Mathematics in the Preschool - Dissertation Example

? Mathematics in the Preschool: Comparing In-service and Pre–service Preschool Teachers’ Knowledge of Early Mathematical Development and their Beliefs about Mathematics Teaching Chapter 1 BACKGROUND OF THE STUDY Preschool is a time when children may first encounter mathematical learning, however, much recent research has indicated that preschool children have the ability to understand a variety of mathematical concepts even prior to entering kindergarten (Klein, Stakey, Clements, Sarama, & Lyer, 2008; Clement & Sarama, 2007). That is why early childhood programs are increasingly expected to implement mathematics instruction in classrooms because of young children’s early mathematical knowledge. Development of early math skills provides the foundation for later learning (Clements & Sarama, 2008; NAEYC & NCTM, 2008; Lee & Ginsburg, 2008). The National Council of Teachers of Mathematics (NCTM, 2000) emphasizes the need to provide high quality early childhood mathematics programs in preschools because a solid mathematical foundation in early childhood is essential. At this age, young children are building their own perceptions of what math is – what it means to know and do math and even about themselves as math learners. “These beliefs influence their thinking about, performance in, and attitudes toward mathematics and decisions related to studying mathematics in later years” (p. 98) According to the NCTM, the quality of mathematical education in early childhood plays an important part in the child’s understanding of possibly difficult math concepts (Loop, 2009). Although there have been many studies pertaining to patterns of mathematical teaching in preschool, there is a lack of research on how teachers of mathematics perceive how young children learn math concepts and even what to teach in accordance to the developmental level of young children. That is why enhancing early childhood education teachers’ confidence and competence in teaching math is considered a route to improvement of mathematics instruction in preschool. Sarama & DiBiase suggests that early childhood teacher training focus both on mathematics content and teachers’ dispositions towards math, reflecting the need for highly qualified professionals (2004). The National Association for the Education of Young Children (NAEYC) Code of Ethics states: “In decisions concerning children’s programs, we shall draw upon the education, experience, and expertise of staff members” (Decker, Decker, Freeman, & Knorpf, 2009, p.359). This principle pertains to the ideal of professional training and development of early childhood educators, “To further the professional development of the field of early childhood care and education and to strengthen its commitment to realizing its core values” (Decker et al, 2009, p.361). Early childhood educators need training for all the tasks expected of them when they begin teach young children. Being an effective teacher encompasses a wide spectrum of responsibilities – from designing an environment conducive to learning, to planning appropriate lessons for young children and implementing them with effective educational strategies, to being able to manage the class well and instilling discipline in the children, to involving parents and coordinating with others regarding the provision of quality education for the children. It takes effort to be able to manage all these skills at the same time, hence professional training with experience is required. The current study has taken inspiration and ideas from the work of Platas (2008) entitled “Measuring Teachers’ Knowledge of Early Mathematical Development and their Beliefs about Mathematics Teaching and Learning in the Preschool Classroom”. Platas was able to measure the knowledge of early mathematical development and beliefs about mathematics teaching and learning in the preschool classroom of preschool teachers through the use of developed and validated survey instruments and statistical analysis (2008). The study found a significant variation in the knowledge and beliefs of early childhood preschool teachers on age-appropriateness of math instruction, classroom locus of generation of mathematical knowledge, math versus socio-emotional development and the comfort level in providing the instruction. The instrument used in the research has been found to be helpful for the preparation and the professional development of current pre-service teachers and in-service preschool teachers (Rosenfeld, ). Platas’ work has raised many issues regarding the developmental knowledge of preschoolers. Such issues include age-appropriateness of mathematical instruction used by teachers in preschools, the extent of learning of such mathematical concepts taught, the purpose of teaching math concepts in preschool and the comfort level of teachers when they introduce such math concepts to preschool children (Clement & Sarama, 2007; Ginsburg & Golbeck, 2004; Platas, 2008). Issues of Teachers Teaching Math in Preschool The National Council of Teachers of Mathematics (2000) identified high quality mathematics programs for early childhood as having the following characteristics: (Brewer, 2001, pp. 319-320) “They build upon and extend children’s intuitive and informal mathematics knowledge” “They are grounded in knowledge of child development” “They provide environments that encourage children to be active learners, eager for new challenges “They develop a strong conceptual framework that provides the foundation for skills acquisition” They nurture and develop children’s inclination to solve problems”. In response to these criteria, many studies have been done to evaluate if early childhood educators are compliant to it. Among these studies, Wilson researched that he surveyed administrators and teacher leaders of their perceptions regarding their knowledge, skills, and confidence in the area of early mathematics. The result of this study revealed that both teacher leaders and non-teacher leaders have a need for professional development in early childhood mathematics (2009). Furthermore, Wilson found that traditionally, teaching of mathematics is associated with computation, memorized rules and rote procedures followed even without understanding the underlying concept. Recently, educational reforms in early childhood educators have emphasized mathematical reasoning, solving problems, connecting mathematical ideas and communicating mathematics to others (Wilson, 2009). Many researchers have studied young children’s mathematical development and have come to understand the developmental progression of children’s mathematical knowledge (Baroody, A. J., Lai, M.-L., & Mix, K.S, 2006; Clement & Sarama 2007; Seo & Ginsburg, 2004). In order to support early mathematical development in young children, researchers recommend that teachers (a) develop a deep understanding of the mathematical content and concepts (Baroody, et al., 2006); (b) develop an understanding of young children’s mathematical development, including the ability to take into consideration the prior knowledge of the child (Baroody et al., 2006; Clement, 2001& Clement & Sarama 2007); and (c) be fluent in the classroom pedagogy that supports and fosters such development including a knowledge of useful representations and strategies as well as common misconceptions and errors (Baroody, et al., 2006; Seo & Ginsburg, 2004). By increasing teachers’ knowledge and understanding of mathematical knowledge development, it is expected that children will also learn better and a more productive classroom atmosphere will be created (Akinsola, 2009). Therefore, it is important for teachers to know how to support children’s mathematical development in the preschool classroom in order to develop effective and appropriate classroom instruction. Preschool teachers also need to understand how children learn mathematical knowledge as well as assessing their understanding, and monitoring their progress (Klein, et al., 2005). Findings in Platas’ previous research indicated that preschool teachers need to have a specific level of competency in their teaching, which can be measured through assessment of their knowledge and beliefs (2008). It was found that kindergarten teachers’ different training experiences and teaching experiences have significant impact on children’s readiness for school (Lin, Lawrence, & Gorrell, 2008). This implies that the level of knowledge and beliefs of preschool teachers have been significantly influenced by their training experiences and level of education received. Preschool teachers have different views on early childhood education with some of them thinking the programs in their schools may either be too advanced or too basic (Hughes & Kwok, 2007). In addition, teachers’ preparation for cultural diversity in preschool is important prior to creating a welcoming learning environment (Hughes, & Kwok, 2007). Skills and knowledge of teachers are important considerations especially on the basic education level. Teachers’ gestures and utterances were found to have a significant role in children’s learning (Klerfelt, 2007, Valenzeno, Alibali, & Klatzky, 2003). In preschool education, high quality teacher-child relationships are expected. Knowing how to establish such good relationships with young children implies that the preschool teacher must have substantial knowledge on relationship building in early childhood education programs (O’Connor, 2010). This also suggests that the teacher is effective and knows how to infuse fun in learning. Preschool teachers must have adequate knowledge and strong beliefs in a particular subject area they are tasked to teach (Lee & Ginsburg, 2007). They should have knowledge of useful representations and strategies and common misconception errors when it comes to mathematical development in young children in particular (Baroody et al., 2006). A development model that coincides with theory and research is needed to create the building blocks for early childhood mathematics (Sarama, & Clements, 2004). A teacher’s math belief is hard to measure but it proves a reliable measure of teacher’s experiences and actions (Muis, Bendixen, & Haerle, 2006). For example, a teacher may believe that young children are capable of understanding quantities because she herself has been successful in teaching preschool children the concept of quantities, of which set has more objects and which set has less. Education and specialized training of preschool teachers is associated with young children’s learning and development (Barnett, 2004). Evaluation of the impact of Early Childhood Education teacher preparation programs and professional development conclude that benefits of high-quality preschool education can more readily be achieved if teachers are professionally prepared and well-compensated (Barnett, 2004). Research shows that a well-prepared early childhood education teacher brings a warm, safe and conducive environment for learning (NAEYC, 2006). In addition, it is important to focus on the quality of classroom activities, child characteristics, teachers’ perceptions of school-related climate and teachers’ perceptions of workload stress (Mantzicopoulos, 2005). It was also found that the amount of a teacher’s math related teaching was highly associated to children’s mathematical knowledge (Klibanoff, Levine, Huttenlocher, Vasilyeva, & Hedges, 2006). All of these issues can essentially be addressed by studying further within the context, knowledge and ability of early educators for early childhood education. SIGNIFICANCE OF THE STUDY This study addresses on Platas’ (2008) suggestion that more research should be done in terms of investigating preschool teachers’ knowledge and beliefs regarding the developmental mathematical knowledge of preschool children. In compliance with the standards of NCTM for high quality math programs, evaluation of teachers’ competencies should be done to ensure the quality of teaching in early childhood programs. This study’s significance lies in ensuring that teachers are adequately skilled and emotionally ready to teach mathematical concepts in the preschool level (Akinsola, 2009). In addition, this study is a contribution to the growing body of knowledge of early childhood mathematics education. Furthermore, new information may be derived from concern ed participants in this research such as fresh ideas coming from the perspective of in-service andpre-service teachers that could eventually be helpful in improving performance in the teaching of mathematics to preschool children. Hence, my study attempts to measure the validity and reliability of the comparisons between the cohorts (cohorts 2 and 3 are all in-service and cohort 1 is pre-service). The importance of assessing this trend of thinking may highly benefit teachers in early childhood education. In particular, there is a need to publish studies such as this in order to provide support and enhancement of skills to new teachers in the field. It is important to study perceptions of early childhood educators because they play a significant role in young children’s learning and development of knowledge and skills (Chien, & Hui, 2010). This study will help not only teachers to better understand the learning process involved in order to maximize mathematical knowledge development of young children. It is also helpful in building a better mathematical foundation for young children. STAEMENT OF PROBLEM The NAEYC Code of Ethics offers guidelines to people who have chosen the path of working with very young children. The core values, ideals and principles consider universal concerns regarding issues on childcare, development and education and advocate for its understanding, acknowledgement and applicability (2005). It encompasses all cultures, religions, abilities of children, family dynamics, economic classes, etc. to deliver a paradigm that is meaningful, relevant and developmentally appropriate for all. It seeks to help young children develop to their full capacity with the help of caring, efficient, reliable and adept adult workers. Many national and state organizations have provided teachers with guidelines to help improve mathematics instruction for young children. Forty-six states have comprehensive learning standards for preschool children (Barett et al., 2008). Furthermore, national organizations such as the National Association for the Education of Young Children (NAEYC) and National Council of Teachers of Mathematics (NCTM) have developed specific recommendations related to high quality mathematics instruction in the early childhood classroom (NAEYC & NCTM, 2006). As the NCTM and NAEYC proclaimed, importance in the learning and development of mathematical skills and knowledge is important in building the foundation of mathematical development of preschool children. However, some early learning programs do not focus on high quality mathematics instruction despite research supporting early mathematics experiences influencing mathematical outcomes later in school and promoting school readiness skills in mathematics (Lopez, Gallimore, Garnier, & Reese, 2007; Slaby, Loucks, & Stelwagon, 2005). Other studies showed that there are many activities involved in teaching mathematics to young children involving knowledge about the subject matter, pedagogical content knowledge, introduction of symbolism, and connecting everyday experience to abstract ideas (Ginsburg & Amit, 2008). This would imply the need to effectively train early childhood teachers of early mathematics. There are many significant studies in early childhood education but little information is revealed about the teaching of early mathematics (Ginsburg & Amit, 2008). This gap in research in mathematics education calls for the need to be filled especially in the area of teachers’ knowledge development and beliefs in teaching math in preschool. The most remarkable research in the area of teachers’ knowledge on preschool mathematical development and their beliefs in teaching mathematics is that of Platas (2008). Platas suggested in her recommendations, further study is still needed in the area taking into consideration a larger sample size and other necessary adjustments (2008). This study is also concerned with the commitment of teachers in teaching mathematics to young children (O’Connor, 2010). It is important to investigate whether teachers know the value of teaching math concepts to preschoolers at a very young age, which math concepts to teach and the value of teaching such concepts. It would also be interesting to know how comfortable these teachers are in teaching mathematics and what their attitudes and insights are towards it. The need for preschool children to be provided with high quality early childhood programs calls for competent educators in general. Specific to this study are the qualities necessary to be effective as preschool teachers concerned with the development of mathematical ability of preschoolers (Rosenfeld, 2010). It is in this regard that early childhood educators in mathematics should create the basic foundation required to be successful in the education system (Rosenfeld, 2010). This can significantly create meaningful understanding on how pre-service teachers and in-service teachers in early childhood education for mathematics should handle and evaluate their teaching abilities and methods. However, it remains uncertain whether pre-service teachers and in-service teachers in early childhood education for mathematics have similar knowledge of mathematical development and belief (Jenkins, 2001). Thus, it is important to evaluate their knowledge of early mathematical development and beliefs about it so as to come up with a clear justification if there is a need to align their level of skills, knowledge and beliefs if necessary. Even though Platas (2008) was able to compare early childhood teachers based on their knowledge of early mathematical development and beliefs about mathematics teaching and learning in the preschool classroom, there was no comparison done between pre-service and in-service preschool teachers in this regard. The researcher believes that this research gap needs to be filled since future and current teachers’ knowledge and beliefs will inevitably create an impact in young children’s mathematical knowledge development. The problem focused on in this study is the investigation of differences in knowledge of early mathematical development as well as beliefs in math teaching of pre-service and in-service teachers. Since it is known that teachers’ knowledge and beliefs about early mathematical learning and development significantly affects their quality of teaching, it will be important to determine if the experience of teaching has an effect on in-service teachers’ knowledge and beliefs concerning mathematical knowledge development or if they remain the same as their pre-service counterparts. Hence this study will analyze and review in-service teachers’ and pre-service preschool teachers’ understanding on mathematical knowledge development and beliefs. PURPOSE OF THE STUDY The purpose of this study is to examine and compare the knowledge of early mathematical development and beliefs in math teaching for preschool children between in-service and pre-service teachers. It aims to determine whether teaching experience will have a differentiating effect on such knowledge and beliefs. The comparison will gauge the need to enhance the teachers’ knowledge of early mathematical development and beliefs in math teaching especially in early childhood education. Recommendations for further development of these teachers’ adeptness in mathematical knowledge as well as fine tuning their beliefs will be more confidently expressed upon the completion of this study. RESEARCH QUESTIONS The researcher will compare and evaluate in-service teachers and pre-service preschool teachers’ knowledge and beliefs of mathematical development and teaching. In order to find significant results for this, the researcher will answer the main research question at of the study: To what extent, if any, is there a difference between in-service teachers and pre- service preschool teachers with regard to their knowledge of early mathematical development and beliefs of how to teach mathematics to preschool children? Specifically, the following sub-questions will be answered by the study: (1) To what extent, if any, do teachers know which teaching activities/materials to incorporate into their teaching of mathematics? (2) To what extent, if any, do pre-school mathematics teachers understand the attitude of their students to a subject like mathematics? (3) To what extent, if any, do teachers understand that learning mathematics is not an isolated activity for preschoolers but also part of their social and emotional development? (4) To what extent, if any, do teachers take into consideration the pre-knowledge about numbers and counting that preschoolers have before starting a mathematics lesson? (5) To what extent, if any, do the preschool mathematics teachers understand their own level of understanding of the subject’s contents as well as their personal limitations? HYPOTHESIS It is hypothesized that there are significant differences between the beliefs of in-service and pre-service preschool teachers on the basis of experience. Those who have experiences in teaching mathematics will have a different perspective based on their own experiences compared to those who have limited or no experience in teaching mathematics to preschool children. It is also hypothesized that there are significant differences in the knowledge of mathematical development between in-service and pre-service preschool teachers. OVERVIEW OF METHODOLOGY Instruments to be used for this study are the instruments developed by Platas (2008) namely the Knowledge of Mathematical Development (KMD) Survey and the Beliefs survey. Demographic questionnaire gathering information about the respondents will also be distributed. The Knowledge of Mathematical Development Survey (Platas) is a set of 20 multiple-choice questions that tests teachers’ knowledge and development in the area of verbal counting sequence, counting, ordinal number of words, addition/subtraction, divisions of sets, written number symbols and words. In each number, the teacher-respondents are to choose which of two math skills preschool children are likely to learn first. If the respondents think that both choices are of equal difficulty, then the choice of “same” may be picked. However if the respondent does not know which skill is easier for a child to learn, he or she can choose the option “do not know”. The Beliefs Survey is a set of 40 beliefs statements about Math teaching and learning. It has a 6-pt. Likert scale with responses that range from Strongly Agree to Strongly Disagree. The belief survey will evaluate the teacher’s perspective in four areas: (1) age appropriateness of mathematics instruction in the early childhood classroom, (2) locus of Generation of Mathematical Knowledge, (3) Social and Emotional vs. Mathematical Development as Primary Goal of Preschool and (4) Teacher comfort in mathematics instruction. The participants involved in the study will be comprised of 60 in-service teachers and 60 pre-service teachers that will be chosen at random. Results will be analyzed using ANOVA as well as correlation analysis. The pre-service participants will be chosen using the stratified purposeful sampling, while the in-service participants will comprise of preschool teachers who have had up to 5 years of experience. Hence, the third group will be made of teachers with 5 or more years of experience. LIMITATIONS OF THE STUDY This study will be limited to exploring the knowledge development and beliefs of pre-service and in-service preschool teachers regarding math. It will not evaluate any other subject area. It is important to note that in terms of experience, the pre-service teachers will have less teaching experience whereas with the in-service teachers’ experience may range from a few months to about 20 years, depending on the sample gathered. Also, the number of in-service participants is much higher than the pre-service participants. These factors may affect the results due to this unbalance and must be taken into account when data is analyzed. DEFINITION OF TERMS: For the purpose of this study, the following definitions were operationally defined as follows. Knowledge of early mathematical development - refers to a teacher’s adeptness and understanding of “the increasingly complex mathematical constructions and goals that young children develop and pursue in their activities” (Platas, 2008, p.3). Incidentally, it is possible to define early mathematical development as the increasingly complicated mathematical constructions and objectives that young children develop and long after in their activities (Saxe, Guberman & Gearhart, 1987). Therefore, KMD survey measures the students’ skills of counting, addition, subtraction etc. Math Belief – what teachers may think as “should and should not be taught, how children will learn the mathematical concepts best and how they will teach each of the objectives” (Jenkins, 2002, p. 2). However, since it is possible for teachers not to be aware of their own beliefs and level of understanding in teaching mathematics at the preschools, it is imperative to discover their quality of teaching by asking questions that get at their construct as far as mathematics is concerned. Invariably, this requires that the teachers’ prior knowledge should be analyzed and their suitability for the job determined beforehand through training and other skill-testing measures (NAEYC & NCTM, 2002, p. 2). Pre-service preschool teachers – aspiring teachers of early childhood education who are qualified to teach but have not yet gained any teaching experience. In-service preschool teachers – teachers who are currently teaching in the preschool level of education. Summary of Chapter 1 The realization that very young children today are capable of early mathematical learning has prompted educators to push for reforms in the early childhood curriculum. This situation has encouraged professional organizations such as NAEYC and NCTM to provide with guidelines that preschool teachers should consult and comply with in their teaching of mathematical concepts and skills (NAEYC & NCTM. (2002). Preschool teachers in early childhood education should basically possess adequate knowledge and skills as well as appropriate beliefs if quality teaching is to be expected. The work of Platas (2008) entitled ““Measuring Teachers’ Knowledge of Early Mathematical Development and their Beliefs about Mathematics Teaching and Learning in the Preschool Classroom” has been a monumental inspiration for this current study. However, this study goes beyond exploring teachers’ knowledge development and beliefs about by comparing knowledge of early mathematical development and beliefs in teaching math of in-service preschool teachers and pre-service preschool teachers. It is presumed that the element of experience will play a significant role in either strengthening or modifying teachers’ knowledge and beliefs about Math in early childhood education (Rosenfeld, 2010, Jenkins, 2001). The significance of the study will be extracted from the quality of knowledge and beliefs held by both pre-service and in-service teachers regarding as it has great implications in the quality of education they will deliver to the young children they will teach. This study also has implications in the improvement of teacher training of early childhood educators as well as the development of mathematical knowledge of pre-service teachers to ensure the high quality of education children rightfully deserve. Chapter 2 Literature Review It is imperative to investigate the differences in ideologies and beliefs as seen in in-service and pre-service preschool mathematics teachers. (A) OVERVIEW OF THE CURRENT PRESCHOOL MATHEMATICS TEACHING The current preschool mathematics teaching follows the status quo whereby teachers are expected to introduce mathematics to the children without being aware or mindful of other factors that could enhance their understanding in the subject (Jennifer &Loyola University, 2008 p.11; Bennett, 2010). This indicates that preschool teachers do not necessarily put into consideration that (i) the pre-knowledge of some mathematical contents like numbers , mathematical signs and sizes etc and (ii) their own (teachers’) levels of understanding of the mathematical contents can definitely affect the quality of understanding in preschool mathematics learning (Jennifer & Loyola University, 2008, p.11). In other words, preschool mathematics teacher do not carry out pre-assessment of the children’s ability as well as that of their own before engaging the young children in the subject. Hence, this approach has been discovered to be inefficient as well as time-consuming. It has not added any value to the issue of teaching mathematics at the preschool age. Preschool teachers have their own beliefs about teaching mathematics to young children (Platas, 2008; Barnett, 2010). And children learn in progression: that is, as they grow up they are able to use their intuition to identify numbers, shapes and undertake simple counting and addition exercises. Hence, preschool teachers are expected to use the knowledge of children’s learning progression to design teaching strategies that would make the young learners like mathematics (Raver et al., 2008; Scott-Little et al., 2006; Farver et al., 2006). But, in reality, preschool teachers are still facing difficulties in designing the appropriate mathematics lesson for their young learners. This problem has been left to persist because of two probable reasons: (i) the preschool teachers are not getting enough training or education during their development periods. Therefore, they lack the appropriate vigor or emotional strength to help the children to understand mathematics as much as they could; (ii) it is also possible that the preschool management may have put in place an old curriculum that do not accommodate modern flexibility in teaching preschoolers (Jennifer & Loyola University, 2008; Burger, 2010; Chien, 2010). As a result of these observations, the preschool mathematics teachers (whether in-service and pre-service) do not have the appropriate knowledge to impart the right mathematical education into the children they teach. This status quo is the reason why many preschoolers do not show strong interest in mathematics at this age and beyond. To correct this major mistake in preschool education, the school management should undergo strategic reforms in both the contents of what are taught to the young children and the methodology used in teaching mathematics at the preschool level. (B) EARLY CHILDHOOD MATHEMATICS IN THE EDUCATIONAL REFORMS Ernest et al. (2009, p. 376) believe that the background of education reforms with respect to early childhood mathematics lies in reforming the two main components of mathematics education in any preschool: which are the curriculum and the teachers’ levels of understanding. Some preschools utilize curriculum that are old and inflexible—this kind of curriculum should be discarded altogether or re-modeled to accommodate the different degrees of understanding among the children. This indicates that every child in a preschool has his or her way of absorbing mathematical knowledge; so, teaching them with a single approach or using a rigid curriculum may be counter-productive (Ernest et al., 2009; Connor et al., 2005). Similarly, preschool mathematics teachers need more training and orientation that would equip them with the right knowledge required to teach mathematics to the preschoolers in a productive manner (Ernest et al., 2009). As described above, any preschool establishment that aims at revolutionizing the procedures for teaching mathematics must decisively tackle the weakness in its current curriculum and encourage teachers’ development. These two approaches are non-negotiable for success and efficient teaching of mathematics at the preschool level. Past reforms in this regard had failed because special attention was not placed on the fact that teachers’ knowledge in a subject determines the quality of their contribution to the understanding of the subject (Cordes $ Gelman, 2005; Decker et al., 2009). Spodek et al. (2006) hinted about the preparedness of children to learn new things in an entirely new environment. This entails that when the method of teaching preschool mathematics is reformed, children can adjust to the new technique as quickly as possible (Spodek et al., 2006; Greenes et al., 2004; Hughes et al., 2005). Young learners should be taught to perceive mathematics as a storytelling exercise whereby the children are allowed to make inferences, based on their ages, to connect numbers, identifying simple and subtraction operations and make right conclusion about what they had learnt (Casey et al., 2004). (C) KNOWLEDGE OF MATHEMATICAL DEVELOPMENT Platas (2008) also reiterated that early mathematical development is an important step in acquisition of mathematical knowledge by preschoolers, and that it can be defined as the increasingly complicated mathematical constructions and objectives that young children develop and work towards in their activities. Hence, KMD Survey is useful in measuring the skills of counting, addition, subtraction etc in these children. And it is a great instrument for both pre-service and in-service teachers to understand the extent of knowledge possessed by the children. In this case, they can design activities that would perfectly suit the children’s learning outcomes. Basically, the two major mathematical operations carried out at the preschool level are number and operations. Under these broad topics, children learn about geometry, algebra, measurement, probability and in spatial thinking (Clements & Samara, 2004; Janus & Brickman, 2010). When designing the KMD Survey, two parameters are strictly considered: (i) the items included in the Survey should show the current established and researched knowledge on the mathematical development in young children; (ii) and the items should show activities that can be done or existing in young children’s environment (Platas, 2008). Therefore, the sub-domains of the Knowledge of Mathematical Development Survey include verbal counting sequence, counting/numerology, ordinal number words, addition/subtraction, division of sets and written number symbols and words (Platas, 2008). It is generally believed that small children progressively learn about the differences in numbers, and this may become their basis for further expansion of their knowledge on number and operations (Platas, 2008). As a result of this, five months infants can differentiate sets of 2 and 3 items; and by the time they are one and a half years old, they can identify which of the two sets is greater. And by age 3, they should be able to do quick enumeration without actually counting between 1 to 4 sets of items. By age 4, children would have developed the skills of addition and subtraction of simple numbers and operations. And division problems can be successfully handled at age 5 (Platas, 2008). However, since children do not normally get exposed to ordinal numbers (second, third, fourth etc) they always find it difficult to resolve any mathematical problem related to this set of numbers. The explanations given above reveal how children learn mathematics in progression; and it also reveals the significance of letting children be exposed to these kinds of activities at the right time as they grow up (National Research Council, 2009; Clements & Samara, 2009). More explanations about these activities are given below: (a) Verbal counting system—this is the process of reciting the verbal counting sequence with the hope of giving the verbal expression of the lexical form of the numbers. Children are likely going to build their counting knowledge on this process (Platas, 2008; Baroody et al., 2006). (b) Counting/numerology—this is simply the ability to recite numbers through rote learning. Preschool children do not need to memorize or theorize what they count before understanding what each number stands for (Platas, 2008; Ginsburg et al., 2006). (c) Ordinal numbers that explain positions—ordinal numbers like second, third, fourth etc describe the position of the item under consideration. Hence, preschool children learn about this through physical identification (Platas, 2008; Bruce & Threlfall, 2004). (d) Addition and subtraction—this deals with the simple mathematical process of adding and subtracting numbers. Preschool children easily understand how to do this in their learning progression (Platas, 2008; Fayol & Seron, 2005). (e) Division of Sets(fair/equal sharing)—Preschool children normally acquire the knowledge of division from home where they are engaged in dividing items among their siblings ((Baroody et al., 2006; Fayol & Seron, 2005). (f) Written Number or Word Recognition—At this age, preschool children also naturally try to identify a certain number, say “3” and a “word” out of other numbers and words on display (Platas, 2008). (D) TEACHERS’ BELIEFS ABOUT MATHEMATICS TEACHING AND LEARNING Past studies or researches have pinpointed that both in-service and pre-service preschool mathematics teachers have their own personal beliefs when it comes to the issue of teaching the young children. This indicates that as a result of their beliefs, they are prevented from seeing the true situation of the young children they teach. If teachers are given beliefs survey, it is possible to discover their individual thoughts and expectations about teaching mathematics to preschoolers. However, most preschool teachers (in-service or pre-service) always harbor these kinds of beliefs in their minds: (i) Age appropriateness—they begin to wonder about which age is the most appropriate for what mathematical activities; (ii) there are some misunderstanding about the locus of mathematical knowledge generation; whether kids normally construct their mathematical knowledge from their play or they need to rely on their teachers to tell them exactly what to do? (iii) some teachers also wonder how to separate the goal of acquiring mathematical knowledge from the social and emotional development of the children; (iv) teachers also worry about their suitability for the job—some thought that they have acquired little or no mathematical knowledge that would qualify them to teach the subject; others worry about the outcomes of not feeling comfortable about the subject in the classroom (Platas, 2008; Ginsburg et al., 2006; Sarama et al., 2004). In order to obtain a useful beliefs survey, some important items must be considered for designing the survey: (i) it should be a research on teacher beliefs that have direct link with the approach of teaching mathematics at preschool; (ii) it should contain actual teachers’ statements and concerns as seen in the anecdotal reports and studies; (iii) and, it should contain real-time activities that had been reported or observed in a preschool environment (Platas, 2008). Due to their years of experiences, more in-service preschool mathematics teachers seem to have a positive outlook on the performance of their young learners. They are able to spot children’s learning abilities based on their level of exposure to mathematics. Hence, some preschool teachers believe that the locus of generation of mathematical knowledge rests with the children, and that teachers are only acting as motivators with appropriate mathematics lessons and activities (Ginsburg et al., 2004; Sarama et al., 2004). (E) TRAINING AND EDUCATION PROGRAMS FOR PRESCHOOL TEACHERS Past studies have linked preschool teachers’ quality of teaching to the quality of education they had received from their universities or training programs (New & Cochran, 2008). Formal educational training is essential to counter the old beliefs pre-service and in-service preschool mathematics teachers have about the subject. In Europe, educators believe that teachers should have a second degree in Education in order to acquire the minimum teaching knowledge that would benefit the educational requirements of their young learners (Dyrfjoro, 2008). The importance of education or training programs can never be over-emphasized: the field of early childhood education is a viable and dynamic one that requires constant modifications on the parts of the teachers and their young learners (Klein et al., 2005; Klibanoff et al., 2006). Invariably, the educational requirements of children in this modern age are quite diverse and more demanding than children that were born centuries ago. As a result of this, preschool teachers in the modern dispensation need practical and useful education or training programs to be able to handle the challenges from the children of nowadays (Kelley & Camilli, 2007). Therefore, teacher education is an important step in achieving success in preschool mathematics teaching because teachers would be exposed to the best practices available in the profession. And they would learn how to approach their job with absolute professionalism, putting aside their personal beliefs and prejudices (Kelley & Camilli, 2007; Dyrfjoro, 2008). Chapter Three Methodology (A) VALIDITY AND RELIABILITY OF THE INSTRUMENT TO BE USED For any research to be successful, the survey instruments used must show a great deal of validity and reliability. Validity requires that the scores\outcomes of the experiment or research must be commensurable with the real-time situations or construct/content it is designed to measure. Hence, it is very important that researchers pay serious attention to validity of their studies so as to develop and evaluate test results that demonstrate high-level reliability (Platas, 2008; Lovelace et al., 2010; NAEYC, 2006). Similarly, reliability of the test results or scores across borders and situations is needed in order to make the research outcomes a consistent one that other researchers could confidently believe in and make references to (Platas, 2008). In every study, attempts must be made to ascertain the validity and reliability of the experiment outcomes in order to test its applicability to the construct/contents tested. I believe working according to the laid-down standardization would help me achieve high level of validity and reliability in this research. And I duly hope that I would be mindful of the limitations of this study so as not to jeopardize the expected outcome of my research (Platas, 2008; Feiler, 2004). (B) THE INSTRUMENTS FOR THIS RESEARCH Two important surveys would be used to obtain the scores or results for this study: namely, the KMD Survey and Beliefs Survey. I will divide the phases of my experiment into four so that I can guarantee quality in the outcomes of my research: (i) Phase 1—a development phase, which involves making the appropriate preparation for the experiment; (ii) Pilot Study A—this is the first set of measurement of the variables and compilation of the results of the surveys; (ii) Pilot Study B—this is the second set of the experiment; (iv) Validation Study—this is the last step aimed at establishing the validity and reliability of the results or outcomes and show consistency in the measurement and computation of the variables indicated in the surveys. This last part is essential so as to detect the usability of the survey in discovering the real-time constructs/contents of preschool mathematics teaching (Platas, 2008; O’Connor, 2010). (C) STATISTICS The results or outcomes from the experiment will be tabulated and analyzed using simple statistical tools like averages, mean, deviation, graphs and charts. 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Unpublished Doctor of Education Dissertation, University of Houston, May, 2009 Read More