TELPS: A method for analysing mathematics pre-service teachers' Pedagogical Content Knowledge

Teaching and Teacher Education 35 (2013) 43e50

Contents lists available at SciVerse ScienceDirect

Teaching and Teacher Education journal homepage: www.elsevier.com/locate/tate

TELPS: A method for analysing mathematics pre-service teachers’ Pedagogical Content Knowledge Anne Prescott a, *, Isabell Bausch b, Regina Bruder b a b

University of Technology, Sydney, PO Box 222, Lindfield 2070, Australia Technische Universität, Darmstadt, Schlobgartenstrabe 7, D-64289 Darmstadt, Germany

h i g h l i g h t s
A survey to analyse pre-service mathematics teachers’ PCK is introduced.
Pre-service teachers analysis of lesson plans is used to determine PCK.
The survey indicates changes in elements of PCK can be determined over time.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 May 2012 Received in revised form 16 May 2013 Accepted 21 May 2013

A pre-service teacher’s Pedagogical Content Knowledge (PCK) and their personal constructs of teaching develop throughout their teacher education program. PCK integrates generic pedagogical knowledge, mathematical teaching methodology and knowledge of the discipline of mathematics and this paper reports on a survey that can be used to assess a pre-service teacher’s PCK. TELPS (Teacher Education Lesson Plan Survey) was developed to determine the PCK of pre-service teachers during their teacher education program. TELPS is shown to analyse pre-service teachers’ PCK with some indication that the pre-service teacher’s development of PCK can be observed. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Pre-service teachers Pedagogical Content Knowledge Lesson plans Repertory-grid theory

1. Introduction Pre-service mathematics teachers begin their teacher education program with beliefs and ideas about teaching mathematics that are well established by their own experience in school (Barkatsas & Malone, 2005). During their teacher education program the preservice teachers are exposed to many new ideas, expanding both their knowledge about teaching mathematics and content knowledge. They generally excelled in the classroom environments of their own education and so, at the beginning of their program, often believe that a good teacher will teach as they were taught (Wilson, Cooney, & Stinson, 2005). University teacher education programs seek to broaden and deepen their assumed model of good teaching. To explore this process we developed the Teacher Education Lesson Plan Survey (TELPS). This survey uses the pre* Corresponding author. Tel.: þ61 2 9514 5406; fax: þ61 2 9514 5556. E-mail addresses: [email protected] (A. Prescott), bausch@ mathematik.tu-darmstadt.de (I. Bausch), [email protected] (R. Bruder). 0742-051X/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tate.2013.05.002

service teachers’ analysis of mathematics lesson plans for insight into their Pedagogical Content Knowledge (Shulman, 1986). While other research projects analyse teachers’ Pedagogical Content Knowledge (PCK) with classical test items (e. g. Krauss, Neubrand, Blum, & Baumert, 2008), TELPS uses an adapted Repertory-Grid Method (Kelly, 1955) which is a psychoanalytical method that establishes peoples’ constructs by comparing objects in a standardised way. The advantage of using Repertory-Grid is the ability to analyse the data in a nomothetic as well as in an ideographical way (Scheer, 1996). Thus, TELPS can explore the development of mathematics teachers’ PCK during their teacher education program to find individual phenomena in their development of PCK or phenomena in connection with their teacher education program. We use TELPS to measure the personal constructs (Kelly, 1955) of Australian and German pre-service teachers about mathematics lesson plans during their teacher education program. Hence, TELPS will help us to answer the following research question: Can TELPS measure pre-service teachers’ PCK through their analysis of lesson plans?

44

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e50

To answer this question TELPS should be able to show a variety of PCK elements that fit into other definitions of PCK (e.g. Ball, Hill, & Bass, 2005; Shulman, 1986). This article focuses on presenting TELPS as a new way of analysing mathematics teacher’s PCK. We will explain the theoretical framework of the TELPS, present first results as an example of data analysis, and discuss the quality criteria of this survey by using the early results of TELPS at two universities in different countries with different teacher education programs. 1.1. Pedagogical Content Knowledge Content knowledge is a necessary but not sufficient condition for good teaching. Mathematics teaching needs more than knowledge of content (e.g. Ball et al., 2005; Goulding & Suggate, 2001; Mewborn, 2001; Shulman, 1986), because teachers need to recognise that an answer is incorrect (or correct), analyse the source of any errors and then work with the student to improve the mathematics. It involves choosing appropriate examples and exercises and sequencing these so that students are guided in their learning. Developing alternative representations of the mathematics is a major part of teaching. Consequently, planning a mathematics lesson includes these requirements of teaching. Shulman (1986) defined the knowledge needed to cope with the challenge of teaching into three categories of teachers’ content knowledge: Subject Matter Content Knowledge, Pedagogical Knowledge and Pedagogical Content Knowledge (PCK). The knowledge of designing ‘good’ mathematic lessons is a part of Pedagogical Content Knowledge (PCK), because Shulman (1986) defined it as “the distinctive bodies of knowledge for teaching. It represents the blending of content and pedagogy into an understanding of how particular topics, problems, or issues are organized, represented and adapted to the diverse interests and abilities of learners, and presented for instruction” (p. 8). PCK is an integration of generic pedagogical knowledge, mathematical teaching methodology and knowledge of the discipline of mathematics (Lim-Teo, Chua, Cheang, & Yeo, 2007; Shulman, 1986; Stacey et al., 2001). Shulman’s theory of teacher knowledge is the foundation of many research projects on mathematics teacher education. For example, the COACTIV project in Germany (Baumert et al., 2010), the ‘Michigan Group’ in USA (Hill, Ball, & Schilling, 2008), Lim-Teo’s group in Singapore (Cheang et al., 2007; Lim-Teo et al., 2007), and the international comparative study, TEDS-M, which examined how different countries prepare their teachers to teach mathematics in primary and lower-secondary mathematics (Schmidt, Blömeke, & Tatto, 2011), each consider different aspects of PCK. The relationship between Subject Matter Knowledge (SMK) and the PCK required for teaching is still not fully understood. Preservice teachers who had several representations for mathematical ideas and whose knowledge was already richly linked were able to draw upon them both in planning and in spontaneous teaching interactions (Huckstep, Rowland, & Thwaites, 2002). However, the boundaries between SMK and PCK may well be blurred and a deep understanding of both is important. Consequently, Kahan, Cooper, and Bethea (2003) believe that mathematicians, whose content knowledge is without question, may not necessarily possess PCK because so many additional attributes are needed. The results of Wong and Lai (2006) further support these findings, because they show no statistical relationship between PCK and SMK. Blömeke, Houng, and Suhl (2011) were also able to differentiate PCK from SMK. Hill et al. (2008) broaden the Shulman definitions by proposing a model of mathematical knowledge for teaching by further

dividing Subject Matter Knowledge and Pedagogical Content Knowledge (Fig. 1). In mathematics, Subject Matter Knowledge includes the Common Content Knowledge (CCK) and the Specialised Content Knowledge (SPK) that one would expect a teacher to know, but also includes Knowledge at the Mathematics Horizon which Ball and Bass (2009) define as “a kind of mathematical ‘peripheral vision’, a view of the larger mathematical landscape, . in which the present experience and instruction is situated” (p. 6). Hill et al. (2008) also further divide Shulman’s PCK (Fig. 1). In order to prepare and teach a lesson, a teacher must be capable of putting in place all the partitions of PCK: namely the Knowledge of Content and Students (KCS), the Knowledge of Content and Teaching (KCT), and the Knowledge of the Curriculum. There are problems in detecting the PCK of the mathematics teachers. A teacher’s ability to know that students often make errors at certain points in a topic is linked with the teacher’s reasoning about what students are thinking or doing. PCK is complicated and Hill et al. (2008) suggest the importance of explicit criteria to measure conceptualisation and development of PCK. A number of researchers analyse mathematics teachers’ PCK using different instruments; for example, COACTIV (Baumert et al., 2010), TEDS-M (Schmidt et al., 2011), and the studies of Hill et al. (2008), with multiple-choice and open questions to learn about teacher’s knowledge. Rowan, Schilling, Ball, and Miller (2001) surveyed teachers’ PCK using a bank of items in reading/language arts and mathematics. They used classroom scenarios in a multiple-choice survey to differentiate between content knowledge, knowledge of students’ thinking and knowledge of pedagogical strategies. Their results were mixed because of difficulties in developing scenarios and writing the items, and their use of a small sample (this study reported a pilot), but they do indicate the possibility of measuring particular facets of teachers’ PCK. Wong and Lai (2006) used direct observation of pre-service primary teachers in schools and the lesson plans they used in the teaching. They found that the pre-service teachers who were further through their course had better results in PCK than those in the early years, suggesting PCK improves as the pre-service teacher progresses through the course. 1.2. Developing a survey to measure PCK PCK is a very complex construct with different levels. Thus, we used lesson planning via TELPS to cover the general ideas of designing a mathematics lesson. These general ideas of

Fig. 1. Domain map for mathematical knowledge for teaching (Hill et al., 2008, p. 377).

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e50

mathematics lessons include the decision about using an inductive or deductive approach, the ways of motivating and activating students etc. Because multiple-choice items mainly cover very specific topics or situations of a mathematics lesson the linkage between these general ideas are difficult. An interview about planning and designing a mathematics lesson could be a possibility, but the analysis of many interviews is very costly. Thus, we looked for a method that is open like an interview and standardised like classical test items, and decided on the Repertory-Grid Method because it has the boundaries of classical test items with the structure of open-ended questions. This standardised structure allows a qualitative and quantitative evaluation in an effective way. Kelly (1955) developed the Repertory-Grid Method to gather individual constructs. At the centre of this method is the comparison of different objects within a given structure. First, the objects to be compared are produced by the interviewer and the constructs are determined by asking the question, “In what important ways are two of them alike but different from the third?” (Kelly, 1955, p. 222). The constructs and objects are then placed in a grid and the participant estimates the importance of the construct for each object. Kelly (1955) defined constructs as the basis for humans’ daily action. In our research, these constructs are a kind of PCK, which is used to plan and design mathematics lessons. The preservice teachers’ constructs on planning and designing a good mathematics lesson are an indicator of their mathematical PCK. Fig. 2 illustrates how the teacher’s knowledge can interweave with their personal constructs in the planning and execution of the lesson by using exploratory learning. For example, the teacher starts out by believing that exploratory learning is a good way of learning mathematics and plans a lesson accordingly. When it does not work as well as expected, the teacher changes his/her constructs of exploratory learning to include more interaction between the teacher and students. Therefore, the personal constructs of this particular teacher are linked to his/her PCK. Lengnink and Prediger (2003) report the use of mathematical tasks as objects in an adapted Repertory-Grid survey to study preservice teachers’ constructs about mathematical tasks. Collet and Bruder (2006) used this Repertory-Grid survey and created three different categories to analyse personal constructs about mathematical tasks during a problem solving teacher professional development program. We designed TELPS based on Legnink’s and Prediger’s Repertory-Grid survey. Instead of mathematical tasks, we used mathematical lesson plans. Lesson plans include aspects of planning mathematics lessons as well as performing a mathematics lesson. Therefore, lesson plans are suitable objects to disclose PCK. The following section explains the structure of TELPS.

45

2. Method 2.1. Teacher Education Lesson Plan Survey (TELPS) The Teacher Education Lesson Plan Survey (TELPS) consists of four elements: I. Questions about the pre-service teacher’s degree program and a code that can be used to identify subsequent surveys but which would maintain anonymity II. Brainstorming about planning a good lesson with the question: Which criteria can be used to analyse the quality of a mathematics lesson? List all the criteria of a good mathematics lesson which come to mind. III. Comparison of lesson plans (Repertory Grid), and IV. Evaluation (assessment) of the compared lesson plans The pre-service teachers focus their thoughts on the features of a ‘good’ mathematics lesson, listing them in no particular order (brainstorming). This initial part of the Survey helps the pre-service teachers get started with the analysis of the lesson plans, and is particularly important for those pre-service teachers in their first teacher education class. The pre-service teachers compare two mathematics lesson plans. For this part of TELPS pre-service teachers could use the list from their brainstorming and could add new constructs that they think are important to describe the similarities and differences between the two lesson plans. The Survey ends with a task where the pre-service teachers have to decide which lesson plan is ‘better’ and they have to explain how they would improve this lesson plan. 2.2. The lesson plans used in TELPS The lesson plans are original mathematics lesson plans used in schools. Two different sets of mathematics lesson plans enable a longitudinal data collection and ensure that the pre-service teachers do not remember their answers from their first survey (testeretest effect). A change in the topics of the lesson plans allows the pre-service teachers to concentrate on the features of a ‘good’ lesson plan rather than their memory of the previous lesson plan. The first set of lesson plans consists of two ways of introducing trigonometry e an historical approach, and the right angle triangle and ratios approach. Each lesson is considered a valid way of introducing trigonometry but emphasises different aspects of trigonometry. The second set of lesson plans are for teaching Newton’s method of approximating the root of a function. One lesson plan is teacher-centred direct instruction, and the other

Fig. 2. The development of a personal construct.

46

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e50

lesson is student-centred with students using a worksheet to discover Newton’s method for themselves. 2.3. The participants TELPS is embedded in the compulsory mathematics teaching programs at both University A (Australia) and University G (Germany). These universities were chosen because of the differences between their teacher education programs. Both programs aim to educate secondary mathematics teachers but are quite different in duration, organisation, structure, and content. Therefore, the influence of different teacher education programs on constructs of a ‘good’ mathematics lesson (and therefore their PCK) can be studied. TELPS is conducted during class time (about an hour) with informed consent to ensure that the pre-service teachers understand that participation is voluntary, as required by ethics at each university. The pre-service teachers are assured of anonymity and that their responses will not influence their assessment during the teacher education program. The first author is the mathematics methodology lecturer at University A, the third author is the mathematics methodology lecturer at University G, and the second author is a post-graduate student at University G. 2.3.1. University A in Australia The University A teacher education program is an 18-month (three semesters) Bachelor of Teaching degree that has an accelerated option of 12 months (two semesters). The pre-service teachers already have a degree in mathematics and the Bachelor of Teaching allows them to teach in secondary schools in Australia. The University A data include the cohorts from 2009, 2010, and 2011. 2.3.2. University G in Germany The teacher education program at University G is divided into two components e nine semesters at the university followed by a two-year internship at school. The pre-service teachers complete the mathematics and the pedagogy in one degree. The University G data include pre-service teachers beginning their teacher education program in 2009, 2010, and 2011, and the final year cohorts finished their teacher education program in 2009, 2010, and 2011. The 2009 cohort will be the first to provide beginning and end of program data. The final year cohorts are not represented in the beginning cohorts as none of the pre-service teachers from 2009 have completed their program yet. It is anticipated that these students will complete their program in twelve months. 2.4. Theoretical scheme to analyse TELPS data To analyse the data from the third element of TELPS, we designed a system of themes describing aspects of mathematics lesson plans. To develop this theoretical scheme for analysing TELPS the results of a number of studies describing the quality of teaching were compared and combined into a theoretical scheme (Fig. 3). This scheme includes ‘Gagné’s Nine Events of Instruction’ (Killpatrick, 2001), Slavin’s ‘QAIT-Model’ (Slavin, 1995), Meyer’s ‘10 points of a good lesson’ (Meyer, 2005), Helmke’s description of good teaching (2009), and the results of TIMSS (Klieme, Schümer, & Knoll, 2001). The scheme in Fig. 3 does not include characteristics of a mathematics teacher, because we are using lesson plans, which cannot show the teachers’ activities during class. We also added characteristics of a written lesson plan to elicit the special qualities of a lesson plan. Based on this scheme we developed a manual to translate students’ constructs into a code.

Fig. 3. Themes used to analyse constructs of a lesson plan.

3. Results This section focuses on the results of the Repertory-Grid data analysis, including the results of the additional tests and analysis to check the quality of TELPS and the early results of the two universities pre-service teachers’ analysis of lesson plans. 3.1. Content validity: Pedagogical Content Knowledge and TELPS In the first instance we conducted a content analysis (Mayring, 2000) of pre-service teachers’ constructs analysing the contextual structure of the constructs used by the pre-service teachers to compare the lesson plans. We found 20 themes, which were in some cases very similar. In a second step we matched these empirical themes to the theoretical framework (Fig. 3) derived from other research literature. Table 1 documents the 20 themes and the results of the match with the derived theoretical themes of a mathematical lesson. In this table, examples of the pre-service teachers’ constructs are also linked to the 20 data-based themes we derived from the content analysis. This documentation of the Repertory-Grid data illustrates the diversity of pre-service teachers’ analysis of the lesson plans. Furthermore, we use the last three columns of Table 1 to compare our themes of mathematics lessons with the mathematics teacher knowledge system of Hill et al. (2008), who divided Pedagogical Content Knowledge into Knowledge of Content and Students (KCS), Knowledge of Content and Teaching (KCT), and Knowledge of Curriculum (KC). The comparison of the results from TELPS with the Hill et al. (2008) analysis of PCK indicates that there are many similarities. Each U indicates that the pre-service teachers’ constructs from TELPS fit into the definition of PCK by Hill et al. (2008). Each of the themes matches at least one of the aspects of PCK in the research by Hill et al. (2008), and all but three themes link to at least two of the partitions of PCK, further indicating the complexity of PCK. 3.2. Reliability of TELPS The reliability of TELPS is considered in two different settings. To prevent test/re-test effects we changed the lesson plan set each time we conducted the survey, and to check the influence of the lesson plan changes we conducted a parallel test. Section 3.2.1 summarises the results of this parallel test. We also secured the coding of the constructs into the themes of mathematics lesson

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e50

47

Table 1 Comparison of the theoretical framework with the results of the content analysis. Literature-based themes

Data-based themes developed by the content analysis

Student constructs from TELPS

KCS

Structure of the lesson plan

- Formal structure of the lesson plan - Instructions for the teacher - Initial situation

List materials, formula work out, topics

U

List prior knowledge, students’ background

U

Initial situation/basic conditions Goal Didactical analysis of the content Structure of the teaching process Motivation Cognitive activation

Internal differentiation Repetition, practice and results Media Ways of teaching and learning

- Goals of the lesson - Mathematical correctness - Reflection and justification of the lesson plan - Phases of teaching - Function of the lesson - Time planning - Motivation - Cognitive activation of students - Self-regulated learning - Clarity, understandable - Internal differentiation -

Tasks Feedback and reflection Prepared material Used media Teaching methods Social forms

Clear goal, sub outcomes to monitor progress, objectives of the syllabus Correct math, Define terminology, explanation of maths principle Provides time guidance, varied techniques, introduction

U

U

U

U

U U

Transparencies, tools instruments, different technology Teacher support, group discussion

U

KC

U U

Practical examples, interesting, contextual relevance Student involvement, engaging to students, different approaches explanations Applies to a variety of abilities, catering for individual needs Tasks, methods to remember, exercises homework

KCT

U

U

U U U

U

U

KCS: Knowledge of Content and Student. KCT: Knowledge of Content and Teaching. KC: Knowledge of Curriculum.

plans by using a database for a standardised categorisation. Once a construct is coded the computer will code this construct in the same way. Section 3.2.2 documents the inter-rater reliability.

the same manual to categorise the constructs as the raters used. The agreement averaged a ¼ 0.68 across-the-board. 3.3. Constructs and themes

3.2.1. Parallel test To ensure that the different lesson plan sets have no effect on students’ constructs, we designed a parallel test. Eleven German students analysed both the trigonometry plans and the Newton lesson plans used in this study. Another eleven students analysed the two lesson plan sets the other way around. The results of the survey correlate with 0.832 whereas the means of the variables (Table 2) are not significantly different. 3.2.2. Inter-rater reliability A manual was devised so that the students’ constructs could be matched with the literature-based themes of a good mathematics lesson (Fig. 3). For example a construct like “clear goals” is coded with the theme “Goal” (cf. Table 1). To check the coding, we ask three people to individually code students’ constructs with the help of the manual describing all the themes. Two of the raters were students who had finished their teacher education program and the third rater was a research assistant. The three raters categorised 780 constructs. To measure the reliability of agreement we calculated Krippendorfs’ alpha across-the-board with an acceptable agreement (a ¼ 0.71) for our research (Krippendorff, 2004). Twenty-seven students completed TELPS and then were given the manual to categorise their own constructs. The students used

Fig. 4 illustrates the number of constructs that were named by the pre-service teachers to describe their analysis of the two compared lesson plans at different times in their teacher education program. We asked students to complete TELPS at the beginning (initial) and at the end (final) of their teacher education program. In Germany 122 first semester and 51 final semester pre-service teachers participated in TELPS. At University A in Australia, TELPS had 53 first and 30 final semester pre-service teachers as participants.

Table 2 Means of the parallel test. Group

Lesson plan

Number of constructs

Number of themes

A (N ¼ 11)

Trigonometry Newton Newton Trigonometry

16.27 14.82 16.82 15.64

8.64 8.36 8.36 8.64

B (N ¼ 11)

Fig. 4. Mean number of constructs at beginning and end of the programs.

48

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e50

Comparing the results between the different levels of education, Fig. 4 shows that the final semester pre-service teachers in both countries found more constructs to explain the differences and similarities of the analysed lesson plans than the first semester preservice teachers do. It appears that the number of constructs increased between the first semester pre-service teachers and the final semester pre-service teachers in both countries. To analyse pre-service teachers’ constructs of mathematics lessons with a focus on content, we categorised all named constructs with the scheme of mathematics lesson plan themes (Fig. 3). Fig. 5 shows the number of themes that were covered by pre-service teachers’ constructs at different times in their teacher education program. This analysis also shows that the first semester preservice teachers’ constructs covered significantly fewer themes than pre-service teachers at the end of their university teacher education program. Therefore, final semester pre-service teachers looked at the lesson plans in a more multifarious way. For example, final semester students at University G covered an averaged of seven themes with their constructs, whereas first semester students at University G covered an average of 5.6 themes of a mathematics lesson plan while naming their constructs. To analyse the data in more detail the following sections describe the results of University G and A in each theme. 3.3.1. Results e University G For first and last semester pre-service teachers at University G the theme ‘Goals’ had the greatest difference (30%), while the difference was only 19% for the theme ‘Internal Differentiation’ (Fig. 6). However, 49% of the final semester pre-service teachers named constructs that linked to internal differentiation, and almost all of the pre-service teachers mentioned constructs linked to motivation (88%, 92%). The theme ‘Cognitive activation’ was not as high at the end of the program (59%) as it was in the beginning (65%). This was the only theme that the University G pre-service teachers thought was less important toward the end of the program. Fig. 6 further illustrates the comparison of the different foci used to analyse a mathematics lesson plan by the first semester preservice teachers and the final semester pre-service teachers. At the beginning of their teacher education program the pre-service teachers focussed on the themes ‘Structure of the lesson’, ‘Motivation’ and ‘Cognitive Activation’, whereas final semester preservice teachers did not show a special focus and named all

Fig. 6. Theme characteristic of University G pre-service teachers.

themes more often. In a principal component analysis we showed that pre-service teachers’ analysis of a lesson plan is linked with the actual lecture and seminars in their university program (Bausch, Bruder, & Prescott, 2011). By the end of the program the University G pre-service teachers had completed their lectures in teaching mathematics and learned more about how teaching mathematics so they had a more general understanding of lesson planning. 3.3.2. Results e University A The theme ‘Goals’ had the greatest difference between the first semester pre-service teachers and the final semester pre-service teachers at University A (Fig. 7). The final semester pre-service teachers most often named the constructs of ‘Goals’, ‘Motivation’, and ‘Cognitive activation’, whereas more first semester pre-service teachers mentioned constructs of ‘Motivation’. Fewer than a third of the final semester pre-service teachers mentioned constructs in the themes ‘Basic conditions’, ‘Internal differentiation’, ‘Media’, ‘Ways of teaching and learning’, and ‘Didactic content analysis’. First semester pre-service teachers focussed on constructs of the themes ‘Motivation’ and ‘Cognitive activation’, while at the end of the second semester this focus was still there, but the theme ‘Goals’ was more important. 4. Discussion The early results of TELPS allow us to discuss the research question.

Fig. 5. Mean number of themes at beginning and end of program.

Fig. 7. Theme characteristic of University A pre-service teachers.

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e50

Can TELPS measure pre-service teachers’ PCK through their analysis of lesson plans? The results of the content analysis show that the constructs of the pre-service teachers have a wide variation and could be summarised within different themes (column two in Table 2). TELPS discloses important constructs of lesson planning because the empirical themes describe characteristics of the theoretically derived scheme of themes of planning a mathematics lesson. In addition, the eleven theoretical themes from the analysis of lesson plans fit into Shulman’s definition (Section 1.1). Table 1 shows how the themes of this study link to the partitioning of PCK by Hill et al. (2008). However, the fact that many of the themes fit into two of Hill et al.’s partitions in turn suggests that the lines between KCS, KCT and Knowledge of Curriculum are not so defined. Just as Shulman believed that content and pedagogy are not mutually exclusive, our data indicate that KCT, KCS and Knowledge of Curriculum are also not mutually exclusive. TELPS could also indicate whether there is a difference between first semester pre-service teacher students’ PCK and final semester pre-service teachers’ PCK. This is in line with the expectation that pre-service teachers develop their PCK during teacher education program. The comparison of the two universities shows that TELPS has the ability to detect differences in pre-service teachers’ PCK. Although the number of credit points in mathematics education is the same at the two universities, the content and length of the particular courses are different and the differences can be depicted by TELPS. Thus, the analysis of lesson plans using the concepts of planning a good lesson has the potential for indicating PCK. The theory of personal constructs (Kelly, 1955) combined with PCK would give teacher educators an indicator of whether new teachers will teach as they were taught even after they have finished their teacher education program. This is a new way to understand PCK. The lesson plans have no individual connection to the participant, thus the analysis is more objective and learned available concepts are used for evaluation. In addition, the results of the parallel test show that the constructs of pre-service teachers are independent of the analysed lesson plans. The constructs of a pre-service teacher are individual and an actual part of his/her knowledge of teaching mathematics. The inter-rater reliability of the coders is also acceptable, so the results of the coding with the manual can be reproduced. To ensure a stable and coherent coding we could use the database for the further coding of the constructs. The inter-rater reliability of the pre-service teachers coding is nearly acceptable. The manual instructions for coding the constructs were very brief, thus the results should be improved if we spend more time explaining the rating system to the coders. By comparing the results of the pre-service teachers at different times TELPS showed differences in the number of constructs and in the number of themes. The pre-service teachers, who are further through their program, found more different aspects to analyse mathematics lessons. Thus, TELPS can show the differences between first and final semester pre-service teachers’ PCK. TELPS can also illustrate different foci of analysing mathematics lesson plans. For example, final semester pre-service teachers differentiated the lesson plans into more themes than the first semester pre-service teachers did, whereas the first semester pre-service teachers at both universities focused on motivating aspects. There appears to be a growth in the development of PCK during the teacher education program because the mean number of themes found by the pre-service teachers in their last semester is higher than mean number of themes in their first semester. Also the focus of analysing lesson plans appears to change while becoming a teacher. This change might be connected with the structure and

49

intensity of the teacher education program. For example the University G pre-service teachers have four years to think about teaching and applying what they learn in the university setting before getting professional experience and name a variety of themes to analyse the lesson plan. The University A pre-service teachers’ program is intensive with professional experience and all university lectures within one year. It remains to be seen whether their PCK continues to develop in their early years of teaching to a broader focus, and to explore these changes in more detail, we will analyse our longitudinal data set in a qualitative as well as in a quantitative way. The first results of a principal component analysis (Bausch et al., 2011) show that the structure of pre-service teachers’ knowledge is linked with their actual courses. We will further need to correlate TELPS results with personal characteristics such as the hours of individual experience in teaching mathematics, the number and content of the mathematics education courses or the number of semesters at university if we are to understand the factors that influence pre-service teachers’ PCK of planning and designing a mathematics lesson. Thus, we have to deepen the inferential statistics to get a deeper understanding of how pre-service teachers’ PCK of teaching mathematics develops. Hill et al. (2008) note the complexities of PCK as well as the difficulty of measuring PCK. TELPS gives an insight into pre-service teachers’ personal constructs as an indicator of the PCK themes determined from the literature (Helmke, 2009; Killpatrick, 2001; Klieme et al., 2001; Meyer, 2005; Slavin, 1995). Further research and inferential statistics are necessary to help us understand the link between pre-service teachers’ analysis of lesson plans and their development of PCK during their teacher education program in a more comprehensive way. At this stage, we have only been able to follow the Australian pre-service teachers through their program from beginning to end. The first cohort of German pre-service teachers will complete their teacher education program in the next semesters, allowing us to further determine the development of PCK in a single cohort. 5. Conclusion In this article, we present a new survey to gather pre-service teachers’ PCK. We determined a variety of constructs concerning designing a ‘good’ mathematics lesson and we also indicated that these constructs are linked with pre-service teachers’ PCK of planning and designing a mathematics lesson. Shulman (1986) originally defined PCK as representing the blending of content and pedagogy into an understanding of how particular aspects of subject matter are organised, adapted, and represented for instruction. When teachers are asked to explain a mathematical problem PCK and CK are linked. Therefore, a more general method of measuring PCK is necessary. Lesson plans solve that problem because they sum up the content and activities of a whole lesson in a clear and concise way e all the different ways students are motivated, specific mathematical terms are introduced, tasks are undertaken, students’ understanding of the mathematics is assessed, and so on. There are problems in detecting the PCK of mathematics teachers because it is complicated and, as Hill et al. (2008) suggest, explicit criteria to measure conceptualisation and development of PCK are important. The development of PCK is an important element of any teacher education program, and TELPS appears to be useful in determining pre-service teachers’ PCK. The division of PCK by Hill et al. (2008) into Knowledge of Content and Students (KCS), and Knowledge of Content and Teachers (KCT) and Knowledge of Curriculum allows deeper analysis of PCK. Using lesson plans, we believe that PCK can be measured using TELPS.

50

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e50

6. Outlook We will continue the study by analysing pre-service teachers’ own lesson plans to see whether the development of PCK that we have derived from TELPS is reflected in their lesson plans. We also intend to follow the University A pre-service teachers into their schools to determine their PCK after a couple of years of teaching. We will analyse the longitudinal data set in a quantitative and qualitative way to explore in more detail the development of PCK during the teacher education program. There are other possibilities for TELPS. We believe that it could be used to support the development of our pre-service teachers’ competence by giving constructive feedback, thereby helping preservice teachers see their own development of PCK and to reflect on their skills. This feedback could be a part of a portfolio to show their cognitive development in analysing lesson plans and help them to reflect on their PCK (Chamoso, Cáceres, & Azcárate, 2012). To improve the implementation of TELPS in teacher education programs more easy, we are concurrently developing an online tool that includes TELPS and gives automatic feedback to the participant when they have completed the survey. TELPS could be a tool for teacher education in the future in different countries since the online tool exists in both German and English. References Bausch, I., Bruder, R., & Prescott, A. (2011). Personal constructs of planning mathematics lessons. In Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 113e120). Ankara, Turkey: PME. Ball, D. L., & Bass, H. (2009). With an eye on the mathematical horizon: Knowing mathematics for teaching to learners’ mathematical futures. Presented at the National Council of Teachers of Mathematics annual meeting, Washington, D.C., April 23, 2009 http://www-personal.umich.edu/wdball/presentations/index.html. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching. American Educator, Fall, 14e17, 20e22, 43e46. Barkatsas, A., & Malone, J. (2005). A typology of mathematics teachers’ beliefs about teaching and learning mathematics and instructional practices. Mathematics Education Research Journal, 17(2), 69e90. Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., & Jordan, A. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47(1), 133e180. Blömeke, S., Houang, R., & Suhl, U. (2011). TEDS-M: diagnosing teacher knowledge by applying multidimensional item response theory and multi-group models. IERI Monograph Series: Issues and Methodologies in Large-Scale Assessments, 4, 109e126. Chamoso, J. M., Cáceres, M. J., & Azcárate, P. (2012). Reflection on the teachinglearning process in the initial training of teachers. Characterization of the issues on which pre-service mathematics teachers reflect. Teaching and Teacher Education, 28(2), 154e164. Cheang, W. K., Yeo, K. K., Chan, C. M., Lim-Teo, S. K., Chua, K. G., & Ng, L. E. (2007). Development of mathematics pedagogical content knowledge in student teachers. The Mathematics Educator, 10(2), 27e54. Collet, C., & Bruder, R. (2006). Evaluation of a teaching concept for the development of problem solving competencies in connection with self-regulation. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. (Vol. 2, pp. 345e352). Goulding, M., & Suggate, J. (2001). Opening a can of worms: investigating primary teacher’s subject knowledge in mathematics. Mathematics Education Review, 13, 41e54.

Helmke, A. (2009). Unterrichtsqualität und Lehrerprofessionalität. Diagnose, Evaluation und Verbesserung des Unterrichts. Seelze-Velber: Klett. Hill, H. C., Ball, D. W., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: conceptualising and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372e400. Huckstep, P., Rowland, T., & Thwaites, A. (2002). Primary teachers’ mathematics content knowledge: What does it look like in the classroom?. Paper given at the annual conference of the British Educational Research Association, University of Exeter. Kahan, J. A., Cooper, D. A., & Bethea, K. A. (2003). The role of mathematics teachers’ content knowledge in their teaching: a framework for research applied to a study of student teachers. Journal of Mathematics Teacher Education, 6, 223e252. Kelly, G. A. (1955). Theory and personality. In The psychology of personal constructs, Vol. 1. New York: Norton. Killpatrick, L. (2001). Gagne’s nine events of instruction. In B. Hoffman (Ed.), Encyclopedia of educational technology. Retrieved from http://coe.sdsu.edu/eet/ Articles/gagnesevents/start.htm. Klieme, E., Schümer, G., & Knoll, S. (2001). Mathematikunterricht in der Sekundarstufe I: “Aufgabenkultur” und Unterrichtsgestaltung. In E. Klieme, & J. Baumert (Eds.), TIMSS e Impulse für Schule und Unterricht. Forschungsbefunde, Reforminitiativen, Praxisberichte und Video-Dokumente (pp. 43e57). Bonn: Bundesministerium für Bildung und Forschung. Krauss, S., Neubrand, M., Blum, W., & Baumert, J. (2008). The professional knowledge of German secondary mathematics teachers: Investigations in the context of the COACTIV project. Online paper for the 11th international congress on mathematics education (ICME 11) in Monterrey, Mexico (TSG 27). Retrieved from http://tsg.icme11.org/document/get/397. Krippendorff, K. (2004). Reliability in content analysis. Human Communication Research, 30, 411e433. Lengnink, K., & Prediger, S. (2003). Development of the personal constructs about mathematical tasks e a qualitative study using repertory grid methodology. In Proceedings of the 27th annual meeting of the International Group for the Psychology of Mathematics Education (PME), Hawaii 2003. Lim-Teo, S. K., Chua, K. G., Cheang, W. K., & Yeo, J. K. (2007). The development of Diploma of Education student teachers’ mathematics pedagogical content knowledge. International Journal of Science and Mathematics Education, 5, 237e261. Mayring, P. (2000). Qualitative content analysis. Forum: Qualitative Social Research [On-line Journal], 1(2)Available at http://qualitative-research.net/fqs/fqs-e/200inhalt-e.htm. Mewborn, D. (2001). Teachers content knowledge, teacher education, and their effects on the preparation of elementary teachers in the United States. Mathematics Education Research Journal, 3, 28e36. Meyer, H. (2005). Was ist guter Unterricht? Berlin: Cornelsen Scriptor. Rowan, B., Schilling, S. G., Ball, D. L., & Miller, R. (2001). Measuring teachers’ pedagogical content knowledge in surveys: an exploratory study Consortium for Policy Research in Education, study of instructional improvement, research note S2. Ann Arbor, MI: University of Michigan. Retrieved from http://www.sii.soe. umich.edu/about/pubs.html. Scheer, J. W. (1996). A short introduction to personal construct psychology. In J. W. Scheer, & A. Catina (Eds.), Empirical constructivism in Europe e The personal construct approach (pp. 13e17). Giessen: Psychosozial Verlag. Schmidt, W. H., Blömeke, S., & Tatto, M. T. (2011). Teacher education matters. A study of the mathematics teacher preparation from six countries. New York: Teacher College Press. Shulman, L. S. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4e14. Slavin, R. E. (1995). A model of effective instruction. The Educational Forum, 59(2), 166e176. Stacey, K., Helme, S., Steinle, V., Baturo, A., Irwin, K., & Bana, J. (2001). Preservice teachers’ knowledge of difficulties in decimal numeration. Journal of Mathematics Teacher Education, 4, 205e225. Wilson, P. S., Cooney, T. J., & Stinson, D. W. (2005). What constitutes good mathematics teaching and how it develops: nine high school teachers’ perspectives. Journal of Mathematics Teacher Education, 8, 83e111. Wong, T.-W., & Lai, Y.-C. (2006). Exploring factors affecting mathematics teaching effectiveness among pre-service primary mathematics studenteteachers. Retrieved from http://www.aare.edu.au/06pap/won06754.pdf.