Differences Between Virtual and Concrete Manipulatives - Essay Example

?LITERATURE REVIEW Durmus and Karakirik (2006) posited that inspirational educational activities and cognitive tools might develop active involvement in the teaching-learning process and promote their reflection on the concepts and relations under investigation. They also argued that usage of manipulatives not only enhances students’ conceptual understanding and problem solving skills but also encourages the development of positive attitudes towards mathematics since they evidently provide “concrete experiences” that focus concentration and increase motivation (p. 117). Physical or real-world features do not define a concrete experience in a mathematical context; it is by how significant the connection is to the mathematical ideas and situations. For example, a student might create the meaning of the concept "four" by building a representation of the number and connecting it with either real or pictured blocks. Virtual manipulatives, also called computer manipulatives, appear to offer interactive environments where students can manipulate computer objects to create and solve problems. Furthermore, perhaps because they are receiving instant feedback about their actions, students then form connections between mathematical concepts and operations. However, whether using physical or virtual manipulatives, it is necessary to connect the use of a specific manipulative to the mathematical concepts or procedures that are being studied (p. 119). Some researchers have observed that some of the constraints inherent to physical manipulatives do not bind virtual manipulatives. Use of models and/or manipulatives gives assessment of mathematical learning a cohesive connection to mathematical instruction (Kelly, 2006). Kelly’s study examines the relationship between mathematical assessment and the use of manipulatives. S/he also noted that teachers who consistently and effectively modeled the use of manipulatives in front of all students were demonstrating their belief that using concrete objects to understand abstract concepts was acceptable and expected of all students. Developing rubric-based assessments for manipulative-based activities with students and colleagues help assure that the assessments actually measures what they are teaching and practicing. The use of such assessments in combination with the use of manipulatives should build strong student investment in the teaching-learning process while developing deeper mathematical learning. Physical Manipulatives Relative to the teaching and learning of mathematics, physical, or concrete, manipulatives are three-dimensional objects used to help students bridge their understanding of the concrete environment with the symbolic representations of mathematics (Clements, 1999; Hynes, 1986; Moyer, 2001; Terry, 1996). There has been historical documentation of the use of manipulatives such as the abacus, counting sticks, and of course fingers, prior to the Roman Empire (Fuys & Tischler, 1979). Examples of teacher-made manipulatives include those that use materials such as beans, buttons, popsicle-sticks, and straws (Fuys & Tischler). Today’s teachers have access to a wide variety of commercially available manipulatives designed to aid in the teaching of most elementary mathematical concepts. Examples include Algebra tiles, attribute blocks, Base-10 materials, color tiles, Cuisenaire rods, fraction strips, geoboards, geometric solids, pattern blocks and Unifix cubes. The appearance of commercially made manipulatives in the United States increased during the 1960s after the work of Zolten Dienes and Jerome Bruner was published (Thompson & Lambdin, 1994). Many educators continue to view manipulatives as teaching tools that involve physical objects that teachers use to engage their students in practical and hands-on learning of mathematics. These manipulatives continue to be instrumental to introduce, practice, or remediate mathematical concepts and procedures. Concrete manipulatives come in a variety of physical forms, ranging from grains of rice to models for our solar system. They can be very simple or sophisticated, purchased or teacher/student-made. The manipulatives support the teaching and learning of mathematics from lessons that address number and operations, algebraic concepts and procedures, geometry, probability and even with college level courses such as calculus and linear algebra. In summary, physical manipulative materials are objects that can relate to mathematical concepts or procedures because they are touchable and movable by learners as they appeal to multiple senses (Heddens, 2005). Their use can help make abstract ideas and symbols more meaningful and understandable to students and widely support mathematics education. Educators have long recommended them (NCTM, 1989, p. 17). Manipulatives should be available for students to “feel, touch, handle, and move” (Reys, 1971). Research Related to the Use of Physical Manipulatives Just as small group instruction benefits students only if the teacher knows when and how to use this teaching practice, the value of using manipulative materials to investigate a concept depends upon how they are in use with students (Douglas A. Grouws and Kristin J. Cebulla, 2000). When selecting a manipulative to support the teaching or leaning of a concept or procedure, Hynes (1986) outlines four pedagogical criteria to consider. It needs to be a clear representation of the mathematical idea; be appropriate for the student’s developmental level - this included both his/her cognitive level and motor development; be of interest to the child; and be versatile. Without proper selection, manipulatives can become set tools that students use to go through the motions of the lesson without understanding the related mathematical topics (Hynes, 1986 p. 11-13). Although kinesthetic experience [using physical manipulatives] can enhance perception and thinking, understanding does not travel through the fingertips and up the arm” (Ball, 1992). “Simply using manipulatives does not guarantee a good mathematics lesson” (Stein & Bavalino, 2001, p.356). Many teachers do not understand how to help their students make the connection between concrete representations and the symbolic representations (Moyer, 2001). Without guidance, students may incorrect or misleading connections (Holt, 1982). For example, it is easy for adults who already understand how to interpret the numerator and denominator in a fraction to see the relationship between Cuisenaire rods and fractions. However, for young students this relationship may not be obvious (Holt). Without proper guidance from a trained teacher, students may make connections that are not beneficial to their understanding (Clements, 1999). Students need to learn how to reflect on their actions while using manipulatives (Clements, 1999). Bohan and Shawaker (1994) claimed that concrete manipulatives were effective, but that “transfer of learning” must take place if students are to reap the full benefits of concrete manipulative use. They also stated that, “Transfer of learning is a situation in which studying topic A will help in understanding topic B” (p.1). Furthermore, they recommended that this transfer should occur on several occasions such as during the concrete, bridging (also referred to as iconic), and symbolic stages of learning. The concrete stage is where mathematical situations are solved strictly using manipulatives without the use of any symbols. Simultaneous manipulation of objects and symbols occur at the bridging stage. At the symbolic stage, we begin working with symbols alone (p.1-2) There have been many writings In the last 15 years by Clements (1999, 2002) and his colleagues (Clements & McMillen, 1996; Clements & Sarama, 2002; Clements & Swaminathan, 1995), concerning rethinking what it means to use a “concrete” material in the mathematics classroom. In numerous articles, Clements has attempted to redefine “concrete” manipulatives. His concern about this topic revolved around the assumption that using physical manipulatives would ensure conceptual understanding. He asked, “… does concrete mean something students can grasp with their hands? Does this sensory character itself make manipulatives helpful?” (Clements & McMillen, 1996, p.270) He concluded that it did not. It is a combination of “… many separate ideas in an interconnected structure of knowledge” (Clements, 1999 p. 48; Clements & McMillen, 1996, p.271) that helps students achieve what he called integrated-concrete knowledge. That is, physical manipulatives may or may not be one of the materials that will help students achieve integrated-concrete knowledge (Clements, 1999). Taylor, (2001) on the other hand, questioned the use of concrete manipulatives as an effective tool in teaching all mathematics concepts, specifically, probability. In fact, Taylor’s study validated her null hypothesis: “There will be no significant difference between students who use concrete manipulatives and students who do not use concrete manipulatives regarding students’ learning skills and concepts in experimental probability”. Taylor’s study also suggested that concrete manipulatives were more useful in teaching certain math concepts than they are in teaching others. While the study refuted the effectiveness of utilizing concrete manipulatives in teaching probability, the manipulatives did help students learn incidental fraction concepts. Taylor also noted that students were comfortable with using concrete manipulatives to help them learn about probability concepts. The application of concrete manipulatives in teaching algebra has not been methodically studied and researched outside of relational word problems (Maccini & Hughes, 2000). The used the STAR algebra problem-solving strategy, in which problems are explored; words are translated into an equation in image form; the problem is represented using concrete manipulatives; the answer is found; and finally the answer is reviewed. Six learning-disabled secondary school students were involved in the study. As noted, the study asked students to generate an iconic representation of the problem as part of the learning process while they were studying how to represent and solve problems that involved the addition, subtraction, multiplication, and division of integers by applying concrete manipulatives as well as pictorial (iconic) displays. On a social validation form, almost all participants suggested that the manipulatives enabled them to understand what it means to solve problems involving integer numbers and recommended its use with other students. In their study, Thompson and Lambdin (1994) used concrete materials for two major purposes, to allow teachers and students to engage in and discuss about something concrete, discussions such as how to think about materials and to interpret the meanings of several actions with them Providing something upon which students can act. They stress that the focus should be mainly on what teachers want students to learn and not what teachers want students to do. “Concrete materials can be an effective aid to students’ thinking and to successful teaching. But effectiveness is contingent on what one is trying to achieve” (p.556-558). Some teachers are likely to view mathematics as isolated rules for manipulating symbols instead of as a unified whole. This leads to students’ misconceptions that stem from incorrectly applying concrete materials and as a weakness of the materials by teachers (Hall, 1998). According to Suydam & Higgins, (1977) physical manipulatives can have a significant positive effect on student achievement. In fact, a meta-analysis of 60 studies conducted by Sowell (1989) concluded that mathematics achievement increased when physical manipulatives were used over an extended period such as a school year or longer. Students who use physical manipulatives outperformed students who did not have access to those manipulatives (Clements, 1999). Another meta-analysis of 64 studies conducted between the years 1960-1982 found that students using manipulatives scored at the 85th percentile on standardized tests as compared to students not using manipulatives who scored at the 50th percentile (Parham, 1983). Hiebert and Wearne, (1988) established that students could use manipulatives in a rote learning behavior, having limited or no understanding of the mathematical concepts related to the procedures. In their study nine students in the fourth grade, ten students in the fifth grade, and ten students in the sixth grade learnt decimal concepts with the aid of Base-10 blocks as the primary tool used to support their understanding of the topics. They found that nearly all students “… established connections between the blocks and symbols that generalized approximately to extended notation” (p.99-122). Along similar lines, Hall (1998) concluded that concrete materials, because of the ease of relating actions on physical objects to mathematical procedures or concepts, could be a useful pedagogical tool that enables learners to move to applying the same operations to icons, characters and symbols. Clements and McMillen (1996) proposed that using manipulatives does not always guarantee conceptual understanding. Jackson (1979) identified several common misconceptions about using manipulative materials including the beliefs that; manipulatives always simplify the learning of mathematical concepts manipulatives are more useful in primary grades than in intermediate and secondary grades, and Manipulatives are mostly applicable for low-ability students and not for high-ability students. In short, employing manipulatives in a class is not straightforward. Good employment requires carefully defining the role of the teacher, the aims of the lesson and the potential that the use of manipulatives has to assist with the tasks involved. Clements and McMillen also claimed that students often fail to link their action with manipulatives to describe the actions. Jackson identified several common mistaken beliefs about the use of manipulative materials, including: that manipulatives do not necessarily simplify the learning of mathematical concepts, that the more manipulatives employed with a single concept, the better the concept is understood, and that manipulatives are more applicable mostly applicable for low-ability students and not for high-ability students. Heddens (2005) stated different arguments. He argued that using manipulative materials in teaching mathematics might help students learn to: link real world situations to mathematical symbols and concepts work as a team and in cooperation to solving problems discuss mathematical ideas and concepts verbalize their mathematical thinking make presentations in front of a large group realize that there are many different ways to solve problems understand that mathematical problems can be symbolized in many different ways understand that they can solve mathematical problems in different ways as from those demonstrated by their teachers Gravemeijer (1990) states that although results differ depending on what and how manipulatives work in learning situations, learning with manipulatives positively correlates with later development of mental mathematics (p.10-32). A good example of this is the use of Base-10 blocks, which is a widely known mathematical manipulative wherein ones are represented by small square tiles, tens by thin rods, hundreds by ten-by-ten flats, and thousands by a large ten-by-ten-by-ten block. The use of Base-10 proved to enhance students’ conceptual understanding of arithmetic operations (Fuson & Briars, 1990). Chassapis (1999) showed that the use of a compass in Geometry to support students’ learning about circles developed a better understanding of center and radius concepts in comparison to the understanding created by the use of conventional circle tracings and templates. In conclusion, physical manipulatives continue to work as important tools that allow students to reach higher levels of thinking. Students can use them to solve problems in non-routine ways. When used properly, by knowledgeable teachers, manipulatives can help students make connections between a concrete understanding of mathematical concepts and the corresponding abstract mathematical ideas (Stein & Bavalino, 2001). Virtual Manipulatives Advances in technology have given birth to a new generation of manipulatives—virtual manipulatives. Virtual manipulatives are computer-based images of objects that are similar to their three-dimensional counterparts. A user in a virtual computer environment can use and move these new objects (Skylar, 2008). That is, virtual manipulatives are hands-on models that students use in a virtual environment to model mathematical objects by entering, clicking, dragging and dropping computer objects into appropriate locations. With the evolution and creation of Java software, virtual manipulatives are becoming easier to create and place on the Internet. Many Internet sites exist and new virtual manipulatives increase regularly. Some of these applets are now available on hand-held devices such as iPads and smart phones. Schools and professional organizations are experimenting with their use in the classroom. Smart Boards now enable teachers and students to interact with virtual manipulatives in a very natural way in their classrooms. Virtual manipulatives are generally “interactive, web-based visual representations of a dynamic object that presents opportunities for constructing mathematical knowledge” (Moyer, Bolyard & Spikell, 2002, p. 373). “Computer based renditions of common mathematics manipulatives and tools …” are virtual manipulatives (Dorward, 2002, p. 329). Steen, Brooks and Lyon (2006) advocated that virtual manipulatives be viewed as more than just electronic replications of their physical counterparts. Their studies showed that virtual manipulatives typically included additional features that expanded on what a physical manipulative could offer. Steen et al. also stated that “some virtual manipulatives are able to present a representation that would not be easily made or even possible with physical manipulatives, an attribute shared with types of computer simulations” (p. 375). Virtual manipulatives exist on the Internet as applets, or smaller versions of application programs. Students can manipulate these dynamic, pictorial objects by moving the computer mouse. A large number and variety of virtual manipulatives are accessible on the Internet, including those found on the websites of the National Council of Teachers of Mathematics (http://illuminations.nctm.org/), the National Library of Virtual Manipulatives (http://nlvm.usu.edu/), and the Shodor Education Foundation (http://www.shodor.org/interactivate/). Mayer & Anderson (1992) point out that, “the general design structure for virtual manipulative applets is to include verbal codes (i.e., letters, numbers, and words) and visual codes (i.e., pictures, movable 2-D and 3-D objects)” presented simultaneously, thereby increasing the effectiveness of multimedia instruction (p.444-452). This indicates that mathematical environments that provide multiple, active systems of codes have a greater potential for enhancing students’ learning capabilities at least two mental representations are available instead of just one. Therefore, virtual manipulatives are an exclusive externalized representational form. Goldin (2003) explained that illustrations are arrangements of symbols, signs, visual characters, icons, or objects that are a representation of something else. Virtual manipulatives perhaps considered an exclusive form of representation or a combination of a number of representations. Goldin and Shteingold (2001) stated that students’ ability to convert the multiple representational systems determines their abilities to model and grasp mathematical constructs. Virtual manipulatives may be an essential element of mathematics learning and teaching as constituents of representational systems. Representations are essential to students' understanding of mathematical concepts and relationships. They enable students to respond to mathematical concepts and arguments, and to share understanding with each other. Additionally, they enable students to familiarize themselves with connections between related concepts and apply those connections to practical mathematical problems. Moyer-Packenham, Salkind, & Bolyard (2008), thought that the skills required to manipulate virtual manipulatives highlighted the teacher’s role as a guide (rather than as a transmitter of facts) when teaching mathematics interactively with virtual manipulatives. Visual representations of concepts and relations help learners to gain insight in mathematics. Virtual manipulatives enable as much engagement as physical manipulatives do since they are actual models of physical manipulatives, such as Tangrams and Geoboards (Dorward & Heal, 1999). According to Durmus & Karakirik (2006), virtual manipulatives are interchangeable with physical manipulatives in mathematical explorations because manipulatives do not make mathematical concepts “touchable” but outline the prominent features of the task under study. They also suggested that virtual manipulatives might offer additional benefits to those provided by physical manipulatives by discarding some of the limitations that physical manipulatives may place on a task, (Durmus & Karakirik). In 2008 Skylar noted that virtual manipulatives allowed students to gain a deeper understanding of complex mathematical concepts and, therefore, facilitated memory retention of those concepts. Haistings (2009) after completing his/her study recommended that when using virtual manipulatives for the first time, students be given sufficient time to become familiar with using the computer-based manipulatives. He/she also noted how important it was for teachers to model the proper use of the virtual manipulatives for all of their students. It was not until the third or fourth day of Hasting’s study that all students were comfortable and independently used the virtual manipulatives with ease. All of the participating teachers commented on the need for more time in the computer lab in order to keep this type of practices consistent. An assistive technology, such as virtual manipulatives, can address a range of learner difficulties. Many manipulative Web-based environments direct students to engage with the material, provide guiding questions, and create multiple opportunities for success. Students with a history of struggling with mathematics will be able to use the virtual manipulatives to verify their thinking and see immediate success. Conversely, students who are struggling with a concept can request a model demonstration, obtain immediate feedback on incorrect answers, or request additional instruction or explanations. It is a belief that development of virtual manipulatives will enhance the environment of learning mathematics. As with any technology, teachers need to plan for the effective use of virtual manipulatives. For example, they must consider how they will support students before, during, and after the instructional activity as well as the different types of support needed for introducing a topic, practicing or applying a skill, or remediating a skill or concept (Zorfass, Follansbee, & Weagle, 2006). Depending upon the instructional goal, teachers can determine how the virtual manipulative is introduced, monitored, and supported. Reimer and Moyer (2005), stated that student questionnaire and attitude surveys demonstrated that virtual manipulatives enabled students to better understand and learn about fractions by offering prompt and specific reviews and feedback enhanced students’ enjoyment while learning mathematics, and were much easier and faster to employ as compared to paper-and-pencil methods They also argued that virtual manipulatives are more effective than physical manipulatives in classroom teaching because physical manipulatives are dependent on the teacher’s ability to make these concrete concepts to abstract symbols connections explicit. Virtual manipulatives are much more than games that integrates into the mathematics curriculum. Manipulatives help students connect abstract ideas with concrete models. Teachers have been incorporating physical manipulatives into their classroom instruction for years, but the greatest barriers to their broader use have been having adequate numbers of manipulatives and the time involved with getting them out, setting them up and putting them away. Another barrier to their full use is the difficulty – or in most cases the impossibility - of sending them home with students for out of school use. Virtual manipulatives can address or limit the impact of several of these barriers. In particular, many virtual manipulatives are accessible by students from their homes and used as a home/school connection for math homework (Lindroth, 2005). Using virtual manipulatives in connection with instruction is a part of the increasing use of technology in connection with mathematics teaching and learning. Although teachers and researchers are still measuring the impact that the use of virtual manipulatives have on students' learning, there is evidence that shows virtual manipulatives offer some unique advantages over the use of physical manipulatives and can be effective in supporting the learning of mathematics. (Crawford & Brown, 2003; Lee & Chen, 2008a; Reimer & Moyer, 2005; Steen, Brooks & Lyon, 2006; Stellingwerf & Van Lieshout, 1999; Suh & Heo, 2005). While using virtual manipulatives, children can apply mathematical concepts and explore processes for representing those concepts (Wright 1994; Clements & Sarama 2003a; Moyer-Packenham et al. (2008). According to Zbiek, Heid, Blume, & Dick, (2007) virtual manipulatives are cognitive technological tools. The characteristics of virtual manipulatives as a cognitive tool are evident as they enables users to act as representations of different objects on the virtual manipulatives, with the consequences of the user’s activities resulting in visual on-screen response from the virtual tool. Even though virtual manipulatives to some extent resemble their relevant physical manipulative, as cognitive tools, virtual manipulatives have inimitable characteristics that go beyond the capabilities of physical manipulatives. In this aspect, Moyer-Packenham et al. (2008) said that their potential is thus improved “… for mathematically meaningful actions by users and influences the user’s learning” (p.203). Theories of learning that include the use of media may provide insight into why researchers are finding constructive initial results in studies involving the application and possible benefits of using virtual manipulatives in classrooms (Moyer-Packenham et al., 2008 p.205). Despite limitations to this base of research on virtual manipulatives, it is important to note that classroom studies and dissertations have manifested the unique features of these tools relative to the teaching of mathematics. If possible, introduction and use of physical manipulatives should be before virtual manipulatives. Students, teachers, and the researcher agreed that students were successful when using the virtual Base Blocks Addition because of similar exercises practiced in the classroom with physical Base-10 blocks. The concept of Base-10 blocks and using them to model numbers was not new when students began to use the virtual version. This allowed students to easily build numbers with virtual Base-10 blocks and focus on the action of combining numbers for multi-digit addition. Teachers in a more recent study (Moyer-Packenham et al. (2008) also expressed a preference for using physical manipulatives prior to virtual manipulatives. Research Related to Using both Physical and Virtual Manipulatives Zacharia, Olympiou, & Papaevripidou (2008), investigated the comparative value of experimenting with physical manipulatives in a sequential combination with virtual manipulatives, by using physical manipulatives prior to the use of virtual manipulatives, and of experimenting with each alone, with respect to changes in students’ conceptual understanding in the domain of heat and temperature. The researchers used a pre–post-comparison study design that involved 62 undergraduate pupils who completed a basic course in physics. There was allotment of each respondent to one control and one experimental group. All students used the same inquiry-oriented study materials. The results of this study indicated that experimenting with a combination of physical manipulatives and virtual manipulatives increased and enhanced students’ conceptual understanding much more than experimenting with just physical manipulatives. The use of virtual manipulatives was identi?ed as the cause of this differentiation. Terry’s (1996) research examined 102 students in grades two through five using Base-10 blocks and attribute blocks and discovered that when learners applied a combination of both physical as well as virtual manipulatives, they showed remarkable improvements during the pre-test and post-test when equated to learners who applied either physical manipulatives or virtual manipulatives alone. Overall, these results have pointed out that learners applying virtual manipulatives, either in combination with physical manipulatives or alone, express gains in mathematics achievement and understanding and appear to be more occupied and on task as mentioned by Moyer-Packenham et al. (2008). However, the application of virtual manipulatives throughout the core of a mathematics lesson appeared to be different from the way other scholars had described teachers’ uses of physical manipulatives. Moyer’s (2001) study reported observing teachers associating use of physical manipulatives with ‘having fun’ and not laying emphasis on the ‘real mathematics’ in the lesson. Even though Moyer’s classroom observations showed that 30% of the lessons in the curriculum employed physical manipulatives for games, it also indicated that games were only 2 out of 95 lessons that involved the use of virtual manipulatives in the curriculum. This drastic difference showed a remarkable gap between the way teachers in Moyer-Packenham et al. (2008) chose to apply virtual manipulatives in their lessons and the way tutors in Moyer’s (2001) study employed physical manipulatives. Of nine studies conducted before 1999, three provided evidence to suggest that students who used virtual manipulatives experienced higher achievement and conceptual understanding in mathematics than those that used physical manipulatives or no manipulatives (Kieran & Hillel, 1990; Smith, 1995; Thompson, 1992). Two studies provided evidence that suggested students who used both virtual and physical manipulatives showed an increase in their conceptual understanding of mathematics (Ball, 1988; Terry, 1996). Four of the nine studies found no statistically significant difference in achievement between students that used physical manipulatives alone, virtual manipulatives alone, a combination of both physical as well as virtual manipulatives, and lastly, no manipulatives (Berlin & White, 1986; Kim, 1993; Nute, 1997; Pleet, 1990). Another recent study compared the use of virtual manipulatives versus concrete manipulatives involved third graders students learning about fractions. The study found that the group using virtual manipulatives significantly outperformed the group using concrete manipulatives (Suh, 2005). However, another study of third graders learning about algebraic relationships found no significant difference in student achievement between those that used either virtual or physical manipulatives (Suh & Moyer, 2007). Brown (2007) designed a study to explore the impact of applying virtual manipulatives and concrete manipulatives on elementary students’ achievement and understanding of concepts about equivalent fractions. The first interest of the study was to detect whether or not students who used virtual manipulatives would out-perform those who used concrete manipulatives across the researcher/teacher-produced post-test. A secondary interest of the study was to examine students’ attitudes concerning the use of manipulatives during mathematics lessons. The study, which involved 48 sixth graders studying fractions, found the group that used concrete manipulatives significantly outperformed the group that used virtual manipulatives (Brown, 2007). Summary In addition to analyzing differences between student performance on pre and posttests, many of these studies looked at qualitative differences between the uses of the two methods. A common theme found among those studies was that students seemed to stay on task more frequently when using virtual manipulatives over other methods (Hunt, Nipper & Nash, 2011) Most of these studies that investigate the use of physical and virtual manipulatives were studies involving elementary school students. Only a few involved middle school students. Howard, Perry and Lindsey (1996) presented some primary baseline data on the use of manipulatives in secondary school mathematics classrooms. They showed that the use of manipulatives in the selected classrooms was particularly low in comparison to such use in primary school mathematics lessons and this was the picture of early stage. Generally, there have not been enough studies conducted with high school students and the use of manipulatives in connection with mathematics courses. In addition, most of the studies concentrated on one content area such as geometry (ref), or only one topic, such as equivalent fractions (ref). Only a few studies investigated instruction using virtual or concrete manipulatives across more than one content area (refs). There seems to be a gap in the available research with regard to these types of comparisons. Although most initial studies involving virtual manipulatives focused on the difference between physical and virtual manipulatives, as additional research was developed and published, the need for more controlled classroom research became evident. After completing a two-classroom study using one virtual manipulative, Moyer, Niezgoda, and Stanley (2005) recognized this need in their conclusion: “…it is important to investigate problems and questions using different technologies and forms of representation in real classrooms and to explore effective ways to use these technologies in teaching mathematics to all”. BIBLIOGRAPHY Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of math education. American Educator, 16(2), 14-18, 46-47. Bohan, H. & Shawakar, P. (1994). Using Manipulatives effectively: A drive down rounding road. Arithmetic Teacher, 41 (5), 246-248. Bolyard, J. J. (2006). A comparison of the impact of two virtual manipulatives on student achievement and conceptual understanding of integer addition and subtraction. Doctoral dissertation, George Mason University, 2006. Dissertation Abstracts International, 66(11), 3960A. Brown, S. E. (2007). Counting blocks or keyboards? 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RESOURCES Association for Middle Level Education - http://www.amle.org/Research/ResearchSummaries/Mathematics/tabid/1832/Default.aspx Contemporary Issues in Technology and Teacher Education - http://www.citejournal.org/vol8/iss3/mathematics/article1.cfm Education Resource Information Center – http://eric.ed.gov/ERICWebPortal/recordDetail?accno=ED496007 National Council of Teachers of Mathematics Illuminations - http://illuminations.nctm.org/ National Library of Virtual Manipulatives - http://nlvm.usu.edu/en/nav/vlibrary.html The National Number Project - http://www.cehd.umn.edu/rationalnumberproject/81_4.html Teacher Vision – http://www.teachervision.fen.com/math/teaching-methods/48952.html Wiley Online Library - http://onlinelibrary.wiley.com/doi/10.1002/tea.20260/abstract Read More