Exploring Kindergarten Teachers Pedagogical Content Knowledge of Mathematics

IJEC (2010) 42:27 41 DOI 10.1007/s13158-010-0003-9 Exploring Kindergarten Teachers Pedagogical Content Knowledge of Mathematics Joohi Lee Published online: 19 February 2010 Ó Springer Science+Business Media B.V. 2010 Abstract The purpose of this study was to assess 81 kindergarten teachers pedagogical content knowledge of mathematics on six subcategory areas such as number sense, pattern, ordering, shapes, spatial sense, and comparison. The data showed participants possessed a higher level of pedagogical content knowledge of number sense (M = 89.12) compared to other mathematics pedagogical content areas. The second highest scores among six subcategories of pedagogical content knowledge of mathematics was for the pedagogical content area of pattern (M = 82.33). The lowest scores among those six subcategories of kindergarten teachers pedagogical content knowledge were obtained from the subcategory of spatial sense (M = 44.23), which involved the means to introduce children to spatial relationships. The second lowest score was obtained for the subcategory of comparison (M = 50.40) which involved the means to introduce the concept of graphing and the use of a balance scale for measurement. Résumé El propósito de este estudio fue el evaluar el conocimiento pedagógico del contenido de matemáticas en 81 maestros de educación infantil en seis subcategorías tales como el sentido de número, patrones, orden, sentido espacial y comparación. Los datos mostraron que los participantes poseen un nivel más alto de conocimiento pedagógico en sentido de número (M = 89.12) en comparación con otras áreas de contenido pedagógico de matemáticas. El segundo resultado más alto entres las seis subcategorías fue el área de contenido pedagógico de patrones (M = 82.33). Las dos subcategorías que muestran más bajos resultados en relación al conocimiento pedagógico del contenido de matemáticas en maestros de educación infantil fue la subcategoría de sentido espacial (M = 44.23), lo cual incluyó J. Lee (&) Department of Curriculum and Instruction-EC4, College of Education and Health Professions, University of Texas at Arlington, Science Hall # 322C, 502 Yeats, Box 19777, Arlington, TX 76019, USA e-mail: joohilee@uta.edu

28 J. Lee los medios por los cuales se introducen o presentan relaciones espaciales a los niños. El segundo resultado más bajo fue obtenido para la subcategoría de comparación (M = 50.40) lo que incluyó el cómo introducir el concepto de graficar y la utilización de una escala de balance para la medición. Resumen Le but de cette étude était d évaluer chez 81 professeurs de jardin d enfants la connaissance de six sous-catégories de contenu pédagogique en mathématiques, soit le sens du nombre, le modèle, la mise en ordre, les formes, le sens spatial, et la comparaison. Les données montrent que les participants connaissaient mieux le contenu pédagogique du «sens du nombre» (M = 89.12) que de celui des autres secteurs. Le deuxième score de connaissance le plus élevé parmi les six sous-catégories de contenu pédagogique en mathématiques se trouve au secteur «modèle» (M = 82.33). Le score le plus bas des professeurs de jardin d enfants à ces six sous-catégories a été obtenu à la sous-catégorie «sens spatial» (M = 44.23), relative aux moyens de présenter les relations spatiales aux enfants. Le deuxième score le plus bas a été obtenu à la sous-catégorie «comparaison» (M = 50.40) qui portait sur les moyens de présenter le concept de graphique et l usage d une balance à fléau pour mesurer. Keywords Pedagogical content knowledge of mathematics Teaching young children mathematics Kindergarten teachers pedagogical content knowledge Introduction This study aims to investigate kindergarten teachers pedagogical content knowledge (PCK) of mathematics which has been identified as a critical factor of children s mathematics achievements. The lack of early mathematics skills is closely associated with children s later mathematics achievement and causes serious disadvantages in their future work and careers (Munn 1997). The association between early mathematics performance and its correlation with later mathematics performance has been well documented (EdudataReports 2008). Thus, the critical importance of early childhood mathematics performance has been highlighted nationally [see national standards of mathematics established by the Council of Teachers of Mathematics (NCTM) (2000); A Nation at Risk reported by The National Commission on Excellence in Education (1983)]. Worldwide, there have been many attempts to improve young children s mathematics skills, such as designing national pre-kindergarten and kindergarten mathematics standards (NCTM 2000), proposing improved methods of teaching and learning mathematics (e.g., the National Numeracy Strategy in England; Kyriacou and Goulding 2009), and producing highly accomplished educators (National Board for Professional Teaching Standards (NBPTS) 2009). Focus has been heavily placed on teacher quality in teaching mathematics by improving teachers knowledge of subject matter knowledge and teaching. Researchers coined the term pedagogical content knowledge (PCK), which is a comprehensive term including both pedagogical knowledge and content knowledge.

Exploring Kindergarten Teachers Pedagogical Content Knowledge 29 PCK has been a critical issue in teacher education, but very few empirical studies have been conducted regarding this matter. Shulman (1987) first used the term publically at a presidential address at the annual meeting of the American Educational Research Association in 1985. Since his speech, several attempts have been made to clarify its meaning. Shulman (1986) defined PCK as the ways of representing and formulating the subject that makes it comprehensible to students (p. 9). Subject matter knowledge is essential in teaching. According to Graeber (1999), subject matter knowledge refers to content knowledge of each subject such as mathematics, science, or biology. Subject matter/content knowledge does not directly impact teachers teaching practices or the quality of their instruction, but it has an indirect effect on teaching practice (Brophy 1992). As Brophy explained, when teachers subject matter knowledge is more explicit, they will tend to teach the subject more fully and to respond to students questions more sensitively. It is true that teachers deep content knowledge of subject is important when they teach, but content knowledge alone is not enough to provide students high-quality teaching in practice (Even 1993). High-quality and expert teachers not only know the subject matter, but also know how to teach certain knowledge which is referred to as PCK (Kennedy 1998). Leinhardt (1986) agreed that being an effective mathematics teacher requires mathematics content knowledge as well as an understanding of the instruction processes needed to efficiently transfer this knowledge to students. He selected seven expert teachers, each of whom had taught for from 12 to 25 years, for a case study. To determine which teachers were expert teachers, Leinhardt traced students improvement on mathematics standardized tests over a 5-year period. Principals and supervisors were also asked to review each mathematics teacher and to suggest outstanding mathematics teachers. In the case study, expert teachers were observed and videotaped to find out what factors made them high-quality teachers in teaching mathematics. Study results showed that teachers PCK was crucial to their success. Areas involved in this study to measure teachers PCK were teachers subject matter knowledge and pedagogical knowledge of mathematics (e.g., patterns used in developing lessons and academic engagement using a variety of instructional materials). Carpenter et al. (1989) supports that a teacher with pedagogical content knowledge of mathematics tends to implement high-quality mathematics education. Their study proved that teachers who had an understanding of how to promote children s problem-solving skills in mathematics tended to promote children s problem-solving strategies more frequently than teachers who didn t. Pedagogical and content knowledge (PCK) is inter-correlated in a complicated manner which goes beyond either content knowledge or pedagogy knowledge. PCK entails how to interpret a certain subject and transfer it to students in an understandable manner. It is the term of formulation of certain subjects. For example, an early childhood teacher has knowledge of basic addition and subtraction, which is considered content knowledge of basic addition and subtraction. However, merely having this knowledge does not mean that the teacher can effectively transfer it to young children. Knowing how to teach or transfer knowledge to the target children is considered pedagogical knowledge. In this case, the teacher should know how to teach or transfer specific knowledge

30 J. Lee about addition and subtraction to young children in an understandable manner. This is considered pedagogical content knowledge of basic addition and subtraction. Feynman (1995) provided a specific example of the use of pedagogical content knowledge to communicate the concept of size of atoms. A teacher may be able to recite the facts related to atom size, which is considered content knowledge. However, this does not ensure that he or she will be able to explain this fact to younger students. Feynman gave the following as an example of how to utilize PCK: If an apple is magnified to the size of the earth, then the atoms in the apple are approximately the size of the original apple (p. 5). Kennedy (1998) also supports the importance of teacher s PCK. In his example, an individual may know the way to the grocery store but be unable to give directions to others. In other words, this person may have deep conceptual understanding of that field and yet have difficulty outlining its major issues to others. How teachers transfer their mathematics knowledge to students in an understandable manner (core of PCK) directly impacts on how students effectively learn mathematics. Therefore, teachers PCK has been addressed as a critical element in being an effective mathematics teacher. Kinach (2002) found that a level of interface existed between student achievement and preservice teachers PCK in mathematics learning. Since the results of national and international assessments (e.g., the Third International Mathematics and Science Studies and several reports of the International Association for the Evaluation of Educational Achievement) indicate that American students show a deficiency in mathematics, teachers knowledge of mathematics has emerged as a major problem (Leinhardt 1986). This is not only a problem in the United States; Stewart (2008) indicated that only one out of 10 primary school teachers in Great Britain earned an A level in math or science PCK. In terms of early childhood settings, a critical problem of discontinuity in pedagogy and curriculum has been indicated when children move from early childhood services to the traditional school system, which utilizes more direct instructional methods (Petriwskyj et al. 2005). This has also been a major issue in recent Australasian and European literature on the subject (Margetts 2002). Regardless of the challenges of applying inconsistent pedagogy, Graeber (1999) argued that it is teachers lack of PCK of mathematics that has caused students lower achievement outcomes in mathematics. Leinhardt (1986) indicated that being an effective mathematics teacher requires both content and pedagogical knowledge. In his case study mentioned above, both mathematics content (mathematics subject matter knowledge) and pedagogical knowledge (e.g., academic engagement, patterns used in developing lessons, effective use of instructional materials) were found to be essential to being an effective teacher. Very little is known about early mathematics teaching focusing on teachers PCK, since historically very few researchers have focused on teachers of children under 5 years old (Ginsburg and Amit 2008). Therefore, in this study I focused on kindergarten teachers PCK of mathematics in six subcategory areas: number sense, pattern, ordering, shapes, spatial sense, and comparison. I also correlated scores of teachers PCK of mathematics with their demographic information including gender, obtained degree, and number of years teaching at the kindergarten level.

Exploring Kindergarten Teachers Pedagogical Content Knowledge 31 Participants Eighty-one kindergarten teachers in the state of Indiana participated in this study. Tables 1, 2, and 3 display participants demographic information involving their age, gender, and race. Among all participants (N = 81), 24.7% ranged from 20 to 30 years of age (N = 20), 16% ranged from 31 to 40 years of age (N = 13), 34.6% ranged from 41 to 50 years of age (N = 28), and 24.7% of them were over 51 years of age (N = 20). Participants were composed of 95% percent female (N = 77) and 5% male teachers (N = 4). Participants in this study were all Caucasian. Table 4 shows the education level of participants. Among participants, 34 (41.9%) teachers held a bachelor s degree, which is required by the state of Indiana to teach kindergarten. The rest of the participants obtained graduate degrees at either the master s or doctoral level. The data showed that 45 (55.6%) teachers obtained master s degrees, and two teachers (2.5%) earned doctoral degrees. Table 5 shows the number of years teachers had worked at the kindergarten level. This number varied from 1 year to more than 16 years. The average number of years teaching at the kindergarten level was 7.8 years. As shown in Table 5, 34.6% of participants had taught for more than 16 years. This indicates that more than one-third of participants in this study were experienced teachers. At the same time, about one-fifth of participants (22.2%) were beginning teachers who had taught only 1 3 years at the kindergarten level. One-third of participants (33.3%) had taught 4 15 years at the kindergarten level. Table 1 Frequency distribution of participants age Age N Percent 20 30 20 24.7 31 40 13 16.0 41 50 28 34.6 Over 51 20 24.7 Total 81 100.0 Table 2 Frequency distribution of participants gender Gender N Percent Female 77 95.1 Male 4 4.9 Total 81 100.0 Table 3 Frequency distribution of participants race Race N Percent Caucasian 81 100 Total 81 100

32 J. Lee Table 4 Frequency distribution of participants acquired degree Acquired degree N Percent Bachelor s degree 34 41.9 Master s degree 45 55.6 Doctoral degree 2 2.5 Total 81 100.0 Table 5 Frequency distribution of years taught at kindergarten level Years taught at kindergarten level N Percent 1 3 years 18 22.2 4 7 years 16 19.8 8 15 years 11 13.5 More than 16 years 28 34.6 No response 8 9.9 Total 81 100.0 Instrumentation For the purpose of this study, the Survey of Pedagogical Content Knowledge in Early Childhood Mathematics (SPECKECM) was utilized (Smith 1998, 2000). The major reason for selecting this particular instrument is the lack of instruments found to assess early childhood teachers PCK in mathematics. In fact, there was no single instrument found in a comprehensive database (EBSCO) to measure early childhood teachers PCK in mathematics. Smith (2000) tested both validity and reliability by administering the survey to 400 early childhood educators and found a good range of reliability (Cronbach s alpha =.70) and validity. Smith also conducted a factor analysis to determine what mathematics content SPECKECM measured. The factor analysis revealed six factors in teaching mathematics: number sense, patterns, ordering, shapes, spatial sense, and comparisons. The operational definitions of the six categories used in SPECKECM are given by Smith (2000) as follows: Number sense refers to the statements in which the ability to compare the quantitative value of groups of object was used (Kennedy & Tipps, 2000). Items grouped by the dimension of patterns can be defined as having the children sort and clarify, analyze simple patterns and make predictions about them, describe change, represent situations mathematically, and develop an intuitive understanding and use of properties of numbers and operations (NCTM 2000, p. 1). The third factor, ordering, is the ability to compare objects or groups of objects based on their differences, for example, from smallest to largest and from lightest to darkest. Ordering also includes counting and numbers. The fourth factor, shapes, refers to those items in which those items in which geometric figures are being taught. Kennedy and Tipps (2000) define spatial sense, the fifth factor, as the relationship between objects and their location in a three-dimensional world (p. 354). The final factor, comparison, is

Exploring Kindergarten Teachers Pedagogical Content Knowledge 33 defined as the ability to distinguish qualitative and quantitative similarities and differences between objects or sets of objects. (p. 95) In the present study, reliability of SPECKECM was examined with 10 kindergarten teachers. A Cronbach s coefficient alpha, which is frequently used to test the reliability of survey items (Cronk 1999), was calculated using the pilot data. The reliability score (.85) of the instrument was categorized in the range of very good reliability. Therefore, based on the pilot study, the instrument had an acceptable amount of reliability to measure kindergarten teachers pedagogical content knowledge of mathematics. Data Analysis Procedures The percentage of teachers PCK scores was calculated to assess kindergarten teachers PCK of mathematics and the correlations between PCK and demographic variables (gender, earned degree, and number of years teaching at the kindergarten level). All variables were examined utilizing an ANOVA and t-test selected from the Statistical Package for the Social Sciences (SPSS). An ANOVA was used to investigate whether each individual variable (demographic variables) accounted for a significant portion of the variance in correlating with the criterion variable (PCK of mathematics). The significant correlations between independent and dependent variables were interpreted by a t-test and determined by the significance level using alpha levels, *p \.05 and **p \.01. Results To assess participants pedagogical content knowledge of mathematics, 35 items on the SPECKECM were used. Possible scores on the 35 items ranged from 0 to 100. The score for pedagogical knowledge of mathematics (M) was obtained and calculated using the following formula: the number of right answers/the number of total questions 9 100. The data are presented in Table 6. The mean scores obtained from the subcategories of kindergarten teachers pedagogical content knowledge were ranked from the highest to the lowest: number sense (M = 89.12), pattern (M = 82.33), ordering (M = 71.25), shapes (M = 68.99), comparison (M = 50.40), and spatial sense (M = 44.23). The data show that participants possessed a higher level of pedagogical content knowledge of number sense (M = 89.12) compared to other mathematics content areas. The subcategory of number sense in this study involved the means to introduce children to the concept of comparing quantitative values of groups of objects. The second highest scores among six subcategories of pedagogical content knowledge of mathematics was for the content area of pattern (M = 82.33). The items to measure teachers pedagogical content knowledge of pattern (how to teach pattern ) included the means to introduce to children to the concept of classifying and sorting, analyzing simple patterns and making predictions about them, and describing pattern change.

34 J. Lee Table 6 Descriptive statistics of pedagogical content knowledge of mathematics Subcategories of pedagogical content knowledge N M SD Number sense 81 89.12 12.32 Pattern 81 82.33 14.74 Ordering 81 71.25 16.43 Shapes 81 68.99 19.00 Spatial sense 81 44.23 24.33 Comparison 81 50.40 21.90 Pedagogical content knowledge (overall mean) 81 67.72 18.12 The lowest scores among the six subcategories of kindergarten teachers pedagogical content knowledge were obtained in the subcategory of spatial sense (M = 44.23), which involved the relationship between objects and their location in a three-dimensional world. The second lowest score was obtained for the subcategory of comparison (M = 50.40), which involved the means to introduce the concept of qualitative and quantitative similarities and differences between objects or set of objects. The results of this study indicate that kindergarten teachers demographic characteristics, such as acquired degree, and the number of years teaching at kindergarten level, affected kindergarten teachers pedagogical content knowledge of mathematics. Table 7 displays these data. The results of ANOVA showed significant correlations between the scores of pedagogical content knowledge of mathematics and teachers acquired degrees. A Tukey HSD, a method of post-hoc comparison, was adopted. The Tukey HSD grouped teachers with graduate degrees (Master s and Doctoral degrees) as a homogeneous group and showed significantly higher scores on PCK of mathematics for these teachers than for teachers with bachelor s degree (see Table 8). Years of teaching experience at the kindergarten level also correlated significantly with PCK of mathematics. The ANOVA showed that teachers with more than 10 years teaching experience at the kindergarten level scored higher than those with fewer than 10 years experience. Table 7 Means and standard deviations of pedagogical content knowledge of mathematics by age, degree, and number of years teaching * The mean difference is significant at the.05 level ** The mean difference is significant at the.01 level Demographics n (%) M (SD) Acquired degree Bachelor s degree 34 (41.9) 58.92 (16.33) Master s degree 45 (55.6) 73.13 (13.98)* Doctoral degree 2 (2.5) 74.12 (13.79)* Number of years teaching at K level 1 4 years 22 (21.2) 64.29 (13.87) 5 10 years 18 (19.8) 65.31 (17.22) Over 10 years 33 (13.5) 69.42 (13.35)** No response 8 (9.9) Not calculated

Exploring Kindergarten Teachers Pedagogical Content Knowledge 35 Table 8 group comparisons of teachers pedagogical knowledge by earned degree (Tukey HSD) * The mean difference is significant at the.05 level ** The mean difference is significant at the.01 level Degree Mean difference SE Significance Bachelor Master s -9.25 4.10.07 Doctoral -40.67 12.59.00** Master Bachelor s 9.25 4.10.07 Doctoral -31.41 12.39.04* Doctoral Bachelor s 40.67 12.59.00** Master s 31.41 12.39.04* Discussion and Conclusions Teachers PCK of mathematics has been emphasized in several studies as a critical element for teachers to acquire in order to implement high-quality mathematics education (Mangione and Maniates 1993; Sherman and Mueller 1996; Stewart 2008). There is a more specific finding correlating early childhood teachers PCK of mathematics with their teaching practice. Lee (2004, 2005) and Lee et al. (2003) indicated that kindergarten teachers with higher scores on PCK of mathematics more frequently implemented higher quality mathematics instruction in their classrooms. Thus, it is necessary to assess early childhood teachers PCK of mathematics because it will eventually impact their teaching practice in mathematics. In this study, the major findings are: (1) kindergarten teachers showed the highest scores of PCK on number sense and the lowest scores on spatial sense; (2) kindergarten teachers with doctoral degrees showed higher scores of PCK of mathematics than teachers with either a Bachelor s or Master s degree; (3) and teachers who have taught for more than 10 years showed higher PCK of mathematics scores than teachers who have taught for less than 10 years. Six subcategories (number sense, pattern, ordering, shapes, spatial sense, and comparison) were used in this study to evaluate teachers PCK of mathematics. In the present study, the scores of kindergarten teachers PCK of mathematics on number sense were highest among all subcategories, while scores obtained for spatial sense were the lowest. When teachers do not possess the knowledge of how to teach certain mathematics content, it obviously hinders implementation of mathematics-related curriculum. In this study, it is interesting to note that kindergarten teachers scored lowest on PCK of spatial sense, which is the foundation of geometry. Previous international assessments indicated that American students performed worst in geometry (Copley 2001). A major reason that has been suggested for this is that geometry content in early and elementary education has largely focused on knowing terms, definitions, and attributes of two- or three-dimensional shapes while neglecting enhancing children s spatial sense (Copley 2000; Lee et al. 2009; NCTM 2000). Doverborg and Samuelsson (2001) recommend exposing children to various experiences of

36 J. Lee exploring shapes in their lives, such as exploring the neighborhood to look for different types of houses, looking for geometric features on houses, investigating objects from different angles, and designing house plans by sketching and constructing a treehouse. Their study showed that children who had a variety of activities were better able to draw two-and three-dimensional houses than children who hadn t. When children explore geometric shapes, it is essential to encourage them to reflect about geometric shapes using various methods such as drawing or verbal expression rather than just teaching the names and characteristics of shapes (Doverborg and Samuelsson 2001). Children are naturally aware of attributes of geometric shapes when they manipulate and play with concrete materials. Providing time for children to explore three-dimensional materials in space would help them build foundations for later geometry learning. Furthermore, allowing children to spend more time exploring geometry in their daily lives is also necessary to make mathematics learning meaningful for them. To better assist early childhood teachers, it is essential for school districts and administrators to provide them with opportunities to develop a better understanding of ways to teach mathematics in general, but especially in geometry. School district administrators can develop programs to help inservice teachers obtain knowledge of how to teach young children geometry content while enhancing children s spatial sense. Teachers are usually required to participate in professional development. To make this more effective in teaching mathematics, more systematic assistance is necessary. These professional development programs can be provided to teachers in several flexible ways. They can be implemented via a group seminar or workshop as well as through individual contact with mathematics mentors or other experienced mathematics teachers. At the very least, a mathematics mentor should always be available to support teaches when they encounter difficulties in teaching certain mathematics content to their classes. In this study, teachers demographic information was significantly associated with their PCK of mathematics (e.g., earned degree and number of years teaching kindergarten level). The findings of this study imply that more education and more teaching experience at the kindergarten level are significant indicators of higher PCK of mathematics. To help new teachers overcome their limited education and teaching experiences, it is necessary to provide in-service professional training on how to teach mathematics to young children. According to Krauss et al. (2008), indepth-mathematical training is significantly associated with high quality of teaching in mathematics. An experimental study also showed positive impacts of professional training for teaching practice in mathematics (Carpenter et al. 1989). In Carpenter et al. study, twenty teachers were assigned to an experimental group that participated in a month-long workshop on how to teach children to solve problems in addition and subtraction. Twenty other teachers were assigned to a control group without training. Carpenter et al. found that the teachers from the experimental group more frequently encouraged their students to use various problem-solving strategies than did teachers from the control group (Carpenter et al. 1989). This study evidences the importance of in-service training for teachers, especially those

Exploring Kindergarten Teachers Pedagogical Content Knowledge 37 with limited teaching experiences. More recently, Darling-Hammond and Richardson (2009) showed that professional development utilizing the Integrated Mathematics Assessment (IMA) approach directly improved teachers PCK in teaching mathematics. The IMA required teachers participation in a 5-day workshop and biweekly meetings throughout the year to discuss insights acquired from teaching practices. These findings are promising as educators and researchers have worked to improve teachers PCK. As existing literature suggests, it is evident that professional development and training are important to improve teaching of mathematics. However, professional development is still lacking in current educational settings. According to the most recent data available, half of teachers surveyed spend a day or less in their professional development (NCES 2001). This barely meets many states minimum licensure requirements. Recommended professional development sessions for inservice teachers are typically 15 days over a 5 year period (NASDTEC 2004). These national data indicate the deficiency of professional training for in-service teachers. This needs to be taken into consideration among school districts and administrators by providing new teachers more feasible ways to attend professional development sessions related to mathematics. At the same time, it is necessary for school districts and administrators to ensure that professional development sessions are high-quality and effective. To make these sessions more effective for their teachers, the contents of sessions should be closely associated with teachers weakness (Hill 2009). Therefore, surveying teachers needs is essential before requiring or recommending teachers to participate professional developments. Additionally, school districts and administrators need to provide a systematic resource to support early childhood teachers attendance at professional development sessions or trainings in teaching mathematics or their taking graduate courses from higher education institutions. School districts also can plan for teachers summer workshops on how to teach young children mathematics by inviting university faculty or mathematics experts who teach early mathematics methodology courses at a university level. There are several recommendations when teaching mathematics to young children. Children learn mathematics more efficiently when using manipulative materials rather than abstract or semi-abstract instructional materials (Heddens 1986) due to their cognitive developmental status (e.g., their lack of understanding of decentration, reversibility, and conservation; Bredekamp and Copple 1997). According to both psychological and educational theorists, children s development of number concept is non-linear and multidimensional based on social and cultural context (Munn 1997). To make mathematics learning meaningful for young children it is necessary to associate mathematics contents with children s real lives. NCTM (2000) recommends for early childhood teachers to promote children s mathematics learning through their everyday activities (NCTM 2000). For example, children can learn mathematical concepts such as sorting or patterning during clean-up time while they are putting away toys. Children learn sense of time based on the sequence of their daily activities (Lee et al. 2009). Thus, using children s daily lives to teach mathematical concepts is essential.

38 J. Lee An essential component when delivering mathematics contents to young children is considering children s interests. Interest appropriateness is a key component in implementing developmentally appropriate practice (Bredekamp and Copple 1997). There has been an experimental study conducted to compare students based on the use of mathematics instructional materials (see Doverborg and Samuelsson 2000 for detail). In this study, when the teachers recognized children s interest in stars, they integrated star cards to promote children s understanding of numbers with the test group. Children in the reference group did not receive intervention (i.e., star card instructional materials). Children from the test group showed significantly higher proficiency with number symbols and quantities (e.g., matching number one with a card with one dot) and number awareness (e.g., picking correct numbers when asked). Thus, assessing and observing children s interest is essential in planning to teach children mathematics in early childhood. As the results of the current study showed, years of teaching experience are also a critical factor in teachers PCK of mathematics. New teachers possess less PCK of mathematics compared to teachers with more than 10 years of teaching experience. To remedy this situation, school administrators can implement mentoring programs to assist these new teachers by pairing them with experienced teachers. Mentoring will provide opportunities for new teachers to hear different ideas about mathematics activities and how to teach certain mathematics content from teachers who have already put these ideas into practice. They would also have opportunities to share their difficulties in teaching mathematics. At the same time, experienced teachers would have time to reflect on their teaching practices as they work with new teachers. Before teachers arrive in the schools, teacher education programs should strengthen teacher candidates PCK of mathematics. In particular, instructors who teach early childhood mathematics education pedagogical courses at the college level should provide teacher candidates with a variety of hands-on activities or mathematics educational materials that can be directly connected to National Council of Teachers of Mathematics content areas (e.g., number and operation, algebra, geometry, data analysis and probability, and measurement). Strawhecker (2005) recommended that integrating field experiences into a mathematics method course is an efficient way to increase pre-service teachers PCK of mathematics. She investigated 96 pre-service teachers enrolled in a mathematics method course and found significantly positive impacts of field-experience components on PCK of mathematics. Integrating technology in mathematics courses is also useful to improve preservice teachers PCK (Cavin 2008; Niess 2005). Limitations Even though there are important educational implications from the findings of this study, a major limitation remains. In this study, I referred mostly to studies conducted in the U.S. to build the foundation of theoretical frameworks for this study. The major reason for this is that there has been a lack of empirical studies associated with pedagogical content knowledge. This lack has been addressed in

Exploring Kindergarten Teachers Pedagogical Content Knowledge 39 several European publications. Alexander (2004) made a commentary on the lack of research related to pedagogy in education in England. Some researchers (e.g., Kansanen 2002; Terhart 2003) argue that this is because of the standards of English language use as the American Educational Research Association suggests. For example, a Finnish research group alternates the use of the terms didactical thinking and pedagogical thinking. Most German empirical studies used the term Didactics/Didaktik instead of pedagogy (Meyer 1997). Kansanen (2009) attempted to clarify the terms of subject-matter didactics which have been associated in Anglo-Saxon countries with pedagogical content knowledge. These are closely related concepts, but the terms cannot be used interchangeably. For this reason, I reviewed literature that included the use of the term pedagogical content knowledge to build the foundation of this study. Furthermore, there has in the past been only one instrument available to measure the PCK of teachers in early childhood programs (i.e., Smith 2000). Some instruments (Even 1993; Fuller 1996) have been designed to assess teachers of more advanced levels of students such as at the elementary or secondary levels. The 35 questions used in this study to assess kindergarten teachers PCK of mathematics were somewhat limited by the need to cover whole content areas as indicated by the NCTM (2000) such as number and operations, algebra, geometry, measurement, data analysis, and probability. This limitation presents a challenge to early childhood educators and other researchers to consider developing more comprehensive instruments to assess early childhood teachers PCK of mathematics, which has been found to be a significant factor. In implementing high-quality mathematics teaching practice. More importantly, there has been a lack of empirical studies on early childhood teachers teaching practice (Ginsburg and Amit 2008). Therefore, more research is necessary to investigate early childhood teachers teaching practice in mathematics. Finally, it is recommended to investigate whether PCK is associated with teaching practice. Turnuklu and Uesildere (2007) argue that for teachers, knowing how to teach certain content is completely different from implementing what they know. They studied 45 primary mathematics teacher candidates in Turkey and found that teacher candidates possess high competency of pedagogical content knowledge of mathematics but showed a lack of high-quality of teaching in mathematics. Therefore, it is necessary to investigate this association between knowledge and practice. References Alexander, R. (2004). Still no pedagogy? Principle, pragmatism and compliance in primary education. Cambridge Journal of Education, 34, 7 33. Bredekamp, S., & Copple, C. (1997). Developmentally appropriate practice in early childhood programs. Washington, DC: NAEYC. Brophy, J. (1992). Conclusion to advances in research on teaching. In J.E. Brophy (Ed.), Advances in research on teaching: Teachers knowledge of subject matter as it relates to their teaching practices (pp. 347 361). Greenwich, CT: JAI Press.

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