Measuring What Counts A Conceptual Guide for Mathematics Assessment (1993) / Chapter Skim
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1 A Vision of School Mathematics
1 A Vision of School Mathematics
Pages 15-28
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From page 15... ...
More than ever before, Americans need to think for a living; more than ever before, they need to think mathematically. n ° opens the first chapter of Everybody Counts: A Report to the Nation on the Future of Mathematics Education, ~ which describes a vision of the mathematics that tJ should guide education so that students will work smarter and think more mathematically.
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From page 16... ...
Outside the classroom, politicians and school administra tors, backed by the public, express dismay over low scores on shou~c~ be mathematics achievement tests. They worry about deteriorating American competitiveness in international markets when students' mathematics skills seem to be declining.3 They want teachers to teach more mathematics to more students while maintaining or increasing test scores.
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From page 17... ...
Lending institutions advertise variously computed interest rates for loans. We listen to music composed and performed with the aid of computers, and we watch the fantastically detailed pictures of imaginary worlds that computers draw.
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From page 18... ...
He argued for tight connections and a blurring of the distinctions between all parts of school mathematics but especially between its pure and applied sides. The prestigious National Committee on Mathematical Requirements of the Mathematical Association of America, reporting in 1 923, formulated the aims of mathematical instruction as practical, disciplinary, and cultural.
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From page 19... ...
contended that the traditional sequence of mathematics courses leading to the calculus is inadequate: Students should be exposed to numerous and varied interrelated experiences that encourage them to value the mathematical enterprise, to develop mathematical habits of mind, and to understand and appreciate the role of mathematics in human affairs; that they should be encouraged to explore, to guess, and even to make and correct errors so that they gain confidence in their ability to solve complex problems; that they should read, write, and discuss mathematics; and that they should conjecture, test, and build arguments about a conjecture's validity.~° The Standards delineate the mathematics students need to learn under various headings, some familiar (measurement, algebra, probability, problem solving) , some perhaps less so (communication, spatial sense, discrete mathematics)
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From page 20... ...
Middle and high school mathematics should continue the development of the strands begun in elementary school mathematics and in addition should include combinatorics, discrete mathematics, logic, number theory, trigo nometry, and some basic ideas from calculus. Fundamental math ematical structures (relations, functions, operational systems)
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From page 21... ...
Trial and error, hypothesis and investigation, and measurement and classification are part of the mathematician's craR and should be taught in school." ~ Moreover, the availability of computers has renewed the emphasis on realistic applications, greatly simplifying the treatment of data in the classroom and permitting dynamic representations of complex processes. With the aid of computers, students can have experiences heretofore impossible in representing patterns, estimating solutions, and exploring how changes in one representation affect another.
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From page 22... ...
Students learn best and most enduringly by reflecting on their experience and by communicating with others about it. A new view of mathematical performance is developing in which the focus is on the concrete tasks students perform in a specific social context rather than on abstract abilities that students are assumed to possesses Learners benefit from performing a MEASURING WHAT COUNTS
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From page 23... ...
Students judge mathematics as harder to learn than other kinds of content, see themselves as less capable of learning mathematics on their own, and feel more dependent on direct instruction from teachers and others. challenging mathematics needed for the future can be These views stem in large part from experiences the students have in school.
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From page 24... ...
Strategies aimed at curriculum and professional development are set out in some detail in recent reports that include Everybody Counts, Curriculum and Evaluation Standards for School Mathematics, Reshaping School Mathematics, Professional Standards for Teaching Mathematics, A Call for Change, and Counting on You.27 These reports agree that goals for student performance are shifting from a narrow focus on routine skills to the provision of a variety of experiences aimed at developing students' mathematical power. They encourage the movement from teaching as the 1 24 MEASURING WHAT COUNTS
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Mathematics teachers have for centuries found it difficult to lead students to a deep understanding of how and why mathematics works as it does. What is different now that makes successful reform in mathematics education more likely?
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From page 26... ...
Mathematical Association of America, National Committee on Mathematical Requirements, The Reorganization of Mathematics in Secondary Education cited in National Council of Teachers of Mathematics, The Teaching of Secondary School Mathematics, NCTM Yearbook (Washington, D.C.: Author, 1970~. 9 College Entrance Examination Board, Commission on Mathematics, Program for College Preparatory Mathematics (New York, NY: Author, 1959~.
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From page 27... ...
, 125-1462. The second is the Cognitively Guided Instruction project, which has been described by Elizabeth Fennema, Thomas Carpenter, and Penelope Peterson [see, for example, Fennema, Carpenter, and Peterson, "Learning Mathematics with Understanding: Cognitively Guided Instruction," in )
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From page 28... ...
Witt, "Design Innovations in Measuring Mathematics Achievement" (Paper commissioned by the Mathematical Sciences Education Board, September 1993, appended to this report)
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