Index

A

AAAS, see American Association for the Advancement of Science

Abstraction, 15, 90, 107-110, 132, 159

Academic education

vocational education and, 11, 12, 15, 24-29, 164

ACT, 61, 69, 71

Adult learners, 11

Advanced Technological Education program, 3, 165

Aesthetic value of mathematics, 4, 35, 36

Agriculture, 26, 27, 39

Algebra, 2, 12, 31-34, 35-41, 69

curricular design, 94-95, 98-100, 103, 104-105

mental mathematics, 83-86

representations, 39-40, 55, 80-82, 87-90, 105, 109-110, 111-114, 124, 133

school-to-work experiences, 2, 12, 37-40

spreadsheets, 32, 80-82, 87-90, 94-95, 105, 111-112, 133, 158

tasks, 32, 33-34, 54-56, 76, 80-82, 83-86, 87-90, 98, 99, 100, 109-110, 111-114, 117, 118, 121-122, 123-124, 133, 158

Algorithms and procedures, 9, 64, 69

adult learners, 11

algebra, 37, 39, 83-86

concepts and understanding, relation to, 32-34, 54, 59, 95, 134

curricular design, 94, 99-100, 102-103, 104-106

mental mathematics, relation to, 83-86

problem solving, relation to, 11, 97

recursion, 62, 75, 76, 80-82, 87-90, 105, 111-114

similarities among, for abstraction, 62, 90, 132

AMATYC, see American Mathematical Association of Two-Year Colleges

American Association for the Advancement of Science, 60

American College Testing, see ACT

American Mathematical Association of Two-Year Colleges, 60

see also Two-year colleges



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--> Index A AAAS, see American Association for the Advancement of Science Abstraction, 15, 90, 107-110, 132, 159 Academic education vocational education and, 11, 12, 15, 24-29, 164 ACT, 61, 69, 71 Adult learners, 11 Advanced Technological Education program, 3, 165 Aesthetic value of mathematics, 4, 35, 36 Agriculture, 26, 27, 39 Algebra, 2, 12, 31-34, 35-41, 69 curricular design, 94-95, 98-100, 103, 104-105 mental mathematics, 83-86 representations, 39-40, 55, 80-82, 87-90, 105, 109-110, 111-114, 124, 133 school-to-work experiences, 2, 12, 37-40 spreadsheets, 32, 80-82, 87-90, 94-95, 105, 111-112, 133, 158 tasks, 32, 33-34, 54-56, 76, 80-82, 83-86, 87-90, 98, 99, 100, 109-110, 111-114, 117, 118, 121-122, 123-124, 133, 158 Algorithms and procedures, 9, 64, 69 adult learners, 11 algebra, 37, 39, 83-86 concepts and understanding, relation to, 32-34, 54, 59, 95, 134 curricular design, 94, 99-100, 102-103, 104-106 mental mathematics, relation to, 83-86 problem solving, relation to, 11, 97 recursion, 62, 75, 76, 80-82, 87-90, 105, 111-114 similarities among, for abstraction, 62, 90, 132 AMATYC, see American Mathematical Association of Two-Year Colleges American Association for the Advancement of Science, 60 American College Testing, see ACT American Mathematical Association of Two-Year Colleges, 60 see also Two-year colleges

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--> Annuities, 109, 111-114 Apprenticeships, see Vocational education Area, geometry, 16, 38, 39, 47, 131, 145-146 Assessment, 3, 16, 59-90, 138, 165 ACT, 61, 69, 71 adequacy for purposes, 61, 70-71 curriculum, influence by, 61-62, 70-74 eighth grade test, national voluntary, 59, 69 international context, 75-76 item development, 97-98 NAEP, 62, 73, 77-79 parents, concerns of, 61, 70 74 SAT, 61, 71-74 standards, relation to, 59, 61 teacher development, role in, 130, 141-144 TIMSS, 4, 95 see also Standards ATE, see Advanced Technological Education program Attitudes, see Motivation Automobiles car purchase, 105, 131, 140, 153 driving, braking distances, 123-124, 125 traffic control, 147-152 B Back-of-the-envelope estimates, see Estimation Baseball, 116, 158 Benchmarks for Science Literacy, 60 Body proportions, 125, 145-146 Business applications, 11-13, 30, 32 tasks, 18-23, 45, 46-47, 49-53, 76, 83, 84-86, 119-121 see also School-to-work experiences; Tasks, mathematical, by context C Calculations, mental, see Mental mathematics Calculators, see Computers and calculators Calculus, 69 high school, in, 31, 104 international context, 75-76 NCTM standards, role in, 28 preparation for, 3, 28, 60, 95, 108, 131, 145 student performance on an algebra problem, 33 tasks, 54-56, 121-122, 145-146 California, 59 Careers, preparation for, 2, 4, 9, 10, 30-31, 34, 67-69, 107-110, 158 see also Business applications; School-to-work experiences; Tasks, mathematical, by context; Vocational education Carl Perkins Vocational and Technical Education Act, 29 Center for Education Research (Wisconsin), 26 Charts and tables, see Graphical representations and tables Clinton, William, 59, 69 Coalition of Essential Schools, 25 Collaborative learning, see Group learning Colleges and universities, see Post-secondary education; Two-year colleges Commission on Achieving Necessary Skills, see Secretary's Commission on Achieving Necessary Skills Communication skills, 1, 27, 30, 32, 34, 54, 107, 130, 138, 140 standards, 60, 62, 64, 66, 68, 69, 78-79, 137 see also Group learning Computers and calculators, viii, 1, 10, 30, 34, 36, 80-82, 87, 105, 109, 133, 135, 153 see also Spreadsheets Conceptual understanding, 2, 9, 10, 30, 37, 62, 95, 159 skills and procedures, relationship with, 9-11, 24-26, 32-34, 54, 59, 62, 66, 68, 75, 79, 83, 107-109, 132-134, 138, 157-158 Cost analysis business plan, 20-23, 68 car purchase, 105, 131, 140, 153 credit, 87-90, 105 optimization, 53 procurement, 45, 46-47, 119-121 sales tax, 99 schooling costs, 16 see also Tasks, mathematical, by context Credit, 62, 87-90, 105 see also Financial applications Crossroads in Mathematics: Standards for Introductory Mathematics Before Calculus, 60 Curriculum and curricular design, 1, 2, 3, 4, 10, 93-126, 157-158, 159 academic/vocational education related, 11, 12, 15, 24-25 algebra, 94-95, 98-100, 103, 104-105 algorithms, 94, 99-100, 102-103, 104-106 Bruner's spiral, 95 calculus, 28 computers and, 10 defined, 93 geometry, 100, 104, 109 habits of mind, 94, 102-106 modeling, 93-94, 99-101, 104

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--> post-secondary education preparation, 93, 107-110 problem solving, 97, 98, 100, 103 standards, 60-62, 63-66, 95, 104, 107-108, 137 tasks, inclusion in, 94-95, 97-101, 104, 108-110, 138, 159 teaching, relationship to, 129-130, 132-133 testing, influence on, 61-62, 70-74 tracking, educational, 11, 12, 15, 24-29, 73 see also Tasks, mathematical, by content Curriculum and Evaluation Standards for School Mathematics, 60, 65, 137 D Data analysis, 4, 12, 21, 22, 68, 75 tasks, 16, 18-23, 27, 42-44, 56, 64, 71-74, 115-116, 153 see also Graphical representations and tables; Spreadsheets; Tasks, mathematical, by content Data representation, see Graphical representations and tables Department of Education, 3 Department of Energy, 56 Department of Labor, 3 Developmental education, 3, 15, 35, 36 Diagrams, see Graphical representations and tables Discrete mathematics, 158 E Economic factors, see Cost analysis;Employment and employers;Financial applications; Wages and salaries The Education Imperative, 2 Employment and employers academic/vocational tracking, 25 see also Business applications; Careers, preparation for; School-to-work experiences; Wages and salaries Energy conservation, 13, 33, 54-56 Engineering, 3, 37-38, 47, 77 Environmental science, 27, 31, 32, 47, 49 Equity issues, 11, 73, 116 Error of measurement, 22, 119, 145, 146 Estimation, 12-13, 21, 30, 31, 76, 131, 145 tasks, 37-38, 45-48, 119-122, 123-126, 139, 145-146 Evaluation, see Assessment; Standards Everybody Counts, ix, 9, 103 Exponential functions, 4, 76, 99, 109-110, 111 F Federal government assessments, 59, 62, 69, 73, 77-79 legislation, 3, 26, 29, 60 see also Carl Perkins Vocational and Technical Education Act; School-to-Work Opportunities Act of 1994; headings beginning ''Department of …" Fermi problems, see Estimation Financial applications, 87-90, 98, 99, 105, 109-110, 111-114, 124-125, 133 Foreign countries, 18-23, 76-78 Formulae, 9, 16, 69, 81, 84-86, 87-88, 90, 112, 113, 117, 121-122, 146, 151 rules of thumb, 39, 95, 123-126 Fractals, 146 Functions, 22, 39, 40, 56, 75-76, 98-99, 110, 112-113, 158 concept of, 37 exponential, 4, 76, 99, 109-110, 111 linear, 33-34, 54-56, 78, 95, 98, 99, 105, 108, 113, 123-126, 133, 147-152, 158 quadratic, 32, 69, 95, 99, 121, 124 trigonometric, 12, 31, 75, 109 G Gender issues, see Equity issues Geometry area calculations, 16, 38, 39, 47, 131, 145-146 curricular design, 95, 100, 104, 109 fractals, 146 representation, 65, 77, 95, 117-118, 119-122, 147-152 mapmaking, 110, 146 ordered pairs, 95, 120 tasks, 39, 64-65, 76, 77, 109, 110, 117-118, 119-122, 145-146, 147-152, 158 topology, 104 vectors, 95, 117-118 Germany, 76 The Goals 2000: Educate America Act, 29(n.5), 60 Government role, see Federal government; State government Graphical representations and tables, 30, 37, 76, 108, 133-134 algebra, 39-40, 55, 80-82, 87-90, 105, 109-110, 111-114, 124, 133 data analysis, 12, 21, 42-44, 115-116, 153 geometry, 65, 77, 95, 117-118, 119-122, 147-152 modeling, 49-53, 55

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--> Group learning, 1, 12, 25, 32, 49, 68, 69 assessment of, 61 business planning, 18-23 teacher involvement, 135-136, 139-140 H Habits of mind, 94, 102-106 Health care, 12, 26, 30, 62 tasks, 39-40, 42-44, 80-82, 115-118, 145-146 Higher education, see Post-secondary education I Incentives, see Motivation Interactive Mathematics Program, 73 International issues, see Foreign countries Interpersonal skills, see Communication skills J Japan, 18-23, 76-78 L Legislation, Federal, 3, 26, 29, 60 Life by the Numbers, 30 Linear functions, 33-34, 54-56, 78, 95, 98, 99, 105, 108, 113, 123-126, 133, 147-152, 158 M Mapmaking, 110, 146 Mathematical Preparation of the Technical Work Force, ix, 2 Mathematical Sciences Education Board, ix, 9 Mathematical tasks, see Tasks, mathematical Measuring Up, ix Medicine, see Health care Memorization, 15, 95, 134 see also Algorithms and procedures; Conceptual understanding Mental mathematics, 62, 83-86 see also Conceptual understanding; Habits of mind; Rules of thumb Minority groups, see Equity issues Models and modeling, 11, 13, 22, 64 curricular design, 93-94, 99-101, 104 energy conservation, 13, 33, 54-56 linear, 54, 133-134 optimization, 49-53 problem solving, 98-99 solar system, 64-65 trigonometry, 31 see also headings beginning "Tasks …" Motivation aesthetic value of mathematics, 4, 35, 36 science, connection with, 25, 26, 64, 66 students, 10-11, 36-37, 77, 79, 98, 108, 113 negative, 12, 15, 24, 36 teachers, 134-135, 136, 140, 141-144 N NAEP, see National Assessment of Educational Progress National Assessment of Educational Progress, 62, 73, 77-79 National Center for Restructuring Education, Schools, and Teaching , 25-26 National Council of Teachers of Mathematics, 28, 137 standards, 28, 61, 63-64, 137 National Science Education Standards, 60, 64-66 National Science Foundation, 3, 168 National Skills Standards Board, 29(n.5), 60 A Nation at Risk, 25, 60 NCTM, see National Council of Teachers of Mathematics NSES, see National Science Education Standards O Open-ended problems, 12, 25-26 assessment using, 61, 70-71, 78 standards, 63-64 teacher involvement in assessment, 130-131, 141-144 Optimization problems, 49-53 Ordered pairs, 95, 120 Order of magnitude, 45 P Partitioning, 145 Post-secondary education, 2, 3, 16, 107-110, 164 admissions tests, see ACT; SAT preparation for, 93, 107-110 school-to-work experiences and, 12 vocational education and, 24-29 see also Two-year colleges Preparing for the 21st Century: The Education Imperative, 1 Probability, 21, 76, 95, 104, 115-118, 119-122, 158 Problem-centered teaching, 9-11, 12, 33, 130, 132-140 research support, 11, 73

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--> Problem solving, 1, 3, 10-11, 12, 16, 27-28, 31, 79, 132 assessment of, 61 business/marketing application, 18-23 classroom environment, 139 curricular design, 97, 98, 100, 103 difficulty level, 138, 139 mental mathematics, 83-84 research support for problem-centered teaching, 11, 73 skills, relationship with, 59, 95, 97, 103 standards, 60, 61, 64, 68 teacher involvement, 130, 132-140 theory, relationship with, 103 types of problems, 98-99 see also Algorithms and procedures; Open-ended problems; Word problems; headings beginning "Tasks …" Procedures, see Algorithms and procedures; Conceptual understanding Professional education, see Teacher education Professional Standards for Teaching Mathematics, 130 Proportional reasoning, 4, 38, 54-56, 78, 94, 104, 115-118 Psychological factors, see Motivation Q Quadratic functions, 32, 69, 95, 99, 121, 124 R Rates, 62, 78, 80-82, 87-90, 95, 100, 110, 111-114, 115-118, 147-152 Reasoning, see Conceptual understanding; Habits of mind; Mental mathematics; Problem solving Recursion, 62, 75, 76, 80-82, 87-90, 105, 111-114 Remedial education, see Developmental education Research science/math integration, 64 supporting problem-centered teaching, 11, 73 Reshaping School Mathematics, ix, 10 Rote learning, see Memorization Rounding off, 76, 90, 95, 119-122 Rules of thumb, 39, 95, 123-126 S Salaries, see Wages and salaries Sampling, 19-20, 22, 44 SAT, 61, 71-74 SCANS, see Secretary's Commission on Achieving Necessary Skills Scholastic Aptitude Test, see SAT School-to-work experiences, 3, 10, 12-13, 26-27 algebra, role in, 2, 12, 35-41, 94 interdisciplinary collaboration, 26, 27 sources of, 164 see also Tasks, mathematical, by context School-to-Work Opportunities Act of 1994, 3, 26 Science education, 10, 47 basis for careers, 1, 9, 12, 30, 34 mathematics, relationship with, 3, 24, 25-27, 31, 75-77 standards, 60, 63-66 tasks, 64-65, 80-82 TIMSS, 4, 95 Secretary's Commission on Achieving Necessary Skills, 60-61, 67-69 Simpson's paradox, 95 Skills, see Algorithms and procedures; Communication skills; Conceptual understandings; Memorization Spreadsheets, 32, 68, 80-82, 87-90, 94-95, 103, 105, 111-112, 133, 153, 158 Standards, 3, 59-90 academic/vocational education integrated, 28 assessments, relation to, 59, 61, 141-144 calculus, 28 communication skills, 60, 62, 64, 66, 68, 69, 78-79, 137 curricular design, 60-62, 63-66, 95, 104, 107-108, 137 definitional issues, 60 national, 59, 61 National Skills Standards Board, 29(n.5), 60 NCTM, 28, 61, 63-64, 137 NSES, 60, 64-66 parents' role in standardized testing, 61, 70, 74 problem solving, 60, 61, 63-64, 68, 130, 137-138 SCANS, 60-61, 67-69 State government role, 59, 60 teaching methods, 130, 137-140, 142 technical education, 2 see also Assessment; Curriculum and curricular design State government assessment and standards, 59, 60 local support for teachers, 130 State-to-state comparisons, SAT testing, 71-74 Statistics, 4, 74, 75, 100, 115-116, 158 sampling, 19-20, 22, 44

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--> tasks, 18-23, 42-44 see also Data analysis; Estimation; Probability Successive approximation, 75, 145 T Tables and charts, see Graphical representations and tables Tasks, mathematical, 4, 9 curriculum, selection of, 94-95, 97-101, 104, 108-110, 138, 159 source of, 163-166 see also Open-ended problems; Problem solving; Tasks, mathematical, by content; Tasks, mathematical, by context Tasks, mathematical, by content, 4, 9 algebra, 32, 33-34, 54-56, 76, 80-82, 83-86, 87-90, 98, 99, 100, 109-110, 111-114, 117, 118, 121-122, 123-124, 133, 158 calculus, 54-56, 121-122, 145-146 data collection and analysis, 16, 18-23, 27, 42-44, 56, 71-74, 115-116, 153 estimation, 37-38, 45-48, 119-122, 123-126, 139, 145-146 fractals, 146 geometry, 39, 64-65, 76, 77, 109, 110, 117-118, 119-122, 145-146, 147-152, 158 optimization, 49-53 probability, 21, 76, 115-118, 119-122, 158 proportional reasoning, 38, 54-56, 78, 115-118 rates, 78, 80-82, 87-90, 100, 110, 111-114, 115-118, 147-152 recursion, 80-82, 87-90, 105, 111-114 representation, 39-40, 42-44, 49-53, 109-110, 115-118, 119-122, 133, 147-152 statistics, 18-23, 42-44 vectors, 117-118 weighted averages, 115-118 see also Functions; Open-ended problems; School-to-work experiences; Spreadsheets Tasks, mathematical, by context agriculture, 39 automobiles, 105, 123-124, 125-126, 140, 147-152, 153 baseball, 116, 158 body proportions, 125, 145-146 business, 18-23, 45, 46-47, 49-53, 76, 83, 84-86, 119-121 construction, 37-38, 39, 109 cooking, 125 energy conservation, 33, 54-56 environmental science, 27, 31, 32, 47, 49 finance, 87-90, 98, 99, 105, 109-110, 111-114, 124-125, 133 health care, 39-40, 42-44, 80-82, 115-118, 145-146 hypothetical, 139 mapmaking, 110, 146 mathematics, 76, 77, 83, 84, 105, 121-122, 158 piano tuning, 45 police protection, 47 salaries, 45, 115, 116 school and education, 16, 33-34, 45, 46, 71-74, 116-117 science, 64-65, 80-82 tax, 47-48, 78, 99, 125 wages and salaries, 45, 115, 116 Tax calculations, 47-48, 78, 99, 125 Taxonomy of Educational Objectives, 97 Teacher education, 3, 4, 127-153 inservice teachers, 141-144 preservice teachers, 130, 134-135, 137, 139 Teachers, 3, 4 assessment, involvement in, 130, 141-144 attitudes, 134, 140 communication skills, 134, 139-140 local support for, 130 motivation, 134-135, 136, 140, 142-143 problem solving, 132-140 student interactions, 10, 129, 135-136, 139-140 Teaching, 3, 14-15, 129-153 curricular design, 129-130, 132-133 feedback, 129, 139 interdisciplinary collaboration, 26, 27 memorization, 15, 95, 134 standards, 63-64, 130, 137-140, 142 see also Curriculum and curricular design; Group learning; Memorization; Models and modeling; Problem solving; Tasks, mathematical Technical applications, see Tasks, mathematical, by context Technical education, 2, 3, 164 Tests and testing, see Assessment Theory, relationship with problem solving, 103 Third International Mathematics and Science Study, 4, 95 TIMSS, see Third International Mathematics and Science Study Tracking, vocational/academic, 11, 12, 15, 24-29, 73 Trigonometry, 12, 31, 75, 109 Two-year colleges, 2, 3, 15, 26, 27, 29, 60, 140, 164

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--> U Uncertainty, 20, 21, 23, 25 error of measurement, 22, 119, 145, 146 Universities and colleges, see Post-secondary education;Two-year colleges University of Chicago School Mathematics Project, 99 V VATEA, see Carl Perkins Vocational and Technical Education Act Vectors, 95, 117-118 Visual aids, see Graphical representations and tables Vocational education, 2, 109, 164 academic education and, 11, 12, 15, 24-29, 164 see also Technical education W Wages and salaries, 26 tasks, 45, 115, 116 Weighted averages, 115-118 What Work Requires of Schools, 60, 67, 68-69 Word problems, 15, 32-33, 69, 98, 108 see also Tasks, mathematical, by content; Tasks, mathematical, by context Workplace, see Careers, preparation for; Employment and employers; School-to-work experiences