Susan K. Peterson and associates (1988) conducted a study on the impact of the use of manipulatives on different kinds of students and concluded that the result of using manipulative was positive for both gifted and disabled students (Peterson, Mercer & O'Shea, 1988). Joseph Martinez (1987) also explained that the use of solid manipulatives made studying math more fun as well as less hectic and demanding for most of the students (Martinez, 1987).
In this modern era where technological advancements are dominating all other spheres of life, the phenomenon of manipulatives has highly benefited. David H. Uttal (1997) and his colleagues in their study focused on the implementation of manipulatives within the primary and secondary schools as well as the use of modern developments with solid manipulatives. The conclusions that were made revolved around the fact that the use of familiar objects was easier for the students to connect with and that it was easier for them to logically use technological equipments on those objects that they were familiar with. Furthermore, they confirmed previous conclusions that the majority of the children formed the connections between math formulas and signs with both the math theories and the real life situations that they could be employed in (Uttal et al. 1997).
Patricia S. Moyer (2001) is an accomplished and revered Director of the Mathematics Education Center who is also an Associate Professor of Mathematics Education in the Graduate School of Education. Moyer carried out a study on the way that the manipulatives were being used by the teachers and to what effect. She concluded that the problematic math formulas and theories were being better explained by teachers and better understood by the students through the purposeful, physical and lucid utilization of solid manipulatives (Moyer, 2001).
Leonard M. Kennedy (1986) conducted a research where he mainly highlighted some of the more important solid manipulatives that were being used. One of the most accurate examples he has given is that of monopoly that allows students to utilize both their fundamental skills of math like addition, multiplication, etc. And the very intricate logics related to the buying and selling of property, mortgaging, loaning, etc. In the game (Kennedy 1986). Brent Denu (1992) along with the other advantages of solid manipulatives in his study also highlighted that the best part about manipulatives was that they were always used in a secure environment without imposing any form of danger on the students. Also, he explained that the students' imagination and the variety of solid manipulatives used in a class were co-dependent and grew together with time (Denu, 1992).
Branch (2006) in his study made two broad categories of the types of manipulatives that have been used within the classroom for the teaching of math. These two categories are: one, the everyday utensils and tools or games that are used by the students in their regular lives or familial settings; and two, the customized manipulatives that are deliberately made to prompt the understanding of one particular formula or a set of formulas. Of course, the use and application of these manipulatives, whether customized or regular, is completely dependent upon the nature and skill of the students who use them. He additionally lists the different and popular manipulatives that are used within the classes along with the pros and cons of each and the range of diversity with which each of these manipulatives can be used. Some of the most common manipulatives that were used in the secondary school setting include Geometric Solids and Relation-shapes (Branch, 2006).
The nature, kind, objective and design of each manipulative is different. In his research, Denman (1984) evaluated a majority of the different manipulatives available at the market that were being used in classes and their individual impacts. The study explained that of all of the different types of manipulatives the ones that most helped the teachers in making students understand the arrangement, chronological placements, associations, number value, fractional relations simultaneously making them approach every problem with reason were that ones that had clusters of colorful and designed squares, triangles, circles, ellipses in an assortment of big and small metric sizes, breadths and widths. Denman also highlights that each manipulative has a different impact in different subjects and should be bought based on the impact that the teachers want to achieve in each subject (Denman, 1984).
Perhaps the most important aspect of the use of manipulatives has been that they have made the algebraic formulas easier for the students to understand. A study conducted by Annette Ricks Leitze and Nancy a Kitt (2000) showed that the algebra tiles or cubes that were custom made at home served as another form of solid manipulatives that, like monopoly, has multi-purposes i.e. they could be used to clarify the simpler algebraic formulas as well as the more difficult ones. Leitze and Kitt (2000) have explained that all algebra tiles mainly have square or rectangular shapes that are different and distinct due to their size, length and breadth proportions. All of the customized home-made algebra tiles are smaller in size than the ones that are available in the market. As a standard, the smaller square is represented by 1 i.e. The unit tile; x [sup2] is the representative of the larger square; and x represents the rectangular shape. In comparison, the side of the unit tile is equal to the total width of the x tile and the side of the x [sup2] is equal to the total length of the x tile (Leitze and Kitt, 2000).
All of the aforementioned studies have revealed the use of manipulatives in teaching mathematics by studying students as their subjects. Very few empirical studies have been carried out on the use of manipulatives and its impact. Therefore this study will fill this gap by interviewing important authorities on this subject and reveal identical patterns in their thoughts.
1. Research Philosophy
All research studies are based on some assumptions that the researcher that the researcher believes to be true. These assumptions make it easy for the researcher to observe his environment from a particular viewpoint, at the same time as ignoring other viewpoints. In this study, the researcher plan to view the subjects as rational individuals who have a scientific viewpoint. They are reasonable in their analysis and their approach and are generally at the level of scientific thinking. In addition, the researcher will assume scientific rationale and commonsense thinking are somewhat similar. In light of these assumptions, this study takes post-positivism as its philosophy as in line with Trochim (2007) post-positivism is the complete and absolute rejection of positivism (which believed that the laws of this world were motorized and deduction was the only method to make sense of this universe) and presuppose that humans reasoning abilities and scientific rationale are more or less similar and to facilitate the truth scientists should use not only deduction but also induction methods (Trochim, 2007). Therefore, post-positivist philosophy will assist this study to achieve its objectives with accuracy, clarity, relevance and precision.
2. Research Approach
The concept of math manipulatives can be considered as a fairly new topic (with regards to its popularity amongst researchers) and its theory in the current literature has not been completely developed. Therefore, this study plans on investigating this phenomenon further by utilizing an inductive method. We plan to collect data by carrying out interviews and subsequently evaluating the obtained results by utilizing coding techniques. The purpose is to uncover consistent and identical patterns in the subjects' thinking so that some preliminary results can be drawn, which help other researchers in their studies. Trochim (2007) asserts that inductive approach is best suitable for situations where the purpose is to develop a theory. The inductive approach is also categorized as a "bottom up" method as the researcher moves from specific and precise interpretations to general themes.
3. Research type and Time line
The research plans to use a cross-sectional viewpoint as it is basically a study about the impact and use of manipulatives in classrooms based on structured interviews conducted in a single point in time. Trochim (2007) points out that that a cross-sectional study takes a portion of whatever the researcher is observing in a single point in time. Cross-sectional studies are different from longitudinal studies which assume two and/or sometimes more measurements in different times (Trochim, 2007). After careful consideration of both budget and time available, the researcher believes that the best choice will be a cross-section study.
4. Data Collection Methods
As noted before, the current literature lacks empirical studies on manipulatives. This study plans to fill this gap by taking in depth and formal interviews from pertinent authorities on this subject. Therefore the data that will be collected will be non-numeric and therefore qualitative in nature. Trochim (2007) asserts that a qualitative studies are different from quantitative studies as they comprise wide varieties of data, which may include sounds, photographs, alphabetical text, videos and/or any other piece of information that is non-numeric…