Introduction

As former primary/elementary students, future teachers and students of mathematics education, we have all been exposed to manipulatives and we understand they are important in the classroom. However, though we are able to name a few manipulatives, most of us probably do not know the history of manipulatives, the many varieties there are, the research supporting proper use of them, and the new developments in virtual manipulatives. For these and many other reasons, I chose to do my personal inquiry on this topic. As both a student of education and a future teacher, I felt it was important to understand these objects and their ability to bridge the gap between abstract concepts and concrete representations for students.

What are Math Manipulatives?

A mathematical manipulative is “an object which is designed so that the student can learn some mathematical concept by manipulating it” (Wikipedia, 2006). Students can use objects as counters, play with geometrical shapes to form other shapes, can build models to understand three dimensional figures, etc. In the manipulation of these materials, students are able to learn abstract concepts in concrete, hands on ways. These materials make even the most difficult mathematical concepts easier to understand for the student (Uttal, Scudder & DeLoache, 1997).

Manipulatives have been around for many years. One of the early versions of a manipulative, the abacus, can be dated back to 300 B.C. (About, 2005). This abacus, known as the Salamis Tablet, was used by Babylonians and was discovered in 1846 (About, 2005). Manipulatives have developed greatly from this early counting device.

A new push for the use of manipulatives occurred in the 19^th century when Pestalozzi lobbied for their use, eventually making manipulatives part of the mathematics curriculum in the 1930’s (Sowell, 1989). In the 1960’s, another resurgence of the use of manipulatives occurred, with a focus on the use of concrete objects and pictorial representations to help children better understand abstract mathematical ideas (Sowell, 1989). Now, manipulatives are available in almost every classroom around the world.

Many varieties of manipulatives are available. These objects can be commercially purchased, downloaded online, or with some creativity, can be made by both the teacher and students. Some examples of manipulatives include tangrams, interlocking blocks, Dienes blocks, counters, geoboards, fraction kits, base ten blocks, pattern blocks, geo strips, and fraction strips. Some of these manipulatives are shown below. For these and many other manipulatives, please visit the National Library for Virtual Manipulatives, located in the links section, as well as other useful websites. Also, manipulatives which can be photocopied are available at the back of our text book.

Examples of Manipulatives

Base Ten Blocks







Counters









Fraction Kit









Geo Strips








Geo Boards









Geometric Blocks











Manipulatives Kit












Tangrams








All images were collected using Google Images.

Values of Manipulatives

As previously stated, “manipulatives help children visualize abstract mathematical ideas (Heuser, 2000, p. 288). Students are able to use hands-on activities to create a knowledge base for mathematical thinking, allowing a greater understanding of the nature of mathematics, and some of its basic concepts, at an earlier age (McCarty, 1998). This is based on the findings of people such as Piaget, who helped prove that young children at the Primary and Elementary grades think at the concrete level. Therefore, the use of concrete objects in teaching abstract ideas would bring these abstract ideas down to the students’ concrete level, making problems tangible and tractable for these young learners (Uttal, Scudder & DeLoache, 1997).

The value of mathematical manipulatives can also be seen in the work of Driscoll, Sowell, and Suydam, who all discovered that students who use manipulatives outperform students who do not use them (Clements and McMillen, 1996). And this is not just true of students at the concrete level of thinking; students in all grade and ability levels, as well as students working in many topics, benefit from the use of manipulatives (Clements and McMillen, 1996). In the Driscoll, Sowell and Suydam study, retention and problem solving test scores were also improved if the students were exposed to manipulatives, and “attitudes toward mathematics [were] improved when students are instructed with concrete materials by teachers knowledgeable about their use” (Clements and McMillen, 1996, p. 270).

The ability for a manipulative to present different situations to students is important. Through the various manipulations they encounter with one form of manipulative, students begin to recognize common mathematical elements (Fennell and Rowan, 2001). This understanding of the concepts students learn can also be enhanced if students can transfer their understanding to various manipulatives using the same idea (Fennell and Rowan, 2001). Also, students using manipulatives to represent concepts “give[s] learners useful tools for building understanding, communicating information, and demonstrating reasoning” (Fennell and Rowan, 2001, p. 289). From this evidence it can be seen that manipulatives can be an asset to any classroom.

Warnings for Manipulative Use

Though the use of manipulatives is encouraged and the benefits are obvious, there are some concerns as concrete manipulatives cannot themselves guarantee meaningful learning. Students may learn to use manipulatives in a rote manner, learning the steps to use them but making no connections between the manipulatives themselves and mathematic concepts (Clements and McMillen, 1996). It is suggested that students cannot just engage with the manipulatives but that they must reflect on their learning and be able to connect the manipulative with the mathematical concept being addressed (Clements and McMillen, 1996).

When introducing and using manipulatives in the classroom, “teachers must take into account how children do (or do not) understand symbolic relations” (Uttal, Scudder & DeLoache, 1997, p. 39). If the student cannot see the relationship between the symbol and its representation of a mathematical idea, the manipulative is almost useless. One way to make the connection easier for students is to link together instruction and manipulative use from the onset, possibly decreasing confusion and increasing the chance the student will relate the manipulative to its referent (Uttal, Scudder & DeLoache, 1997). Also, the teacher may need to provide extensive instruction and practice for students before the use of manipulatives is used, explaining the idea that an object can represent an idea.

The form of the manipulative is also a concern for teachers. Some educators may use highly colorful and attractive manipulatives in the hopes they will engage students and hold their focus. However, the opposite effect may occur. Children are likely to “focus more on the manipulatives as objects per se rather than on the relation of the objects to a concept or an alternate form of expression” (Uttal, Scudder & DeLoache, 1997, p. 49). Aside from interesting objects, it is also suggested that you not use objects that children deeply care about. For example, if you were to use legos as counters, the student may focus on the lego as their play toy rather than on the lego as a tool for representation. Therefore, teachers may want to choose manipulatives that are used only for mathematics learning and that are not overly distracting or entertaining to students.

Virtual Manipulatives

In the every present growth of technology in our world, nothing goes unaffected. This is also true regarding manipulatives, as virtual manipulatives are now available. Virtual manipulatives can be defined as “interactive, web-based representations of a dynamic object that presents opportunities for constructing mathematical knowledge” (Moyer, Bolyard & Spikell, 2002, p. 373). These manipulatives are still concrete, though they are not “physical” (Clements and McMillen, 1996). Though students are not able to physically pick up virtual manipulatives they can still move the objects on the computer screen and interact with them. Teachers can integrate these representations into their classroom because they can “be more manageable, “clean”, flexible, and extensible” (Clements and McMillen, 1996, p. 271). The biggest advantage of virtual manipulatives is their interactive capabilities. These manipulatives allow students to see mean and relationships based on the results of their actions (Moyer, Bolyard & Spikell, 2002).

Virtual manipulatives can be both static and dynamic. Static manipulatives, or those which are simply “virtual”, offer pictorial or other forms of representation, however no interaction is permitted for students (Moyer, Bolyard & Spikell, 2002). Dynamic, or true virtual manipulatives, however, allow the students to move, count, change and work with objects on the screen to build an understanding of the concept presented (Moyer, Bolyard & Spikell, 2002).

Virtual manipulatives can also be defined as being static images or representations, computer-manipulated images or representations, or virtual manipulative websites. Those which are static images show a pictorial representations which may change, but the user is unable to change or manipulate the image (Moyer, Bolyard & Spikell, 2002). Computer-manipulated images allow the image or object to be manipulated in response to a students answer or action (Moyer, Bolyard & Spikell, 2002). Finally, a true virtual manipulative offers the physical movement of objects in virtual form, allowing for the greatest amount of manipulation and interaction for the user (Moyer, Bolyard & Spikell, 2002).

According to Clements and McMillen (1996), computer based manipulatives, or virtual manipulatives, offer many advantages. These are:

Computer manipulatives allow for changing the arrangement or representation.

Computers store and later retrieve configurations.

Computers record and replay students' actions.

Computers change the very nature of the manipulative; Students can do things that they cannot do with physical manipulatives.

Computer manipulatives build scaffolding for problem solving.

Computer manipulatives may also build scaffolding by assisting students in getting started on a solution.

Computer manipulatives focus attention and increase motivation. (p. 272-273)

Another great value for virtual manipulatives is their availability and free expense. Often manipulatives can be expensive and many teachers may not be able to afford access to many varieties of manipulatives. By simply searching online teachers can be provided with many manipulatives to use at their choosing (Hodge and Brumbaugh, 2003). Also, these manipulatives can be made available at home, helping both the struggling student and parents who would normally not have access to such tools. Teachers will be able to send home homework with manipulatives that may be of assistance for students if they know they will have access to the materials at home (Moyer, Bolyard & Spikell, 2002). This may also help older students who look at using physical manipulatives as “playing with blocks”. Older students can view virtual manipulatives much like a computer game, aiding the student while retaining confidence in the learner (Moyer, Bolyard & Spikell, 2002).

Research

Due to the popularity of manipulatives over the last twenty years, many researchers have conducted studies to test their true value in the classroom. This began in the 1960’s when Dienes and Bruner theoretically justified the use of manipulatives through their research (McCarty, 1998). Since these early studies “[m]any studies of the effectiveness of using concrete learning materials have been conducted since then, giving rise to agreement that effective mathematics instruction in the elementary grades incorporates the liberal use of concrete materials” (McCarty, 1998, p. 368). Many of the studies which have been conducted have focus on this idea of the value and importance of manipulatives.

A study completed by Resnick and Omanson investigated the difficulty children have when trying to form a connection between manipulatives and the mathematical concepts they represent (Uttal, Scudder & DeLoache, 1997). This research “addressed whether and how third graders established connections between different forms of mathematical expression” (Uttal, Scudder & DeLoache, 1997). Resnick and Omanson studied how children comprehended mathematical information in both written and manipulative form. Children were evaluated in both domains throughout the school year and most showed progress in using Dienes blocks and other manipulatives (Uttal, Scudder & DeLoache, 1997). According to this research “most children could interpret the Dienes blocks expressions for numbers involving hundreds, tens, and ones [and] moreover, they could solve multidigit addition and subtraction problems with the blocks” (Uttal, Scudder & DeLoache, 1997, p. 45).

However, when children were asked to interpret written expressions of problems which were similar to the block problems, they did not show an advantage (Uttal, Scudder & DeLoache, 1997). Students could not solve problems which were much simpler than those practiced (ex. students could solve 103 + 52 but had difficulty with 12+14) (Uttal, Scudder & DeLoache, 1997). This showed students did not understand the concept of addition but could work with their manipulative to solve the previous problem. Use of the Dienes blocks was almost separate from written solutions as seen in results in which children who performed best with Dienes blocks performed the lowest in written problems (Uttal, Scudder & DeLoache, 1997).

Students had learned how to use Dienes blocks but could not relate the manipulatives to other problems of a similar nature, showing how important it is for students to draw a connection between mathematical concepts and manipulatives. If children do not learn how to make this connection they will have to learn how to solve a problem using two separate systems, creating more confusion than aiding in learning (Uttal, Scudder & DeLoache, 1997). Teachers must therefore provide instruction which is guided and constrained when teaching students to work with manipulatives, ensuring students understand concepts and not simply how to work with manipulatives (Uttal, Scudder & DeLoache, 1997).

Choosing Manipulatives

Once we, as teachers, understand the importance of manipulatives and the cautions we must take in using them, we need to learn how to select the best manipulatives for our classroom and our students. Clements and McMillen (1996) make many suggestions to help teachers when choosing manipulatives:

Select manipulatives primarily for children’s use. Teacher demonstrations with manipulatives can be valuable; however, children themselves should use the manipulatives to solve a variety of problems.

Select manipulatives that allow children to use their informal methods. Manipulatives should not prescribe or unnecessarily limit students' solutions or ways of making sense of mathematical ideas. Students should be in control.

Use caution in selecting "prestructured" manipulatives in which the mathematics is built in by the manufacturer, such as base-ten blocks as opposed to interlocking cubes.

Select manipulatives that can serve many purposes.

Choose particular representations of mathematical ideas with care. Perhaps the most important criteria are that the experience be meaningful to students and that they become actively engaged in thinking about it.

To introduce a topic, use a single manipulative instead of many different manipulatives.

Select computer manipulatives when appropriate. (p. 278).

Also, I feel it is important to select manipulatives which are appropriate for your class. Only you will know how your students learn, what manipulatives will work best for them and how you can introduce and use manipulatives to meet the needs of your many students. Getting to know your students and their needs will help accomplish this.

Using Manipulatives Effectively

Clements and McMillen (1996) also offer some suggestions for teachers on how to use manipulatives effectively in the classroom:

Increase the students' use of manipulatives. Most students do not use manipulatives as often as needed. Thoughtful use can enhance almost every topic. Also, short sessions do not significantly enhance learning. Students must learn to use manipulatives as tools for thinking about mathematics.

Recognize that students may differ in their need for manipulatives. Teachers should be cautious about requiring all students to use the same manipulative. Many students might be better off if allowed to choose their manipulatives or to use just paper and pencil.

Encourage students to use manipulatives to solve a variety of problems and then to reflect on and justify their solutions. Such varied experience and justification help students build and maintain understanding. Ask students to explain what each step in their solution means and to analyze any errors that occurred as they use manipulatives--some of which may have resulted from using the manipulative.

Become experienced with manipulatives. Attitudes toward mathematics, as well as concepts, are improved when students have instruction with manipulatives, but only if their teachers are knowledgeable about their use.

Some recommendations are specific to computer manipulatives:
Use computer manipulatives for assessment as mirrors of students' thinking.
Guide students to alter and reflect on their actions, always predicting and explaining.
Create tasks that cause students to see conflicts or gaps in their thinking.
Have students work cooperatively in pairs.
If possible, use one computer and a large-screen display to focus and extend follow-up discussions with the class.
Recognize that much information may have to be introduced before moving to work on computers, including the purpose of the software, ways to operate the hardware and software, mathematics content and problem-solving strategies, and so on.
Use extensible programs for long periods across topics when possible. (p. 278).

Conclusion

While completing my personal inquiry, my understanding and appreciation for manipulatives has grown. I originally thought manipulatives were simply physical objects which students used to count or see three dimensional figures. I assumed they would be given to students as a last resort, used only when students could not understand a mathematical concept. Thanks to this project, my opinion has changed.

I have learned the incredible value of manipulatives and how they can make abstract concepts concrete for learners, allowing them to understand ideas which would otherwise have been unattainable. However, I also discovered students do not automatically link a mathematical idea with the manipulative they are using. Students must be given time to reflect on their learning and intensive instruction may be required to introduce students to manipulatives and what they represent.

One great personal gain I made through this research was the collection of manipulatives I can eventually use in my classroom. I have found printable manipulatives, virtual manipulatives and those which I can make by hand in my classroom. I will be able to create tools for learning for my students with little to no cost. Thankfully, I was also able to find information on how to select and effectively use manipulatives. This will hopefully take much of the guess work out of choosing manipulatives to use in my classroom.

Though I have learned a great deal from this personal inquiry, I still have question I would like to answer if the time was available. These include:
What manipulatives are most appropriate at each grade/developmental level?

Are there manipulatives which should be avoided depending on learning disabilities?

What manipulatives are popular in Special Education classrooms?

Do schools supply manipulatives or are they a teacher’s responsibility?

What should I expect to be supplied with in my first year?

What are students’ opinions on the use of manipulatives?

Do students have favorite manipulatives?

References

About. (March 20, 2005). A brief history of the abacus. Retrieved March 22, 2007 from http://math.about.com/gi/dynamic/offsite.htm?zi=1/XJ/Ya&sdn=math&cdn=education&tm=30&gps=147_12_1020_615&f=00&tt=14&bt=1&bts=1&zu=http://www.ee.ryerson.ca:8080/~elf/abacus/history.html.

Clements, D. H., & McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2, 270-279.

Fennell, F., & Rowan, T. (2001). Representation: An important process for teaching and learning mathematics. Teaching Children Mathematics, 7(5), 288-282.

Heuser, D. (2000). Mathematics workshop: Mathematics class becomes learner centered. Teaching Children Mathematics, 6(5), 288-295.

Hodge, T., & Brumbaugh, D. (2003). Web-based manipulatives. Teaching Children Mathematics, 9(8), 461.

McCarty, D. (1998). Books + Manipulatives + Families = A Mathematics Lending Library. Teaching Children Mathematics, 4, 368-375.

Moyer, P.S., Bolyard, J. J., & Spikell, M. A. (2002). What are virtual manipulatives? Teaching Children Mathematics, 8(6), 372-377.

Sowell, E. J. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20(5), 498-505.

Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach. Journal of Applied Developmental Psychology, 18, 37-54.

Wikipedia (2006, December 1). Mathematical manipulatives. Retrieved March 23, 2007, from http://en.wikipedia.org/wiki/Math_manipulatives.