Suppose you were told that you must learn the “fact” ΩTM∂_. How would you learn it? You would probably need to repeat the sequence to yourself several times before you would be able to reproduce it without looking at it. Are all of the symbols used familiar to you? If not, you might make connections between unfamiliar symbols and objects that are familiar to you, perhaps calling Ω “horseshoe” or calling ∂ “backward six.” Without some logic or meaning to the sequence of symbols, it would be difficult to commit this sequence to memory for more than a short period.
Constructivism and Conceptually Based Mathematics
Constructivism is a view of learning influenced by the works of Brownell, Piaget, Vygotsky, Dienes, and Bruner that asserts that children build or construct their own knowledge by integrating new concepts and procedures into existing mental structures. Children must create or recreate mathematical relationships in their own minds. For example, before a child learns the value of various U.S. coins, the child may believe that a nickel is more valuable than a dime because of the relative sizes of the coins. Once the child has assimilated the idea of coin value and has learned that a nickel is worth five cents while a dime is worth ten cents, the child will agree that a dime has more value than a nickel.
To process information, one must interpret it in relation to what is already known or believed. Children should be active in the learning experience in order to fully internalize the experience. Teachers who use the constructivist method of teaching encourage discovery learning and active experimentation by their students. With constructivism the emphasis is not on teaching, but on learning. Even very young children are capable of constructing mathematics concepts and algorithms (Cobb, 1994; Kamii & Ewing, 1996).
Procedural Learning and Conceptual Learning
Various mathematics educators have compared two types of learning: procedural learning and conceptual learning (Hiebert & Lindquist, 1990; Skemp, 1971). Procedural learning involves learning processes or algorithms by rote. You may have learned how to divide one fraction by another by learning a rule like, invert the divisor and multiply. Conceptual learning involves understanding the concepts and meanings underlying the operations as opposed to merely applying rules. The main tenet of conceptually based mathematics is that when students understand the concepts and reasoning underlying a process, they are more likely to be able to correctly apply that process. They are also more likely to be able to apply that process in learning related new skills and procedures. Let’s use the example 3 × 5 = 15 to illustrate these ideas.
Three Levels of Representation of Knowledge
You probably do not recall learning that three times five is fifteen. This is a fact that you learned at approximately the second or thirdgrade level. Yet you are able to retain this fact in your memory. Why? For one thing, this fact has been used hundreds, maybe thousands of times in your everyday life. Also, the symbols used in the representation of this fact (3 × 5 = 15) are familiar to you. Most important, the fact makes sense. If you were to combine 3 piles of 5 blocks each, you would have a pile of 15 blocks—which explicitly shows that three times five is fifteen.
In learning this particular fact, you probably did not begin with the symbolic representation 3 × 5 = 15. You may have begun with a concrete representation such as the 3 piles of 5 blocks. Many elementary textbooks introduce addition or multiplication facts pictorially. For this particular example an elementary textbook might show a picture of 3 rows with 5 blocks in each row, and ask “How many blocks in all?” Skip counting can be used to see that in the 3 sets of 5 blocks there are 5, 10, 15 blocks in all. The fact 3 × 5 = 15 makes sense. Research indicates that, as this example illustrates, human beings tend to represent knowledge in basically three ways (Bruner, 1964; Piaget, 1952).

Concrete: The key component of the concrete level of representation is the performance of an action on an object or objects. At the concrete level, students use handson manipulatives to physically arrive at the solution. Consider the following problem: “My three dogs each have five fleas. How many fleas do they have all together?” A student at the kindergarten or firstgrade level might solve this problem by first setting out three paper plates or three pages of paper to represent three dogs. They could then distribute five counters or “fleas” to each “dog” to solve the problem.
The solution described above would be possible only in a classroom where handson experimentation was encouraged. The idea of using plates or papers to represent dogs and counters to represent fleas shows great imagination and creativity on the part of the student.
Pictorial: At the pictorial level of representation the child no longer needs to physically manipulate objects to solve problems such as the one given above. The student may solve such a problem by drawing a picture of three dogs and drawing five fleas on each dog. In some elementary textbooks, several pictures are already drawn and the student is asked to choose the picture that matches a given problem situation. Often in a disposable textbook, part of a picture is drawn and the student is instructed to complete the drawing and use it to solve the given problem. The student may also solve the problem mentally by visualizing three sets of five and counting.

Symbolic: At the symbolic level of representation the student has acquired enough experience with the given problem type to represent it symbolically without using concrete operations or pictures or images. For the dog and flea problem discussed above, the student would merely write or state that 3 × 5 = 15.
When encountering a new problem type, students will need experiences at the concrete and pictorial levels before they are introduced to the symbolic representation of the problem. The concrete and pictorial experiences let the student acquire meaning for the concepts and operations involved in the problem. Many activities use a combination of the three levels of representation described. Students often use pictures as concrete objects, combining the levels concrete and pictorial. The pictorial level is combined with the symbolic or abstract level when students use pictures to arrive at an answer and then write or state a corresponding equation.

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