Chances are, when you learned elementary mathematics, you learned to perform mathematical procedures. Known to mathematicians as algorithms, these procedures enabled you to find answers to problems according to set rules. If, for example, you think of division in terms of "divide, multiply, subtract, bring down" then you learned a division procedure (or algorithm). For another example, if you think only in terms of cross multiplying as a way of approaching problems involving proportions, chances are you learned only a procedure for solving mathematical proportions. At this point you may be wondering, "What else is there? What else would a person learn in a mathematics class?" The answer is, there is a great deal more to mathematics! These mathematical procedures are much like recipes that efficiency experts have developed to enable people to go straight to specific kinds of answers when confronted with particular kinds of well-defined problems. If we thought cooking was nothing more than following recipes developed by experts, we would be missing out on much of the joy of cooking! If there were no room in cooking for serendipitous combining of favorite ingredients, then cooking would be boring indeed.
As important as it is to be able to follow a recipe to quickly and efficiently obtain a certain kind of answer to a certain kind of problem, this is not the essence of mathematics. A true mathematician understands that an operation like division is a mental operation that may be performed by using any number of different procedures. A division problem may be solved by repeated subtraction, by repeated addition, by use of a numberline, or even by using objects and modeling the action of division. The procedure using "divide, multiply, subtract, bring down" is only one of many possible procedures. The underlying idea of division as an action is something quite different from anyone of the procedures for solving division problems. This underlying idea is referred to as the concept of division. The concept of division and the procedure of solving division problems are not the same thing. In today's mathematics classrooms we are teaching concepts first and foremost. Procedures are learned too, but not without a conceptual understanding. One of the benefits to emphasizing conceptual understanding is that a person is less likely to forget concepts than procedures. If conceptual understanding is gained, then a person can reconstruct a procedure that may have been forgotten. On the other hand, if procedural knowledge is the limit of a person's learning, there is no way to reconstruct a forgotten procedure. Conceptual understanding in mathematics, along with procedural skill, is much more powerful than procedural skill alone.
This different emphasis results in a different use of class time in mathematics classes. In a procedurally oriented mathematics class, almost all of the student's time must be spent practicing procedures so that the student can carry out the procedures flawlessly. In a conceptually oriented mathematics class, the bulk of time is spent helping the students develop insight. Activities and tasks are presented to provide learners with experiences that provide opportunities for new understandings. Once the students gain understanding, then there is a need for some time to be spent on practice. Research has shown that students in a conceptually oriented mathematics class outperform students in a procedurally oriented mathematics class on tests and on measures of attitude toward mathematics. (See, for example, Boaler, 1998; Cain, 2002; Fuson et al., 2000; Masden and Lanier, 1992.) During the 1990s and into the twenty-first century, schools have been focusing more on conceptual knowledge than ever before. The results have been encouraging, as documented by the National Assessment of Educational Progress (NAEP), which has shown a steady increase in student mathematical performance since 1990.
For teachers to be able to focus on the conceptual teaching of mathematics, they must have conceptual understandings themselves. One of the biggest challenges as we have been moving from procedurally oriented teaching to conceptually oriented teaching has been ensuring that the teachers have the necessary mathematical understandings. This is the reason for this book.
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