Math curriculum development and implementation

Math Curriculum

Science and its modes of studies are very much reliant on the mathematical techniques which have been heavily evaluated over the past few years. Numerous studies like the ones conducted by Mike Cass et al. (2003) and Lynn T. Goldsmith and June Mark (1999) have analyzed both the teaching techniques used for mathematics as well as the overall sections that are included in its course. In these studies the common conclusions that were made regarding fundamental knowledge of mathematic formulas and techniques is that these allow the students to develop essentially their primary aptitudes in the subject. Other studies also highlight that by solving the simpler questions through the use of fundamental formulas allows the students to find vague and difficult answer to questions by using merely their own analysis and logic (Cass, Cates, Smith, and Jackson 2003; Goldsmith and Mark 1999).

Margaret Brown et al. (1998) conducted a research where they studied that relevance and association between the faculties of science and math as well as their teaching standards and techniques. The conclusions made were clear in pointing out that the initiation of math's association with science was in the decade of the 1960s when the main focus of most scientists was to figure out a way to reach the moon. The use of mathematical formulas was believed to be one of the most important aspects of this objective of reaching the moon and most students of the time, along with environmental demographics, were taught math so that they could learn to use their common sense in different situations. The most common method of teaching, that Brown and associates defined as practically inapplicable and un-researched and the cause of the loss of interest in the subject was the hierarchical approach of skill-drill-kill technique before actually solving the problem.

The fact of the matter is that the overall academic achievements being recorded in the department of math have been on a declining pattern. The National Council of Teachers of Mathematics NTCM (2004) highlighted in their research that the overall teaching of the subject was becoming void in the modern era and the overall effect of that is grievous on the students. The stats shown in the study exhibited that the majority of the high school students were below average math students and that the overall student body in the U.S. were not more then average in math. They demanded that they U.S. government should take steps to rectify this situation because it was leaving the majority of their nation in dire states (NCTM 2004).

The hierarchical approach of skill-drill-kill technique to teach math has not seen much success anywhere out of U.S. either. Douglas H. Clements (1989) carried out researches in UK where he mainly concentrated on the historical records of math abilities in high school graduates and stats showed that most of the students did not have much practical knowledge about math. Furthermore, numerous researches in Europe have also highlighted the need for fresh techniques to teach math and develop interest in the subject (for details, see Anghileri, 2001; Beishuizen, 1999; Buys, 2001; and Bieshuizen, 1999; van den Heuvel-Panhuizen, 2001). A good example of new teaching techniques was shown in the research by Julia Anghileri (2001) where she highlighted some successful techniques being employed in Netherlands that were mainly a hit because they were realistic and truth based approaches to teach math so that the students could see its relevance in real life.

There is no doubt about the fact that the overall techniques that have been employed up until now are not up to the par and that there need to be amendments made so that the results can improve. In the study conducted by Shelley Dole, Bob Wright and Robyn Zevenbergen (2004) there were two very important recommendations made for the alterations that were needed in the teaching of mathematics. Firstly, they highlighted that the topics and their extent should be revised to provoke more intelligent and authentic links of math with other subjects so that the students can use their logic to get more networked learning preparation. They also added that all math techniques should be taught with the help of instances from real life so as to spark the interest of the student. Secondly, they said that the practicality of the subject should be focused on. One of the ways that they suggested this could be done was through assortment of students from different backgrounds working together as one collective unit. They further highlighted that the key for finding practicality in math was the attitude of the teachers and their approach towards inviting the students towards implementing their math formulas and techniques in real life situations. In this study Dole, Wright and Zevenbergen (2004) concluded by stressing on the need for the utilization of solid manipulatives as tests show that the manipulatives help the students understand the concepts more thoroughly and makes teaching a lot simpler for the teachers. Also, they highlighted, that manipulatives were useful tools for teaching math in the current era and even though manipulatives have been around for about 2 centuries, they have only recently received mass acceptance by mathematical researchers in the world.

Clements (1999) is Professor at the University of Buffalo where he teaches early childhood, mathematics, and computer. He conducted a study on the use and affects of solid manipulatives on student comprehension and concluded that most of the teaching techniques especially in math are heavily reliant on the utilization of solid manipulatives for better understanding. Clements explains that it's the solid nature of the manipulatives that makes it easier for the students to understand the tricky theoretical concepts of math.

Thesis Statement

Since, most researchers have produced facts that manipulatives are useful tools for teaching math, we, in this study will evaluate the impact of their use as well as focus on their standard traits and their best use by taking a survey and interviewing pertinent authorities on the subject.

Significance of the study

The importance of this study is the information that it will add to previous knowledge in this area and the understanding that will be generated in relation to the development of math teaching techniques and their necessity in the modern era with respects to the interest of students in the subject. Without knowledge of basic mathematical formulas the understanding of or the ability to find numerical and statistical connections between the normal events society, politics, and economics is diminished for the learner. It is necessary that students are taught to logically separate and link the different math formulas from real life incidents so that they can not only develop interest in the subject but also be able to compare and contrast events and focus at certain connections of the past and present events in mathematical terms. Furthermore, the students being exposed to higher and more intricate technology gadgets will need to know the ins and outs of these in both practical and mathematical terms.

Literature Review

As mentioned before, the theory and use of manipulatives has existed for a long time but under the radar, and it is only in the past decade or two that it has increased in recognition and importance amongst many educational institutions. A research conducted by Matthew Branch (2006) is the most informative source on how and why the use and importance of manipulatives has increased in recent years. He highlights that the "No Child Left Behind" Act as well as numerous yearly government-authorized or government-approved assessments have been the main advocates of the use of manipulatives.

Deborah Loewenberg Ball (1992) has highlighted in her study that another reason for the growth of popularity of manipulatives has been the demand for finding new successful methods of teaching math. Manipulatives and their application in the teaching of math was one of the experimental methods utilized to see if the impact on student comprehension was positive. The end result was that the solid nature of the manipulatives made it easier for the students to learn basic formulas as well as see the connection of those formulas in different circumstances. Ball explains that manipulatives are basically the solid objects that an individual comes across to on a daily basis and since the individual is already familiar with these objects, the overall understanding of their mathematical dynamics becomes easier.

Since, its initiation, the manipulative tools have statistically seen a consistent increase in their use within the classroom settings especially for the teaching of math. One research conducted by Leonard Kennedy and Steve Tipps (1994) explained that the teachers were regularly using the solid manipulative tools to simplify and explain the more complex mathematical formulas to the students. In the same study the researcher explained that the use of solid manipulatives made it easier for the students to identify the connection of math in real life situations like billing. Dala Ramsey Tooke et al. (1992) in their study have also argued that by utilizing manipulatives teachers are easily and more proficiently teaching mathematics. 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.

Methodology

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 (Trochim, 2007). It is the researcher's belief that qualitative data collection methods fit the purpose of this study.