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From page 171... ... :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 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 Read the entire page →
From page 172... ... ........ 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 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 Read the entire page →
From page 173... ... j 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 Read the entire page →
From page 174... ... Japan, 18-23, 76-78 ...... 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 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 head ings 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 ............. Read the entire page →
From page 175... ... . ." 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 Quadratic functions, 32, 69, 95, 99, 121, 124 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 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) Read the entire page →
From page 176... ... tasks, 18-23, 42-44 see also Data analysis; Estimation; Probability Successive approximation, 75, 145 ......... 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 Read the entire page →
From page 177... ... 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 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 ............... 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-towork experiences Read the entire page →

From page 171...

... :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 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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From page 172...

... ........ 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 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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From page 173...

... j 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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From page 174...

... Japan, 18-23, 76-78 ...... 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 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 head ings 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 .............

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From page 175...

... . ." 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 Quadratic functions, 32, 69, 95, 99, 121, 124 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 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)

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From page 176...

... tasks, 18-23, 42-44 see also Data analysis; Estimation; Probability Successive approximation, 75, 145 ......... 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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From page 177...

... 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 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 ............... 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-towork experiences

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Index Pages 171-177

Index
Pages 171-177