Manipulatives in Mathematics Instruction

Manipulatives in Mathematics Instruction

Descriptions and Definitions

Manipulatives are classified as any concrete object which can be moved and used in a way to represent abstract concepts in a physical fashion. The meaning of "concrete manipulatives" commonly implies a touchable and movable object which can provide children something real to reflect on. They can be complex and ornate or as a simple as desired; their importance lies in being able to represent mathematical situations which are generally abstract. Students can then use their intuitive knowledge of how objects work concretely, or make observations of how those objects work, as a method of discovering, inventing, or coming to understand how the parallel abstract situations work.

Examples of Manipulatives

Unit blocks or Cuisenaire Rods - These are rods of different lengths, the longest usually being 10cm long and the shortest is 1cm long. These are used to help young childred develop different math concepts. Here are some examples of how to use cuisenaire rods: http://www.lifelearning.org/2007/09/18/101-uses-for-cuisenaire-rods-cuisenaire-rod-math-games-and-more/

Algebra Tiles - These are colored tiles (usually green for positive and red for negative) that represent variable values. An 'x' tile is set up to be 1 unit tall by 2 units wide while an 'x-squared' tile is set up as a square of sides 2 units. Students can visually represent and solve algebraic equations by manipulating the each side of the equation in the exact same way. Here is a great resource for learning how to effectivly use Algebra Tiles: http://www.phschool.com/professional_development/teaching_tools/pdf/using_algebra_tiles.pdf

Computer Based Manipulatives

Math manipulatives are not limited to just physical/concrete objects that students can touch with their hands. Recently, there have been many software titles that can act as virtual manipulatives for the students. Although the students are not able to physically touch the objects on the screen, they do control the objects through the inputs given into the software. In this sense, these virtual manipulatives can be just as effective as real ones. Below are some examples of software based virtual math manipulatives:

The Geometer's Sketchpad. http://www.dynamicgeometry.com/ Mathematica. http://www.wolfram.com/ Tinkerplots. http://www.keypress.com/x5715.xml Fathom. http://www.keypress.com/x5656.xml

Internet-Based Manipulatives

These manipulatives, as with computer-based manipulatives, are accessed using a computer. The activities are web-based rather than downloaded to the computer. These interactive activities are usually created with the Java programming language. Many math sites offer a few virtual manipulatives. There are a few websites that are clearinghouses for all types of math-related online interactivity.

Shodor Interactivate Activities

http://www.shodor.org/interactivate/activities/

Shodor is a national resource for computational science education. They have offered online education tools such as Interactivate and the Computational Science Education Reference Desk (CSERD) since 1994. The activities are sorted from Grade 3 through Undergraduate.

National Library of Virtual Manipulatives

http://nlvm.usu.edu/

Utah State University has offered this collection of internet-based manipulatives since 1999. The activities are sorted from Pre-Kindergarten through High School.

Illuminations: Activities

http://illuminations.nctm.org/ActivitySearch.aspx

Illuminations has been found on a section of the website for the National Council of Teachers of Mathematics since 2000. Students and teachers from Pre-Kindergarten through High School can use these interactivities.

MSTE at the University of Illinois

The Office for Mathematics, Science, and Technology Education (MSTE) at the University of Illinois at Urbana-Champaign has two good resources.

MSTE Online Resource Catalog

http://mste.illinois.edu/resources/

Offered by the Department of Mathematics, Science and Technology Education at University of Illinois since 1994.

M2T2

http://mste.illinois.edu/m2t2/appletslist.html

According to their website:

Mathematics Materials for Tomorrow's Teachers (M2T2) are a set of mathematics modules created in the spring of 2000 by a team consisting of teachers, administrators, university researchers, mathematicians, graduate students, and members of the Illinois State Board of Education.

They are five modules. Each module is connected to one of the goals for mathematics in the Illinois Learning Standards. The content is at a middle school level. The M2T2 materials were developed under the direction of the Office for Mathematics, Science, and Technology Education (MSTE) at the University of Illinois at Urbana-Champaign. It was our goal to create modules that are interesting, mathematically rich, challenging, and appropriate for students and teachers in the middle grades. We have also worked to connect these modules to the Illinois Learning Standards for mathematics with a special view toward preparation for the Illinois Standards Achievement Test (ISAT).

Application in classrooms and similar settings

One of the most controversial questions connected to manipulatives is the specific ways in which manipulatives should be used and introduced to students. The great temptation with manipulatives is to use them as another vehicle for direct instruction. In this model, a teacher will tell students exactly how to use the manipulatives, what each one represents, and what to do with each one. This is unfortunate, as it tends to not be any more helpful to students than direct instruction with the original abstract quantities, because students still are not thinking through how and why to use the objects in a way that makes sense. The strength of manipulatives is that it allows students to draw upon their informal and intuitive knowledge, but using manipulatives in this way does not allow students to use or construct their own knowledge, because they are still trying just to do what they are told.

The other way in which manipulatives can be used is to give them as tools for students to use while solving a problem which they have some context for. For example, before teaching division with fractions you can give students fraction tiles or circles (bars or portions of circles that represent different fractions of a whole) and ask them to consider a problem such as the following: Susie has 1 1/2 cups of sugar. She has a recipe for a batch of cookies which requires 3/8 of a cup. How many batches of cookies can Susie make if she has plenty of all other ingredients? Before starting this problem, students should be familiar with the name and size of each fraction piece, and the teacher could lead them through a qualitative reasoning activity such as "Would you expect Susie to be able to make less than 1 batch, exactly one batch, or more than 1 batch of cookies?". Students should be able to estimate that she can make more than one batch, since only 3/8 is required and she has 1 1/2 cups.

At this point, students must be left alone to decide how to model the situation as they see fit. This will lead to some incorrect methods and answers, but these are sources of learning and shouldn't be avoided. Students will inevitably arrive at a correct procedure if given the opportunity to think through their answers and discuss with each other. In cases where students get "stuck", teachers should first encourage students to challenge and talk through processes with others of differing views. Eventually, if students are won over by incorrect logic, the teacher should create cognitive conflict. One example of this would be in the case where students use Cuisenaire Rods (colored blocks of different lengths) to model 1/2 + 1/6. A student stacked the 1-block on top of the 2 block, and then another 1 block on top of a 6 block, and combined the top blocks and then the bottom blocks to create 2 blocks on top of 8. S/he therefore arrives at the answer: 1/2 + 1/6 = 2/6= 1/3. One way to create cognitive conflict in this situation is to ask students if 2/6 (or 1/3) is more or less than 1/2. How could it be that you start with 1/2, add a little bit more, and get an answer that is less than 1/2? This illustrates a flaw in the concepts which led to the answer, but does not destroy students' abilities to arrive at the answer themselves. Now, students can go back and try to create a new possibility for solving the problem, and will eventually discover that you must be comparing and adding pieces of the same size, which means the whole must be partitioned equally. This is where the formal concept of common denominators comes from, but it is often taught simply as doctrine. Students rarely understand WHY you can't add fractions without the same denominator, and the answer most frequently given seems to be "because I said so". This forces the math classroom to a behaviorist approach, where students must obey blindly, and teachers must provide sufficiently high incentives, rewards, and bribes, to get students to do what they want.

In the example with Susie's cookies above students can physically compare the fraction tiles and see how many groups of 3/8 fit in the 1 1/2. They should find that 12 1/8 pieces take up the same area as 1 1/2, so Susie can make 4 batches of cookies with her present amount of sugar. Obviously, we do not want students using manipulatives forever, as it slows students down. But it lets them see important patterns and draw their own conclusions which can be suprisingly insightful. This is especially important as it is the type of work in which "real" mathematicians are engaged. They find a problem, propose possible methods, look at it how it works with simple cases, derive ways it would work abstractly in all cases (e.g. proof), and a few of them even look at further applications of that mathematics. In this view, it is the process of learning how to solve a problem, rather than knowing that y = mx+b that is critical. While algorithms, formulas, and equations do save time, and can be quite useful when applied correctly, far too many students have never understood mathematics, and spend their time instead trying to remember the correct formula. This leads to many disasters, including many instances of misapplying formulas. There are many instances in which informal processes are vastly superior to formal equations. One example of this would be the probability of pulling a red and a yellow marble (in any order, with replacement) from a bag containing two reds, a yellow and a blue marble. Many teachers have taught their students that "and" in probability always means multiply. Following this logic, I multiply the probability of a red (2/4 = 1/2) by the probability of a yellow (1/4) and get the probability of a red and a yellow being 1/8. Making a chart, I can quickly see all the possibilities that include one red and one yellow marble.

	Red	Red	Yellow	Blue
Red			**
Red			**
Yellow	**	**
Blue

I have 4 possibilities, out of the total 16 in the chart, so 4/16 or 1/4 is the correct answer. Where did I go wrong formally?

The problem is that there is a condition attached to the algorithm of multiplying the independent probabilities. It has to do with the order of the events. The probability of a 1st event, then a second event = p(1st event) times p(second event). This means to solve this formally I must consider two situations:

1.) yellow, then red.

2.) Red, then yellow.

p(yellow then red) = p(yellow)* p(red)= 1/4*1/2=1/8

p(red, then yellow)= p(red)*p(yellow) = 1/2*1/4=1/8

I'm sure everyone remembers this and remembers that "or" in probability means "add". So then we do:

p(yellow then red or red then yellow) = p (yellow then red) + p (red then yellow)

=p(yellow)*p(red)+p(red)*p(yellow)=1/4*1/2+1/2*1/4

Of course, remembering order of operations gives us:

1/8 + 1/8 = 2/8=1/4

Personally, I think the little chart above is a perfectly viable, quick, simple, and refined way to solve this sort of problem. The formal approach, while powerful for difficult problems, tends to confuse students who lack an informal understanding of probability and what "makes sense", and is a little like using a sledgehammer to unlock a door; it may get the job done, but a little key would be a whole lot easier, and require much less clean-up after the fact!!

Evidence of effectiveness

There are two main domains in which effectiveness of strategies are generally examined for mathematics instruction: affect and achievement. Affect deals with how students feel about mathematics, and achievement deals with a perceivable difference between mathematics learned and able to be completed by students before and after a given treatment.

Manipulatives have been found to have a great effect on affect. There are several reasons for this. The use of manipulatives causes teachers to emphasize and validate students' informal knowledge of math, which all students have and which can promote a sense of mathematical and personal efficacy. Also, students tend to perceive manipulation of objects as something fun and interesting, different than the "dull" manipulation of numbers and letters on paper. So students also tend to enjoy mathematics more when they are using manipulatives. Thirdly, all students are able to succeed, and so it is especially helpful for students who traditionally fail or struggle in mathematics class.

While affect is important, and there is much research regarding the role affect plays in encouraging students to continue learning mathematics, or to succeed in the mathematics they do take, many are skeptical of using affect as a determiner for success. Instead, these argue, we need to see hard proof that students' mathematics achievement is actually influenced.

An important thing to remember is that mathematicians who originally discovered components of mathematics did so in a context much more similar to students working with manipulatives in small groups than the normal math class is today. There was no one giving them formulas to memorize, for those formulas had not yet been discovered. Those who are touted as the "geniuses" of mathematics did what we should be having our students do...observing patterns and inventing shortcuts to make them more mathematically powerful.

One of the chief weaknesses of American mathematics students is the phenomenon of "forgetting". Students memorize formulas for tests, but when they need to use that formula, they don't remember it and are unable to recall it to use. It is interesting to note that countries in the East (e.g. China and Japan) emphasize understanding, not forgetting. If a student is not able to perform a mathematical task previously done, the assumption is that the student did not really understand it, and remediation focuses on the breakdown of cognitive processes that occurred. The strength of this emphasis on understanding is it allows for students to be able to re-create missing pieces, rather than forcing them to memorize endless formulas. While a student may still forget the formula for area of a regular polygon (1/2 the perimeter times the apothem...do you remember what an apothem is?! ), if they understand where that formula comes from in a meaningful manner, they will be able to re-create it if and when they need it. Very few mathematicians memorize lots of formulas; they may remember them by frequent use, but they are usually able to re-derive them if necessary, because they possess a strong command of the systems which produces those formulas to begin with.

The use of manipulatives can have a positive effect on students' mathematical achievement. It does not do so in every case, but it is usually due to one of two reasons. First, teacher misuse of manipulatives can cause lots of harm and prevent students from making meaningful progress. Secondly, students' problems may be due to previously learned formulas which do not make sense. For example, if I teach 5th grade students fractions and decimals in meaningful ways, but they have never really understood place value, they will still struggle, and the use of manipulatives may not help them terribly much. I need to trace back to the last concept that was truly understood, and work forward from that point. This will allow students to build a strong conceptual base upon which they can rely. Gaps and holes in cognitive understanding can prevent a solid foundation from being laid, but these gaps can often be identified through the solution of meaningful problems with manipulatives.

Critics and their rationale

There are a few main objections of critics of manipulative use in mathematics classes. One objection is the idea of spending large amounts of time on a single problem. In many math classes, students are given worksheets full of 30+ problems, practicing a single skill over and over. The idea of taking an entire class period to complete only 1, 2 or maybe 3 problems is disturbing to some teachers, who tend to judge learning by volume. However, when students understand what they are doing, they have less need for extensive practice. Instead, they can focus on understanding the concepts involved thoroughly. Afterwards, they will be able to complete practice problems in much less time, and don't run the risk of "forgetting" processes. They are not trying to force themselves to "remember" processes, but instead to UNDERSTAND what is occurring. Understanding is permanent, and makes the learner truly the possessor of knowledge.</br>

Another objection to manipulatives is that they are difficult to learn to use, too time-consuming or confusing for students. Manipulative use does require a certain flexibility of thinking, because students must work with the constraints of the physical material presented to them. The most challenging part of using the manipulatives is to decide what each piece will represent. While with some manipulatives this is always the same, many times it is dependent on the constraints of the problem. This is a vitally important part of the problem solving process. Teachers who try to "protect" their students from this component do them no favors; students must be able to examine a problem, examine their resources, and find the ways in which resources can be used to meet the demands of the problem. Learning how to think flexibly and allowing the same object to represent different quantiites in different situations is an important part of growing in problem solving skills and developing abstract reasoning. This takes time, but is important. </br>

Teachers must also allow students to think through these issues themselves. Students are more likely to be confused by someone randomly telling them "let the blue block be 10 this time", "this time let the blue block be 100", than by the process of thinking through the questions themselves. The reason for using manipulatives is to empower students to correctly arrive at answers themselves. To do it "for them" negates the purpose, and actually is MORE confusing, because students can't see the purpose for the little blue block that really isn't helping them at all! </br>

One reason many teachers raise this objection is because the teachers often find the manipulatives difficult to understand or use themselves. This is due to the same problems which manipulatives are trying to combat for students: dependence on memorized algorithms rather than a deep and rich conceptual understanding. Teachers at all levels are often stuck with a purely algorithmic knowledge of the math they teach. Those with particularly rich mathematics experiences may be able, perhaps after several years, to deepen their understanding of the concepts they teach enough to foster conceptual understanding in their students and so be able to use manipulatives in a meaningful way. This is true at every grade level, from K all the way up to 12, and perhaps beyond. Using manipulatives, and especially for the teacher who analytically and carefully considers in what ways to use manipulatives, requires a rich understanding of the meaning behind the numbers and the letters. Otherwise, the teacher can't figure out how to appropriately model a situation, or how to judge if a model, and the answer produced by that model is accurate or not. It is not only an elementary school problem, nor only a secondary school one; it is in every level! Teachers are in the unenviable position of having to rise to the challenge of teaching in ways they never had the opportunity to learn in. This is yet another call for learning communities for teachers, small groups such as those proposed by lesson study and others which examine the mathematics behind the lesson in meaningful ways, and allow teachers to grow and develop in mathematically meaningful sense in the safe zone of their peers. Additionally, it is a caution to professional developers who aim to teach teachers "how" to use manipulatives. Just as telling a student what each block represents doesn't ensure understanding, so telling a teacher what to do and say doesn't mean either they or their students will comprehend the significance of what is occurring. </br>

Another criticism, heard especially at the middle and secondary levels is that kids are too old to be "playing with blocks". The logic behind this is that manipulatives are only for young children and don't really serve much instructional purpose. Instead, older children need to do "real math", not play around. While the committment to rigor is admirable, it is important to build a conceptual understanding, regardless of students' age. If a 20 year old still doesn't understand "invert and multiply" with division of fractions, they still need to build a conceptual basis. Ignoring it or telling them "they should be past this" by now will not help them build that conceptual basis. Rather, students of all ages will have holes in their understanding and instead of pointing the blame at previous teachers or denying that those holes exist in an effort to make ourselves feel better about education as a whole, we need to do whatever we can to fill those gaps so students can succeed. Age does not make misunderstandings or confusion dissipate. </br>

It has been my practice at the start of each year to spend some time with manipulatives allowing students to experiment with many concepts: place value, naming and operating on fractions, decimal division and multiplication, story problems related to the basic four operations, and simple probability and statistics. I taught not elementary or even middle school, but high school. But many of those students never really got a grasp of those concepts until they had the opportunity to "play" with them. If you have ever had the experience of someone giving you step by step instructions for a computer or other piece of technology while you were just listening or listening and watching them do it versus when you could listen and do it as you went, you will understand the distinction a little bit. Hearing just the steps, we are overwhelmed with trying to remember it all. But once we've done it we may even come to understand a little better how things are organized, what is in which folder, etc., and are better equipped to either repeat the action later or perhaps even make more reasonable conjectures about how to solve future similar, but slightly different problems. The situation is exactly the same with manipulatives; it gives students something concrete to make sense of, producing an understand which can then be applied to more abstract problems without the use of manipulatives.

Alternative explanations due to Diversity considerations

It is encouraging to teachers of traditionally underserved students to find that manipulatives possess potential to reach out to students for whom mathematics has held little attraction. White, middle to upper class students are usually distinct from other students in more easily making connections with abstract ideas, more able to sit and listen quietly for long periods of time, and more able to connect abstract ideas conceptually to their everyday experiences.

Are manipulatives helpful for children beyond kindergarten or first grade? Research illustrates that using manipulative can help chidlren of all ages construct a deeper understanding of mathematics(Clements & McMillen, 1996). Manipulatives allow students to take a more active role in classroom experiences, which answers a major challenge to especially African-American students who often struggle in the austere and passive environment of many traditional mathematics classrooms.

Many minorities struggle to link the abstract concepts of mathematics to anything they have real experience with, and so mathematics becomes a question of memorizing an absurd number of formulas, equations, and facts. Memorizing is quite simply terribly efficient, and the student who can rely on his or her informal knowledge and develop his or her own strategies of problem solving will always outperform the student who has everything memorized. Manipulatives simply provide a convenient way of helping all students learn to use their informal knowledge!

Much research backs up the use of manipulatives with diverse students.

Signed "life experiences", testimonies and stories

When used for whole-class instruction, I have always found manipulatives to be a bit cumbersome. Managing a whole class with manipulatives in front of them can be daunting. As an elementary math teacher, I have had my best results when using math manipulatives in small group situations or when working with individual students to reteach a complicated math concept or introduce a new concept. - E. Remington

As a math major and pre-teacher, I've found that manipulatives help engage a number of types of learners. Even though I've always been able to learn from lecture, manipulatives engage another part of the brain, so that students can learn and see things in a number of ways. Also, the use of manipulatives in any subject is connected to discovery learning, which is extremely effective in all classroom settings. -Maggie Schlosser

Considering that I have never heard that a visual representation of a math problem has confused a student, I think that manipulatives are a very good resource for any classroom. It lets the students discover and see how the mathematics work as opposed to accepting that it is true from the teacher's lecture. The only problem is that some students may not be responsible and either fool around with the manipulatives, break them, or lose them. It can be a distraction and could cause many problems in the classroom. On the other hand, if a normally rowdy student is a strong visual learner, then that student may behave because that student is not having to sit still during a lecture. Overall, I think the benefits outweigh the negatives. Thus, I think Mathematical Manipulatives need to be used as much as possible. - John N. Janowiak

Based on my observations of chidlren's mathematics learning, children have some implicit knowledge learned form everyday life. Teachers can take the advantage of chidlren's existing knowledge and encourage chidlren to apply everyday life experiences to figure out their solutions by using manipulatives. Chidlren may discover the underlying relationships through using real manipulatives in a personally meaningful way.-Hsin-Mei Huang

As a math teacher, I use as many manipulatives as I can. There are so many different learning styles out there and it is my job as an educator to appeal to as many of them as I can. Manipulatives appeal to the visual learner. For example, when I teach my junior high students about probability I get out cards, coins, die and spinners. I don't just make them use their imagination, I physically show them. I also allow the students to take part in the lesson. I feel that when they are actively engaged, they learn so much better. Students like anything that is new and not the same old lecture and take notes. Since there are so many resources available to math teachers, there is no reason why one cannot use manipulatives. ---Jodi Herrmann

The following web site National Library of Virtual Manipulatives offers a wonderful set of manipulatives for students to explore or teachers to use in class. As a teacher of gifted math students, we would allow the students access to this site after completing their class work. It gave them the opportunity to explore and learn. Many reported going home and continuing to work with the site from home. ~BSmall

Manipulative are a great form of learning in certain circumstances, but I think it is more because it is something new and different for the students. I feel if you were to overuse manipulative they just become toys and the learning curve is lost to a certain extent. – Dale Donner

I remember a second grade teacher bringing out manipulatives -- colored blocks of different sizes. I thought they were cool just because of the colors (I was a child who played the game Risk with my brother just because I was attracted to the brightly colored pieces -- especially the yellow and pink in the clear plastic box). I'm not sure that I learned anything from the math manipulatives. They were only used once or twice and we had a huge class of about 36 students. -- E Bearden

I used manipulatives in my Algebra class this week. On the first day of a new topic I presented the material in a lecture format on the smartboard. Students were struggling to understand the concept. On the second day I had note cards with letters and numbers written on them for students to manipulate to create equations. As described about in the "critics" section I got through fewer problems than I hoped. But at the end of class my students said they completely understood the topic now, and it was all because of using the note cards. I gave my note cards to another teacher and she said her students had the same reaction. It was a simple lesson, but being able to physically work with the problem helped students understand. R. Fruin

Using manipulatives in math is a great way to get the students to learn the material. A lot of people are hands-on learners and it helps a lot for the students to be able to work with something they can understand. Manipulatives work well in cooperative learning because it empowers students to not only learn, but teach also. I teach with manipulatives every chance I get. There are however, those students that don't enjoy working with their hands and they would just prefer to do the work without. Fortuantely for me, these students are far and few between. It gives me chills to be able to observe the learning physically more often than seeing it on paper. I love the discussions that come from the use of manipulatives. The students work very hard to help and understand each other. ~ R. Hayes

As a math teacher, I want my students to understand the concept and function behind the math. Manipulatives certainly help here. Interestingly, my honors students tend to dislike the manipulatives. These are students for whom recalling processes comes easily. Working with manipulatives is difficult for them as it requires them to exercise a different part of their brains and deepen their understanding. If they know how to do it, they don't necessarily want to know the why part. For low performing students, we have to be very careful with manipulatives. They can sometimes lead to cognitive overload. Keep the number of items on the desk to a minimum, and break the lesson down into small bites. T. Stilts

I have begun to use to use more hands on activities in my geometry class to promote understanding. I do not use block as manipulatives, but I have found that activities with paper, scissors, string and rulers can be powerful. I also find that it has the flexibility that is mentioned in this article. For instance, when I guide the students to draw a triangle, I only want them to use a straight edge, but do not tell them how big, or what type of triangle. We collect individual data, then compile the classes data and analyze it for generalities. So the pictures are "manipulative"-like in that the students can see, fold, measure, and analyze the triangle, while preserving the less directed aspect of assigning specific values to blocks. I then reinforce the idea that because certain properties or rules hold for all of the different triangles the students created then we can generalize those aspects, at that point the theorem we are investigating is given and explained in terms of our activity. I do not have any definitive data showing effectiveness, but the students are engaged. My honors class was able to predict the actual theorem we were going to discuss, so it does seem effective. G. Van Hoorn

As a geometry teacher I am always trying to show different ways of looking at a 3D object even thought it is drawn in 2D on the screen. It is invaluable to have manipulatives in these classes so students have something concrete to hold on to. I can have students memorize formulas over and over but if they do not have a conceptual knowledge of what lateral area means then they will not be albe to make connections in their head about those formulas. D. Hohman

Once students reach upper levels in school it is often difficult to encourage them to "play" math. However, I have found that students who struggle with math topics really connect with the manipulatives. I find them drawing their own pictures of the manipulatives and moving them around the paper rather than relying solely on the final product that was taught. While the still attempt the core math structure, they rely on the manipulatives to check their work. It is a nice way to make math more tangible. -M. Pule

Manipulatives in Math I really don’t know if there is a better way to teach math than with the inclusion of manipulatives. Regardless of a students’ particular learning style I have not met a student that did not enjoy the use of manipulatives. In my math class I used manipulatives as tools to build understanding and often times would lead students to the very powerful process of synthesizing their own ideas about a particular concept. Using manipulatives to prove simple formulas, develop fraction and mixed number understanding, and promote whole and decimal place value have been some basic concepts in which manipulatives have been used. Using manipulatives to build understanding can be tricky in that a dependence on manipulatives is not going to help students achieve high test scores. There always needs to be a follow up of a concept with practice in working through problems without the use of manipulatives. M Reichert

References and other links of interest

Baroody, A. & Coslick, R.T. (1998). Fostering children's mathematical power: An investigative approach to K-8 mathematics instruction. Mahwah, NJ: Erlbaum Associates.

Base 10 blocks on the web: http://www.arcytech.org/java/b10blocks/b10blocks.html

Cass, M., Cates, D., Jackson, C. & Smith, M. (2003). Effects of Manipulative Instruction on Solving Area and Perimeter Problems by Students with Learning Disabilities. Learning Disabilities: Research & Practice, 18(2), 112 – 120.

Virtual math manipulatives: http://www.ct4me.net/math_manipulatives.htm

Reference

Chester, J. (1991). Math Manipulatives Use and Math Achievement of Third-Grade Students. (ERIC Document Reproduction Service Number ED 339 591).

Clements, D. H., & mcMillen, S. (1996). Rethinking "concrete" manipulatives. Teaching Children Mathematcis, 2(5), 270-279.

Leitze, A. R.& Kitt, N. A. (2000). Using Homemade Algebra Tiles To Develop Algebra and Prealgebra Concepts. Mathematics Teacher, 93(6), 462-66.

Pandiscio, E.A. (2002). Alternative Geometric Constructions: Promoting Mathematical Reasoning. Mathematics Teacher, 95(1), 32-36.

Stein, M.K. & Bovalino, J. W. (2001). Manipulatives: One Piece of the Puzzle. Mathematics Teaching in the Middle School, 6(6),356-59.

Whitenack, J. W., Knipping, N., Novinger, S. & Underwood, G. (2001). Second graders circumvent addition and subtraction difficulties. Teaching Children Mathematics, 8(4), 228 – 233.

Other Links of Interest

Hands on Equations

Algebra Tiles

Math Playground