Basic Algebra

3 Key excerpts on "Basic Algebra"

• 

  eBook - ePub

  Learning and Teaching Mathematics

  An International Perspective

  • Peter Bryant, Terezinha Nunes(Authors)
  • 2016(Publication Date)
  • Psychology Press

    (Publisher)

  The different interpretations will lead to different ways of tackling the problem and to different solutions: in the former case one finds the roots of the equation using the quadratic formula, in the latter the values of parameters p and q are sought for which the coefficients of the same powers of x in both expressions are equal (p + 2 q = 5 and 3 p - q = 1). From this first section of the chapter, one might be led to conclude that there could be as many different perspectives on algebra as there are researchers one might choose to listen to. However, underlying all these perspectives—even a functional one—one can find a few common threads. All use algebra at least as a notation, a tool whereby we not only represent numbers and quantities with literal symbols but also calculate with these symbols. One could stretch this definition to the extent of saying that algebra is also a tool for calculus (in fact, historically, algebra led to calculus). However, that would miss the point, which is that the symbols have different interpretations depending on the conceptual domain (i.e. the letters represent different objects in school algebra than they represent in the calculus courses). In addition, some new symbols are generated and the procedures for calculating with the symbols also change in subtle ways as one moves from one mathematical sub-discipline to another. Algebra can thus be viewed as the mathematics course in which students are introduced to the principal ways in which letters are used to represent numbers and numerical relationships—in expressions of generality and as unknowns—and to the corresponding activities involved with these uses of letters—on the one hand, justifying, proving, and predicting, and, on the other hand, solving. These uses of algebraic letters all require some translation from one representation to another; furthermore, the activities associated with these letter-uses all require some symbol manipulation

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  eBook - ePub

  Algebra in the Early Grades

  • James J. Kaput, David W. Carraher, Maria L. Blanton, James J. Kaput, David W. Carraher, Maria L. Blanton(Authors)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  algebra means the content of traditional algebra I and the courses that follow it. For authors of algebra I textbooks, algebra is a tightly integrated system of symbolic procedures, each of which is closely connected with a particular problem type. The procedures are often introduced as the mathematical means to solve specific types of problems, but the focus quickly becomes learning how to manipulate symbolic expressions. These procedures are then practiced extensively and later applied to specific problem situations (i.e., word problems). Teaching this content involves helping students to interpret various commands—solve, reduce, factor, simplify—as calls to apply memorized procedures that have little meaning beyond the immediate context. For many students, this reduces algebra to a set of rituals involving strings of symbols and rules for rewriting them instead of being a useful or powerful way to reason about situations and questions that matter to them. Consequently, many students limit their engagement with algebra and stop trying to understand its nature and purpose. In many cases, this marks more or less the end of their mathematical growth.

  Many mathematics educators have recognized the deep problems of content and impact of algebra I and have made introductory algebra a major site in curricular reform efforts (Chazan, 2000; Dossey, 1998; Edwards, 1990; Fey, 1989; Heid, 1995; Phillips & Lappan, 1998). In one class of proposals, algebra is presented as a set of tools for analyzing realistic problems that outstrip students’ arithmetic capabilities. In contrast to algebra I, problem situations involving related quantities serve as the true source and ground for the development of algebraic methods, rather than mere pretext (Chazan, 2000; Lobato, Gamoran, & Magidson, 1993; Phillips & Lappan, 1998). These introductions to algebra aim to develop students’ abilities to use verbal rules, tables of values, graphs, and algebraic expressions to analyze the mathematical functions embedded in the problem situations, and centrally involve computer-based tools and graphing calculators to achieve these goals (Confrey, 1991; Demana & Waits, 1990; Heid, 1995; Schwartz & Yerushalmy, 1992).

  Other proposals have emphasized the abstract and formal aspects of mathematical practice, suggesting that introductory algebra should develop students’ abilities to identify and analyze abstract mathematical objects and systems. For example, Cuoco (1993, 1995) characterized algebra as the study of numerical and symbolic calculations and, through the development of a theory of calculation, the study of operations, relations among them (e.g., distributivity), and mathematical systems structured by those operations. Cuoco’s proposal reflects mathematicians’ interest in the study of increasing abstract and general algebraic systems.

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  eBook - ePub

  How to Solve Mathematical Problems

  • Wayne A. Wickelgren(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  11

  Problems from Mathematics, Science, and Engineering

  This chapter is designed to establish the generality of the problem-solving methods discussed throughout the book. In previous chapters, the problems used to illustrate the methods were deliberately selected so that they could be solved by the reader with no more background than a high school student with one year of algebra and one year of plane geometry. Many of the problems were of the puzzle (or brain teaser or recreational mathematics) variety, which require no specialized knowledge of mathematics, science, or engineering. Although methods for solving such problems have some recreational interest, there is also a serious practical reason in mastering them, for they are also useful for solving serious problems in all areas of mathematics, science, and engineering. This chapter is designed to demonstrate this applicability and to give the reader some experience in it.

  ALGEBRA

  The solution of systems of simultaneous linear equations provides a simple example of the use of evaluation functions, hill climbing, and subgoals. As an example, consider the following system of three linear equations:

  (E1)

  (E2)

  (E3)

  The operations available for solving such a system are essentially the following. We can (a) multiply both sides of an equation by the same number, (b) add equals to equals (or subtract equals from equals), and (c) substitute equals for equals. As an example of the first, consider the action of multiplying both sides of equation (E2) by the number —2. This yields the equation —2x —4y —10z

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