The Nature and Role of Algebra in the K-14 Curriculum Proceedings of a National Symposium (1998) / Chapter Skim
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Developing a Coherent and Focused K-12 Algebra Curriculum (E. Phillips)
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It eliminates the most pernicious curricular element of today's school mathematics late, abrupt, isolated, and superficial high-school algebra courses. An early-start, strands approach to algebra also fits well with an inclusive, big-idea strands oriented approach to the curriculum at large, which contrasts with the layer cake-filter structure that delays and ultimately denies access to powerful ideas for all students but a few.
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While 3. is a school mathematics topic strand found today mainly at the advanced levels, it is also an important domain of mathematics in its own right abstract algebra.
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Continued demonstration of students' lack of basic skills despite two decades of a skill-and-drill curriculum should convince us to look for a new way of bringing students and algebra together. In the search for new ways of teaching and learning algebra, we must seek answers to the following questions: How do we build a coherent algebra curriculum across the grades that develops understanding of important algebra concepts and skills and students' quantitative reasoning abilities?
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Taken altogether, the intuitive uses of algebraic thinking in a problem-based curriculum plus the specific development of the concepts of variable and relationships among variables constitute a powerful, sense-making approach to algebra in the K- 12 curriculum for all students. However, these practices are a radical departure from the traditional algebra curriculum that has focused almost exclusively on manipulating expressions and solving symbolic equations.
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Washington, DC: National Center for Educational Statistics. National Council of Teachers of Mathematics Algebra Working Group.
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For a student with this "algebra sense," symbol manipulation is a meaningful activity, and unreasonable answers are apparent. In this interpretation, algebraic reasoning includes the ability, through pattern recognition, to conjecture, even to have confidence in, symbolic generalizations.
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They also are supposed to introduce students to a complex cultural tradition in the liberal arts and the sciences, a tradition that now includes the entire world, not simply Western civilization. They further are charged with developing individual skills relating to such nonacademic areas as vocation, leisure-time activity, and citizenship, as well as providing such specialized programs as driver education, sex education, drug education, and parent education.
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The collective "Mathematics as Reasoning" standards (National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics, 1989) call for students to justify their answers and solutions, to make and evaluate mathematical conjectures, to use counter examples effectively, to draw logical conclusions through deductive and inductive reasoning, and to appreciate the role and power of reasoning in mathematics.
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( 1989.) Curriculum and Evaluation Standards for School Mathematics.
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I will summarize the research under four headings: the importance of a functional approach with contextual problems, the use of multiple representations, the importance of student methods and interactions, and the systemic or community approach to change. A FUNCTIONAL AND CONTEXTUAL APPROACH Most textbooks still begin algebra with solving for unknowns.
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The context-based part of the approach means that the algebraic activities emerge from problems and tasks that relate frequently but not necessarily exclusively to referents from everyday activity. Researchers have offered a multitude of arguments for this, including the ease of identifying situations to which students can relate; the value of motivating math with a need to explain, predict, and model and thus actively to involve students in useful mathematics; the value of position, time, and rate to make the idea of variation accessible, to anticipate calculus, and to lessen dependence on symbol strings and to vary representations with graphs and tables, for example; and how making algebraic manipulation skills instrumentally rather than intrinsically valuable and using new technologies to carry out the algorithms of traditional algebra allows students to focus on using problem solving, communication, and learning technologies to engage in rich modeling-based activities.
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Finally, research in the use of new technologies and project-based learning allows us to question many peoples' assumptions about the required developmental sequences. It appears that when students operate in complex settings with powerful tools, the sequences of learning vary, and the insights that are gained are more a function of the tasks and tools than any rigid developmental sequence.
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For this reason, participation and success in it has been cast by some researchers as an issue of civil rights. It is at this critical juncture that our decisions about content, sequence, pace, curricular approach, use of technology, forms of interaction, types of help, and feedback to students carry out a political agenda that has ramifications way down the line in terms of participation and success.
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The algebra that most people use on a regular basis consists primarily of generalized and specific examples of function relationships that illustrate the systematic dependence of one quantity on another. This perspective of algebra is different from the perspective evident in traditional school algebra, but it is equally important.
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The College Board also reported that, in general, majority students who complete high school algebra attend college at a much higher rate than underrepresented minority students. For example, at the national level in 1993, 58% of all white students as compared to only 47% of all African American students attended college.
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We ask campus leaders to join mathematics educators in rethinking the purpose of algebra, in verbalizing the importance of algebra for every student, and in supporting efforts to create such a program. Counting on You: Actions Supporting Mathematics Teaching Standards (National Academy Press, 1991)
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(1991.) Counting on You: Actions Supporting Mathematics Teaching Standards.
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, and equitable as well as appropriate grading practices are non-negotiable components. Additionally, sessions on technology, both the calculator and computer, critical thinking, reading and mathematics, appropriate use of manipulatives, curriculum alignment, developmentally appropriate instructional practices, learning styles, and cultural diversity as they relate to and support algebra instruction should be included in all professional development activities.
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At all times, however, instructional decisions are made on the basis of what is best for meeting the needs of the students and careful analysis of student performance. In conclusion, we must reflect once more on the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics.
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(1989.) Curriculum and Evaluation Standards for School Mathematics.
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