How Children Learn Mathematics

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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 third-grade 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.