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## "Concrete" Manipulatives, Concrete IdeasBy Douglas H. Clements

The notion of "concrete," from concrete manipulatives to pedagogical sequences such as "concrete to abstract," is embedded in educational theories, research, and practice, especially in mathematics education. While such widely accepted notions often have a good deal of truth behind them, they can also become immune from critical reflection. In this article, I will briefly consider research on the use of manipulatives and offer a critique of common perspectives on the notions of concrete manipulatives and concrete ideas. From a reformulation of these notions, I re-consider the role computer manipulatives may play in helping students learn mathematics, providing illustrations from our empirical research.

Research on Concrete Manipulatives

Students who use manipulatives in their mathematics classes usually outperform those who do not, although the benefits may be slight . This benefit holds across grade level, ability level, and topic, given that use of a manipulative "makes sense" for that topic. Manipulative use also increases scores on retention and problem solving tests. Attitudes toward mathematics are improved when students have instruction with concrete materials provided by teachers knowledgeable about their use.

However, manipulatives do not guarantee success . One study showed that classes not using manipulatives outperformed classes using manipulatives on a test of transfer . 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 beansticks used the (one) bean as ten and the beanstick (with ten beans on it) as one.

Similarly, students often fail to link their actions on base-ten blocks with the notation system used to describe the actions . For example, when asked to "select a block to stand for one, then put blocks out to represent 3.41," one fourth-grader put out 3 flats, 4 longs, and 1 single after reading the decimals "three hundred forty one."

Finally, teachers often use manipulatives as a way to reform their mathematics teaching, without reflecting on their use of representations of mathematical ideas or on the other aspects of their instruction must be changed . Both teachers and parents often believe that reform in mathematics education indicates that "concrete" is good and "abstract" is bad. In contrast, professional standards suggest students have a wide range of understandings and tools.

In summary, although research might suggest that instruction begin "concretely," it also warns that manipulatives are not sufficient to guarantee meaningful learning. To understand the role of concrete manipulatives and any concrete-to-abstract pedagogical sequence, we must further define what we mean by "concrete."

The Meaning of "Concrete"

Most practitioners and researchers argue that manipulatives are effective because they are concrete. By "concrete," they probably mean objects that students can grasp with their hands. This sensory nature ostensibly makes manipulatives "real," connected with one's intuitively meaningful personal self, and therefore helpful. There are, however, problems with this view.

Problems Attributing Benefits of Concrete Manipulatives to their Physical Nature

First, it cannot be assumed that concepts can be "read off" manipulatives. That is, the physical objects may be manipulated meaningfully without the concepts being illuminated. Worked with Cuisenaire rods, John Holt 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, even if children begin to make connections between manipulatives and nascent ideas, physical actions with certain manipulatives may suggest different mental actions than those we wish students to learn. For example, researchers found a mismatch among students using the number line to perform addition. When adding 5 + 4, the students located 5, counted "one, two, three, four," and read the answer. This did not help them solve the problem mentally, for to do so they have to 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 . These researchers also found that students' external actions on an abacus do not always match the mental activity intended by the teacher.

Therefore, although manipulatives have an important place in learning, their physicality does not carry the meaning of the mathematical idea. They can even be used in a rote manner, as did the student who used the bean as ten and the beanstick as one to illustrate place value. Students may require concrete materials to build meaning initially, but they must reflect on their actions with manipulatives to do so. They need teachers who can reflect on their students' representations for mathematical ideas and help them develop increasing sophisticated and mathematical representations. "Although kinesthetic experience can enhance perception and thinking, understanding does not travel through the fingertips and up the arm."

Further, when we speak of concrete understanding, we are not always referring to physical objects. Teachers of later grades expect students to have a concrete understanding that goes beyond manipulatives. For example, we like to see that numbers-as mental objects ("I can think of 13 + 10 in my head")-are "concrete" for older students. It appears that there are different ways to think about "concrete."

Two Types of Concrete Knowledge

We have Sensory-Concrete knowledge when we need to 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 things. Consider Brenda, a primary grade student. The interviewer had covered 4 of 7 squares with a cloth, told Brenda that 4 were covered, and asked how many in all. Brenda tried to raise the cloth, but was thwarted by interviewer. She then counted the three visible squares.

 B: 1, 2, 3 (touches each visible item in turn) I: There's four here (taps the cloth). B: (Lifts the cloth, revealing two squares) 4, 5. (touches each and puts cloth back). I: Ok, I'll show you two of them (shows two) . There's four here, you count them. B: 1, 2 (then counts each visible): 3, 4, 5 I: There's two more here (taps the cloth). B: (Attempts to lift the cloth.) I: (Pulls back the cloth.) B: 6, 7 (touches the last two squares).

Brenda's attempt to lift the cloth indicates that she was aware of the hidden squares and wanted to count the collection. This did not lead to counting because she could not yet coordinate saying the number word sequence with items that she only imagined. She needed physically present items to count. Note that this does not mean that manipulatives were the original root of the idea. Research tends to indicate that is not the case.

Integrated-Concrete knowledge is built as we learn. It is knowledge that is connected in special ways. This is the root of the word concrete-"to grow together." What gives sidewalk concrete its strength is the combination of separate particles in an interconnected mass. What gives Integrated-Concrete thinking its strength is the combination of many separate ideas in an interconnected structure of knowledge. While still in primary school, Sue McMillen's son 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 . For students with this type of interconnected knowledge, physical objects, actions performed on them, and abstractions are all interrelated in a strong mental structure. Ideas such as "75," "3/4," and "rectangle" become as real, tangible, and strong as a concrete sidewalk. Each idea is as concrete as a wrench is to a plumber-an accessible and useful tool. Jacob's knowledge of money was in the process of becoming such a tool for him.

Therefore, an idea is not simply concrete or not concrete. Depending on what kind of relationship you have with the knowledge , it might be Sensory-Concrete, abstract, or Integrated-Concrete. Further, we as educators can not engineer mathematics into Sensory-Concrete materials, because ideas such as number are not "out there." As Piaget has shown us, 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 physical or pictured blocks . As Piaget's collaborator Hermine Sinclair says, "�numbers are made by children, not found (as they may find some pretty rocks, for example) or accepted from adults (as they may accept and use a toy)."

What ultimately makes mathematical ideas Integrated-Concrete is not their physical characteristics. Indeed, physical knowledge is a different kind of knowledge than logical/mathematical knowledge, according to Piaget . Also, some research indicates that pictures are as effective for learning as physical manipulatives . What makes ideas Integrated-Concrete is how "meaning-full"-connected to other ideas and situations-they are. John Holt reported 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" . Good manipulatives are those that aid students in building, strengthening, and connecting various representations of mathematical ideas. Indeed, we often assume that more able or older students' greater facility with mathematics stems from their greater knowledge or mathematical procedures or strategies. However, it is more often true that younger children possess the relevant knowledge but cannot effectively create a mental representation of the necessary information . This is where good manipulatives can play an important role.

Comparing the two levels of concrete knowledge, we see a shift in what the adjective "concrete" describes. Sensory-Concrete refers to knowledge that demands the support of concrete objects and children's knowledge of manipulating these objects. Integrated-Concrete refers to concepts that are "concrete" at a higher level because they are connected to other knowledge, both physical knowledge that has been abstracted and thus distanced from concrete objects and abstract knowledge of a variety of types. Ultimately, these are descriptions of changes in the configuration of knowledge as children develop. Consistent with other theoreticians, I do not believe there are fundamentally different, incommensurable types of knowledge, such as "concrete" versus "abstract."

The Nature of "Concrete" Manipulatives and the Issue of Computer Manipulatives

Even if we agree that "concrete" can not simply be equated with physical manipulatives, we might have difficulty accepting objects on the computer screen as valid manipulatives. However, computers might provide representations that are just as personally meaningful to students as physical objects. Paradoxically, research indicates that computer representations may even be more manageable, "clean," flexible, and extensible than their physical counterparts. 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 to a physical bean-stick environment, this computer environment offered equal, and sometimes greater control and flexibility to students . The computer manipulatives were just as meaningful and easier to use for learning. Both computer and physical beansticks were worthwhile. However, addressing the issues of pedagogical sequencing, work with one did not need to precede work with the other. In a similar vein, students who used physical and software manipulatives demonstrated a much greater sophistication in classification and logical thinking than did a control group that used physical manipulatives only . Finally, a study of eighth graders indicated that a physical, mechanical device did not make mathematics more accessible, though a computer microworld did.

Like beauty, then "concrete" is, quite literally, in the mind of the beholder. It is ironic that Piaget's period of concrete operations is often used, incorrectly, as a rationalization for objects-for-objects' sake in elementary school. Kamii , one of the staunchest of Piagetians, eschews much traditional use of manipulatives. Good concrete activity is good mental activity . Good manipulatives are those that are meaningful to the learner, provide control and flexibility to the learner, have characteristics that mirror, or are consistent with, cognitive and mathematics structures, and assist the learner in making connections between various pieces and types of knowledge-in a word, serving as a catalyst for the growth of integrated-Concrete knowledge. Computer manipulatives can serve that function.

Shapes: A Computer Manipulative

Let us consider several (concrete!) examples. Shapes is a computer manipulative, a software version of pattern blocks, that extends what children can do with these shapes (see Fig. 1). Children create as many copies of each shape as they want and use computer tools to move, combine, and duplicate these shapes to make pictures and designs and to solve problems.

Shapes was designed on theoretical and research bases to provide children with specific benefits. These are described in the following sections in two categories: Practical and pedagogical benefits and mathematical and psychological benefits. For several, we provide illustrations from our participant observation research with kindergarten-age children.

Practical /pedagogical benefits. This first group includes advantages that help students in a practical manner or provide pedagogical opportunities for the teacher.

1. Providing another medium, one that can store and retrieve configurations. Shapes serves as another medium for building, especially one in which careful development can take place day after day (i.e., physical blocks have to be put away most of the time-on the computer, they can be saved and worked on again and again, and there's an infinite supply for all children).

We observed this advantage when a group of children were working on a pattern with physical manipulatives. They wanted to move it slightly on the rug. Two girls (four hands) tried to keep the design together, but they were unsuccessful. Marisssa told Leah to fix the design. Leah tried, but in re-creating the design, she inserted two extra shapes and the pattern wasn't the same. The girls experienced considerable frustration at their inability to get their "old" design back. Had the children been able to save their design, or had they been able to move their design and keep the pieces together, their group project would have continued.

2. Providing a manageable, clean, flexible manipulative. Shapes manipulatives are more manageable and clean than their physical counterparts. For example, they always snap into correct position even when filling an outline and-also unlike physical manipulatives-they stay where they are put. If children want them to stay where they're put no matter what, they can "freeze" them into position. We observed that while working on the Shapes software, children quickly learned to glue the shapes together and move them as a group when they needed more space to continue their designs.

3. Providing an extensible manipulative. Certain constructions are easier to make with the software than with physical manipulatives. For example, trying to build triangles from different classes. That is, we have observed children making non-equilateral triangles by partially occluding shapes with other shapes, creating many different types of triangle. Making right angles by combining and occluding various shapes is a similar example.

As an illustration, Matthew was taking his turn working with the physical manipulatives (off-computer) and was trying to fill in an outline of a man using all blue diamonds. At the end he was left with a space that only a green triangle would fit. He said, "If I was on computer I could make it all blue." Upon further questioning, it was revealed that the child knew that the two shapes were not the same, but that two green triangles is the same as one blue diamond, so half would go on the other shapes. The flatness of the screen allows for such "building up" and thus explorations of these types of relationships.

4. Recording and extending work. The printouts make instant record-your-work, take-it-home, post-it paper copies. (Although we are also in favor of kids recording their work with templates and/or cut-outs, but this is time consuming and should not be required all the time.)

Mathematical/psychological benefits. Perhaps the most powerful feature of the software is that the actions possible with the software embody the processes we want children to develop and internalize as mental processes.

1. Bringing mathematical ideas and processes to conscious awareness. The built-in turn and flip tools are a good way to bring geometric motions to an explicit level of awareness and explicate these motions. For example, when Mitchell worked off-computer, he quickly manipulated the pattern block pieces, resisting answering any questions as to his intent or his reasons. When he finally paused, a researcher asked him how he had mad a particular piece fit. He struggled with the answer and then finally said that he "turned it." When working on-computer he again seemed very sure of himself and quickly manipulated the shapes, avoiding answering the questions. However, he seemed more aware of his actions, in that when asked how many times he turned a particular piece, he said," Three," without hesitation. Thus, he was becoming explicitly aware of the motions and beginning to quantify them.

2. Changing the very nature of the manipulative. Shapes' flexibility allows children to explore geometric figures in ways not available with physical shape sets. For example, children can change the size of the computer shapes, altering all shapes or only some. The example of Matthew wanting to make an all blue man, recognizing that he could overlap the diamonds and be able to exactly cover a triangle space, illustrates this benefit.

3. Allowing composition and decomposition processes. Shapes encourages composition and decomposition of shapes. The glue tool allows children to glue shapes together to make composite units. The hammer tool allows the decomposition of those shapes. In addition, the hammer tool allows children to decompose one shape (e.g., a hexagon) into other shapes (e.g., two trapezoids), a process difficult to duplicate with physical manipulatives.

Mitchell started making a hexagon out of triangles. After placing two, he counted with his finger on the screen around the center of the incomplete hexagon, imaging the other triangles. He announced that he would need four more. After placing the next one, he said, "Whoa! Now, three more!" Whereas off-computer, Mitchell had to check each placement with a physical hexagon, the intentional and deliberate actions on the computer lead him to form mental images (decomposing the hexagon imagistically) and predict each succeeding placement.

4. Creating and operating on units of units. With tools such as the glue and hammer, Shapes allows the construction of units of units in children's tilings and linear patterns. Identify the unit of units that forms the core. Show how the glue tool in the software can be used to actually make such a unit and then slide, turn and flip it as a unit. It also makes building such patterns much easier (and more elegant). A set of ungrouped objects, for example, can be turned together. However, in that case each shape turns separately. Only grouped shapes turn as a unit. Thus, the actions children perform on the computer are a reflection of the mental operations we wish to help children develop.

5. Connecting space/geometry learning to number learning. One of the most powerful benefits of the computer is to help children link their ideas and processes about number and arithmetic to their ideas about shape and space. Geometric models are used ubiquitously to teach about number. Shapes helps in that it dynamically links spatial and numerical representation. As an example, Shapes' manipulative sets include on-screen base-ten blocks. Whereas physical blocks must be "traded" (e.g., in subtracting, students may need to trade 1 ten for 10 ones), students can break a computer base-ten block into 10 ones. Again, such actions are more in line with the mental actions that we want students to learn. The computer also links the blocks to the symbols. For example, the number represented by the base-ten blocks is dynamically linked to the students' actions on the blocks, so that when the student changes the blocks the number displayed is automatically changed as well. This 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.

Benefits of Other Computer Manipulatives

Other programs we have developed illustrate-better than Shapes does-some additional advantages of computer manipulatives.

Recording and Replaying Students' Actions

Computers allow us to store more than static configurations. Once we finish a series of actions, it's often difficult to reflect on them. But computers have the power to record and replay sequences of our actions on manipulatives. We can record our actions and later replay, change, and view them. This encourages real mathematical exploration. Computer games such as "Tetris" allow students to replay the same game. In one version, Tumbling Tetrominoes , students try to cover a region with a random sequence of tetrominoes (see Figure 2). If students believe they could improve their strategy, they can elect to receive the same tetrominoes in the same order and try a new approach.

Linking the Concrete and the Symbolic with Feedback

Other advantages go beyond convenience. For example, we already established that a major advantage of the computer is the ability to link active experience with objects to symbolic representations. The computer connects objects that you make, move, and change to numbers and words. For example, students can draw rectangles by hand, but never go further thinking about them in a mathematical way. In Logo, however, students must analyze the figure to construct a sequence of commands (a procedure) to draw a rectangle (see Figure 3). So, they have to apply numbers to the measures of the sides and angles (turns). This 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, the actions of the turtle object, and the figure are direct and immediate. Studies confirm that students' ideas about shapes are more mathematical and precise after using Logo.

Some students understand certain ideas, such as angle measure, for the first time using Logo. They have to make sense of what it is that is being controlled by the numbers they give to right and left turn commands. The turtle helps them link 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 . Fortunately, students are not surprised that the computer does not understand natural language, so they have to formalize their ideas to communicate them. Students formalize about fives times as often using computers as they do using paper.

Is it too restrictive or too hard to have to operate on symbols rather than 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 . 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.

In computer environments such as computer base-tens blocks or computer programming, students can not overlook the consequences of their actions, whereas that is possible to do with physical manipulatives. So, computer manipulatives 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.

Encouraging and Facilitating Complete, Precise, Explanations

Compared to students using paper and pencil, students using computers work with more precision and exactness . In one study, we 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 led to more discussion of the motions themselves, rather than just the shapes

Final Words: Concrete Manipulatives and Integrated-Concrete Ideas

Manipulatives can play a role in students' construction of meaningful ideas. They should be used before formal symbolic instruction, such as teaching algorithms. However, other common perspectives on using manipulatives should be re-considered. Teachers and students should avoid using manipulatives as an end-without careful thought-rather than as a means to that end. A manipulative's physical nature does not carry the meaning of a mathematical idea. Manipulatives alone are not sufficient-they must be used in the context of educational tasks to actively engage children's thinking with teacher guidance. In addition, definitions of what constitute a "manipulative" may need to be expanded to include computer manipulatives, which, at certain phases of learning, may be more efficacious than their physical counterparts.

With both physical and computer manipulatives, we should choose meaningful representations in which the objects and actions available to the student parallel the mathematical objects (ideas) and actions (processes or algorithms) we wish the students to learn. We then need to guide students to make connections between these representations . We do not yet know what modes of presentations are crucial and what sequence of representations we should use before symbols are introduced . We should be careful about adhering blindly to an unproved concrete � pictorial � abstract sequence, especially when there is more than one way of thinking about "concrete." There have been, to my knowledge, no studies that actually have evaluated the usefulness of this sequence, as opposed to the reverse sequence or-as I suspect may be best-all three in parallel. When students connect manipulative models to their intuitive, informal understanding of concepts and to abstract symbols, when they learn to translate between representations, and when they reflect on the constraints of the manipulatives that embody the principles of a mathematics system , they build Integrated-Concrete ideas. This should be the goal of our use of manipulatives.

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Time to prepare this material was partially provided by the National Science Foundation under Grants No. MDR-9050210, MDR-8954664, and ESI-9730804. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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