# Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers

## Synopsis

This supplement to Teaching Fractions and Ratios for Understanding is not merely an answer key but a resource that includes in-depth discussions of all the problems in the text; develops and extends discussion of the issues, teaching problems, and other considerations raised in the chapters; and contains additional problems with solutions that instructors may find helpful for assessment purposes and a set of problems that require both good reasoning and computational skills.

## Excerpt

All too often, children's disenchantment with mathematics begins late in elementary school or in middle school when, even after years of practice, they cannot remember how to "do" fractions after summer vacation, or when they can perform steps, but are totally bored because they do not know what the steps mean or why they are doing them. Understanding fractions marks only the beginning of the journey toward rational number understanding and, by the end of the middle school years, as a result of maturation, experience, and instruction in mathematics, it is presumed that students are capable of a formal thought process called proportional reasoning. This form of reasoning opens the door to high school mathematics and science, and eventually, to careers in the mathematical sciences. The losses that occur because of the gaps in conceptual understanding about fractions, ratios, and rational numbers are incalculable. The consequences of "doing," rather than understanding, directly or indirectly affect a person's attitudes toward mathematics, enjoyment and motivation in learning, course selection in mathematics and science, achievement, career flexibility, and even the ability to fully appreciate some of the simplest phenomena in everyday life.

For this reason, the National Council of Teachers of Mathematics asserted in their Curriculum and Evaluations Standards (1989), that proportional reasoning "is of such great importance that it merits whatever time and effort must be expended to assure its careful development" (p. 82). Unfortunately, until recently, we have had little understanding of how proportional reasoning develops. By the time one reaches middle school, mathematics and human cognition are of sufficient complexity that studying the development of understanding in fractions, ratios, and rational numbers is a challenging research site. Without a research base to inform decision making about the important conceptual components of proportional reasoning, textbook approaches have unintentionally encouraged simplistic, mechanical treatment of ratios and proportions, highlighting the algebraic representation of a proportion and the manipulation of the symbols. The rules for solving problems using proportions were indelibly printed into our memories: put like term over like term, cross multiply, then divide. For most people, the rule is a proxy for reasoning about quantities and their relationships.

This book represents an attempt to shorten the inevitable time lag between the completion of research and the translation of that research into usable ideas for the classroom. In recent years, those who do research in the teaching and learning of fractions, ratios, and rational numbers have developed a deeper understanding of the complexity of the mathematics, the ways in which these domains are related, and some of the critical components of understanding. While there is still much to be learned about children's thinking and learning in relation to rational numbers, there are some fundamental ideas that . . .

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