Strengthening Intermediate-Level Mathematics Teaching Using Manipulatives:
A Theory-Backed Discourse
Jennifer Lira, M.Ed.
Vice Principal
Greater Essex County District School Board
Ontario, Canada
E-mail: jennifer.lira@gecdsb.on.ca
Anthony N. Ezeife, PhD
Professor of Math/Science Education
Faculty of Education
University of Windsor
E-mail: aezeife@uwindsor.ca
Stressing the key role that mathematics plays in human affairs and activities, the Report of the Expert Panel on Student Success in Ontario’s Leading Math Success: Mathematical Literary Grades 7 – 12 states that “mathematics is a fundamental human endeavour that empowers individuals to describe, analyse, and understand the world we live in” (2004, p. 9). In the same vein, the National Council of Teachers of Mathematics (NCTM) draws attention to the increasing importance of math in today’s number-governed, technology-oriented society when it observes that, “In this changing world, those who understand and can do mathematics have significant opportunities and options for shaping the future” (NCTM, 2000, p. 5). To be able to fit into this changing world where every individual comes face to face with numbers and calculations on a daily basis, students must acquire mathematical literacy and related skills right from the early stages of their academic career. One of the recommendations of the Expert Panel on Student Success in Ontario (2004) for creating mathematically literate students is to actively engage students in math teaching and learning through the use of manipulatives, both visual and tactile. It is the view of the Panel that manipulatives can be used to effectively bridge the gap between the abstract ideas for which math is known and dreaded by many, and familiar concrete experience that form part and parcel of students’ day-to-day life in the environment.
Mathematical Literacy
The Organisation for Economic Cooperation and Development (OECD) Programme for International Student Assessment (1999) defines mathematical literacy as “an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded mathematical judgments and to engage in mathematics in ways that meet the individual’s current and future life”(p. 41). Similarly, Bussière, Cartwright, Crocker, Ma, Oderkirk and Zhang (2001) maintain that mathematical literacy “revolves around the wider use of mathematics in people’s lives rather than being limited to mechanical operations” (p. 86). Essentially, a mathematically literate person is able to see the broader benefit of mathematics education, realizing in the process that mathematics is not a random set of rules but activities that are used in everyday life. It is not the tiny mathematical operations that are most important, but rather the mathematical ideas and concepts that permeate our everyday lives.
What are Manipulatives?
Concrete materials or manipulatives, as they are commonly referred to in math literature, are defined as “objects designed to represent explicitly and concretely mathematical ideas that are abstract. They have both visual and tactile appeal and can be manipulated by learners through hands-on experiences” (Moyer, 2001, p. 176). Some examples of these materials include tangrams, base ten blocks, algebra tiles, connecting cubes, pattern blocks, playing cards, etc. Teachers use them to teach abstract mathematical concepts that ordinarily may be difficult for students, such as adding and subtracting integers, solving equations, and determining the value of fractions. Moyer (2001) states that this is necessary because “students’ abstract thinking is closely anchored in their concrete perceptions of the world; actively manipulating these materials allows learners to develop a repertoire of images that can be used in the mental manipulation of abstract concepts” (p. 176).
Theoretical Framework
Swiss psychologist Jean Piaget suggested, “that children do not have the mental maturity to grasp mathematical concepts presented in words or symbols” (cited in Moyer, 2001, p. 175). Students are viewed as active learners that need to interact with their environment. This interaction with the environment allows students to create their own understanding and meaning about the world. Piaget’s theories also represent a constructivist view of learning. This view of learning posits that children are not blank slates; rather they interact with their environment as they attempt to learn. Learning results when children create their own knowledge as a result of this interaction (Van de Walle, 2001). The basic foundation of this theory is that children construct their own knowledge. Therefore, mathematical concepts cannot be simply presented to students; rather these learners must be actively involved and able to extract meaning out of mathematical concepts through experience (Moyer, 2001).
Piaget’s Concrete Operational stage of cognitive development lasts from age 7 to about 11. This stage is characterized by children being able to form relationships among objects. In terms of mathematics instruction, this means that students need to have concrete references and examples on which to hinge and relate their learning experiences in math, and thereby solve problems. This can be achieved through the use of manipulatives. After the Concrete Operational stage, students move on to the Formal Operational stage, which according to Piaget, starts from about the age of 11 and extends into adulthood. At this stage, children are expected to be able to think hypothetically and use metaphorical relationships. In terms of mathematics instruction, this means that students are capable of abstract thinking, and are able to generalize concepts to other concepts.
Ideally, intermediate-level students from age 12 to 15, should be operating in the Formal Operational stage. However, Hewitt (1994) draws attention to the fact that many adolescents in this age range are actually in transition between the Concrete and Formal Operational stages. In the major work, Teaching Teenagers: Making Connections in the Transition Years, Hewitt (1994) cites several studies that indicated 80% or more of young adolescents operate exclusively in the Concrete Operational stage rather than the Formal Operational stage. This has implications for mathematics instruction for students in this age range. The highly abstract concepts such as operations with integers, square roots, and solving equations, etc., are all topics covered in math starting around grade 7 or age 12. If students are still thinking in terms of concrete operations, intermediate teachers need to address this by using appropriate instructional materials such as manipulatives to facilitate learning, and make math meaningful to the students.
In Ediger’s (1999) review of the psychological foundations in the teaching and learning of mathematics, a description of Piaget’s developmental stages was explained. The author states that in Piaget’s research he found that children in the Concrete Operational stage were able to “emphasize reversibility” (p. 13). For instance, a student is able to reverse addends in an addition problem and still realize that the sum remains the same. Also, students at this stage are capable of associativity. For example, students are able to add three or more numbers in any order and come out with the same result, or multiply numbers in any order and the product remains the same. At this stage, it is very important for teachers to use concrete materials to solidify students’ understanding of these concepts as the students prepare to move into the Formal Operations stage where they would encounter these concepts in more abstract mathematical situations. Without a firm mastery of the concepts using concrete operational tools at the Concrete Operational stage, it would be difficult for the students to operate at the abstract level associated with the Formal Operational stage of cognitive development.
Ediger (1999) also draws attention to a situation that often arises in a typical intermediate-level class. Thus, in a class of 30 grade seven students, half may still be in the Concrete Operational stage while the other half may be entering the stage of Formal Operations. This suggests the need for teachers of students in a class of this age bracket to be aware of the maturational level of all students in the class, and instruct accordingly. The obvious implication of this is that concrete materials should not be jettisoned because intermediate-level students, by virtue of their chronological age, are expected to have matured into the Formal Operational stage of intellectual development. Cautioning against this misleading line of reasoning, the author emphasizes that teachers cannot move their students into a maturational level that they are not in yet. This means that there may be some concepts that may be beyond some students in a given grade level, while their colleagues in the same grade level may not find the same concepts as challenging. If a student’s cognitive development is not ready, learning will not occur. For a subject like math that usually poses a challenge for a significant number of learners, measures have to be taken by the intermediate-level teacher to ensure that when difficult concepts are being introduced, adequate supporting manipulatives would be used to cushion and facilitate learning, especially for those students still viewing the world solely through a Concrete Operational lens, and those in transition between the Concrete and Formal Operational stages of cognitive development. Supporting this position in a related work, Reys, Lindquist, Lambdin, Smith and Suyduam (2003) maintain that students at adolescence may not exhibit the same readiness for learning, and that cognitive development varies greatly. It is up to teachers to “play a crucial role in judging the developmental stage” (p. 26) at which students may be operating and teach them accordingly. The authors also contend that students need to be exposed to a rich learning environment. Essentially, this is an environment that allows students to experience mathematics at the appropriate developmental level. As the environment becomes more stimulating, the greater the math learning that would occur.
Brain Research and Math Learning
In the book, Brain Matters: Translating Research into Classroom Practice, Wolfe (2001) points out that learning is the process of building neural networks. The author goes on to discuss the three levels of learning brought into play in the strengthening of the networks. The first level of learning is concrete experience. The second level is representational or symbolic while the third is abstract. It is the interconnection between the three types of learning that strengthens neural networks. Concrete experience provides sensory experience and meaning, since “without the concrete experiences the representational or symbolic may have little or no meaning, no matter how much someone explains it to you” (Wolfe, 2001, p. 137). This gives evidence to support the notion that in the classroom students must actually engage in activities that encourage them to use the three levels of learning rather than merely listening or watching someone do mathematics. Wolfe (2001) also states that, “many of our strongest neural networks are formed by actual experience” (p. 138). The types of actual mathematics experience that would be relevant or appropriate here are projects and the use of manipulatives. These activities increase students’ understanding of mathematical concepts and the motivation of students to learn math. Jensen (1998) also adds that physical activity, such as hands-on activities promote student learning.
The connection between emotions and learning is also important to both the teacher and learner of mathematics. Many studies have found a positive correlation between attitude and achievement in mathematics (Expert Panel on Early Math in Ontario, 2003; Expert Panel on Student Success, 2004). If students are in part unsuccessful in mathematics because they have negative attitudes toward the subject, then understanding the role of the brain and emotion is important. Three areas of the brain – the hypothalamus, amygdala, and hippocampus – are involved in the regulation of emotions. These areas are the most primitive and earliest developed in the brain. The hypothalamus is the internal chemical regulator, that produces hormones and is responsible for the flight or fight response. Essentially, the hypothalamus is responsible for basic emotional responses. The amygdala contains specialized nerves that measure emotional content or emotional experience. The hippocampus filters long-term memories. The location of hippocampus and amygdala next to each other helps individuals to associate emotions with memories. What does this mean for mathematics instruction? Ross, Hogaboam-Gray, and McDougall (2002) stress that math education reform needs to focus on teachers helping students to develop self-confidence in math, which is just as important as achievement. Therefore, adolescent students need to have favourable experiences with mathematics in order to create memories and feelings that lead to positive emotional responses. Further, students must feel safe when dealing with mathematics, or else the natural tendency of the hypothalamus to flee stressful situations will result. If this happens, students may develop math anxiety. To avoid this situation, teachers must make mathematics learning environments inviting, engaging and safe for these adolescent learners in the intermediate grades who are experiencing so much change at this point in their lives. A sure way to make math inviting to adolescent learners is to demystify math content by making math lessons real, true to life, and hence meaningful – through the use of readily available real life concrete objects or manipulatives.
Evidence from Research Literature
Several research studies discuss the overall benefits that accrued from the use of manipulatives in mathematics teaching, and in many cases the improved student performance that resulted. For example, Sowell (1989) looked at 60 studies to determine if mathematics instruction with concrete and pictorial manipulatives was an effective method of instruction. The students in the 60 studies ranged from kindergarten to college-age, and mathematics content was varied. Sowell discovered that with long-term use of manipulatives, students’ mathematics achievement improved regardless of age. In addition, students’ attitudes towards mathematics also improved as a result of the use of manipulatives. However, it was noted that this positive effect was evident only when teachers were knowledgeable about manipulative usage.
Hinzman (1997) investigated 27 grade eight algebra students’ mathematical scores when hands-on manipulatives and group activities were used to teach. Over an eighteen-week period, students in one class participated in activities using concrete materials that related to the curriculum and were given pre- and post-tests. These were compared to students that did not use concrete materials. It was found that grades were enhanced by the use of manipulatives. More interesting findings were the survey results regarding students’ perceptions and attitudes toward mathematics. Students that participated in hands-on activities developed more positive feelings toward mathematics than they had in previous years when they did not use hands-on activities. In addition, when these students were asked on the survey if they were willing to take higher levels of mathematics in school, close to 90% of both females and males indicated that they would. This provides evidence that hands-on learning not only helps to increase students’ scores in mathematics, but also their feelings toward the subject and the possibility that they would take higher level mathematics-related courses in their future careers.
Garrity’s (1998) action research on middle-class high school students, also shows positive evidence that when students are allowed to use concrete materials to explore mathematical concepts, mathematics scores improve. A program for increasing adolescent understanding of geometry problems was created to help students visualize geometry problems more easily. The study found that students had difficulty with geometry for several reasons. Some of these reasons included inadequate middle school instruction, lack of motivation and negative feelings toward mathematics. In addition, students claimed that they had problems with visualization because their current geometry instruction focused too highly on the memorization of facts and that the opportunity for using hands-on materials were few, if at all. Students used geoboards to explore and solve geometry problems in small co-operative groups. Results, through interviews and journal entries by students, indicated that students enjoyed this methodology and preferred hands-on learning opportunities. Furthermore, it was found that paper and pencil test scores also improved, thereby showing student understanding of the concepts taught.
Clements (1999) cites research to support the proposition that students who use manipulatives outperform those who do not use manipulatives on retention and problem solving tasks. The author points to the fact that when students are allowed to use manipulatives there are also attitudinal gains because these students develop more positive attitudes toward mathematics.
A summary of the Handbook of Research on Improving Student Achievement published by Grouws and Cebulla (2000) looked at the effective strategies necessary to improve student learning. The authors summarized the findings on manipulatives and stated that, “long-term use of concrete materials is positively related to increases in student achievement and improved attitudes towards mathematics” (p. 27). In addition, they cited a study by Suydam and Higgins that suggested the use of manipulatives “produce greater gains than not using them” (p. 27). They suggested, therefore, that teachers should regularly use manipulatives in mathematics instruction so as to give students “hands-on experience that helps them to construct useful meanings for mathematical ideas” (p. 28).
Moyer (2001) maintains that manipulatives are a tool that students can use to construct their own insights regarding math concepts. In addition, Moyer cites several studies that show students who use manipulatives achieve better results than those who do not. Furthermore, this effect is greater for students when their teachers are experienced in the use of manipulatives. Ten middle grade teachers’ use of manipulatives for mathematics instruction over a school year was examined. The teachers participated in a workshop to teach them how to use various manipulatives. Throughout the school year, these teachers were observed in class as well as interviewed. The teachers reported that they found teaching more enjoyable because their students enjoyed the experience as well. As a result of using manipulatives during lessons, these teachers also reported that their students were more active and engaged in learning. Moyer found overall, that manipulatives were effective for instructional use when teachers chose appropriate manipulatives for their lessons and when teachers had a firm grasp on mathematical concepts, as opposed to mathematical procedures. This finding points to the need for further teacher training in this area if students are to enjoy the full potential of the use of manipulatives.
Touch Math is a math series that uses manipulatives. In an action research by Wisniewski and Smith (2002), the effectiveness of this program in improving mathematics achievement of special needs students in third and fourth grades was examined. Over a 14-week period, students received 45 minutes of mathematics instruction a day, of which 20 minutes were spent on the use of Touch Math. All students that participated in the study improved on accuracy from pre- to post-tests of the popular Math Mad Minute, a paper and pencil math test that measures procedural knowledge. If the use of manipulatives can improve the achievement of students that struggle with learning, it must also be effective for all learners.
Cass, Cates, Jackson and Smith (2002) using a multiple baseline design, studied the performance of students learning the concepts of area and perimeter. Students were having difficulty completing problem-solving tasks with area and perimeter. Rural students with learning difficulties in grades seven to ten, were trained for 15 – 20 minutes daily with geoboard strategies to solve problems involving area and perimeter. During baseline study, no student could successfully solve problems. After intervention, there was an increase in the number of problems students could solve. Once students could solve 80% of the problems correctly in three straight days, researchers deemed this evidence of improvement. Maintenance checks were done over a three-week period, which included a gap over Christmas break. Students continued to display knowledge and understanding of the concepts of area and perimeter by scoring between 90 and 100 % on paper and pencil tests. The researchers point to this as evidence that concrete materials “resulted in a fairly rapid acquisition and maintenance of basic perimeter and area problem-solving skills” (p. 107).
Middle school students with mathematics disabilities were studied by Butler, Miller, Crehan, Babbitt, and Pierce (2003) during the instruction on equivalent fractions. Fifty students in grades six through eight in a large urban setting were separated into two treatment groups, concrete-representational-abstract (CRA) and representational-abstract (RA). Ten lessons were given to both groups where the CRA students were instructed using manipulatives and the RA group simply used drawings to show equivalent fractions. Pre- and post-tests were administered at the end of the study. Data analysis revealed that students in the CRA group had better knowledge and understanding of equivalent fractions than the RA group.
Classroom Examples and Benefits of Manipulatives
For the effective use of manipulatives in math teaching, teachers must be familiar with specific manipulatives and to what lessons they are best suited. Explained below are some of the common manipulatives found in the intermediate math kits used in the Greater Essex County District School Board in Ontario, and the math strands that are best taught through their use.
1. Algebra Tiles™ – These are rectangular shapes that provide models of variables and integers. They usually consist of x and y tiles that use different colours to represent positive and negative values (see Appendix A). These tiles are helpful to intermediate students because they bridge the gap between the abstract ideas of equations, integers and algebraic expressions and a concrete visual model. Appropriate math strands addressed by algebra tiles are Number Sense and Numeration, and Patterning and Algebra. The tiles are particularly effective as concrete math teaching resources because they display a visual representation of a variable. From our classroom experience, we observe that many Grade 8 students struggle with the idea that a variable can be any number. Appendix B provides an example of a lesson that can be used for grade 7 or 8 students as they start to solve simple equations. The benefit of these tiles is that students can actively manipulate or move tiles from one side of the equation to the other.
Algebras Tiles™ are also useful for the teaching of integer concepts. We find that most Grade 8 students struggle with the concept that there are numbers that are less than zero. Drawing relationships to temperature and debt are helpful to show that numbers can exist below zero, but students benefit most by seeing a visual representation of this concept. The different colours of the tiles represent positive and negative values. We find that one concept that comes across most successfully with grade 8 students using Algebra Tiles™ is the zero principle, or the sum of equal opposite integers is zero. When students see a -1 and +1 tiles in front of them, they can see that their sum is zero. Often, teachers use a number line to show this, but our experience indicates that some students struggle with the idea of movement on an integer number line. A strong understanding of the zero principle makes it easier for students to add and subtract integers. Students can use the tiles to add, for example, +7 and -9 and see that the answer is –2, because of the zero principle.
2. Connecting Cubes - These are interlocking cubes in an array of colours (see Appendix C). They may be of various heights and weights. These cubes help students develop a spatial sense. The cubes are helpful in a variety of strands such as Number Sense and Numeration, Geometry and Spatial Sense, and Measurement. They are useful as a non-standard measurement for ratio explorations and are also easy to use in probability experiments, and patterning exercises.
Connecting Cubes can be used to create and show topographical models, or top, side and front views of three-dimensional objects. Many students struggle with spatial problems, so this is a way to facilitate learning for them and other students who find it difficult to form mental images. Connecting Cubes can be used to show area and volume situations concretely. The cubes are particularly beneficial for showing the concept of “volume” because students can see clearly that volume takes up three dimensions of space rather than the two dimensions that “area” takes. These cubes can be useful to show how a fixed perimeter can be manipulated into different forms as well. During actual classroom use, we also found that these cubes are very helpful in showing proportions and equivalent fractions as students easily see how 1/3 can equal 2/6. They also see the pattern, and figure out that 3/9 is the same as 1/3 and 2/6. Similarly, students can see the pattern and continue it to create the next proportion.
3. Geoboards - This plastic square tool with pegs is used to hold elastics into different shapes and sizes (see Appendix D). This tool is valuable for a number of reasons. It is well suited for explorations in Geometry and Measurement, as students easily carry out investigations on area and perimeter using the board. Students see how shapes of objects can change even when the perimeter or area is held constant. Additionally, known concepts such as the area of a square or rectangle can be used on a Geoboard to show how the area of a triangle can be calculated.
Also, students can manipulate variables such as the lengths of the sides of a right triangle to demonstrate the Pythagorean theorem. We have found that this is useful also for helping students to see that the areas of the squares of the legs of a right triangle when added together equal the area of the square of the hypotenuse. This is useful because not all students understand the traditional Pythagorean formula: (a² + b² = c²). In addition, there is a variety of Geoboards that has an additional side that is in the form of a circle. This side is useful for the investigation of relationships between the circumference, diameter, and radius of a circle. Again, by manipulating the sizes of circumference and diameter, students can draw the important conclusion that when they divide the circumference by the diameter, the answer is always 3.14 or pi, a constant.
4. Pattern Blocks – These wooden blocks have different colours and shapes. Included in the set are squares, equilateral triangles, trapezoids, hexagons and rhombi (see Appendix E). The shapes are related as well, in that six equilateral triangles equal one hexagon. These tools are helpful for investigations in fractions, angles and geometric transformations. We have used these blocks to teach addition and subtraction of fractions. It is helpful because of the relationships between shapes – students can see how denominators must be the same when adding and subtracting fractions.
Sources of Manipulatives: For Teacher and Student Guidance
In addition to the relevant books and articles in the research literature referenced in this paper, we would like to highlight some crucial Internet sites with brief descriptions of their usefulness to both teachers and intermediate-level students.
1. http://matti.usu.edu – National Library of Virtual Manipulatives from Utah State University (1999-2007) offers tutorials and provides teachers and students with a set of virtual manipulatives that are interactive. Funded by the National Science Foundation since 1999, the site gives activities for students from kindergarten to grade 12. The philosophy of the site is that students of mathematics are not meant to be spectators, but should be active participants in mathematics lessons. The site is organized efficiently and is broken down into mathematical strands and age levels. In addition, each activity is linked to National Council of Teachers of Mathematics standards. Additional information is provided for teachers and parents for each concept. This site is easy for intermediate students to use independently as well as through formal instruction.
2. http://www.ct4me.net/math_manipulatives.htm - “Computing Technology for Math Excellence” provides resources for the teaching and learning of mathematics from kindergarten to grade 12. It also provides resources for calculus, in line with the standards movement in education. This site offers links to various mathematics journals and a host of papers, professional development opportunities, initiatives and software. In addition, there is a link to other sites that provide students with virtual manipulative activities. This site is extensive and is a large source of information for teachers of mathematics.
3. http://www.shodor.org/interactivate/index.html - The Shodor Education Foundation’s (1994-2008) mission is to advance science and math education through the use of computational science, modeling and technology. The foundation created the Project Interactive site to provide teachers and students information and opportunities to engage in authentic use of technology in the study of mathematics. The site affirms that student learning is improved and teachers’ confidence is enhanced by using this site. The site follows the NCTM standards as well. There is a link to many activities meant specifically for intermediate students.
4. http://illuminations.nctm.org/About.aspx - This site was created by the National Council of Mathematics Teachers, a public voice of mathematics education that provides vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. Interactive activities are provided for students from kindergarten to grade 12. Most important in this site is the link to the NCTM’s Principles and Standards for School Mathematics.
5. http://www.mathsnet.net/index.html - MathsNet (1996-2008) features concepts in number, geometry, algebra, graphs, and data handling. There are curriculum materials, articles, books, and a section for download, including free software to explore topics interactively. The site seems overwhelming at first, but the activities and access to software makes it a worthwhile site for intermediate students.
6. http://arcytech.org/java/java.shtml - The Educational Java™ Site was created in 1997 with the purpose of providing Java™ Applets that can be used as tools to help and enhance the education of children both at school and at home. There are activities for students of all ages. The number of activities is limited, yet they are valuable. There is currently a link to science activities, and the site claims that music activities will be included in the future.
7. The Geometer’s Sketch Pad® - This software is widely used in most Ontario School Boards. The program allows students to create situations to model and show relationships in geometry. For example, students can create intersecting lines and manipulate variables to discover alternating and supplementary angle theories.
8. Students’ ingenuity and creativity – Having been exposed to the use of manipulatives in the classroom, students can take the process further by creating their own manipulatives from readily available household objects, or even ordinary paper. They can then use these self-made tools to help them with their assignments after regular school hours. It is common knowledge that children, especially young adolescents in intermediate-level classes, tend to tinker with objects and things around them, often surprising adults, and even themselves, by producing useful gadgets and equipment.
Barrier Factors and Limitations
Some factors that militate against the use of manipulatives in math teaching are cited and briefly discussed here.
Teacher reluctance: A major stumbling block to the incorporation of manipulatives into math lessons is that of teacher reluctance or the unwillingness of some math teachers to use them. This reluctance is due to the fact that some of these teachers are unwilling to break from the past since they are used to teaching their lessons the old, traditional way – the talk and chalk method. Also, some teachers believe that manipulatives are “toys” and that older students have no business with them. In addition, a good number of teachers are not knowledgeable about how to actually use these instructional tools, and are unaware of research that supports their use. Furthermore, some teachers are resistant to using manipulatives because of the classroom management issues involved. These teachers are afraid of the noise and movement that can result in a classroom when students are actively engaged in a learning situation using concrete materials, especially if students work in discussion or interactive groups. They feel that the apparent elevated noise level indicates that they have no class control. Also, it might appear to others that students are not doing “real math” when they use manipulatives. Traditionally, it is more acceptable to have students sit at their desks and work as individuals completing activities that build procedural knowledge, rather than working with peers in an active and engaging learning environment, even if research points to the value of this for adolescent learners. In addition, the management of the manipulatives themselves can be cumbersome and difficult. If there are not enough tools available for a class, the teacher has to devise a method to share or find additional tools in the school. This is added stress that many teachers may want to avoid.
Time constraints: There is no doubt that the use of manipulatives entails a good degree of investment in time. First, a teacher has to devote ample time for selecting appropriate materials for specific lessons, set up the materials, and then has to actively involve the students in using these materials in class. Also, the teacher who is conscious of covering the syllabus may find that the use of manipulatives creates a strain on the overall teaching time available.
Issue of cost or availability: Economic reasons also contribute to the lack of use of manipulatives in schools. These resources cost money and so school boards may not be affluent enough to afford manipulatives on an adequate scale. To ameliorate this situation, some school boards have resorted to buying some essential manipulative kits and storing them in their central resource centres from where the kits are signed out to teachers who ask for them. However, this arrangement intended to solve the problem of insufficient equipment, introduces two other problems of its own. First, many teachers frown at the idea of having to compete for the scarce resources, maintaining that if they are supposed to use manipulatives in their classes, the resources should be made readily available to them in their individual schools to save them the time and administrative encumbrance of having to apply for, and go to collect the resources from a central location. Secondly, teachers raise the key point that they need to get familiar with these materials before introducing them to their students, and so if the resources are not handy and readily available, they would not be able to acquire the necessary expertise and confidence that would enable them use the manipulatives effectively.
Lack of adequate user guides: Another limitation to the consistent use of manipulatives in math classes is the non-availability of adequate documents to guide teachers who want to use these resources. In some situations, where helpful guides have been provided, the issue of in-sufficiency again crops up because teachers do not have ready access to guides that provide grade-specific math lessons that include manipulatives. For example, as a follow-up to the Expert Panel’s Report in Ontario, instructional guides have been provided to help teachers with their math lessons. For intermediate grades, this guide is entitled, Targeted Implementation Planning Strategies, and offers manipulative-based lessons that are constructivist in orientation. The problem is that all intermediate teachers in one school must share the sole document provided for the school, since there are not enough of the documents for every teacher. How do four teachers in one school, for example, use one document to teach students mathematics all on the same day, especially when math lessons may be going on simultaneously in different intermediate classes during the same teaching slot?
Discussion and Recommendations
Having highlighted some impediments to the use of manipulatives in math classes, the logical question to ask is: how does the school system tackle these problems so that effective classroom practice is ensured and students achieve meaningful and transferable math knowledge? In our opinion, it is necessary that further research be conducted to find out the reasons why teachers do not implement sound research-based practices on a more regular basis. In particular, a survey of math teachers should be conducted regarding the reasons why they use or do not use manipulatives in their practice. It is possible that valuable information, and suggestions for improving the status quo, would emerge from this practitioner-based input and perspective, since as the saying goes, it is the one that wears the shoe that knows where it hurts. As Ezeife (2006) rightly noted, teachers are the key actors in the success or failure of any curriculum, so it is acutely important that we seek and listen to their views in all matters concerning educational practice, especially in an area such as math that causes headaches for many students in the school system. Furthermore, we recommend that research studies that measure the difference in student achievement in mathematics between schools that use and those that do not use manipulatives should be embarked upon, even at local board levels. Findings of such studies should be widely disseminated using all available teacher-administrator interaction forums, and educational media.
Since the issue of inadequate teacher preparation was highlighted as a limiting factor, school boards must be willing to train all their teachers the use of manipulatives. If boards invest money on the kits, they must also invest money in training teachers to use these resources properly and effectively. Also, manipulative kits must include only tools that are absolutely necessary. An effective kit should not include so many tools that it becomes overwhelming for teachers to use, and thereby they do not get used at all. This would amount to a waste of money and resources. Curricular documents that include lessons that link the curriculum to effective practice have to be made available to every math teacher. Fortunately, some school boards are beginning to see the necessity to provide support for teachers to enable them understand and implement research-based practices. These boards have hired specialist proficient teachers to coach teachers on site. They provide support to teachers within the school setting to implement curricular and instructional changes. This is a positive way to give teachers current and timely access to professional development. Also, it allows administrators the ability to target the needs of a school towards the goal of greater student achievement.
Finally, we would like to stress the need for a level of mathematics proficiency that elementary teachers must have in order to teach in our educational system. In our opinion, this may be the easiest and most immediate requirement to implement. This stance is borne out of the finding of Ross, et. al. (2002) who cite research evidence indicating that many elementary teachers are not equipped to teach math appropriately because of the generalist orientation of their training. The National Research Council (1989) had unambiguously directed attention to this deficiency, stating that elementary teachers “are drawn primarily from three-quarters of the population who dropped math after two or more courses in high school” (p. 38). The report goes on to mention that these teachers did not have positive experiences with math in high school, and that these feelings are eventually transferred to their students. Further, in a review of the cultural reasons why women shy away from math, Kenschaft (2003) indicates that many elementary teachers themselves received poor instruction in school. Again, if teachers themselves have not had positive experiences with math, how can they possibly teach this subject confidently and effectively to their students? Bearing all this in mind, we strongly recommend that Faculties of Education should start to require a certain level of mathematics proficiency for candidates who want to enter teacher preparation programs. If students fall below a certain level, they must take math courses, not about the math curriculum, or methodology of math teaching, but on actual mathematics content or subject-matter, in order to successfully obtain certification to teach. With well grounded and mathematically competent math teachers, we can work towards eliminating math phobia in young elementary school students in the first instance, and ultimately improving mathematical literacy in the general student population as students progress through the rungs of the educational ladder.
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Appendices
APPENDIX A
Algebra Tiles
The large red square represents the variable squared. The other of the square will be a different colour and represent a variable in the negative.
The red rectangle represents the variable. The other side represents the variable as a negative.
The small squares represent actual units or numbers, red being positive and yellow in this case negative. Therefore, one red square equals +1.
APPENDIX B
An example of using Algebra Tiles to solve equations:
- Retrieved May 13, 2006 from http://www.accd.edu/spc/
APPENDIX C
Connecting Cubes
APPENDIX D
Geoboards
APPENDIX E
Pattern Blocks
Relationships: Six green triangles create one hexagon. Two trapezoids equal one hexagon. Three rhombi create one hexagon.
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