Developing “Function Sense”Arithmetic in the early grades can be characterized as the mathematics of number. We teach our students about numbers of different sorts [e.g. whole numbers, decimals, fractions] and the operations that can be done with these mathematical objects. We hope they develop a certain level of fluency and ease and comfort with numbers and their operations. If we are successful our students are said to have developed a reasonable level of “number sense”. Despite calculators it is still important that students know the standard computational algorithms and understand why they work. However, the widespread availability of calculators has made it possible for us to go beyond endless computational drill and shift our attention to such arithmetic life skills as estimation and approximation, which are at the heart of what we normally mean by “number sense”. The teaching and learning of mathematics in the middle and secondary grades centers on algebra. In school algebra we teach our students about functions of different sorts. Functions are the mathematical objects of algebra. Early on, attention is paid most heavily to linear, quadratic and to some extent absolute value functions. Later, attention turns to exponential, logarithmic, trigonometric and rational functions. Throughout the algebra curriculum, we teach our students to manipulate and transform the functions they are learning about. We hope they develop a certain level of fluency and ease and comfort with functions and their manipulations and transformations. Most important, however, is the goal of having our students use the tools of algebra as part of their life skills in the analysis of quantitative relationships among the elements of the world in which they live. This last goal readily explains the emphasis on extending the study of functions to functions of quantity and their use in modeling the world in which we all live. In the case of early mathematics, i.e., arithmetic, the technology of calculators has allowed us to shift our emphasis so that less effort needs be focused on having students develop automaticity with computational skills and more emphasis can be placed on arithmetic life skills as estimation and approximation. In algebra, the growing availability of graphing software and dynamic visual representations makes it possible for us to shift our instructional emphasis away from the development of automaticity with the manipulation of symbols and the plotting of graphs and place more emphasis on the algebra life skills of modeling and graph interpretation. If we are successful in getting our students to develop a certain level of fluency and ease and comfort with functions, and their graphs and interpretations we can say that they have developed a reasonable level of “function sense”. Judah L. Schwartz
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