Reshaping School Mathematics A Philosophy and Framework for Curriculum (1990) / Chapter Skim
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A Framework for Change
A Framework for Change
Pages 35-52
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From page 35... ...
· An available curriculum, that can be taught with existing teaching materials and currently trained teachers. · An adopted curriculum, that a school district says should be taught.
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From page 36... ...
Students who achieve significant mathematical power during their school years will be able to use mathematics in their careers and in everyday lives, They will be intelligent users of mathematical ideas, accepting or rejecting claims that are ostensibly based on mathematical arguments, They will see things mathematically, recognizing when mathematical analyses help to explain events. They will have sufficient mathematical knowledge to pursue a profession or vocation of their choice and to undertake further study of subjects that require mathematical proficiency.
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From page 37... ...
Students will achieve mathematical power only if they see mathematics as a modern, relevant subject. New curricular materials must be designed in the expectation of continuous change resulting from further scientific And technological clevelopments.
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From page 38... ...
Even for important priorities that do remain, modern applications or technology may suggest quite different approaches. Often a fresh approach can avoid the rigidity of thought that inhibits desirable change, In many clistricts, the secondary mathematics curriculum especially is already full of topics, many treated too quickly, Nevertheless, there is much mathematics that could be made accessible and interesting to students, Unless traditional topics contribute directly to curricular goals, they clutter the curriculum, Neither the number of
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From page 39... ...
It will no longer be appropriate for most mathematics instruction to be in the traditional mode where teachers present material to a class of passive students. No single teaching method nor any single kind of learning experience can develop the varied mathematical abilities implied under the definition of mathematical power (Fey, 1979; Mathematical Sciences Education Board (MSEB]
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From page 40... ...
Goals By themselves, general principles provide insufficient direction to help focus curriculum development. More specific goals, related to the new Curriculum and Evaluation Stanclarcis for School Mathematics (NCTM, 1989)
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From page 41... ...
Student abilities to reason effectively with numerical information requires experience with: · Representation-the ability to use numbers to express quantitative data and relations, · Operations mastery of single-digit arithmetic; ability to determine appropriate arithmetic procedures; facility in estimation; experience in select ing appropriate means to carry out complex calculations. Interpretatio~the ability to draw inferences from data and check both the data and the inferences for accuracy and reasonableness.
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From page 42... ...
If students are to be better prepared mathematically for vocations as well as for everyday life, the elementary school mathematics must include substantial subject matter other than arithmetic · Geometry, including properties of two- and threedimensional objects, symmetry and congruence, constructions of geometric figures, and transformations of geometric figures; · Measurement, including units of measure, telling time, reading temperatures, and computing with money; · Data analysis, including collection, organization, representation, and interpretation of data; construction of statistical tables and diagrams; and the use of data for analytic and predictive purposes; Probability, introduced with simple experiments and data-gathering; · Discrete mathematics, including basic combinatorial thinking and the use of graphs to model problems. Each of these topics can play a distinctive role in making the elementary school mathematics curriculum more interesting and relevant to stuclents.
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From page 43... ...
Children of ail ages must constantly explore the relation between the relatively pristine patterns studied in school mathematics and the messier reoiity of worldly clata. Real data are more convincing than contrived data.
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From page 44... ...
. Understanding the concepts of elementary school mathematics is essential for the study of secondary school mathematics; however, proficiency in the procedures of hand arithmetic computation should no longer be the critical factor in judging student readiness for advanced study.
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From page 45... ...
Developing fluency with symbols and other abstract entities which can be geometric, algebraic, or algorithmic-must be a central aim of secondary school mathematics. Student ability to reason effectively with symbols requires experience with: · Representation-the ability to represent mathematical problems in symbolic form and to use these symbolic representations in relations, expressions, and equations; · Operations -- the ability to determine appropriate symbolic procedures and to select appropriate means to solve problems expressed in symbolic form; · Interpretatio~the ability to ciraw inferences by reasoning with symbolic systems to check these results for accuracy and reasonableness.
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From page 46... ...
In teaching mathematics it is important to illustrate the unity and integrity of the discipline. For example, fractal geometry is quite accessible to high school students and involves aspects of algebra, geometry, and discrete mathematics, as well as providing fascinating uses of computers, Data analysis Reacts directly to algebraic and geometric methods, while algebra and geometry themselves are joined in analytic geometry.
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From page 47... ...
Both algebra and discrete mathematics provide excellent opportunities for arguments expressed in oaragraph form; , even flowcharts and spreadsheets can be used to ...........
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From page 48... ...
Enabling Conditions One cannot separate curriculum and instruction from the brooder context of education, To improve mathematics education, change must occur simultaneously in curriculum, in teaching, in professional development, in textbooks, and in assessment practices. Although our focus in this Framework is principally on the content of the curriculum, there are important implications of our recommendations for other parts of the educational context.
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From page 49... ...
Therefore, particular attention must be paid to in-service training of elementary school teachers, as well as to the use of specialist mathematics teachers throughout the elementary school Oracles. Secondary and middle school teachers who are already prepared to some extent with a specialization in mathematics will need extensive continuing education both in new areas of content that are not part of their present repertoire, and in styles of teaching better suited to active student participation.
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From page 50... ...
Textbooks must reflect fully a new conception of mathematics education, integrating into the main subject matter of the text all important principles of mathematics curricula and educational research. Unless textbooks help teachers to use actively calculators, computers, and genuine problems, new emphases such as technology and applications will continue to have insignificant impact on the curriculum, Unless textbooks contain engaging projects and group activities, few teachers will have time to create them on their own, And unless textbooks include suitable assignments to enhance students' experience with reading and writing in the language of mathematics, students will remain deficient in their ability to communicate effectively.
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From page 51... ...
But just as careful assessment of writing cannot be accomplished without having students write real essays, neither can mathematical power be assessed unless students have to solve nonroutine problems. Mathematics for the Future We are entering a decade in mathematics education of transition from entrenched precomputer traditions to new structures appropriate to the twenty-first century.
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