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.
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.
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.
A version of this article is published as: Clements, D. H. (1999). 'Concrete' manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45-60
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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