Virtually all of mathematics is related to reasoning in one form or another. Perhaps you use the word reasoning in the same way you use the word thinking. However, we will be examining several specific kinds of mathematical reasoning, and each of these kinds of reasoning will have a name. Recent research on the brain suggests that each different kind of reasoning is handled by different parts of the brain or by different combinations of parts of the brain. Therefore, each of these different kinds of reasoning will probably have a different feel to the student of mathematics. It is likely that each learner will find some forms of reasoning to be easier than others, and each person will differ in which areas he or she finds to be difficult. One of the most powerful features of mathematics is that many mathematical problems can be solved using various forms of reasoning. One person will easily find a visual way to solve a problem, while another person will easily find a symbolic way to solve the same problem. By studying and becoming very familiar with many forms of reasoning, the student of mathematics will discover which ways are her preferred ways and will know to use those ways deliberately whenever she is faced with a difficult problem.
Students who have concluded that mathematics is too difficult for them need to realize that reasoning abilities are not out of their reach. Reasoning, in the various forms we will present, is not totally dependent on a person's innate, intellectual ability. Reasoning can be taught and learned; it can be practiced, and reasoning ability can be improved. As you study mathematics during your study of this text, you will learn about inductive and deductive reasoning, and you will be given exercises to help you improve your ability in both areas. You will identify spatial, or geometric, reasoning as a particular kind of reasoning, and you will have the opportunity to improve your geometric reasoning ability through practice. You will have opportunities to focus attention on proportional reasoning and improve your proportional reasoning ability through practice in various contexts. You will have many opportunities to develop numeric reasoning, often referred to as number sense, through a variety of exploratory and practice problems.
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