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Rethinking Concrete Manipulatives

Douglas H. Clements
Sue McMillen

Close your eyes and picture students doing mathematics. Like many educators, the mental pictures may include manipulative objects, such as cubes, geoboards, or colored rods. Does the use of such concrete objects really help students learn mathematics? What is meant by "concrete"? Are computer displays concrete and can they play an important role in learning? By addressing these questions, the authors hope to change the mental picture of what manipulatives are and how they might be used effectively.

Are Manipulatives Helpful?

Helpful, yes... Students who use manipulatives in their mathematics classes usually outperform those who do not (Driscoll, 1983; Sowell, 1989; Suydam, 1986). This benefit holds across grade level, ability level, and topic, given that using a manipulative makes sense for the topic. Manipulative use also increases scores on retention and problem-solving tests. Finally, attitudes toward mathematics are improved when students are instructed with concrete materials by teachers knowledgeable about their use (Sowell, 1989).

...But no guarantee. Manipulatives, however, do not guarantee success (Baroody, 1989). One study showed that classes not using manipulatives outperformed classes using manipulatives on a test of transfer (Fennema, 1972). In this study, all teachers emphasized learning with understanding.

In contrast, students sometimes learn to use manipulatives only in a rote manner. They perform the correct steps, but have learned little more. For example, a student working on place value with beans and bean sticks used the bean as 10 and the bean stick as 1 (Hiebert and Wearne, 1992).

Similarly, students often fail to link their actions on base-ten blocks with the notation system used to describe the actions (Thompson and Thompson, 1990). For example, when asked to select a block to stand for 1 then put blocks out to represent 3.41, one fourth grader put out three flats, four longs, and one single after reading the decimal as "three hundred forty-one."

Although research suggests that instruction begin concretely, it also warns that concrete manipulatives are not sufficient to guarantee meaningful learning. This conclusion leads to the next question.

What Is Concrete?

Manipulatives are supposed to be good for students because they are concrete. The first question to consider might be, What does concrete mean? Does it mean something that students can grasp with their hands? Does this sensory character itself make manipulatives helpful? This view presents several problems.

First, it cannot be assumed that when children mentally close their eyes and picture manipulative-based concepts, they "see" the same picture that the teacher sees. Holt (1982, 138-39) said that he and his fellow teacher "were excited about the rods because we could see strong connections between the world of rods and the world of numbers. We therefore assumed that children, looking at the rods and doing things with them, could see how the world of numbers and numerical operations worked. The trouble with this theory is that [my colleague] and I already knew how the numbers worked. We could say, "Oh, the rods behaved just the way numbers do. But if we hadn't known how numbers behaved, would looking at the rods enable us to find out? Maybe so, maybe not."

Second, physical actions with certain manipulatives may suggest mental actions different from those that teachers wish students to learn. For example, researchers found a mismatch when students used the number line to perform addition. When adding 5 and 4, the students located 5, counted "one, two, three, four," and read the answer. This procedure did not help them solve the problem mentally, for to do so they must count "six, seven, eight, nine" and at the same time count the counts -- 6 is 1, 7 is 2, and so on. These actions are quite different (Gravemeijer, 1991, 59). These researchers also found that students' external actions on an abacus did not always match the mental activity intended by the teacher.

Although manipulatives have an important place in learning, they do not carry the meaning of the mathematical idea. They can even be used in a rote manner. Students may need concrete materials to build meaning initially, but they must reflect on their actions with manipulatives to do so. Later, they are expected to have a "concrete" understanding that goes beyond these physical manipulatives. For example, teachers like to see that numbers as mental objects -- "I can think of 13 + 10 in my head" are "concrete" for sixth graders. It appears that "concrete" can be defined in different ways.

Types Of Concrete Knowledge

Students demonstrate sensory-concrete knowledge when they use sensory material to make sense of an idea. For example, at early stages, children cannot count, add, or subtract meaningfully unless they have actual objects to touch.

Integrated-concrete knowledge is built through learning. It is knowledge that is connected in special ways. This concept is the root of the word concrete -- "to grow together." Sidewalk concrete derives its strength from the combination of separate particles in an interconnected mass. Integrated-concrete thinking derives its strength from the combination of many separate ideas in an interconnected structure of knowledge. While still in primary school, Jacob read a problem on a restaurant place mat asking for the answer to 3/4 + 3/4. He solved the problem by thinking about the fractions in terms of money: 75¢ plus 75¢ is $1.50, so 3/4 + 3/4 is 1 1/2. When children have this type of interconnected knowledge, the physical objects, the actions they perform on the objects, and the abstractions they make all are interrelated in a strong mental structure. Ideas such as "four," "3/4," and "rectangle" become as real and tangible as a concrete sidewalk. Each idea is as concrete to a student as a ratchet wrench is to a plumber -- an accessible and useful tool. Jacob's knowledge of money was such a tool.

An idea, therefore, is not simply concrete or not concrete. Depending on what kind of relationship the student has with it (Wilensky 1991), an idea might be sensory-concrete, abstract, or integrated-concrete. The catch, however, is that mathematics cannot be packaged into sensory-concrete materials, no matter how clever our attempts are, because ideas such as number are not "out there." As Piaget has shown, they are constructions -- reinventions -- of each human mind. "Fourness" is no more "in" four blocks than it is "in" a picture of four blocks. The child creates "four" by building a representation of number and connecting it with either real or pictured blocks (Clement, 1989; Clement and Battista 1990; Kamii 1973, 1985, 1986).

Mathematical ideas are ultimately made integrated-concrete not by their physical or real-world characteristics but rather by how "meaningful" -- connected to other ideas and situations -- they are. Holt (1982, 219) found that children who already understood numbers could perform the tasks with or without the blocks. "But children who could not do these problems without the blocks didn't have a clue about how to do them with the blocks. ... They found the blocks ... as abstract, as disconnected from reality, mysterious, arbitrary, and capricious as the numbers that these blocks were supposed to bring to life."

Are Computer Manipulatives Concrete?

The reader's earlier mental picture of students using manipulatives probably did not feature computer technology. But, as has been shown, "concrete" cannot be equated simply with physical manipulatives. Computers might supply representations that are just as personally meaningful to students as real objects; that is, they might help develop integrated-concrete knowledge. These representations may also be more manageable, "clean," flexible, and extensible. For example, one group of young students learned number concepts with a computer-felt-board environment. They constructed "bean stick pictures" by selecting and arranging beans, sticks, and number symbols. Compared with a real bean-stick environment, this computer environment offered equal, and sometimes greater, control and flexibility to students (Char 1989). The computer manipulatives were just as meaningful and were easier to use for learning. Both computer and physical bean sticks were worthwhile, but work with one did not need to precede work with the other.

The important point is that "concrete" is, quite literally, in the mind of the beholder. Ironically, Piaget's period of concrete operations is often used, incorrectly, as a rationalization for objects-for-objects' sake in elementary school. Good concrete activity is good mental activity (Clements 1989; Kamii 1989).

This idea can be made more concrete. Several computer programs allow children to manipulate on-screen base-ten blocks. These blocks are not physically concrete. However, no base-ten blocks "contain" place-value ideas (Kamii 1986). Students must build these ideas from working with the blocks and thinking about their actions.

Actual base-ten blocks can be so clumsy and the manipulations so disconnected one from the other that students see only the trees -- manipulations of many pieces -- and miss the forest -- place-value ideas. The computer blocks can be more manageable and "clean."

In addition, students can break computer base-ten blocks into ones or glue ones together to form tens. Such actions are more in line with the mental actions that students are expected to learn. The computer also links the blocks to the symbols. For example, the number represented by the base-ten blocks is usually dynamically linked to the students' actions on the blocks; when the student changes the blocks, the number displayed is automatically changed as well. This process can help students make sense of their activity and the numbers. Computers encourage students to make their knowledge explicit, which helps them build integrated-concrete knowledge. A summary of specific advantages follows.

Computers offer a manageable, clean manipulative. They avoid distractions often present when students use physical manipulatives. They can also mirror the desired mental actions more closely.

Computers afford flexibility. Some computer manipulatives offer more flexibility than do their noncomputer counterparts. For example, Elastic Lines (Harvey, McHugh, and McGlathery 1989) allows the student to change instantly both the size, that is, the number of pegs per row, and the shape of a computer-generated geoboard (fig. 1). The ease of accessing these computer geoboards allows the software user many more experiences on a wider variety of geoboards. Eventually, these quantitative differences become qualitative differences.

Fig. 1: Elastic Lines (Harvey, McHugh, and McGlathery 1989) allows a variety of arrangements of "nails" on its electronic geoboard; size can also be altered.

Computer manipulatives allow for changing the arrangement or representation. Another aspect of the flexibility afforded by many computer manipulatives is the ability to change an arrangement of the data. Most spreadsheet and data base software will sort and reorder the data in numerous different ways. Primary Graphing and Probability Workshop (Clements, Crown, and Kantowski 1991) allows the user to convert a picture graph to a bar graph with a single keystroke (fig. 2).

Fig. 2: Primary Graphing and Probability Workshop (Clements, Crown, and Kantowski 1991) converts a picture graph to a bar graph with a single keystroke.

Computer store and later retrieve configurations. Students and teachers can save and later retrieve any arrangement of computer manipulatives. Students who had partially solved a problem can pick up immediately where they left off. They can save a spreadsheet or database created for one project and use it for other projects.

Computers record and replay students' actions. Computers allow the storage of more than static configurations. Once a series of actions is finished, it is often difficult to reflect on it. But computers have the power to record and replay sequence of actions on manipulatives. The computer-programming commands can be recorded and later replayed, changed, and viewed. This ability encourages real mathematical exploration. Computer games such as Tetris allow students to replay the same game. In one version, Tumbling Tetrominoes, which is included in Clements, Russell et al. (1995), students try to cover a region with a random sequence of tetrominoes (fig. 3). If students believe that they can improve their strategy, they can elect to receive the same tetrominoes in the same order and try a new approach.

Fig. 3: When playing Tumbling Tetrominoes (Clements, Russell et al. 1995), students attempt to tile tetrominoes -- shapes that are like dominoes except that four squares are connected with full sides touching. Research indicates that playing such games involves conceptual and spatial reasoning (Bright, Usnick, and Williams 1992). Students can elect to replay a game to improve their strategy.

Computer manipulatives link the concrete and the symbolic by means of feedback. Other benefits go beyond convenience. For example, a major advantage of the computer is the ability to associate active experience with manipulatives to symbolic representations. The computer connects manipulatives that students make, move, and change with numbers and words. Many students fail to relate their actions on manipulatives with the notation system used to describe these actions. The computer links the two.

For example, students can draw rectangles by hand but never go further to think about them in a mathematical way. In Logo, however, students must analyze the figure to construct a sequence of commands, or a procedure, to draw a rectangle (see fig. 4). They have to apply numbers to the measures of the sides and angles, or turns. This process helps them become explicitly aware of such characteristics as "opposite sides equal in length." If instead of fd 75 they enter FD 90, the figure will not be a rectangle. The link between the symbols and the figure is direct and immediate. Studies confirm that students' ideas about shapes are more mathematical and precise after using Logo (Clements and Battista 1989; Clements and Battista 1992).

Fig. 4: Students use a new version of Logo, Turtle Math (Clements and Meredith 1994), to construct a rectangle. The commands are listed in the command center on the left (Clements, Battista et al. 1995; Clements and Meredith 1994).

Some students understand certain ideas, such as angle measure, for the first time only after they have used Logo. They have to make sense of what it is being controlled by the numbers they give to right- and left-turn commands. The turtle immediately links the symbolic command to a sensory-concrete turning action. Receiving feedback from their explorations over several tasks, they develop an awareness of these quantities and the geometric ideas of angle and rotation (Kieran and Hillel 1990).

Fortunately, students are not surprised that the computer does not understand natural language and that they must formalize their ideas to communicate them. Students formalize about five times more often using computers than they do using paper (Hoyles, Healy, and Sutherland 1991). For example, students struggled to express the number pattern that they had explored on spreadsheets. They used such phrases as "this cell equals the next one plus 2; and then that result plus this cell plus 3 equals this." Their use of the structure of the spreadsheet's rows and columns, and their incorporation of formulas in the cells of the spreadsheet, helped them more formally express the generalized pattern they had invented.

But is it too restrictive or too hard to operate on symbols rather than to operate directly on the manipulatives? Ironically, less "freedom" might be more helpful. In a study of place value, one group of students worked with a computer base-ten manipulative. The students could not move the computer blocks directly. Instead, they had to operate on symbols -- digits -- as shown in figure 5 (Thompson 1992; Thompson and Thompson 1990). Another group of students used physical base-ten blocks. Although teachers frequently guided students to see the connection between what they did with the blocks and what they wrote on paper, the physical-blocks group did not feel constrained to write something that represented what they did with blocks. Instead, they appeared to look at the two as separate activities. In comparison, the computer group used symbols more meaningfully, tending to connect them to the base-ten blocks.

Fig. 5: A screen display of the base-ten blocks computer microworld (Thompson 1992).

In computer environments, such as computer base-tens blocks or computer programming, students cannot overlook the consequences of their actions, which is possible to do with physical manipulatives. Computer manipulatives, therefore, can help students build on their physical experiences, tying them tightly to symbolic representations. In this way, computers help students link sensory-concrete and abstract knowledge so they can build integrated-concrete knowledge.

Computer manipulatives dynamically link multiple representations. Such computer links can help students connect many types of representations, such as pictures, tables, graphs, and equations. For example, many programs allow students to see immediately the changes in a graph as they change data in a table.

These links can also be dynamic. Students might stretch a computer geoboard's rectangle and see the measures of the sides, perimeter, and area change with their actions.

Computers change the very nature of the manipulative. Students can do things that they cannot do with physical manipulatives. Instead of trading a one hundred-block for ten ten-blocks, students can break the hundred-block pictured on the screen into ten ten-blocks, a procedure that mirrors the mathematical action closely. Students can expand computer geoboards to any size or shape. They can command the computer to draw automatically a figure symmetrical to any they create on the geoboard.

Advantages Of Computer Manipulatives for Teaching and Learning

In addition to the aforementioned advantages, computers and computer manipulatives possess other characteristics that enhance teaching and learning mathematics. Descriptions of these features follow.

Computer manipulatives link the specific to the general. Certain computer manipulatives help students view a mathematical object not just as one instance but as a representative of an entire class of objects. For example, in Geometric Supposer (Schwartz and Yerushalmy 1986) or Logo, students are more likely to see a rectangle as one of many that could be made rather than as just one rectangle.

This effect even extends to problem-solving strategies. In a series of studies, fourth-grade through high school students who used Logo learned problem-solving strategies better than those who were taught the same strategies with noncomputer manipulatives (Swan and Black 1989). Logo provided malleable representations of the strategies that students could inspect, manipulate, and test through practice. For example, in Logo, students broke a problem into parts by disembedding, planning, and programming each piece of a complex picture separately. They then generalized this strategy to other mathematics problems.

Computer manipulatives encourage problem posing and conjecturing. This ability to link the specific to the general also encourages students to make their own conjectures. "The essence of mathematical creativity lies in the making and exploring of mathematical conjectures" (Schwartz 1989). Computer manipulatives can furnish tools that allow students to explore their own conjectures while also decreasing the psychological cost of making incorrect conjectures.

Because students may themselves test their ideas on the computer, they can more easily move from naive to empirical to logical thinking as they make and test conjectures. In addition, the environments appear conducive not only to posing problems but to wondering and to playing with ideas. In early phases of problem solving, the environments help students explore possibilities, not become "stuck" when no solution path presents itself. Overall, research suggests that computer manipulatives can enable "teaching children to be mathematicians vs. teaching about mathematics" (Papert, 1980, 177).

For example, consider the following dialogue in which a teacher was overheard discussing students' Logo procedures for drawing equilateral triangles.

Teacher:	Great. We got the turtle to draw bigger and smaller equilateral triangles. Who can summarize how we did it?
Monica:	We changed all the forward numbers with a different number -- but all the same. But the turns had to stay 120, 'cause they're all the same in equilateral triangles. (See fig. 6.)
Chris:	We didn't make the biggest triangle.
Teacher:	What do you mean?
Chris:	What's the biggest one you could make?
Teacher:	What do people think?
Rashad:	Let's try 300.

The class did (see fig. 7).

 

Fig. 6: Students use Turtle Math (Clements and Meredith 1994) to construct an equilateral triangle.

Fig. 7: When commands are changed, the figure is automatically changed -- here, to an equilateral triangle with sides of length 300 turtle steps.

Monica:	It didn't fit on the screen. All we see is an angle.
Teacher:	Where's the rest?
Beth:	Off here [gesturing]. The turtle doesn't wrap around the screen.
Tanisha:	Let's try 900!

The student typing made a mistake, and changed the command to FD 3900.

Teacher:	Whoa! Keep it! Before you try it, class, tell me what it will look like!
Ryan:	It'll be bigger. Totally off the screen! You won't see it at all!
Jacob:	No, two lines will still be there, but they'll be way far apart.

The children were surprised when it turned out the same! (See fig. 8.)

Fig. 8: Why did the figure not change when the side lengths were changed to 3900 turtle steps?

Teacher:	Is that what you predicted?
Rashad:	No! We made a mistake.
Monica:	Oh, I get it. It's right. It's just farther off the screen. See, it goes way off, there, like about past the ceiling.

The teacher challenged them to explore this and other problems they could think of.

Jacob:	I'm going to find the smallest equilateral triangle.
Rashad:	We're going to try to get all the sizes inside one another.

Computer manipulatives build scaffolding for problem solving. Computer environments may be unique in furnishing problem-solving scaffolding that allows students to build on their initial intuitive visual approaches and construct more analytic approaches. In this way, early concepts and strategies may be precursors of more sophisticated mathematics. In the realm of turtle geometry, research supports Papert's (1980) contention that ideas of turtle geometry are based on personal, intuitive knowledge (Clements and Battista 1991; Kynigos 1992). One boy, for example, wrote a procedure to draw a rectangle. He created a different variable for the length of each of the four sides. He gradually saw that he needed only two variables, since the lengths of the opposite sides are equal. In this way, he recognized that the variables could represent values rather than specific sides of the rectangle. No teacher intervened; Logo supplied the scaffolding by requiring a symbolic representation and by allowing the boy to link the symbols to the figure.

Computer manipulatives may also build scaffolding by assisting students in getting started on a solution. For example, in a spreadsheet environment, typing headings or entering fixed numbers might help students organize their ideas.

Computer manipulatives focus attention and increase motivation. One group of researchers studied pairs of students as they worked on computers and found that the computer "somehow draws the attention of the pupils and becomes a focus for discussion," thus resulting in very little off-task talk (Hoyles, Healy, and Sutherland 1991). Although most children seem to enjoy working on the computer, such an activity can be especially motivating for some students who have been unsuccessful with mathematics. For example, two such third graders were observed as they eagerly worked in a Logo environment. They had gone forward twenty-three turtle step, but then figured out that they needed to go forward sixty turtle steps in all. They were so involved that both of them wanted to do the necessary subtraction. One grabbed the paper from the other so he could compute the difference.

Computer manipulatives encourage and facilitate complete, precise explanations. Compared with students using paper and pencil, students using computers work with more precision and exactness (Butler and Close 1989; Clements and Battista 1991; Gallou-Dumiel 1989). For example, students can use physical manipulatives to perform such motions as slides, flips, and turns. However, they make intuitive movements and corrections without being aware of these geometric motions. Even young children can move puzzle pieces into place without a conscious awareness of the geometric motions that can describe these physical movements. In one study, researchers attempted to help a group of students using noncomputer manipulatives become aware of these motions. However, descriptions of the motions were generated from, and interpreted by, physical motions of students who understood the task. In contrast, students using the computer specified motions to the computer, which does not "already understand." The specification had to be thorough and detailed. The results of these commands were observed, reflected on, and corrected. This effort led to more discussion of the motions themselves, not just the shapes (Butler and Close 1989).

Firming Up Ideas about the Concrete

Manipulatives can play a role in students' construction of meaningful ideas. They should be used before formal instruction, such as teaching algorithms. However, teachers and students should avoid using manipulatives as an end -- without careful thought -- rather than as a means to that end.

The appropriate use of representations is important to mathematics learning. In certain topics, such as early number concepts, geometry, measurement, and fractions, proper use of manipulatives is especially crucial. However, manipulatives alone are not sufficient -- they must be used to actively engage children's thinking with teacher guidance -- and definitions of what constitute a "manipulative" may need to be expanded. Research offers specific guidelines for selecting and using manipulatives.

How Should Manipulatives Be Selected?

The following guidelines are offered to assist teachers in selecting appropriate and effective manipulatives.

Select manipulatives for children's use. Teacher demonstrations with manipulatives can be valuable; however, children 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. They can become what the colored rods were for Holt's students -- "another kind of numeral, symbols made of colored wood rather than marks on paper" (1982, 170). Sometimes the simpler, the better. For example, educators from the Netherlands found that students did not learn well using base-ten blocks and other structured base-ten materials. A mismatch may have occurred between trading one base-ten block for another and the actions of mentally separating a ten into ten ones or thinking of the same quantity simultaneously as "one ten" and "ten ones." The Netherlands students were more successful after hearing a story of a sultan who often wants to count his gold. The setting of the story gave students a reason for counting and grouping: the gold had to be counted, packed, and sometimes unwrapped -- and an inventory had to be constantly maintained (Gravemeijer 1991). Students, therefore, might best start by using manipulatives with which they create and break up groups of tens into ones, such as interlocking cubes, instead of base-ten blocks (Baroody 1990). Settings that give reasons for grouping are ideal.

Select manipulatives that can serve many purposes. Some manipulatives, such as interlocking cubes, can be used for counting, place value, arithmetic, patterning, and many other topics. This versatility allows students to find many different uses. However, a few single-purpose devices, such as mirrors or Miras, make a significant contribution.

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. One theory held that students had to see an idea presented by several different manipulatives to abstract the essence of this idea. However, in some circumstances, using the same material consistently is advantageous. "Using the tool metaphor for representations, perhaps a representation becomes useful for students as they handle it and work with it repeatedly" (Hiebert and Wearne 1992, 114). If the tool is to become useful, perhaps an advantage accrues in using the same tool in different situations rather than in using different tools in the same situation. Students gain expertise through using a tool over and over on different projects.

Should only one manipulative be used, then? No, different children may find different models meaningful (Baroody 1990). Further, reflecting on and discussing different models may indeed help students abstract the mathematical idea. Brief and trivial use, however, will not help; each manipulative should become a tool for thinking. Different manipulatives allow, and even encourage, students to choose their own representations. New material can also be used to assess whether students understand the idea or just have learned to use the previous material in a rote manner.

Select computer manipulatives when appropriate. Certain computer manipulatives may be more beneficial than any physical manipulative. Some are just the sort of tools that can lead to mathematical expertise. The following recommendations and special considerations pertain to computer manipulatives. Select programs that --

• have uncomplicated changing, repeating, and undoing actions;
• allow students to save configurations and sequences of actions;
• dynamically link different representations and maintain a tight connection between pictured objects and symbols;
• allow students and teachers to pose and solve their own problems; and
• allow students to develop increasing control of a flexible, extensible, mathematical tool. Such programs also serve many purposes and help form connections between mathematical ideas.

Select computer manipulatives that --

• encourage easy alterations of scale and arrangement,
• go beyond what can be done with physical manipulatives, and
• demand increasingly complete and precise specifications.

How Should Manipulatives Be Used?

The following suggestions are offered to assist teachers in effectively using manipulatives in their classrooms.

Increase 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 might be better off if allowed to choose their manipulatives or to just use paper and pencil. Some students in the Netherlands were more successful when they drew pictures of the sultan's gold pieces than when they used any physical manipulative. Others may need manipulatives for different lengths of time (Suydam 1986).

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 helps 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 (Sowell 1989).

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.

Final Words

With both physical and computer manipulatives, teachers should choose meaningful representations, then guide students to make connections between these representations. No one yet knows what modes of presentations are crucial and what sequence of representations should be used before symbols are introduced (Baroody 1989; Clements 1989). Teachers should be careful about adhering blindly to an unproved, concrete-pictorial-abstract sequence, especially when more than one way of thinking about "concrete" is possible It is known that students' knowledge is strongest when they connect real-world situations, manipulatives, pictures, and spoken and written symbols (Lesh 1990). They should relate manipulative models to their intuitive, informal understanding of concepts and translate between representations at all points of their learning. This process builds integrated-concrete ideas.

Now when teachers close their eyes and picture children doing mathematics, manipulatives should still be in the picture, but the mental image should include a new perspective on how to use them.

References

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Bright, George, Virginia E. Usnick, and Susan Williams. Orientation of Shapes in a Video Game. Hilton Head, S.C.: Eastern Educational Research Association, 1992.

Butler, Deirdre, and Sean Close. "Assessing the Benefits of a Logo Problem-Solving Course." Irish Educational Studies 8 (1989):168-90.

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Clements, Douglas H., and Michael T. Battista. "Learning of Geometric Concepts in a Logo Environment." Journal for Research in Mathematics Education 20 (November 1989):450-67.

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Clements, Douglas H., Michael T. Battista, Joan Akers, Virginia Woolley, Julie Sarama Meredith, and Sue McMillen. Turtle Paths. Cambridge, Mass.: Dale Seymour Publications, 1995. Includes software.

Clements, Douglas H., Warren D. Crown, and Mary Grace Kantowski. Primary Graphing and Probability Workshop. Glenview, Ill.: Scott, Foresman & Co., 1991. Software.

Clements, Douglas H., and Julie Sarama Meredith. 1994. Turtle Math. Montreal: Logo Computer Systems (LCSI), 1994. Software.

Clements, Douglas H., Susan Jo Russell, Cornelia Tierney, Michael T. Battista, and Julie Sarama Meredith. Flips, Turns, and Area. Cambridge, Mass.: Dale Seymour Publications, 1995. Includes software.

Driscoll, Mark J. Research within Reach: Elementary School Mathematics and Reading. St. Louis: CEMREL, 1983.

Fennema, Elizabeth. "The Relative Effectiveness of a Symbolic and a Concrete Model in Learning a Selected Mathematics Principle." Journal for Research in Mathematics Education 3 (November 1972):233-38.

Gallou-Dumiel, Elisabeth. "Reflections, Point Symmetry and Logo." In Proceedings of the Eleventh Annual Meeting, North American Chapter of the International Group for the Psychology of Mathematics Education, edited by C. A. Maher, G. A. Goldin, and R. B. Davis, 140-57. New Brunswick, N.J.: Rutgers University Press, 1989.

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