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Social and Behavioral Sciences

Procedia Social and Behavioral Sciences 2 (2010) 2222-2226

WCES-2010

Sufficiency of undergraduate education in developing mathematical

pedagogical content knowledge: Student teachers' views

Esra Bukova-Guzela, Semiha Kulaa, I§ikhan Ugurela, Zekiye Ozgurb

aFaculty of Buca Education, University of Dokuz Eylul, Izmir, 35160, Turkey bSchool of Education, University of Wisconsin-Madison, 53706, USA

Received October 19, 2009; revised December 28, 2009; accepted January 11, 2010

Abstract

The aim of this study is to determine the views and evaluations of high school mathematics student teachers regarding undergraduate courses aimed at developing mathematical pedagogical content knowledge, and to reveal their suggestions as to improve those courses. From four different state universities, in total 36 secondary school mathematics student teachers who were in their senior year participated in this study. The data were gathered through four open-ended questions asked to student teachers. Student teachers' responses were analyzed by means of content analysis. In the study, student teachers stated that these courses were necessary but there are some problems and limitations in the development of PCK. We hope that the study will contribute to the revision of courses aimed at developing pedagogical content knowledge in Turkey.

© 2010 Elsevier Ltd. All rights reserved.

Keywords: teacher education; mathematics student teacher; pedagogical content knowledge; sufficiency; undergraduate program.

1. Introduction

Shulman (1987) emphasizes 'Pedagogical Content Knowledge (PCK)' in his study regarding the knowledge bases that teachers should possess. The idea that knowing a subject matter is not adequate to teach it has produced emphasis on mathematical PCK besides mathematical content knowledge. PCK comprises presenting a topic to others in an understandable way, and understanding the ways which facilitate or complicate the learning of a topic (Shulman, 1986). Strong PCK is positively correlated with student achievement (Carpenter et al.; 1989; Smith & Neale, 1989; Gudmundsdottir, 1988; Rovegno, 1992; Wilson & Winwberg, 1989). Because teachers with strong PCK present the content more accurately and meet the diverse needs of students by using various teaching strategies, focusing on students' understanding, explaining the topic and/or material according to students' cognitive development, and providing appropriate examples and analogies. Ball (1988) defines mathematics teachers as a person who can distinguish the difference between knowing mathematics for himself/herself and teaching mathematics to others. Therefore, being knowledgeable in a subject matter does not guarantee to teach it effectively. Thus it is essential for mathematics student teachers to develop PCK. This knowledge base has evoked the revision of teachers' training courses in many countries since the mid 1980s. Such courses which aimed at developing PCK has existed in teacher education programs since 1990s and still today this knowledge base continues to evoke changes in programs. Special Teaching Methods, Examination of Textbooks, and Instructional Technologies and Material Development are some examples of courses in our teacher education programs which oriented to developing PCK, though their content may vary depending on each university. The aim of those courses is to make student teachers get to know about the curriculum, the instructional materials that are aligned with curriculum, misconceptions and various assessment approaches, and to help student teachers put their theoretical knowledge into practice. It is important to determine the quality of those courses offered in teacher education programs and then to reveal how those courses can be further improved in order to create more qualified teaching force. Accordingly, the purpose of the study is to determine the views and evaluations of secondary school mathematics student teachers regarding undergraduate courses aimed at developing mathematical PCK, and to reveal their suggestions as to improve those courses.

1877-0428 © 2010 Published by Elsevier Ltd.

doi:10.1016/j.sbspro.2010.03.312

2. Method

This study is a qualitative research in which case study design is used to determine the views and evaluations of secondary school mathematics student teachers regarding the courses they have taken which aimed at developing PCK, and regarding their own PCK.

2.1. Participants

36 secondary school mathematics student teachers in their last year of teacher education program participated in this study in 2006-2007 spring term. In Turkey, secondary school teacher education is an integrated program which lasts five years since 1998. The first 3.5 years mostly focuses on mathematics, and the last 1.5 years focuses on teacher training. In this context, all participants were in the above-mentioned program, and were about to complete their program one month later. The participants were from four different state universities from three different regions of Turkey (Aegean, Central Anatolia, and Marmara Region). In the data collection, we found out that there are eight universities throughout Turkey which had fifth grade secondary school mathematics student teachers at that time. Instruments were sent to those universities, and four of them agreed to administer the instruments. Instead of drawing conclusion based on findings from one university, we took advantage of findings from four different universities; so that the comprehensiveness of the study was increased. In this way, the maximum variation sampling technique which is a purposeful sampling technique was used in this research. We received 8 to 15 instruments from each of four participant universities. However, we excluded some of them if a student teacher did not respond some questions or if responded mostly as 'yes-no'. After this elimination, our final sample consists of 5 participants from A and B universities (out of 40 student teachers) and 13 participants from C and D universities (out of 65 student teachers). Each university was randomly assigned a letter (A, B, C and D) when presenting the research findings.

2.2. Instruments

Student mathematics teachers were asked to write their responses to four open-ended questions regarding their experiences in the courses they had taken which were intended for developing PCK such as Special Teaching Methods, Examination of Textbooks, and Instructional Technologies and Material Development. Those questions comprise the evaluations of student teachers about how these courses contributed to the student teachers' PCK development, and the student teachers' views regarding the necessity and quality of such courses.

2.3. Procedures

In order to carry out the research we communicated with a willing instructor from each of three participant universities (except the university we work), and explained the structure and the aim of the research. Then each instructor posed the open-ended questions to the participants who were willing to participate in the study. The aim of the study was explained to the student teachers as we wanted them evaluate their training in regard to the undergraduate courses they have taken. In one of the participant universities in the Central Anatolian Region and in the university where the authors work, candidate mathematics teachers were asked to write their responses on the papers that were provided them by the instructor. On the other hand, teacher candidates from other participant universities sent their responses directly to the authors via e-mail. Responses received via email were printed without any change. This study constitutes the third part of a comprehensive study which consists four parts. In each part of the overall study, it is aimed respectively to examine the views of student teachers regarding 1-Subject Matter Knowledge, 2-Pedagogical Knowledge, 3- PCK, and 4- Professional Practicum Experience. The findings and conclusions regarding the fourth part of the study is presented in another article which is accepted to be published in December-2009 issue of the Journal of Inonu University Education Faculty (Ozgur, Bukova-Guzel, Kula, & Ugurel, 2009). This study focuses only on the third part of the broader study, in which it is aimed at identifying student teachers' evaluations toward their training in terms of the courses and applications with respect to PCK development. Therefore, in this study only pertinent findings are presented and discussed.

2.4. Data Analysis

Student teachers' responses to the open-ended questions were analyzed by means of content analysis. Firstly, each response of each participant was examined by three authors individually. Then, common or very similar categories in each author's analysis report were identified by constantly comparing student teachers' responses with each other in the frame of the themes emerged. The student teachers' self-assessment regarding their PCK were examined by means of the PCK framework that is developed by Bukova-Guzel & Kula (2009) by examining and synthesizing the literature to PCK (Grosman, 1990; Magnusson, Krajcik & Borko, 1999; Rowland, Huckstep, & Thwaites, 2005; Shulman, 1987; Kovarik, 2008; Schoenfeld, 2000; Wagner, 2003). It is found that their self-assessment is in line with the Teaching Strategies and Multiple Representation Knowledge

(TSMRK) and the Curriculum Knowledge (CK) components of the PCK framework (Table 1), and the findings are presented below in Table 2. On the other hand, except one student teacher, it is seen that they did not touch on the Learner Knowledge (LK) component of the PCK framework. Findings are categorized as Self-Assessment in terms of PCK and Views regarding PCK Courses, and each category is presented in detail below in the tables constructed based on the frequency of student teachers' responses at each participant university. The column named Total in the tables refers to the percentage of participants who mentioned that particular code in their response regardless of their university.

Tablel. The PCK Framework

_TSMRK_LK_CK_

Using appropriate mathematical activities when presenting Knowing prior knowledge of students Recognizing all components of mathematics

mathematical concepts Knowing the difficulties where students curriculum

Using real life examples and simulations when presenting may encounter when teaching a topic Recognizing and using the instructional tools to

mathematical concepts be used in teaching a topic

Taking advantage of various teaching strategies when Knowing the potential misconceptions Recognizing and using assessment tools can be

presenting mathematical concepts students may have used in assessing learning

Taking advantage of various mathematical representations Knowing student's differences Having both horizontal and vertical curriculum

when presenting mathematical concepts knowledge for any topic

3. Results (Findings)

The participants' agreement on each code is presented in percentages derived from the frequency of their statements. Excluding the codes stated under 5%, other codes are presented in the Table 2, 3. Following the tables, some excerpts from participants' responses which underpin the context of the related codes are given.

3.1. Evaluations Regarding PCK

Based on the participants' responses, subcategories of Category-1 "Self-Assessment in terms of PCK" are generated with the help of the PCK framework given in the Table-1. In general, it is seen that the student teachers did not evaluate the contributions of the undergraduate courses they had taken in terms of the learner knowledge.

Table 2. Category-1: Self-Assessment in terms of PCK

Category—1: Percentages of Statements by the Universities

Self -Assessment in terms of PCK A B C D Total

I- Teaching Strategies and Multiple Representation Knowledge

I-a In terms of using appropriate mathematical activities in presenting mathematical concepts

Considering sufficient %20 %60 %69 %23 %44

Considering insufficient %80 %40 %15 %62 %44

I-b In terms of using real life examples and simulations in presenting mathematical concepts

Considering sufficient %20 %20 %62 %23 %36

Considering insufficient %80 %80 %15 %62 %50

I-c In terms of taking advantage of various teaching strategies when presenting mathematical concepts

Considering sufficient %20 %80 %62 %31 %47

Considering insufficient %80 %20 %15 %54 %39

I-d In terms of taking advantage of various mathematical representations when presenting mathematical concepts

Considering sufficient %20 - %62 %23 %33

Considering insufficient - %20 %8 %8 %8

II- Curriculum Knowledge

Il-a In terms of recognizing all components of mathematics curriculum

Considering sufficient %20 %20 %62 %23 %36

Considering insufficient %80 %80 %15 %62 %50

Il-b In terms of recognizing and using the instructional tools to be used in teaching a topic

Considering sufficient %20 %40 %69 %38 %47

Considering insufficient %80 %20 %15 %46 %36

II-c In terms of recognizing and using assessment tools can be used in assessing learning

Considering sufficient %20 %40 %62 %38 %44

Considering insufficient %80 %40 %15 %54 %42

II-d In terms of having both horizontal and vertical curriculum knowledge for any topic

Considering sufficient %40 %40 %69 %31 %47

Considering insufficient %60 %40 %15 %54 %39

While we cannot find significant difference in student teachers' qualifications when we look at the total column in Table-2, we found out that most of the student teachers at University A and University B consider themselves insufficient when we examined the data seperately for each university. It is apparent that student teachers at University C generally find themselves sufficient in two components of the PCK framework which are CK and TSMRK. To illustrate, some excerpts from student teachers' responses are given below.

"I can prepare lesson plans appropriate to the level of class, but I don't think I can create diverse learning environments. I don't think I will be qualified enough for preparing materials and to using them. " (20th student teacher- University C)

"Since mathematics is an abstract science, it somewhat constrains the techniques to be used, however; I can use interesting and necessary methods and techniques. " (9th student teacher- University D)

" I believe that I can make use of different learning methods and approaches according to the circumstances of class." (16th student teacher- University C)

It is evident from the Table-3 that all student teachers regards the courses intended for developing PCK as necessary. However, a considerable number of student teachers stated that they are not satisfied how those courses are currently carried out. It is found that student teachers at University C have the higgest satisfaction level with the way those courses are carried out. This finding can be explained by the correlation between the high level of self-confidence of student teachers at University C for their PCK development and their satisfaction level with the way the courses are carried out.

Table 3. Category-2: Views regarding PCK Courses

Category—2: Percentages of Statements by the Universities

Views regarding PCK Courses A B C D Total

I- Regarding the Way Courses are Carried out

Satisfied %40 %40 %77 %54 %58

Unsatisfied %60 %60 %15 %38 %36

II-Regarding the Necessity of the Courses

Considering necessary %100 %100 %100 %100 %100

Considering unnecessary - - - - -

4. Discussion

In this study, the sufficiency of undergraduate courses in developing PCK is examined through the lens of mathematics student teachers, and it is found out that student teachers hold the idea that they do not possess all indicators of PCK. In particular, student teachers did not mention about the contributions of the PCK courses they have taken to their learner knowledge development. On the other hand, especially student teachers at University C consider themselves sufficient to a great extent in the other two components of pedagogical content knowledge except that learner knowledge. The fact that all student teachers stated that they consider the courses meant to develop PCK as necessary can be inferred as an indicator of student teachers' awareness concerning the significance of PCK among knowledge bases that teachers should possess.

5. Conclusion and Recommendation

In conclusion of responses of participant mathematics teacher candidates in four Turkish state universities, it appears that there are some problems and limitations in the development of PCK. Those problems and limitations emerged both in the student teachers' self-assessments and also in their views regarding the way those courses are carried out. In this sense, an in-depth analysis of the PCK development process in the frame of its different aspects, especially in its implementation dimension, would be very beneficial. We suggest a further research with a large sample by employing both quantitative and qualitative research methods. In addition, in order for student teachers to be more confident about their PCK development, we suggest designing new undergraduate courses aimed at discussions and applications of some subject related issues such as the potential misconceptions students may have, potential difficulties where students may face when learning a topic, and different representations that can be used in teaching.

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