Enhancing Math Learning with Manipulatives

Chapter 1

1.0 Introduction

The main aim of this study is to determine the impact of concrete mathematics learning aids on the cognitive development and enthusiasm of pupils and teachers respectively during math leaning. Education is increasingly becoming a subject of interest of many researchers, many of them trying to understand how an ideal school learning environment can be created. According to Ediger (2013), an ideal school learning environment is one that students can achieve the best performance they can. However, failure to establish various elements within the learning environment that facilitate students’ achievement of the best they can not result in the desired outcomes (Ediger, 2013). Therefore, achieving the best results among pupils require the efforts of all education stakeholders and this is particularly relevant when we consider mathematics pedagogy (Kusuma & Sulistiawati, 2014). Developing effective mathematics skills among pupils is crucial to the functioning of today’s society (Tat & Bulut, 2010). Mathematics skills are not only important in the classroom but also the execution of daily life activities. However, whereas it is common knowledge that mathematics is important, many pupils still have poor math skills, and this necessitates a change in teaching methods – thus the growing interest in and popularity of concrete mathematics learning aids (also known as manipulatives). This study seeks to identify the effectiveness of manipulatives on pupils’ mathematics performance and teachers’ attitudes towards math teaching.

1.2 Background of the study

Because primary school learning lays pupil’s educational foundation, pupils’ experience of learning is crucial not only for developing fundamental skills but also in influencing positive motivators to various school subjects (Kinchin, Chadha, & Kokotailo, 2008). Hence, it is monumental pupils should enjoy the subject whenever they are taught to an extent that they not only understand the subject but also apply the subject in real-life situations (Denggao, et al., 2011). Against this backdrop, research by Piaget (1970), which developed the learning theory confirm the idea that children can easily learn mathematics and understand the concepts with the use of concrete objects. Believers in the learning theory argue that learning occurs through constructivism, whereby learners acquire knowledge through continuous self-construction (Beswick & Muir, 2013). Besides, according to o Discroll (2005), learning theory stipulates that through constructivism, learners can construct knowledge by making sense of their experiences. Thus, manipulatives can be applied based on the premise that pupils find it challenging to interpret abstract mathematical concepts through lectures and explanations (Golfashani, 2013). Instead, pupils need experiences with instruments and models to understand mathematical concepts. This study seeks to establish whether the instruments and models can not only improve pupil’s performance but also enhance teachers’ attitudes towards math pedagogy.

1.3 Guide to methodology

The study relied on a mixed research design consisting of qualitative and quantitative research approaches. With regards to qualitative research approaches, the study relied on survey questionnaires to seek data from teachers about their attitudes towards math pedagogy. Furthermore, the questionnaires will facilitate the gathering of information regarding teachers’ confidence levels of teaching mathematics. On the other hand, quantitative methods will be used to collect and analyze data on students’ mathematics performance pre and post introduction of manipulatives.

1.4 Dissertation layout

After this introductory chapter, the author will move to the literature review chapter, which contains a review of relevant literature that is useful in informing the study’s conceptual framework. After the literature review chapter, the author will move to the study methodology chapter, which entails the identification and justification of various research methods and approaches used in achieving the study’s objectives. The next chapter will be the findings section, will entail a details analysis and presentation of study findings. The next chapter will be the discussion chapter, which entails an in-depth discussion and interpretation of study findings concerning the study aims. The last chapter will be the conclusion stage, which will focus on concluding the study findings, with a specific focus on both the discussion chapter and the other preceding chapters.

1.5 Conclusion

This chapter has presented a justification of the study by arguing that from a theoretical standpoint, pupils are likely to effectively learn mathematical skills with the use of manipulatives and that it would be important to test the hypothesis that manipulatives can promote mathematics learning and improve teachers’ attitude towards mathematics pedagogy. The next chapter gives san expansive exploration of existing literature on manipulatives and their established impact on mathematics pedagogy. Through the literature review section, the researcher will identify the existing research gaps, what is already known and what is yet to be known regarding the use of manipulatives in mathematics pedagogy.

Chapter 2

2.0 Literature review

The use of manipulatives to teach mathematics in schools dates to several years ago. Many pieces of research (Burns & Hamm, 2011, Swan & Marshall, 2010, and Freer, 2006) have explored the use of manipulatives in mathematics teaching intending to help students have a better understanding of mathematics concepts. In many cases, teachers have been tasked by students to reveal the importance and use of mathematics in real life. One of the mean reasons why most students ask this question is that mathematics is a subject that consists of abstract concepts (Nabie, Raheem, Agbemaka, & Sabtiwu, 2016). Moreover, students at elementary levels of education might be interested in understanding how mathematics applies to real-life because at this stage, they find themselves in between formal operational and concrete operational stages as stipulated in Piaget’s theory of cognitive stages of development (1965). Typically, children in the concrete operational stage (age 7-11) rely on concrete experiences to develop generalizations and reasoning skills while those in the formal operational stage can make assumptions about concepts through their reasoning skills. Therefore, considering the features of mathematics as a subject and the developmental characteristics of children at this stage, it is important to investigate whether concrete learning materials can be effective in helping the students to effectively learn mathematics. The word manipulative learning can emerge from an old French term “manipüle” which means to “handle”. In the English language, to manipulate means to be in control of, manage or operate with one’s hands. Hence, a literature review of the concept of manipulatives is conducted, several definitions of manipulatives are found. For instance, according to Cope (2015), manipulatives can be used to denote concrete materials that facilitate pupil’s understanding of abstract concepts by concretizing them, helping them to understand the relationship between abstract mathematical concepts and the manipulatives to have a concrete experience of the concepts, thereby equipping them with long-term mastery of mathematical skills (Cass et al 2003). Manipulatives enable pupils to integrate their knowledge with their thought to have a better understanding of mathematical concepts (Fyfe, McNeil, Son, & Goldstone, 2014). In doing so, they facilitate the pupil’s communication with their mathematical thinking and elevate their understanding of the concepts to a higher cognitive level (Parhan, Sauri, Majid, & Nurihsan, 2014). They also improve the enjoyability of the teaching process by amusing both the pupils and teachers through active participation (Poon, Yeo, & Zanzali, 2012). Consequently, the pupils develop permanent learning because they have equal opportunities with their colleagues (Tarabuzan & Popa, 2015). In their research, Tunc et al (2011) observed that manipulatives are effective tools that teachers can use to enhance both the teaching process and learning because they enhance the students’ interpretation and conceptualization of the learning concepts. Nonetheless, manipulatives do not only enhance the pupil’s cognitive abilities but also improve the learner’s psychomotor skills (Bussi, Taimina, & Isoda, 2010). Particularly, manipulative is said to improve learners’ psychomotor skills by enhancing their hearing, touch and sight senses (Figueira-Sampaio, Santos, Carrijo, & Cardoso, 2013). However, according to Bussi, Taimina, & Isoda (2010), manipulative should not be perceived as a remedy for every lack of mathematical skills a student might lack. Instead, teachers are advised to make use of manipulatives based on their professional judgment and common sense (Furinghetti & Menghini, 2014). When used otherwise, manipulatives would only be used as a source of entertainment to the learners and not as a means of enhancing learning. That is why it is emphasized that before using manipulatives, teachers should first have a proper understanding of how it is applied in a teaching-learning process (Bozza, 2016). In the same regard, Fanta & Boubacar (2016) emphasizes that teachers should have proper information about when, how and where to use manipulatives. Furthermore, it is recommended by many studies that before applying manipulatives in the classroom environment, teachers should explicitly explain the objectives to learners and give them the necessary information during the learning process (Fanta & Boubacar, 2016). It is only when such information is given to learners that they will go beyond entertaining the learners in enhancing the learners’ learning process. Nevertheless, studies also emphasize that manipulative should be used continuously rather than periodically as this is the only way it can positively enhance the learner’s conceptualization (Zhang, Zhang, Stafford, & Zhang, 2013). In the same note, it is observed that continuous and long-term use of manipulatives facilitates effective learning by enabling the learners to assimilate the abstract mathematical concepts by observing models thereby improving the learners’ skill acquisition (White & Mitchelmore, 2003). Thus, for teachers to develop productive and effective use of manipulatives, they should be informed of the characteristics and components of those manipulatives within the learning environment, the steps and processes of preparing them, the advantages and limitations as well as the necessary skills needed to have the instruments clearly explained to the learners (Ediger, 2013). Such an appropriate application of manipulatives within the classroom environment provides leads to learners’ improved performance in class, improves their exam grade, raises their cognitive abilities, decreases their anxiety against mathematics, and helps them develop a positive attitude towards mathematics (Kusuma & Sulistiawati, 2014). However, little information is known about how manipulatives could help improve teachers’ attitudes towards mathematics pedagogy. To fill this research gap, the current study seeks to explore whether manipulatives can also improve teachers’ attitudes towards teaching mathematics.

Whereas several scholars mentioned above (Ediger, 2013, Kusuma & Sulistiawati, 2014, Fanta & Boubacar (2016) emphasize on the importance of using manipulatives, teachers are likely to employ traditional methods of teaching as a result of the biases they hold against manipulatives including the perception that they lack the skills for using manipulatives, or that manipulatives are time and money-consuming, that the time limits do not allow for the use of manipulatives, that manipulative can cause confusion within the classroom or that they do not help meet the teaching objectives (Tat & Bulut, 2010). All these excuses for not using manipulatives decrease the application of manipulatives especially in higher grades where students can understand abstract concepts as opposed to lower levels of learning where students require manipulatives more often (Kinchin, Chadha, & Kokotailo, 2008). To add, (Mutodi, 2014) asserts that teachers who lack adequate information and skills on how to use manipulatives are less likely to adopt their use in the classroom, regardless of the level of learning (Fyfe, McNeil, Son, & Goldstone, 2014). A further review of previous literature reveals that the use of manipulative as a tool for teaching mathematics is associated with positive results. Particularly, research in the 1970s and 1980s (Raphael & Wahlstrom, 1989, Suydam & Higgins, 1979 and Suydam & Higgins, 1979) seem to agree on the effectiveness of manipulatives in teaching mathematics. After observing a pattern of most studies agreeing to the effectiveness of manipulatives, research by Suydam and Higgins (1977) concluded that far more of earlier studies favor the use of manipulatives than the non-use of manipulative sin teaching mathematics. When this study conducted a more-detailed literature review, the acceptance and use of manipulatives in mathematics pedagogy seemed to have followed a historical pattern, which impacts the current study. Therefore, in the next few paragraphs, this study will give a chronological account of perspectives given by several pieces or research regarding the importance of manipulatives. In 1978, Friedman (1978) conducted a literature review research on 15 studies conducted elementary school teaching in the 1970s. While the literature review by Friedman (1978) is a bot outdated, its findings are still relevant and useful in the current study. For instance, the study found that while manipulatives were beneficial to young children, it conferred no significant benefits to older children. Similarly, during the same decade, Suydam and Higgins (1977) conducted a comprehensive review of 20 studies on activity-based learning in mathematics pedagogy among children in kindergarten to grade eight classes. While the study was similarly outdated, the results indicate that using manipulatives had more benefits than not using them at all in elementary mathematics teaching. In the 1980s, Praham (1983) conducted a study supporting the findings by Suydam and Higgins on the use of manipulatives. During the same decade, Sowell (1989) conducted a similar study albeit with a different perspective of evaluating students’ attitudes towards mathematics when manipulatives were used in the teaching process. The study found that students had a positive attitude towards mathematics when concrete materials were used by teachers how had a good understanding of how to use those materials. In another study by DeLoache et al (1997), the researchers supported the use of manipulatives by concluding that when concrete materials are used in teaching mathematics among young children, the children are more able to understand mathematical concepts that otherwise may remain a mystery.

The literature review also reveals that manipulatives can be used to teach mathematics among children of all ages and learning levels. In the literature review study by Suydham and Higgins (1977), the authors found that the use of manipulatives was important at both elementary and upper elementary school levels. Similar remarks were made by Driscoll (1983) whose study results revealed a positive impact of manipulative activity lessons at every grade level. Particularly, the author noted that when using in intermediate grades, learners develop better capabilities of acquiring new mathematical concepts and benefit from remedial help., especially those who are struggling. The application of manipulatives in mathematics pedagogy has evolved. In a study by Golfashani (2013), the author narrates how mathematics pedagogy has developed from the use of counters or beans to fraction circles, linking cubes and other forms of technology. With the increased use of manipulatives, mathematics has evolved from a set of rules to be followed to a way of problem-solving and critical thinking. Through many pieces of research, manipulatives have been proven to have a positive impact on the mathematics teaching process. For instance, Ralphael and Wahlstrom (1989) observed that using instructional aids, teachers were able to cover more topics and the aids were found to be significantly related to students’ better performance in abstract topics such as proportions, ratios, and percentages. In a study by Moyer (2001), the author observed that it is the teachers’ responsibility to provide a proper environment that consists of representatives to enhance the learner’s thinking. On the same note, Vinson (2001) asserted that the use of concrete materials is an effective way of ensuring that learners understand abstract mathematical concepts. Three years later, Moyer and Jones (2004) conducted a study among 10 female teachers from a summer institute, the study found that the benefits of using manipulative inspired teachers and enhanced their attitudes towards teaching mathematics. The study also found that using manipulative also aided student learning. Particularly, Moyer and Jones (2004) concluded that students perceived manipulative as one of the important tools within their classroom environment that they could spontaneously use to ease their learning process. More research is needed on the effectiveness and teacher knowledge of manipulatives.

In a more recent study, Marshall (2010) re-evaluated the use of manipulatives in schools by examining the different ways in which teachers use manipulatives to teach mathematics. Their results showed that systematic use of manipulatives had potential benefits on learners’ ability to understand mathematics. One year after the study by Marchall (2010), Burns & Hamm (2011) conducted another study on grade 3 & 4 students and concluded that manipulatives enhanced the learners’ ability to understand math concepts and achieve averagely better performance. In a more recent study by Björklund (2014), the authors found that manipulatives were an important tool for enhancing the understanding capabilities of children as young as one and two years old. Björklund (2014) further concluded that some types of manipulatives seem to encourage some occupations by emphasizing on certain learning objects than others. Interestingly, there are pieces of research that have found no correlation between the use of manipulatives and learners’ mathematical performance as a result of teachers’ prejudices against manipulatives (Drickey, 2006), and that manipulatives have no positive impact on mathematics learning (Sowel, 1989); leading to more studies declining to support the use of manipulatives in mathematics instruction (Winograd & Flores 1986). In one study conducted by Fennema (1972), the authors included 95 participants at the age of 7 to 8 years old. The participants were divided into two groups of concrete and symbolic learners. While the concrete group used a manipulative device to answer some mathematics questions, the symbolic group did not use any manipulative tool. Results showed that the pupils leaned better without the use of any manipulatives. Also, the authors noted that there were no indications that symbolic models were less effective than concrete models, leading to a conclusion that it was difficult to tell whether manipulatives had any benefits in mathematics teaching. Similar conclusions were made in another earlier study conducted by Eastman & Barnett (1979) although the population involved university students. Nonetheless, Eastman & Barnett (1979) conducted a study on 78 university students at elementary levels of their studies, of whom 36 were placed in the experimental group with manipulatives while the rest were placed on a control group with no manipulatives. From their results, the authors concluded that there was no evidence of the experiments group performing better than the control group either on group tests that involved the use of manipulatives or on paper tests that did not require any use of manipulatives. This study seemed to conclude that manipulatives did not have any impact on students’ mathematics performance.

Seemingly, researchers hold different views concerning the benefits of manipulatives as a tool in mathematics learning. While one group of researchers claim that manipulative scan improves learners’ ability to understand mathematics, another group of researchers argue that this is not the case. Thus, the question of whether manipulative can help improve mathematics learning among children remains largely unanswered. The existence of conflicting study results on the use of manipulatives and their benefits to mathematics learning increases the uncertainty of whether manipulatives are effective in enhancing learner’s performance in mathematics. It is, therefore, nonocclusive to say that if teachers use manipulatives during mathematics lessons, their students will have improved performance in mathematics. Therefore, the purpose of this study is to investigate the impact of manipulative son learners’ mathematics performance. The study will, therefore, answer the research question: what is the impact of manipulatives on learners’ performance in mathematics?

Chapter 3

3.0 Research Methods

This chapter outline the research methods used in achieving the research objectives. It seeks to identify and justify the various research approaches and methods applied to facilitate the achievement of research objectives. Thus, in this chapter, the author mentions various components of research methods including research paradigm, research design, research instruments, population sampling methods and data analysis methods used within the study and why they were selected for use.

3.1 Research paradigm

Research paradigm refers to the researchers’ belief of how data should be gathered, interpreted and used (Roberts & Povee, 2014). The choice of research philosophy depends on the kind research design (i.e. qualitative, quantitative or mixed research design) as this determines the method of data analysis and interpretation used in the study. For qualitative research design, researchers rely mostly on interpretivism research paradigm because the researcher will have to conduct some data interpretation while for quantitative research design, researchers mostly adopt positivism research paradigm because the data is measurable or observable in some way (Logan, 2008). Ideally, interpretivism research philosophy entails the assessment of a phenomenon through the perception of the people experiencing it (Crowe, Sheppard, & Sheppard, 2012). On the other hand, the positivist research paradigm entails a systemized data interpretation through quantification to enable a precise description of various research variables and a dan identification of any relationships among them (Schulze, 2009). The main aim of the current study was to explore whether the use of concrete materials in teaching mathematics contribute to positive outcomes among learners and improved teachers’ attitudes towards teaching mathematics. Because the study took a mixed research design, both positivist and interpretive research paradigms were applied. Interpretivism was useful in analyzing interview responses from teachers regarding how the use of manipulatives impacted on their attitudes towards teaching. On the other hand, the positivist research paradigm was applied in the analysis of quantitative data derived from students’ performance before and after the introduction of manipulatives.

3.2 Research Design

The research design refers to the method of collecting, analyzing data through various approaches including qualitative, quantitative and mixed approaches (Wilkins & Woodgate, 2008). Quantitative research design entails the collection and analysis of quantifiable data to answer narrow or specific research questions (Wilkins & Woodgate, 2008). The quantifiable data are often numbers that are analyzed through statistical approaches to develop clear and objective results about the phenomenon under investigation. Thus, through quantitative research design, the researcher can quantify the variables under investigation and identify any relationship between them. On the other hand, qualitative research design entails the collection and analysis of subjective data (e.g. opinions, perspectives, experiences or attitudes) from participants (Wilson, 2013). The data is mostly collected in words and the words are analyzed to develop themes (Wilson, 2013). Lastly, mixed research methodology entails the use of both qualitative and quantitative research approaches in one study (Roberts J. M., 2014). in the current study, the researcher intended to investigate whether the usage of manipulatives in teaching mathematics contribute to positive learning outcomes for students and whether teachers’ attitudes towards teaching mathematics improves with the use of manipulatives. These aims justify the use of mixed research design because while the former aim can be achieved through quantitative research approaches, the latter aim can be achieved through qualitative approaches. Particularly, the impact of manipulatives on pupil’s mathematics performance requires a quantitative analysis of performance while the impact of manipulatives on teachers’ attitudes towards mathematics pedagogy can be evaluated through qualitative approaches.

3.3 Research Instruments

The study relied on both qualitative and quantitative data collection instruments because it adopted a mixed research design. To gather information from teachers regarding the impact of manipulatives on their attitudes, comfort and confidence towards mathematics teaching, the researcher relied on questionnaires (appendix 6). The 20 questionnaires (to match 20 teachers participating in the study) were designed to ascertain whether the use of manipulatives contributed to the improvement of teachers’ attitudes, confidence and comfort towards teaching mathematics. On the other hand, the impact of manipulatives on students’ performance was evaluated by collecting and analyzing data on students’ performance before and after the introduction of manipulatives. Because a standardized test had not been developed and was not readily available to use among pupils of multiple grades, the researcher together with two teachers developed an assessment tool (consisting of specific questions) for students in each grade. To enhance the study’s validity, the questions were asked depending on the students’ grade.

3.4 Sampling Method

Research sampling refers to the method of selecting a subset of the entire population to be included in a research study (Farquhar, Ewing, & Booth, 2011). While a study can take a probability or non-probability sampling approaches, the current study relied on non-probability sampling because the target population were from a select Irish elementary school. Thus, convenient sampling (a type of non-probability sampling) was selected for use in the current study. Perhaps the easiest sampling method, convenient sampling was selected particularly because it relies on the willingness and availability of the population to participate (Robinson, et al., 2011). Therefore, because the researcher could access participants from the selected school, 20 teachers across all grade, including some multi-grade teachers and 43 students were selected to participate depending on availability to partake the test.

3.5 Data analysis

The study relied on a two-thronged data analysis consisting of quantitative and qualitative data analysis. Regarding data analysis, the author conducted an ANCOVA analysis conducted on IBM SPSS 22. Particularly, the ANCOVA analysis enabled the researcher to capture the impact of one or more factors on a variable i.e. the conduction of pre and posttest analysis while eliminating the effect of another variable. ANCOVA is a common analysis technique for pre and post-test research design, whereby the emphasis is made on the dependent variable while at the same time controlling for the effect of the covariate as well as other confounding factors including ethnicity and gender (Loehnert, 2010). While the study could also have relied on repeated ANOVA, multiple regression and mean grade scores, we believed that repeated measures of ANOVA could have shown unclear results (Berk, 2007). furthermore, whereas gain scores would have been easier and more straightforward, covariance analysis proved to be more powerful in this case (Jakobsson, 2004). There were various assumptions made before the use of ANCOVA. For instance, the study assumed that a reasonable correlation (for example between .3 and .9) existed between the dependent variable (post-test) and the covariate (pre-test). Furthermore, an assumption was made that there and post-test scores had a linear relationship particularly based on the observations made on the visual scatterplot. Nonetheless, the researcher realized that the regression slopes have a weak homogeneity even through no interaction was evident. There was a violation in tests for normality in both the dependent variable and the covariate as depicted in the Shapiro-Wilk’s test (p < .05) even though a significant level of more than .05 was needed. Whereas visual observation of the scatterplot revealed heteroscedasticity, a Levene’s test revealed homogeneity of variance (p = .081) including both pre-test and post-test variables. Moreover, an outlier data (i.e. too low scores) was observed on both pre-test and post-test even though the student was included in the study because he was part of the class. With regards to the qualitative data gathered from the interviews, the author conducted a simultaneous data collection and analysis to develop a better understanding of the response to the interview questions. The simultaneous collection and analysis of data were done repetitively until no more fresh categories of data emerged from the interviews (i.e. point of data saturation), indicating the need to end the interviews. Consequently, the interviews yielded a large amount of data that seemingly could be overwhelming to analyze. Hence, to identify specifically useful data for answering the research question, the author used thematic analysis method. The thematic framework was enabled a systematic synthesis of data by organizing the data into themes and sub-themes for a clear understanding of how they answer the research question. Apart from helping to organize a large volume of data into meaningful bits, the thematic analysis also helped in developing the scope of the study beyond just the respondents’ experiences by putting themes into context (Loehnert, 2010). The thematic analysis was conducted in 6 major steps. First, the author read the entire interview transcript to familiarize with it. This was characterized by repeated reading of the interview responses while making notes on the emerging ideas from the text. The second step was to manually generate codes. Here, the researcher used tools such as colored pens and highlighters to identify as many codes as possible form the data while collecting related ones to develop robust and well-analyzed themes. Next, the researcher developed initial codes by evaluating broader levels of themes and sorting most of them for a more refined rendition of the data. It was also in this stage that some sub-themes were developed from the main themes. The fourth stage of the thematic analysis process was the review of developed themes. This was done in two stages namely a review of the coded data to evaluate whether they fit into the major themes, and a review of the themes to ensure that the codes within the theme have a logical relationship. The penultimate step was to name the themes while the ultimate step was to develop the thematic analysis report.

3.6 Ethical considerations

Throughout the study, the author held with high regard to various ethical consideration sin academic research because they formed a fundamental basis of ensuring the participants’ safety. Furthermore, it was necessary to make various ethical considerations because this defined the moral principles within which the study was conducted. Thus, some of the most important ethical considerations made in this study included issues of participant dignity, confidentiality, informed consent, and privacy. The study relied on various ethical principles that bordered on the relationship between the teachers, pupils and the researcher. The various precautionary measure was taken to protect the well-being and safety of participants especially considering that some of the participants were children. For instance, the study sought informed consent from the teachers and the school authority before setting out to collect data. Thus, it was an important ethical consideration to make because failure to gain consent before collecting data is tantamount to fraud (Boakes, 2009). Gaining informed consent ensured that the participants took part in the study on their own volition. To do so, each participant was given a consent form (Appendix 1) that they were to fill and return before participating. Furthermore, the study gained permission from the university ethics committee as required by the university research regulations. This was obtained by submitting an ethics form (Appendix 2), which was reviewed before receiving permission to go ahead with the study. A ratification by the university’s ethics committee is an indication that the study passed all the available ethics tests and maintained high ethics standards throughout until completion. The study acknowledged the risk of children being exposed to emotional harm such as anxiety, embarrassment or shame in case they did not perform well in the tests. Therefore, tow professional teachers were selected to administer the tests in a way that there was no interference with the emotional well-being of the children participants. Though, the study relied on the ethical principles of truthfulness, non-maleficence, justice and beneficence. By observing the principle of beneficence and non-maleficence, the study ensured that any component or activity of the study that threatened the safety and well-being of participants was eliminated. For instance, all the sensitive data was stored in a password-locked memory stick and they were deleted upon completion of the study. Besides, the researcher observed the principle of impartiality, whereby all participants were treated as equal regardless of their mathematics performance. More importantly, the researcher ensured that no participant was discriminated based on race, age or gender. In conclusion, this chapter built on the methodology section of a previously completed research proposal to outline in detail, the different methodological approaches that were taken to complete the study. Specifically, this chapter has illustrated how mixed research methodologies were applied and why they were considered the most appropriate research approaches for the study. In the next chapter, the study presents its findings.

Chapter 4

4.0 Findings

This chapter presents an in-depth analysis of the study findings. The quantitative section of this chapter outlines the key results of the study through tables, numbers and pictures while the qualitative section of the chapter presents the results thematically with quote extracts from the interview transcripts. Besides, all the themes from the qualitative analysis will be summarized in a table depending on the categories within which they occur.

4.1 Quantitative Results

The participants of this study consisted of 43 grade 3 & 4 and 20 teachers from a rural school in Ireland. Appendix 3 presents the demographic information of each participant in terms of their ethnicity and gender. Both male and female gender in the control and experimental group were slightly around the 50% midpoint. Participant in the treatment group had an average age of 6.77 while those in the control group had an average age of 6.80, thus the control group were on average slightly younger than the treatment group. The descriptive analysis reveals standard deviation and means of the pre-test scores (i.e. covariate) as well as the post-test scores (dependent variable). Appendix 4 illustrates the pre-test and post-test scores, standard deviation and mean by ethnicity and gender. From the results, there is an average increase in post-test scores. Averagely, the treatment group has an average post-test score of 19.4 out of 20 (i.e. 97%) while the average post-test score of the control group was 15.8 out of 20 (79%). It was however interesting that randomly selected students from the treatment group achieved higher pre-test score than those selected from the control group. This is an important observation because there was a need to control for pre-test scores when evaluating students’ performance and establish the math score before of each participant before examining a difference in the students’ performance with or without manipulatives. Appendix 3 illustrates the post-test average score of the control and treatment groups. The findings indicate that the post-test mean scores for the treatment group are higher than those of the control group. Nonetheless, our analysis revealed that gender did not have any significant impacts on the scores and thus was removed from further analysis. Meanwhile, the ANCOVA analysis evaluated the effect of manipulatives compare to the controls in post-test results while controlling for ethnicity. In this regard, the results showed that the use of manipulatives had a significant effect (1, 42 = 28.33, p = 0.001) on the post-test scores, accounting for 42% (partial eta2=.42) of the post-test scores even after controlling for ethnicity as illustrated in appendix 5. The model made up for 845 of the post-test score variance. These findings supported the hypothesis that students who used manipulatives had higher scores (post-test) in mathematics. As illustrated in appendix 4, the average score of the treatment group (97%) was higher than that of the control group when ethnicity and pre-test scores are taken into consideration. This implies that the percentage score post-test scores of students who used manipulatives were 18% higher than those of the students who did not use manipulatives. To further validate these results, an analysis of covariance (ANCOVA) indicated a significant value of less than 0.001 between the post-test treatment and control groups. This implies that the probability of getting such a value by chance is 1 out of 1000. Thus, the probability falls below this study’s established alpha cut off a range of acceptability (α = 0.05), supporting the hypothesis of group difference. The descriptive results of this study, therefore, support that the post-test treatment group (i.e. those who used manipulatives) would be significantly higher than that of the control group who did not use manipulatives. In the following section, we present the qualitative section of the study, whereby interview data are analyzed to evaluate how the use of manipulatives impact on teachers’ confidence, attitude and confidence in mathematics pedagogy.

4.2 Qualitative results

Apart from the descriptive results, the study also gathered qualitative results from interview questionnaires that were distributed to and self-administered by 20 teachers. Particularly, the questionnaires were meant to gain the teachers’ opinion on how the use of manipulatives impacted their teaching experience, the challenges they experienced with manipulatives and some of the factors that enhanced their use of manipulatives. Ultimately, this data would be used to gauge a teacher’s attitude and confidence in the use of manipulatives in mathematics pedagogy. Appendix 7 illustrates a compilation of the themes.

4.2.1 Challenges of Teaching Mathematics

First, it was important to know whether the teachers experienced any difficulties when teaching mathematics and whether these challenges persisted even with the use of manipulatives. Most of the respondents referred to mathematics as a relatively difficult subject to teach especially when handling topics with abstract concepts such as geometry. Besides, some respondents remarked that the abstract nature of mathematics makes it a more demanding subject to teach because they must use creative ways to facilitate the pupil’s understanding. For instance: T: “…teaching mathematics is not just like teaching these other subjects because math has certain underlying rules and concepts that the students must understand…” D: “…the difficulty in teaching mathematics comes in a situation when you have to explain to the learner a concept that they might not directly see how it applies in real life situation…” A: “…One of the most challenging scenarios in my math teaching career is when I have to help the child to understand abstract math concepts and apply them in real world scenarios…” B: “…Teaching mathematics is sometimes difficult because children sometimes find complete difficulty in generalizing wider applications and applying them to different scenarios…. some of the most difficult experience I have encountered when teaching mathematics include fractions, integers, vectors matrices and complex numbers...”

4.2.2 Role of manipulative sin influencing teachers’ attitudes

4.2.2.1 Facilitating effective teaching of abstract concepts

Part of the questions also targeted to understand whether the manipulatives assisted in teaching mathematics. With examples, the responses revealed various ways in which manipulatives facilitated effective teaching of abstract concepts in math and how they could use various ways to facilitate the pupil’s understanding of the concepts. For instance, J: “…There is always a big difference in my teaching experience when I use manipulatives and when I do not use them. Mostly, the students ask fewer questions, are more sedated and quieter than when I do not use manipulatives…” D: “…. even when they have questions, most of the questions are directed at me and are asked in a manner that triggers less discussion because they are mostly asked in Yes/No format…. For example, students mostly ask questions like what the difference between a rhombus is and a parallelogram?” These responses demonstrate how lessons taught without manipulatives tend to be teacher-centered rather than learner-centered. Furthermore, the responses demonstrate how the lack of manipulatives within the classroom setting puts the teacher in total control of the class making the students to follow the teachers’ lead instead of having an interactive learning session as illustrated further in the following extract: K: “…In most cases when I do not use manipulatives in my classrooms, the children are less interactive and more sedated…. they make less noise and hardly move. While this is acceptable, the students only tend to be mentally engaged and not physically nor socially…” G: “…whenever I use manipulatives, the students tend to find it easier to memorize them because manipulatives facilitate a better delivery of the information…”

4.2.2.2 Enhancing the Teaching Process

There were more positive responses regarding the use of manipulatives and how they enhanced the teaching process. For example, some responses illustrated how manipulates made the children more inquisitive, excited and talkative as opposed to lessons when manipulatives were not used. J: “….as opposed to when there are no manipulatives, my students always tend to be more talkative, excited and inquisitive whenever I use manipulatives…the students tend to ask questions that spark discussion among them. Similarly, whereas some questions are directed at me, others are often directed at their fellow students, a phenomenon that I hardly experience when I do not use manipulatives” From these responses, it is possible to extrapolate the use of manipulative facilitates a conversation between teachers and students as well as between students and their colleagues. This demonstrate show the use of manipulatives facilitates engagement and knowledge sharing within the classroom environment to enable an easier understanding of mathematics concepts. An analysis of the above responses reveals that when students are taught using manipulative, they develop the habit of thinking more and being more inquisitive about the concepts. Manipulative snot only facilitates the absorption of information into the students’ brains but also the application of such knowledge into concrete activities. It is also possible to extrapolate that because the manipulative helps the students to apply those question in real life activities, the application process leads to emergence of more questions about the concepts thus creating a conversation among the students as they share the knowledge. The questions and comments that students make to each other as they use manipulatives help them to learn not only from the teacher but also from each other. As the students work with manipulatives, they develop additional meaning to the concepts which are then assimilated into their knowledge. This was illustrated by one respondent who noted that: Y: “…sometimes after teaching a lesson on polygons, I organize the children into groups of two and give them geoboards and rubber bands to construct different shapes of octagons, during that time, o overhear some students claims that “ oh, so that is also an octagon? I only knew octagons take the shape of a stop sign” ….” Based on the above response, it is possible to extrapolate that manipulative helps students to add a new meaning to a concept based on a prior knowledge they already had on a subject matter.

4.2.3 Enabling factors for the use of manipulatives

4.2.3.1 Availability of Manipulatives

The questionnaire interviews also revealed that the teachers’ attitudes and confidence towards the use of manipulatives were affected by both enabling and disabling factors for the use of manipulatives. For instance, it an attempt to explain what enhances their attitudes towards the use of manipulatives, one of the respondents noted that they found it easier to use manipulatives because the set of manipulatives were always readily available in the class: H: “…what makes it easier for me to use manipulative sin may lessons is that they are readily available in the class and therefore I could just pull out one or two of the materials at may convenience whenever I feel I really need them…”

4.2.3.2 Adequate knowledge of manipulatives uses

Another respondent noted that using manipulatives were always a challenge until the school took them through a training on how to use them: N: “…. using those materials are always not an easy task because they involved a high level of technical application especially when you have to understand the child’s knowledge needs and how the material would help address that need. Therefore, I always avoided using them until we were taken for a training sponsored by our school administration on how to use them...”

4.2.4 Disabling Factors for the Use of Manipulatives

The response also hinted to certain factors that affected teachers’ attitudes and confidence towards the use of manipulatives. For instance, while explaining the nature of their attitude towards manipulatives, one of the respondents noted that in situations when manipulative was not readily available within the classroom, it was difficult to use them because he lacked the time to prepare them and practice their use: S: “…I only feel inclined to use manipulatives when they are readily available in the classroom otherwise it would be hard for me to fetch, organize ands prepare them for use. I might also need considerable time to practice how I would use them because without adequate practice, it is highly likely that I might mislead the students...” Apart from lack of time to set up and practice the use of manipulatives, the responses also hinted to lack of space as a significant factor determining teachers’ attitude and confidence in the use of manipulatives. When explaining why they would choose to or not to use manipulatives in their lessons, one of the respondents complained that he is often limited by the lack of space within the classroom and therefore choose not to use manipulatives most of the time: E: “…Manipulatives are some of the most effective tools for enhancing students’ understanding of math concepts. However, I am sometimes limited by space especially when I am teaching grade three students who are large in number and sometimes must squeeze in their small class…” These responses indicate that administrative issue such as the size of classroom may affect teachers’ attitude and use of manipulatives. This implies that the use of manipulatives is largely influenced by the environment in terms of space and the materials within the space. This chapter has identified, analyzed and presented the study results in both descriptive and qualitative form. The descriptive data emerged from a quantitative analysis of pre-test and post test results of children to gauge the impact of manipulatives in their mathematics performance. On the other hand, qualitative data emerged from a questionnaire interview administered on teachers to evaluate their experience of attitude and confidence toward the use of manipulatives.

Chapter 5

5.0 Discussion

This chapter selectively interprets the collected data with regards to how they help answer the research question. It gives an analysis of what the gathered evidence says about the topic and how this evidence relates to existing research. Thus, this section will integrate the study findings with existing secondary data to develop a concrete conclusion over the research question. In doing so, the author will extract meaning from the data and the implications of findings in real life teaching practice. One of the questions this study sought to answer is whether the usage of concrete materials in teaching mathematics contribute to positive learning outcomes for students. Therefore, the study sought to investigate whether there was a difference in students’ performance in mathematics when the teacher used manipulatives and when manipulatives were not used. Through descriptive statistics and data analysis, this study provided results supporting that manipulative shave a positive impact on grade 2 children’s mathematics performance and capabilities, even when other confounding factors such as gender and ethnicity were controlled for. Thus, there is evidence to claim that manipulatives can benefit students. The study provides empirical evidence that students who used manipulatives during their mathematics lessons averagely performed better that their counterparts who did not use manipulatives on a post-test experiment. These results complement the findings of other previous studies that have explored the use of manipulatives in mathematics pedagogy. For instance, research by Gauthier et al., (2004), Freer (2006), Driscoll (1983), DeLoach, Sudder and Uttal (1997), Hamm (2011), Swan & Marshall (2010), Sowell (1989) and Raphael and Wahlstrom (1989)have evaluated the use of manipulative sin elementary schools and found that it resulted to an increase in students’ mathematics achievement. However, the current study is unique in the sense tat while a minimal intervention was introduced, strong results were still achieved. Therefore, the findings of this study support the argument that if students use manipulatives when learning mathematics concepts, it does not only improve their mathematics performance but also gives them additional strategies and methods that are useful in developing problem-solving skills. The findings of this study suggest that manipulatives can be used as a tool to improve students’ development and learning capabilities. Similar observations were made by Golafshani (2013) because every classroom has students with different math understanding capabilities and therefore teachers must develop strategies and techniques to help all the students achieve their best in the subject. This justifies why teachers should consider the use of manipulatives when teaching mathematics to enhance students’ success. Through manipulatives, students will not only be prepared to handle the curriculum requirements for mathematics but also eventually develop effective foundational math skills for middle or secondary school. In the end, this study has established that students who can use manipulatives during math lessons end up performing better than students who do not use manipulatives during math lessons.

The qualitative results also revealed that the use of manipulatives has appositive impact on children’s mathematics performance as well as a generally positive attitude of teachers towards the use of manipulatives in math pedagogy. The findings of this study indicate that when students use manipulatives during their math lessons, they make more movements and noise within the classroom, a phenomenon that was viewed by some respondents as a sign of active engagement in learning, meaning that the use of manipulatives help develop an active and engaging class or learners who share knowledge through interaction. While the study found that students who use manipulatives discussed concepts with each other, displayed more excitement, asked more questions and remained active throughout the lessons, there is a lack of other existing studies that support this finding. Therefore, more studies are required on how manipulatives impact on students’ behavior and interaction in classroom as the current study was more focused on the impact of manipulatives in students’ performance. Teachers’ comments and responses indicated that using manipulatives, students assimilated new concepts into the ideas they had prior knowledge on. Whereas the current study did not test on students’ retention, the active engagement and leaning process the students underwent using manipulatives enabled their retention of new concepts because they could easily assimilate those new concepts in the knowledge they already had. Several other earlier studies (Yağcı, 2010, Gürbüz, 2007, and Enki, 2014) have found similar results about how manipulatives enhance students’ activity and engagement in the classroom, thereby improving their attitude towards learning and consequently boosting their performance. These studies also agree with the current study on how manipulatives promote students’ math achievement by enabling an easier understanding of abstract concepts through concrete demonstrations. Considering the current study’s qualitative results, its empirical findings indicate that students’ achievement in mathematics can be increased through mathematics prepared teaching materials through the function of manipulatives that help to concretize abstract mathematical concepts. Similar results have been found by Yıldız & Tüzün (2011), Yeniçeri (2013), Yağcı (2010), Tunç, Durmuş, & Akkaya (2011), Tuncer (2008), Şahin (2013), and Strom (2009). With regards to comparing the post-test performance of both the control and experiment groups, the higher performance of the experiment group in the use of manipulatives during geometry, circle and sphere lessons indicate the effectiveness of manipulatives in the learning of abstract mathematics topics and concepts. However, there are some subjects such as White (2012), Baran, Işık, Kal, & Hazer (2013), Enki (2014), Enki (2014) and Boakes (2009).

Based on the performance of students on the control and experimental group before and after the use of manipulatives, it is possible to extrapolate that manipulatives help students to perform better in mathematics and enhance teachers’ as well as students’ attitudes towards teaching and learning mathematics respectively. There are existing studies that have found a positive impact of manipulatives on students’ attitudes towards learning and consequently enhancing their performance (Sowell 1989, Enki, 2014). For instance, studies by Enki (2014) illustrated how the students acknowledged that as opposed to traditional forms of learning, learning through manipulatives increased their motivation and gave them pleasure in learning mathematics because it made math become more fun. Such positive feedbacks, like those given by teachers in the current study point to a positive attitude possessed by both teachers and students when manipulatives are used in math pedagogy. The qualitative results indicate that lack f time to practice the use of manipulatives affected their confidence in using them during mathematics lessons. Similarly, the lack of time hindered the teachers’ ability to introduce the concept of manipulative within the classroom, yet the introduction is an important component of manipulative use because the students must be given the rules and instructions of using the objects. Another important concern raise by the teachers was the lack of enough space within the classroom within which the students can freely use the manipulatives. This implies that establishing a specious workstation is important to allow teachers enough workspace for using manipulatives. Besides, these findings indicate the indirect relationship between classroom space and students’ performance in mathematics. Obviously, a teacher would not use manipulatives if they do not have enough space and time. School administrations should therefore ensure that they classrooms have enough space for teachers to execute the use of manipulatives. Teachers mostly believe that the use of manipulatives can help students understand abstract math concepts Enki (2014), and manipulatives have always been an alternative to consider when there is need to teach and make students understand abstract concepts. However, this study reveals that certain factors such as lack of adequate space and time may make teachers consider manipulative as a teaching option and not a necessity. Yet, if the teachers believed that the use of manipulatives can enhance students’ understanding of abstract concepts in mathematics, then they should consider manipulatives as a necessity. Nonetheless, the findings of this study indicate that whether teachers perceive manipulative as a necessity or an option, they agree that the use of manipulatives have a significant impact on the ease of teaching mathematics because they facilitate both abstract and visual learner’s ability to understand and acquire math knowledge in different ways. Similar results were found by Yağcı (2010) who asserted that for abstract learners, manipulatives provide a new practical approach to solving math problems through calculations and use of numbers. Similarly, according to Enki (2014), manipulatives can help visual learners to learn concepts and write those concepts I abstract mathematical terms. An analysis of teacher’s responses on factors that promoted their positive attitude and confidence towards the use of manipulatives, some of the most prominent responses entailed the availability of manipulatives in the classroom and administration’s support towards the use of manipulatives through training that enabled their persistent use of manipulatives. This persistence use of manipulative could be attributable to the fact that upon gaining adequate knowledge on how to prepare and use manipulatives, teachers develop more positive attitude because they can easily use the objects. Meanwhile, it also possible to extrapolate from the study results that the use of manipulatives completely changes the classroom environment to make it more exciting and engaging. Students also discovered that they could learn more about previously held knowledge about certain concepts, and this made mathematics more of a pleasure subject to learn. Similar results were eminent in the study by Baran, Işık, Kal, & Hazer (2013), which found that using manipulatives to learn mathematics, students would find it enjoyable to learn mathematics because they begin to have a better understanding of concepts they previously could not understand.

Chapter 6

6.0 Conclusion and recommendations

During this time when there is an increasing demand for better schools, higher quality learning and greater teacher and school accountability towards students’ learning process, any factor that contribute sot students’ positive learning must be taken seriously. An important objective of schools s to enhance students’ leaning abilities for short-term and long-term knowledge acquisition. In this regard, the current study supports the belief that the use of manipulatives can help improve students’ mathematics score. With good mathematics scores, students are more capable of developing effective decision-making and problem-solving skills. Despite these positive findings about manipulatives, there are several methodological limitations to this study that affects the generalization of its findings. For instance, the study violated the steps for measuring the normality of participants as this is a significant assumption made when conducting ANCOVA analysis. The study also lacked a standardized test that could be used in repeating a similar study, even though one was created based on a single division assessment test. Thus, future studies should consider the development of a more difficult test with more test items to facilitate the scores distribution. The second limitation of this study is that both the treatment and control group were few (i.e. only 43 elementary school pupils and 5 teachers) and this limits the generalization of the study findings to a larger population. Moreover, the sample population did not have any student with disabilities. Thus, future studies should look at the potential effect of use of manipulative son students with disabilities.

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Meanwhile, from the study findings, teachers face unique challenges the use of manipulatives that might affect their attitudes and confidence towards materials. These challenges might be multidimensional due to the diversity of teachers’ demands. More importantly, teachers’ confidence with and knowledge of the use of manipulatives, as well as time and space for the use of manipulatives are objects of major concern that can be resolved through various ways. Therefore, the following recommendations might help enhance the use of manipulatives in elementary schools and consequently promote students’ performance in mathematics: There needs to be further research on the other challenges affecting teachers’ use of manipulatives within the classroom setting and develop effective interventions for those challenges Schools should provide more resources to teachers that not only aligns the curriculum content to the use of manipulative but also provides a variety of choices to teachers on which kind of manipulatives they should use of different math concepts. Teachers should also be exposed to new and emerging trends in the use of manipulatives through training programs and workshops that keep them up to date with current practices. Practitioners should also acknowledge that teachers may adopt a more constructivist approach to teaching whenever they use manipulatives Schools should develop a coordinated approach towards the use of manipulatives by allowing teachers to share their strategies and techniques for using manipulatives.

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