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Teaching and Teacher Education 35 (2013) 43e50

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Teaching and Teacher Education

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

Anne Prescott a,*, Isabell Bausch b, Regina Bruder b

aUniversity of Technology, Sydney, PO Box 222, Lindfield 2070, Australiab 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

Article history:Received 19 May 2012Received in revised form16 May 2013Accepted 21 May 2013

Keywords:Pre-service teachersPedagogical Content KnowledgeLesson plansRepertory-grid theory

* Corresponding author. Tel.: þ61 2 9514 5406; faxE-mail addresses: Anne.prescott@uts.edu.au

mathematik.tu-darmstadt.de (I. Bausch), bruder@(R. Bruder).

0742-051X/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.tate.2013.05.002

a b s t r a c t

A pre-service teacher’s Pedagogical Content Knowledge (PCK) and their personal constructs of teachingdevelop throughout their teacher education program. PCK integrates generic pedagogical knowledge,mathematical teaching methodology and knowledge of the discipline of mathematics and this paperreports on a survey that can be used to assess a pre-service teacher’s PCK. TELPS (Teacher EducationLesson Plan Survey) was developed to determine the PCK of pre-service teachers during their teachereducation program. TELPS is shown to analyse pre-service teachers’ PCK with some indication that thepre-service teacher’s development of PCK can be observed.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Pre-service mathematics teachers begin their teacher educationprogram with beliefs and ideas about teaching mathematics thatare well established by their own experience in school (Barkatsas &Malone, 2005). During their teacher education program the pre-service teachers are exposed to many new ideas, expanding boththeir knowledge about teaching mathematics and content knowl-edge. They generally excelled in the classroom environments oftheir 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 educationprograms seek to broaden and deepen their assumed model ofgood teaching. To explore this process we developed the TeacherEducation Lesson Plan Survey (TELPS). This survey uses the pre-

: þ61 2 9514 5556.(A. Prescott), bausch@

mathematik.tu-darmstadt.de

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service teachers’ analysis of mathematics lesson plans for insightinto their Pedagogical Content Knowledge (Shulman, 1986). Whileother research projects analyse teachers’ Pedagogical ContentKnowledge (PCK) with classical test items (e. g. Krauss, Neubrand,Blum, & Baumert, 2008), TELPS uses an adapted Repertory-GridMethod (Kelly, 1955) which is a psychoanalytical method that es-tablishes peoples’ constructs by comparing objects in a stand-ardised way. The advantage of using Repertory-Grid is the ability toanalyse the data in a nomothetic as well as in an ideographical way(Scheer, 1996). Thus, TELPS can explore the development of math-ematics teachers’ PCK during their teacher education program tofind individual phenomena in their development of PCK or phe-nomena in connection with their teacher education program. Weuse TELPS to measure the personal constructs (Kelly, 1955) ofAustralian and German pre-service teachers about mathematicslesson plans during their teacher education program. Hence, TELPSwill help us to answer the following research question:

Can TELPS measure pre-service teachers’ PCK through their anal-ysis of lesson plans?

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) 43e5044

To answer this question TELPS should be able to show a varietyof 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 ana-lysing mathematics teacher’s PCK. We will explain the theoreticalframework of the TELPS, present first results as an example of dataanalysis, and discuss the quality criteria of this survey by using theearly results of TELPS at two universities in different countries withdifferent teacher education programs.

1.1. Pedagogical Content Knowledge

Content knowledge is a necessary but not sufficient conditionfor good teaching. Mathematics teaching needs more than knowl-edge of content (e.g. Ball et al., 2005; Goulding & Suggate, 2001;Mewborn, 2001; Shulman, 1986), because teachers need to recog-nise that an answer is incorrect (or correct), analyse the source ofany errors and then work with the student to improve the math-ematics. It involves choosing appropriate examples and exercisesand sequencing these so that students are guided in their learning.Developing alternative representations of the mathematics is amajor part of teaching. Consequently, planning a mathematicslesson includes these requirements of teaching. Shulman (1986)defined the knowledge needed to cope with the challenge ofteaching into three categories of teachers’ content knowledge:Subject Matter Content Knowledge, Pedagogical Knowledge andPedagogical Content Knowledge (PCK). The knowledge of designing‘good’ mathematic lessons is a part of Pedagogical ContentKnowledge (PCK), because Shulman (1986) defined it as

“the distinctive bodies of knowledge for teaching. It representsthe blending of content and pedagogy into an understanding ofhow particular topics, problems, or issues are organized, rep-resented and adapted to the diverse interests and abilities oflearners, and presented for instruction” (p. 8).

PCK is an integration of generic pedagogical knowledge, math-ematical teaching methodology and knowledge of the discipline ofmathematics (Lim-Teo, Chua, Cheang, & Yeo, 2007; Shulman, 1986;Stacey et al., 2001).

Shulman’s theory of teacher knowledge is the foundation ofmany research projects on mathematics teacher education. Forexample, the COACTIV project in Germany (Baumert et al., 2010),the ‘Michigan Group’ in USA (Hill, Ball, & Schilling, 2008), Lim-Teo’sgroup in Singapore (Cheang et al., 2007; Lim-Teo et al., 2007), andthe international comparative study, TEDS-M, which examinedhow different countries prepare their teachers to teach mathe-matics in primary and lower-secondary mathematics (Schmidt,Blömeke, & Tatto, 2011), each consider different aspects of PCK.

The relationship between Subject Matter Knowledge (SMK) andthe PCK required for teaching is still not fully understood. Pre-service teachers who had several representations for mathemat-ical ideas and whose knowledge was already richly linked wereable to draw upon them both in planning and in spontaneousteaching interactions (Huckstep, Rowland, & Thwaites, 2002).However, the boundaries between SMK and PCK may well beblurred and a deep understanding of both is important. Conse-quently, Kahan, Cooper, and Bethea (2003) believe that mathema-ticians, whose content knowledge is without question, may notnecessarily possess PCK because so many additional attributes areneeded. The results of Wong and Lai (2006) further support thesefindings, because they show no statistical relationship between PCKand SMK. Blömeke, Houng, and Suhl (2011) were also able todifferentiate PCK from SMK.

Hill et al. (2008) broaden the Shulman definitions by proposinga model of mathematical knowledge for teaching by further

dividing Subject Matter Knowledge and Pedagogical ContentKnowledge (Fig. 1). In mathematics, Subject Matter Knowledgeincludes the Common Content Knowledge (CCK) and the Speci-alised Content Knowledge (SPK) that one would expect a teacher toknow, but also includes Knowledge at the Mathematics Horizonwhich Ball and Bass (2009) define as “a kind of mathematical ‘pe-ripheral vision’, a view of the larger mathematical landscape, . inwhich the present experience and instruction is situated” (p. 6).

Hill et al. (2008) also further divide Shulman’s PCK (Fig. 1). Inorder to prepare and teach a lesson, a teacher must be capable ofputting in place all the partitions of PCK: namely the Knowledge ofContent and Students (KCS), the Knowledge of Content andTeaching (KCT), and the Knowledge of the Curriculum.

There are problems in detecting the PCK of the mathematicsteachers. A teacher’s ability to know that students often make er-rors at certain points in a topic is linked with the teacher’sreasoning about what students are thinking or doing. PCK iscomplicated and Hill et al. (2008) suggest the importance of explicitcriteria to measure conceptualisation and development of PCK. Anumber of researchers analyse mathematics teachers’ PCK usingdifferent 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’sknowledge.

Rowan, Schilling, Ball, and Miller (2001) surveyed teachers’ PCKusing a bank of items in reading/language arts and mathematics.They used classroom scenarios in a multiple-choice survey todifferentiate between content knowledge, knowledge of students’thinking and knowledge of pedagogical strategies. Their resultswere mixed because of difficulties in developing scenarios andwriting the items, and their use of a small sample (this study re-ported a pilot), but they do indicate the possibility of measuringparticular facets of teachers’ PCK.

Wong and Lai (2006) used direct observation of pre-serviceprimary teachers in schools and the lesson plans they used in theteaching. They found that the pre-service teachers who werefurther through their course had better results in PCK than those inthe early years, suggesting PCK improves as the pre-service teacherprogresses through the course.

1.2. Developing a survey to measure PCK

PCK is a very complex construct with different levels. Thus, weused lesson planning via TELPS to cover the general ideas ofdesigning a mathematics lesson. These general ideas of

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mathematics lessons include the decision about using an induc-tive or deductive approach, the ways of motivating and activatingstudents etc. Because multiple-choice items mainly cover veryspecific topics or situations of a mathematics lesson the linkagebetween these general ideas are difficult. An interview aboutplanning and designing a mathematics lesson could be a possi-bility, but the analysis of many interviews is very costly. Thus, welooked for a method that is open like an interview and stand-ardised like classical test items, and decided on the Repertory-GridMethod because it has the boundaries of classical test items withthe structure of open-ended questions. This standardised struc-ture allows a qualitative and quantitative evaluation in an effec-tive way.

Kelly (1955) developed the Repertory-Grid Method to gatherindividual constructs. At the centre of this method is the compar-ison of different objects within a given structure. First, the objectsto be compared are produced by the interviewer and the constructsare determined by asking the question, “In what important waysare two of them alike but different from the third?” (Kelly, 1955, p.222). The constructs and objects are then placed in a grid and theparticipant estimates the importance of the construct for eachobject. 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 pre-service teachers’ constructs on planning and designing a goodmathematics lesson are an indicator of their mathematical PCK.Fig. 2 illustrates how the teacher’s knowledge can interweave withtheir personal constructs in the planning and execution of thelesson by using exploratory learning. For example, the teacherstarts out by believing that exploratory learning is a good way oflearning mathematics and plans a lesson accordingly. When it doesnot work as well as expected, the teacher changes his/her con-structs of exploratory learning to includemore interaction betweenthe teacher and students. Therefore, the personal constructs of thisparticular teacher are linked to his/her PCK.

Lengnink and Prediger (2003) report the use of mathematicaltasks as objects in an adapted Repertory-Grid survey to study pre-service teachers’ constructs about mathematical tasks. Collet andBruder (2006) used this Repertory-Grid survey and created threedifferent categories to analyse personal constructs about mathe-matical tasks during a problem solving teacher professionaldevelopment program.

We designed TELPS based on Legnink’s and Prediger’s Reper-tory-Grid survey. Instead of mathematical tasks, we used mathe-matical lesson plans. Lesson plans include aspects of planningmathematics lessons as well as performing a mathematics lesson.Therefore, lesson plans are suitable objects to disclose PCK. Thefollowing section explains the structure of TELPS.

Fig. 2. The development of

2. Method

2.1. Teacher Education Lesson Plan Survey (TELPS)

The Teacher Education Lesson Plan Survey (TELPS) consists offour elements:

I. Questions about the pre-service teacher’s degree programand a code that can be used to identify subsequent surveysbut which would maintain anonymity

II. Brainstorming about planning a good lesson with thequestion:Which criteria can be used to analyse the quality of a mathe-matics lesson? List all the criteria of a good mathematics lessonwhich come to mind.

III. Comparison of lesson plans (Repertory Grid), andIV. Evaluation (assessment) of the compared lesson plans

The pre-service teachers focus their thoughts on the features ofa ‘good’ mathematics lesson, listing them in no particular order(brainstorming). This initial part of the Survey helps the pre-serviceteachers get started with the analysis of the lesson plans, and isparticularly important for those pre-service teachers in their firstteacher education class. The pre-service teachers compare twomathematics lesson plans. For this part of TELPS pre-serviceteachers could use the list from their brainstorming and couldadd new constructs that they think are important to describe thesimilarities and differences between the two lesson plans. TheSurvey ends with a task where the pre-service teachers have todecide which lesson plan is ‘better’ and they have to explain howthey would improve this lesson plan.

2.2. The lesson plans used in TELPS

The lesson plans are original mathematics lesson plans used inschools. Two different sets of mathematics lesson plans enable alongitudinal data collection and ensure that the pre-serviceteachers do not remember their answers from their first survey(testeretest effect). A change in the topics of the lesson plans allowsthe 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 introducingtrigonometry e an historical approach, and the right angle triangleand ratios approach. Each lesson is considered a valid way ofintroducing trigonometry but emphasises different aspects oftrigonometry. The second set of lesson plans are for teachingNewton’s method of approximating the root of a function. Onelesson plan is teacher-centred direct instruction, and the other

a personal construct.

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e5046

lesson is student-centred with students using a worksheet todiscover Newton’s method for themselves.

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

2.3. The participants

TELPS is embedded in the compulsory mathematics teachingprograms at both University A (Australia) and University G (Ger-many). These universities were chosen because of the differencesbetween their teacher education programs. Both programs aim toeducate secondary mathematics teachers but are quite different induration, organisation, structure, and content. Therefore, the in-fluence 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) withinformed consent to ensure that the pre-service teachers under-stand that participation is voluntary, as required by ethics at eachuniversity. The pre-service teachers are assured of anonymity andthat their responses will not influence their assessment during theteacher education program. The first author is the mathematicsmethodology lecturer at University A, the third author is themathematics methodology lecturer at University G, and the secondauthor is a post-graduate student at University G.

2.3.1. University A in AustraliaThe University A teacher education program is an 18-month

(three semesters) Bachelor of Teaching degree that has an accel-erated option of 12 months (two semesters). The pre-serviceteachers already have a degree in mathematics and the Bachelorof Teaching allows them to teach in secondary schools in Australia.The University A data include the cohorts from 2009, 2010, and2011.

2.3.2. University G in GermanyThe teacher education program at University G is divided into

two components e nine semesters at the university followed by atwo-year internship at school. The pre-service teachers completethe mathematics and the pedagogy in one degree. The University Gdata include pre-service teachers beginning their teacher educationprogram in 2009, 2010, and 2011, and the final year cohorts finishedtheir teacher education program in 2009, 2010, and 2011. The 2009cohort will be the first to provide beginning and end of programdata. The final year cohorts are not represented in the beginningcohorts as none of the pre-service teachers from 2009 havecompleted their program yet. It is anticipated that these studentswill complete their program in twelve months.

2.4. Theoretical scheme to analyse TELPS data

To analyse the data from the third element of TELPS, wedesigned a system of themes describing aspects of mathematicslesson plans. To develop this theoretical scheme for analysing TELPSthe results of a number of studies describing the quality of teachingwere 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 ‘10points of a good lesson’ (Meyer, 2005), Helmke’s description ofgood teaching (2009), and the results of TIMSS (Klieme, Schümer, &Knoll, 2001).

The scheme in Fig. 3 does not include characteristics of amathematics teacher, because we are using lesson plans, whichcannot show the teachers’ activities during class. We also addedcharacteristics of a written lesson plan to elicit the special qualitiesof a lesson plan. Based on this scheme we developed a manual totranslate students’ constructs into a code.

3. Results

This section focuses on the results of the Repertory-Grid dataanalysis, including the results of the additional tests and analysis tocheck the quality of TELPS and the early results of the two uni-versities 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 contextualstructure of the constructs used by the pre-service teachers tocompare the lesson plans. We found 20 themes, which were insome cases very similar. In a second step we matched theseempirical themes to the theoretical framework (Fig. 3) derived fromother research literature. Table 1 documents the 20 themes and theresults of the match with the derived theoretical themes of amathematical lesson. In this table, examples of the pre-serviceteachers’ constructs are also linked to the 20 data-based themeswe derived from the content analysis. This documentation of theRepertory-Grid data illustrates the diversity of pre-service teachers’analysis of the lesson plans.

Furthermore, we use the last three columns of Table 1 tocompare our themes of mathematics lessons with the mathematicsteacher knowledge system of Hill et al. (2008), who divided Peda-gogical Content Knowledge into Knowledge of Content and Stu-dents (KCS), Knowledge of Content and Teaching (KCT), andKnowledge of Curriculum (KC). The comparison of the results fromTELPS with the Hill et al. (2008) analysis of PCK indicates that thereare many similarities. Each U indicates that the pre-serviceteachers’ constructs from TELPS fit into the definition of PCK byHill et al. (2008). Each of the themes matches at least one of theaspects of PCK in the research by Hill et al. (2008), and all but threethemes link to at least two of the partitions of PCK, further indi-cating the complexity of PCK.

3.2. Reliability of TELPS

The reliability of TELPS is considered in two different settings. Toprevent test/re-test effects we changed the lesson plan set eachtime we conducted the survey, and to check the influence of thelesson plan changes we conducted a parallel test. Section 3.2.1summarises the results of this parallel test. We also secured thecoding of the constructs into the themes of mathematics lesson

Table 1Comparison of the theoretical framework with the results of the content analysis.

Literature-based themes Data-based themes developedby the content analysis

Student constructs from TELPS KCS KCT KC

Structure of the lesson plan - Formal structure of the lesson plan- Instructions for the teacher

List materials, formula work out, topics U

Initial situation/basicconditions

- Initial situation List prior knowledge, students’ background U U

Goal - Goals of the lesson Clear goal, sub outcomes to monitor progress,objectives of the syllabus

U U

Didactical analysisof the content

- Mathematical correctness- Reflection and justificationof the lesson plan

Correct math, Define terminology, explanationof maths principle

U U

Structure of the teachingprocess

- Phases of teaching- Function of the lesson- Time planning

Provides time guidance, variedtechniques, introduction

U U

Motivation - Motivation Practical examples, interesting, contextual relevance U U

Cognitive activation - Cognitive activation of students- Self-regulated learning- Clarity, understandable

Student involvement, engaging to students, differentapproaches explanations

U

Internal differentiation - Internal differentiation Applies to a variety of abilities, cateringfor individual needs

U U

Repetition, practiceand results

- Tasks- Feedback and reflection

Tasks, methods to remember, exercises homework U

Media - Prepared material- Used media

Transparencies, tools instruments,different technology

U

Ways of teachingand learning

- Teaching methods- Social forms

Teacher support, group discussion U U

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

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e50 47

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

3.2.1. Parallel testTo ensure that the different lesson plan sets have no effect on

students’ constructs, we designed a parallel test. Eleven Germanstudents analysed both the trigonometry plans and the Newtonlesson plans used in this study. Another eleven students analysedthe two lesson plan sets the other way around. The results of thesurvey correlate with 0.832 whereas the means of the variables(Table 2) are not significantly different.

3.2.2. Inter-rater reliabilityA manual was devised so that the students’ constructs could be

matched with the literature-based themes of a good mathematicslesson (Fig. 3). For example a construct like “clear goals” is codedwith the theme “Goal” (cf. Table 1). To check the coding, we askthree people to individually code students’ constructs with the helpof the manual describing all the themes. Two of the raters werestudents who had finished their teacher education program and thethird rater was a research assistant. The three raters categorised780 constructs. To measure the reliability of agreement we calcu-lated Krippendorfs’ alpha across-the-board with an acceptableagreement (a¼ 0.71) for our research (Krippendorff, 2004).

Twenty-seven students completed TELPS and then were giventhe manual to categorise their own constructs. The students used

Table 2Means of the parallel test.

Group Lesson plan Number of constructs Number of themes

A (N¼ 11) Trigonometry 16.27 8.64Newton 14.82 8.36

B (N¼ 11) Newton 16.82 8.36Trigonometry 15.64 8.64

the same manual to categorise the constructs as the raters used.The agreement averaged a¼ 0.68 across-the-board.

3.3. Constructs and themes

Fig. 4 illustrates the number of constructs that were named bythe pre-service teachers to describe their analysis of the twocompared lesson plans at different times in their teacher educationprogram. We asked students to complete TELPS at the beginning(initial) and at the end (final) of their teacher education program. InGermany 122 first semester and 51 final semester pre-serviceteachers participated in TELPS. At University A in Australia, TELPShad 53 first and 30 final semester pre-service teachers asparticipants.

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

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

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e5048

Comparing the results between the different levels of education,Fig. 4 shows that the final semester pre-service teachers in bothcountries found more constructs to explain the differences andsimilarities of the analysed lesson plans than the first semester pre-service teachers do. It appears that the number of constructsincreased between the first semester pre-service teachers and thefinal semester pre-service teachers in both countries.

To analyse pre-service teachers’ constructs of mathematics les-sons with a focus on content, we categorised all named constructswith the scheme of mathematics lesson plan themes (Fig. 3). Fig. 5shows the number of themes that were covered by pre-serviceteachers’ constructs at different times in their teacher educationprogram. This analysis also shows that the first semester pre-service teachers’ constructs covered significantly fewer themesthan pre-service teachers at the end of their university teachereducation program. Therefore, final semester pre-service teacherslooked at the lesson plans in a more multifarious way. For example,final semester students at University G covered an averaged ofseven themes with their constructs, whereas first semester stu-dents at University G covered an average of 5.6 themes of amathematics lesson plan while naming their constructs.

To analyse the data in more detail the following sectionsdescribe the results of University G and A in each theme.

3.3.1. Results e University GFor first and last semester pre-service teachers at University G

the theme ‘Goals’ had the greatest difference (30%), while the dif-ference was only 19% for the theme ‘Internal Differentiation’(Fig. 6). However, 49% of the final semester pre-service teachersnamed constructs that linked to internal differentiation, and almostall of the pre-service teachers mentioned constructs linked tomotivation (88%, 92%). The theme ‘Cognitive activation’ was not ashigh at the end of the program (59%) as it was in the beginning(65%). This was the only theme that the University G pre-serviceteachers thought was less important toward the end of theprogram.

Fig. 6 further illustrates the comparison of the different foci usedto analyse a mathematics lesson plan by the first semester pre-service teachers and the final semester pre-service teachers. Atthe beginning of their teacher education program the pre-serviceteachers focussed on the themes ‘Structure of the lesson’, ‘Moti-vation’ and ‘Cognitive Activation’, whereas final semester pre-service teachers did not show a special focus and named all

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

themes more often. In a principal component analysis we showedthat pre-service teachers’ analysis of a lesson plan is linkedwith theactual lecture and seminars in their university program (Bausch,Bruder, & Prescott, 2011). By the end of the program the Univer-sity G pre-service teachers had completed their lectures in teachingmathematics and learned more about how teaching mathematicsso they had a more general understanding of lesson planning.

3.3.2. Results e University AThe theme ‘Goals’ had the greatest difference between the first

semester pre-service teachers and the final semester pre-serviceteachers at University A (Fig. 7). The final semester pre-serviceteachers most often named the constructs of ‘Goals’, ‘Motivation’,and ‘Cognitive activation’, whereas more first semester pre-serviceteachers mentioned constructs of ‘Motivation’.

Fewer than a third of the final semester pre-service teachersmentioned constructs in the themes ‘Basic conditions’, ‘Internaldifferentiation’, ‘Media’, ‘Ways of teaching and learning’, and ‘Di-dactic content analysis’. First semester pre-service teachersfocussed on constructs of the themes ‘Motivation’ and ‘Cognitiveactivation’, while at the end of the second semester this focus wasstill there, but the theme ‘Goals’ was more important.

4. Discussion

The early results of TELPS allow us to discuss the researchquestion.

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

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e50 49

Can TELPS measure pre-service teachers’ PCK through their anal-ysis of lesson plans?

The results of the content analysis show that the constructs ofthe pre-service teachers have a wide variation and could be sum-marised within different themes (column two in Table 2). TELPSdiscloses important constructs of lesson planning because theempirical themes describe characteristics of the theoreticallyderived scheme of themes of planning a mathematics lesson.

In addition, the eleven theoretical themes from the analysis oflesson plans fit into Shulman’s definition (Section 1.1). Table 1shows how the themes of this study link to the partitioning ofPCK by Hill et al. (2008). However, the fact that many of the themesfit into two of Hill et al.’s partitions in turn suggests that the linesbetween KCS, KCT and Knowledge of Curriculum are not so defined.Just as Shulman believed that content and pedagogy are notmutually exclusive, our data indicate that KCT, KCS and Knowledgeof Curriculum are also not mutually exclusive.

TELPS could also indicate whether there is a difference betweenfirst semester pre-service teacher students’ PCK and final semesterpre-service teachers’ PCK. This is in line with the expectation thatpre-service teachers develop their PCK during teacher educationprogram. The comparison of the two universities shows that TELPShas the ability to detect differences in pre-service teachers’ PCK.Although the number of credit points in mathematics education isthe same at the two universities, the content and length of theparticular courses are different and the differences can be depictedby TELPS.

Thus, the analysis of lesson plans using the concepts of planning agood lesson has the potential for indicating PCK. The theory of per-sonal constructs (Kelly,1955) combinedwith PCKwould give teachereducators an indicator of whether new teachers will teach as theywere taught even after they have finished their teacher educationprogram. This is a newway to understand PCK. The lessonplans haveno individual connection to the participant, thus the analysis ismoreobjective and learned available concepts are used for evaluation. Inaddition, the results of the parallel test show that the constructs ofpre-service teachers are independent of the analysed lesson plans.The constructs of a pre-service teacher are individual and an actualpart of his/her knowledge of teaching mathematics.

The inter-rater reliability of the coders is also acceptable, so theresults of the coding with the manual can be reproduced. To ensurea stable and coherent coding we could use the database for thefurther coding of the constructs. The inter-rater reliability of thepre-service teachers coding is nearly acceptable. The manual in-structions for coding the constructs were very brief, thus the resultsshould be improved if we spend more time explaining the ratingsystem to the coders.

By comparing the results of the pre-service teachers at differenttimes TELPS showed differences in the number of constructs and inthe number of themes. The pre-service teachers, who are furtherthrough their program, found more different aspects to analysemathematics lessons. Thus, TELPS can show the differences be-tween first and final semester pre-service teachers’ PCK. TELPS canalso illustrate different foci of analysing mathematics lesson plans.For example, final semester pre-service teachers differentiated thelesson plans into more themes than the first semester pre-serviceteachers did, whereas the first semester pre-service teachers atboth universities focused on motivating aspects.

There appears to be a growth in the development of PCK duringthe teacher education program because the mean number ofthemes found by the pre-service teachers in their last semester ishigher thanmean number of themes in their first semester. Also thefocus of analysing lesson plans appears to changewhile becoming ateacher. This change might be connected with the structure and

intensity of the teacher education program. For example the Uni-versity G pre-service teachers have four years to think aboutteaching and applying what they learn in the university settingbefore getting professional experience and name a variety ofthemes to analyse the lesson plan. The University A pre-serviceteachers’ program is intensive with professional experience andall university lectures within one year. It remains to be seenwhether their PCK continues to develop in their early years ofteaching to a broader focus, and to explore these changes in moredetail, we will analyse our longitudinal data set in a qualitative aswell 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 islinked with their actual courses. We will further need to correlateTELPS results with personal characteristics such as the hours ofindividual experience in teaching mathematics, the number andcontent of the mathematics education courses or the number ofsemesters at university if we are to understand the factors thatinfluence pre-service teachers’ PCK of planning and designing amathematics lesson. Thus, we have to deepen the inferential sta-tistics 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 thedifficulty of measuring PCK. TELPS gives an insight into pre-serviceteachers’ personal constructs as an indicator of the PCK themesdetermined from the literature (Helmke, 2009; Killpatrick, 2001;Klieme et al., 2001; Meyer, 2005; Slavin, 1995). Further researchand inferential statistics are necessary to help us understand thelink between pre-service teachers’ analysis of lesson plans and theirdevelopment of PCK during their teacher education program in amore comprehensive way. At this stage, we have only been able tofollow the Australian pre-service teachers through their programfrom beginning to end. The first cohort of German pre-serviceteachers will complete their teacher education program in thenext semesters, allowing us to further determine the developmentof PCK in a single cohort.

5. Conclusion

In this article, we present a new survey to gather pre-serviceteachers’ PCK. We determined a variety of constructs concerningdesigning a ‘good’ mathematics lesson and we also indicated thatthese constructs are linked with pre-service teachers’ PCK ofplanning and designing a mathematics lesson. Shulman (1986)originally defined PCK as representing the blending of contentand pedagogy into an understanding of how particular aspects ofsubject matter are organised, adapted, and represented for in-struction. When teachers are asked to explain a mathematicalproblem PCK and CK are linked. Therefore, a more general methodof measuring PCK is necessary. Lesson plans solve that problembecause they sum up the content and activities of a whole lesson ina clear and concise way e all the different ways students aremotivated, specific mathematical terms are introduced, tasks areundertaken, students’ understanding of the mathematics isassessed, and so on. There are problems in detecting the PCK ofmathematics teachers because it is complicated and, as Hill et al.(2008) suggest, explicit criteria to measure conceptualisation anddevelopment of PCK are important.

The development of PCK is an important element of any teachereducation program, and TELPS appears to be useful in determiningpre-service teachers’ PCK. The division of PCK by Hill et al. (2008)into Knowledge of Content and Students (KCS), and Knowledge ofContent and Teachers (KCT) and Knowledge of Curriculum allowsdeeper analysis of PCK. Using lesson plans, we believe that PCK canbe measured using TELPS.

A. Prescott et al. / Teaching and Teacher Education 35 (2013) 43e5050

6. Outlook

We will continue the study by analysing pre-service teachers’own lesson plans to see whether the development of PCK that wehave derived from TELPS is reflected in their lesson plans. We alsointend to follow the University A pre-service teachers into theirschools to determine their PCK after a couple of years of teaching.

We will analyse the longitudinal data set in a quantitative andqualitative way to explore in more detail the development of PCKduring the teacher education program.

There are other possibilities for TELPS. We believe that it couldbe used to support the development of our pre-service teachers’competence by giving constructive feedback, thereby helping pre-service teachers see their own development of PCK and to reflecton their skills. This feedback could be a part of a portfolio to showtheir cognitive development in analysing lesson plans and helpthem to reflect on their PCK (Chamoso, Cáceres, & Azcárate, 2012).To improve the implementation of TELPS in teacher educationprogramsmore easy, we are concurrently developing an online toolthat includes TELPS and gives automatic feedback to the participantwhen they have completed the survey. TELPS could be a tool forteacher education in the future in different countries since theonline tool exists in both German and English.

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