Enhancing Kindergartners' Mathematical Development: Effects of Peer-Assisted Learning Strategies - Essay Example

Critical evaluation of Enhancing Kindergartners' Mathematical Development: Effects of Peer-Assisted Learning Strategies Introduction The development of mathematics knowledge during the early stages of a child’s life is critical for any child to succeed at any stage of life. So, it is very important to understand the learning behavior of children during the first six years of life in order to understand and improve their learning techniques. Such methods would help in improving children who are disabled and are lacking basic mental qualities of a normal human being. (Pagani et. al. 2006) Considering all these, there is growing importance placed on understanding those stages of life. This report aims at critically analyzing one of such researches done on children’s mathematical development. In this process, views of external critics are taken into account and the choice of the method applied is critically analyzed along with its results and outcomes. The other aspect that is discussed in this report is the benefits that are obtained in implementing this method and the probable ethical issues that might arise if this is to be implemented in practical scenarios. Research problem The background for this research is based on the results obtained from the analysis done by Griffin and Case in 1997. (Griffin & Case, 1997) According to them, children, when they attain the age of six, should have developed a ‘mental number line representation’ by combining the two basic concepts of mathematics – counting and comparison. (Fuchs et. al. 2001) Once they achieve such a line, the required ground work gets established and with that as base, they can build on it to solve further problems in mathematics and understand the concepts better. But, many of the children who come from poor financial backgrounds and from utter poverty fail to achieve this line of understanding and hence wound up facing long term mathematical failures. The research under discussion aims to improve the students from high poverty background to achieve this mental line representation before they move on to higher studies. The research is actually an extrapolation of the research works done by the authors, Griffin and Case in the year, 1997. (Griffin & Case, 1997) Based on the three research exercises that they had done on a select number of students grouped under experimental and control group categories, they received high percentage of success, with many of the guided students who underwent this study were able to cope up with the middle level students, thus allowing them to compete equally and subsequently improve their mathematical knowledge. (Fuchs et. al. 2002) Research methodology Peer Assisted Learning Strategies of (PALS) is the method followed here to achieve the goal of this research. It is a method where the teachers deploy a program that allows them to identify and assist students who lack in specific set of areas. (Ashwin, 2003) Did their hypothesis/research problem correspond to the adopted collection and analysis of the research design? The aim of their research is to improve the understanding level of the students in mathematics. The purpose of the research problem is to extrapolate the previous results in three ways. 1. To expand the research area. That is, to involve more numbers of students, whose levels of understanding varies drastically. This is to present the teachers with details about how to proceed with the treatment in diversified classrooms. This dimension of the problem was addressed during the design phase of the research by identifying and including students with different performance levels and from both disabled and nondisabled sections. 2. To evaluate the capacity of the teachers. The need for this result was to identify the response from students, to explore the fidelity of their teaching method and to generate an informative feedback about the treatment. This dimension of the problem was addressed by choosing teachers who were from different demographical regions and teaching in naturalistic kindergartens. 3. To validate the results obtained through researcher developed tests against the standardized achievement tests. This is to identify a practical measure of the students’ understanding of global mathematics as compared to researcher oriented test results. In addition, the research design was conceptualized for a larger sample and the selection of teachers for classrooms was done in a random manner. The chosen research method, PALS, did not replace the existing curriculum of mathematics but ably complemented it by allowing the students to experiment on what they learnt from the teachers. (Pasnak et al. 2006) Hence, the choice of adopting PALS for this research is in correspondence with the research problem. What was their research design? Was it reliable and valid measure(s)? Considering the requirements of the research problem, their research design involved a large number of students, precisely 248, including those with disabilities and a group of 20 teachers. Initially, the students were examined by administering a mathematics achievement test and based on the results, the students were categorized as high, middle and low achieving students. The formulas that were designed to categorize them are given below: High-achieving (HA): 1.5(SD) > mean of z scores Middle-achieving (MA): 0.75(SD) > mean of z scores Low-achieving (LA): 1.5(SD) < mean of z scores where SD is the Standard Deviation Considering the fact that the difference between the mean of scores of the pretest for PALS and No-PALS was significantly small and, the test that was conducted was a proven and standardized test, clearly justifies the measures taken to categorize the students. (Fuchs et. al. 2001) The base treatment followed by the No-PALS teachers was based on the Grade K Basal math. (Fuchs et. al. 2001) Since the set of assumptions followed in this basal are standardized and form the basic curriculum of math, there are no deviations or experimentation related to this treatment. The initial pitfall of the PALS treatment would have been the duration of its course. If it had taken up additional slots, adding to the general mathematical curriculum, the treatment would have had serious shortcomings. It is important to highlight on the authors’ ability to manage the sessions within the curriculum and only for a period of 15 weeks, 2 days a week. The PALS treatment applied the binomial theory in creating pairs from the bifurcated groups of students. Another aspect of this treatment was that the pairing of students was constantly changed allowing a student to get exposed to different perspectives of the same concept from different pairs. One disadvantage of such strategy could be the mismatch of wavelength of students and the time taken for creating an understanding between the pairs, each time they are changed. But, this was clearly overcome as the end results show more than expected results for mathematical development. (Amanda et. al. 2006) Research method – a critique Measures and Data Collection Fidelity of treatment The treatment’s fidelity was measured keeping teachers as the unit of measure. It indicates that the fidelity of the students as well as of the teachers was measured based on the aspects of how a teacher responds and observes the reactions of the students. The parameters of this calculation were the number of features of PALS that were accomplished in each sitting of the treatment. As the features were divided into teacher and student components, fidelity was measured. An accurate prediction of fidelity is measured on how well the features are implemented. This was calculated by dividing the observed by the sum of the observed and non-observed. The fact that 90% agreement was achieved with the observed exemplifies the fidelity levels with respect to student and the teachers. (Arnold et. al. 2003) Student learning As mentioned before, the grouping of students were done based on the pretest conducted - the SESAT test. The statistics that were used to rate the mathematical skills and knowledge of the students was the Kuder-Richardson formula and the standard error of measurement. (Fuchs et. al. 2001) Since the above statistical formulas were proven over large number of students, it was expected to be more or less accurate for this treatment’s group selection. Although the PALS program was planned more or less similar to SESAT, there were a lot of discrepancies in results due to high performing students and hence an attempt at accurate analysis of those results would have been futile. The attempt to include a posttest based on Primary Stanford test as well as SESAT test balanced the results and allowed for a more detailed analysis of the results. Data analysis The authors have used descriptive statistics to describe fidelity, through the data collected in the treatment. Hence, it is just a basic explanation of the collected data. There are no detailed descriptions on the variability as the primary focus has been on gauging the improvement factor of the students. The “two way between-subjects analysis of variance” (Fuchs et. al. 2001) allows for grouping the students based on comparisons of their performance levels. As the students are made into pairs, each picked from two groups that are formed by dividing an initial arrangement of the whole batch in descending order of their scores, the compatibility is easily measured. ANOVA is run for both the students’ data as well as teachers’ data to evaluate their perceptions. (Fuchs et. al. 2001) Research results – a critique Fidelity of treatment The percentage of fidelity of teachers varied in the range from 77% to 100% with many teachers achieving the higher end of the range when compared to the few number of teachers who performed the components correctly. The fidelity of the students also varied on the same range but for one teacher for whom the student fidelity was 40%. Student learning The primary achievement in student learning is the achievement of the expected results in the ANOVA done on SESAT scores. The results clearly indicate that the grouping of the students were done correctly as the students grouped under low achievement performed less than the mid and high achieving students and similar expected results were obtained for mid and high achieving students. The results clearly prove the third expected result of this research design. On analyzing the results of the ANOVA tests run on both the pre and post test scores of the SESAT, the obtained mean and standard deviation of the scores of the PALS students was evidently higher than that of the contrasting students. The remaining statistics of the ANOVA run on the two SESAT tests indicated two critical points, proving the relevance and validity of the research undertaken. They are: Expected performances on the pretest from LA, MA and HA groups. Improved performance of the LA and DIS groups over MA and, MA over HA groups in the posttest. The HA groups also significantly bettered their performance than the pretest. Teacher Perceptions of Treatment Efficacy and Feasibility The teachers’ response for the question on the effectiveness of PALS on the three groups (LA, MA and HA) indicate that they were not entirely satisfied about the improvement shown in the LA group students. However, their responses were pretty high for MA and HA group students. This indicates that the research method succeeded in improving the mathematical knowledge of students who were performing better on average and indicates that there is still some scope for improving on the program implemented for PALS. On rating their own experience in implementing PALS, their ratings show that they felt highly comfortable and this indicates that the capabilities of the teachers can be easily groomed to implement PALS in school curriculum. Interpretation of the results Based on the results obtained about the fidelity of treatment from the teachers’ perspective, the authors state that the PALS program can be applied by real teachers in standard curriculum. Their interpretations are only partially correct as they do not offer any proof regarding the variations in the fidelity where one student scored 40% and two teachers, 77%. Regarding the question about PALS improving the learning levels, the authors have provided a positive answer. Their interpretations about the results prove them to be correct and the feedback from the teachers also indicate that they are ready to involve in such future programs. Piaget and Vygotsky interpretation of children’s mathematical development Piaget and Vygotsky are two renowned constructivists and played an important role in explaining about the children’s mathematical development. Both of them, although offered varying theories in regard to this component of psychology, concentrated on identifying the concepts that could help teachers to attend to the specific requirements that every child needs. They believed that a child’s cognitive development is based on two things: the way the inputs are fed to them and their societal needs. The social needs play a part here because most of the children’s learning is done through how they perceive their surroundings. According to Piaget, there are four stages involved in the development of the cognitive thinking within children. The four stages are spread across the first two years of the child. Vygotsky, on the other hand is of the opinion that there are no stages involved in a child’s mathematical development but it is dependent more on the aspect of a person’s thinking within themselves. In the end, both of them succeeded in providing their own ways for developing methods that could help a child foster in his/her mathematical development. (Geary & Brown, 1991) Ethical Issues An important ethical issue to be considered while implementing this research is about the role played by the teachers. Unless the teachers are specifically mentioned about their roles, there is every chance of an ethical dilemma to arise in their minds. The Mathematic teachers should undergo a specialized kind of training that deals with the counseling and psychological aspects of the human mind. It would enable them to handle the situation with ease as there is always mounting pressure with low performing students. (National Research Council 2009) Another issue is to clear the presumed beliefs that the teachers and the students might have about mathematics. Since the teachers and students, especially in the field of mathematics have their own assumptions about the things which they believe to be correct and at any time, support those views. It is important to clear the participants of any such assumption that may be against the approach taken up in PALS. Finally, the teachers should take up a sense of ethical responsibility to implement the program successfully. Since, they have volunteered to the program there is every possibility for the participant to lose interest as the program progresses. (Ponder et. al. 2009) Strengths The easy implementation procedure and affordability in terms of cost and expenditure is one of the biggest assets of this approach. From a student perspective, it gives an opportunity for them to get actively involved in completing tasks and even perform them successfully. (DfEE 1999) The potential of the students to learn mathematics is evidently improved. The available resources to provide instructions are greatly increased allowing for an increased level of motivation among the students. The liberal nature of the environment and its settings allows the disabled students to get used to the natural school curriculum and gives them an opportunity to feel in same league as normal students. Last but not the least, the program gives an opportunity for students lacking in mathematical abilities to come to terms with the expected level and prepares them to play an active role in any similar events in future. (Bridges et. al. 2003) Weaknesses to consider One of the issues that could deter the success of this method is the selection of students by teachers. Although the programme does not allow for any manual intervention during the control designs, the selective strategy is a pretty manual one and there is every possibility for a biased selection to take place when the teachers are allowed for selection of students. (Askew & Wiliam 1995) Another issue that could hamper the success is the selection of teachers. Since most of the teachers had volunteered themselves for this program, there is no evaluation criteria associated with selecting the teachers. Also, as they are the ones who implement the program and even their feedback plays a critical role, the results may not go well with those teachers who are not eager enough to involve in these programs. Hence, the research and its results lack a sense of ubiquitous nature in them. (Scliemann et. al. 1985) Conclusion The critical analysis of the PALS program done on children’s mathematical development has been detailed. The views of external critics have been taken into account and the proposed method is analyzed along with its results and outcomes. In addition, the benefits that are obtained in implementing this method and the probable ethical issues that might be faced if this program has to be implemented have also been listed. Works Cited Pagani, Linda et. al. “Does preschool enrichment of precursors to arithmetic influence intuitive knowledge of number in low income children?” Early Childhood Education Journal, 34(2), (2006): 133-146. Print Pasnak, Robert., et al. “Applying principles of development to help at-risk preschoolers develop numeracy”. The Journal of Psychology, 140(2), (2006):155-173. Print Amanda M, et. al. “Further development of measures of early math performance for preschoolers.” Journal of School Psychology, 44(6), (2006): 533-553. Print Arnold, David, et. al. “Accelerating math development in Head Start classrooms”. Journal of Educational Psychology, 94(4), (2003): 762-770.Print. National Research Council (U.S.). “Mathematics learning in early childhood: Paths toward excellence and equity.” (2009). Washington, DC: National Academies Press. Ponder, Bentley. Et. al. Evaluation of the pre-k summer readiness pilot program. (2009) Atlanta: Georgia State University, Andrew Young School of Policy Studies. Bridges et. al Early childhood education and school readiness: Conceptual models, constructs, and measures: Profiles of early childhood measures. (2003). Washington, DC: Child Trends. Askew, M. & Wiliam, D. Ofsted Reviews of Research: Recent research in mathematics education 5-16. (1995) London: HMSO Scliemann, et. al “mathematics in the streets and in schools.” British Journal of Developmental Psychology. 3, (1985) : 21-29. Print DfEE The National Numeracy Strategy: Framework for teaching mathematics from Reception to Year 6. (1999a) Sudbury: DFEE Publications Geary DC, & Brown SC. “Cognitive addition: Strategy choice and speed-of-processing differences in gifted, normal, and mathematically disabled children”. Developmental Psychology;27(3), (1991):398-406. Print Fuchs et. al. “Enhancing kindergarteners’ mathematical development: Effects of peer-assisted learning strategies”. Elementary School Journal,101(5), (2001):495-510. Fuchs et. al. “Enhancing first-grade children’s mathematical development with peer-assisted learning strategies. “ School Psychology Review;31(4), (2002):569-583 Ashwin, P. “Peer Support: Relations between the context, process and outcomes for students who are supported.” Instructional Science, 31, (2003a): 159-173. Griffin, S., & Case, R. “Re-thinking the primary school math curriculum: An approach based on cognitive science.” Issues in Education, 3, (1997a):1-49. Griffin, S., & Case, R. “Wrapping up: Using peer commentaries to enhance models of mathematics teaching and learning” Issues in Education, 3, (1997b):115-134. Read More