open question at this time and should be the subject of further research. Acknowledgment This study was funded by the Deutsche Telekom Stiftung. The views expressed in this paper are those of the authors. References Adams, R. J., Wilson, M., & Wang, W. C. (1997). The multidimensional random coefficients multinomial logit model. Applied Psychological Measurement, 21, 1e23. Anderson, L. W., & Krathwohl, D. R. (Eds.). (2001). A taxonomy for learning, teaching and assessing. New York: Longman. Anderson, J. R., & Lebière, C. (1998). The atomic components of thought. Mahwah, NJ: Erlbaum. Ball, D. L., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: the unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed.). New York: Macmillan. Baumert, J., & Kunter, M. (2006). Stichwort: Professionelle Kompetenz von Lehrkräften. Zeitschrift für Erziehungswissenschaft, 9, 469e520. 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, 133e180. Bigalke, H.-G. (1974). Sinn und Bedeutung der Mathematikdidaktik. Zentralblatt für Didaktik der Mathematik, 6, 109e115. Blömeke, S. (2014). Vorsicht bei Evaluationen und internationalen Vergleichen: Unterschiedliche Referenzrahmen bedrohen die Validität von Befragungen zur Lehrerausbildung. Zeitschrift für Pädagogik, 1/2014 (in press) Blömeke, S., Suhl, U., & Döhrmann, M. (2012). Zusammenfügen was zusammengehört: Kompetenzprofile am Ende der Lehrerausbildung im internationalen Vergleich. Zeitschrift für Pädagogik, 58, 422e440. Blömeke, S., Suhl, U., Kaiser, G., & Döhrmann, M. (2012). Family background, entry selectivity and opportunities to learn: what matters in primary teacher education? An international comparison of fifteen countries. Teaching and Teacher Education, 28, 44e55. Bollen, K. A., & Curran, P. J. (2006). Latent curve models: A structural equation perspective. Hoboken, NJ: Wiley. Bromme, R. (1995). Was ist ‘pedagogical content knowledge’? Kritische Anmerkungen zu einem fruchtbaren Forschungsprogramm. In S. Hopmann & K. Riquarts (zus. m. W. Klafki & A. Krapp) (Eds.), Didaktik und/oder Curriculum. Zeitschrift für Pädagogik (Beiheft 33, pp. 105e115). Weinheim: Belz. Bromme, R. (1997). Kompetenzen, Funktionen und unterrichtliches Handeln des Lehrers. In F. E. Weinert (Ed.), Bd. 3: Psychologie des Unterrichts und der SchuleEnzyklopädie der Psychologie (pp. 177e212). Göttingen: Hogrefe. Bromme, R. (2005). Thinking and knowing about knowledge: a plea for and critical remarks on psychological research programs on epistemological beliefs. In M. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and sign e Grounding mathematics education (pp. 191e201). New York: Springer. Clarke, D. J., & Hollingsworth, H. (2002). Elaborating a model of teacher professional growth. Teaching and Teacher Education, 18, 947e967.

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