Index

A

AAAS, see American Association for the Advancement of Science

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

Academic education
      vocational education and, 11, 12, 15, 24­29, 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, 31­34, 35­41, 69
      curricular design, 94­95, 98­100, 103, 104­105
      mental mathematics, 83­86
      representations, 39­40, 55, 80­82, 87­90, 105, 109­110, 111­114, 124, 133
      school-to-work experiences, 2, 12, 37­40
      spreadsheets, 32, 80­82, 87­90, 94­95, 105, 111­112, 133, 158
      tasks, 32, 33­34, 54­56, 76, 80­82, 83­86, 87­90, 98, 99, 100, 109­110, 111­114, 117, 118, 121­122, 123­124, 133, 158

Algorithms and procedures, 9, 64, 69
      adult learners, 11
      algebra, 37, 39, 83­86
      concepts and understanding, relation to, 32­34, 54, 59, 95, 134
      curricular design, 94, 99­100, 102­103, 104­106
      mental mathematics, relation to, 83­86
      problem solving, relation to, 11, 97
      recursion, 62, 75, 76, 80­82, 87­90, 105, 111­114
      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

Annuities, 109, 111­114

Apprenticeships, see Vocational education

Area, geometry, 16, 38, 39, 47, 131, 145­146

Assessment, 3, 16, 59­90, 138, 165
      ACT, 61, 69, 71
      adequacy for purposes, 61, 70­71
      curriculum, influence by, 61­62, 70­74
      eighth grade test, national voluntary, 59, 69
      international context, 75­76
      item development, 97­98
      NAEP, 62, 73, 77­79
      parents, concerns of, 61, 70 74
      SAT, 61, 71­74
      standards, relation to, 59, 61
      teacher development, role in, 130, 141­144
      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, 123­124, 125
      traffic control, 147­152

B

Back-of-the-envelope estimates, see Estimation

Baseball, 116, 158

Benchmarks for Science Literacy, 60

Body proportions, 125, 145­146

Business applications, 11­13, 30, 32
      tasks, 18­23, 45, 46­47, 49­53, 76, 83, 84­86, 119­121
      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, 75­76
      NCTM standards, role in, 28
      preparation for, 3, 28, 60, 95, 108, 131, 145
      student performance on an algebra problem, 33
      tasks, 54­56, 121­122, 145­146

California, 59

Careers, preparation for, 2, 4, 9, 10, 30­31, 34, 67­69, 107­110, 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, 78­79, 137
      see also Group learning

Computers and calculators, viii, 1, 10, 30, 34, 36, 80­82, 87, 105, 109, 133, 135, 153
      see also Spreadsheets

Conceptual understanding, 2, 9, 10, 30, 37, 62, 95, 159
      skills and procedures, relationship with, 9­11, 24­26, 32­34, 54, 59, 62, 66, 68, 75, 79, 83, 107­109, 132­134, 138, 157­158

Cost analysis
      business plan, 20­23, 68
      car purchase, 105, 131, 140, 153
      credit, 87­90, 105
      optimization, 53
      procurement, 45, 46­47, 119­121
      sales tax, 99
      schooling costs, 16
      see also Tasks, mathematical, by context

Credit, 62, 87­90, 105
      see also Financial applications

Crossroads in Mathematics: Standards for Introductory Mathematics Before Calculus, 60

Curriculum and curricular design, 1, 2, 3, 4, 10, 93­126, 157­158, 159
      academic/vocational education related, 11, 12, 15, 24­25
      algebra, 94­95, 98­100, 103, 104­105
      algorithms, 94, 99­100, 102­103, 104­106
      Bruner's spiral, 95
      calculus, 28
      computers and, 10
      defined, 93
      geometry, 100, 104, 109
      habits of mind, 94, 102­106
      modeling, 93­94, 99­101, 104
      post-secondary education preparation, 93, 107­110
      problem solving, 97, 98, 100, 103
      standards, 60­62, 63­66, 95, 104, 107­108, 137
      tasks, inclusion in, 94­95, 97­101, 104, 108­110, 138, 159
      teaching, relationship to, 129­130, 132­133
      testing, influence on, 61­62, 70­74
      tracking, educational, 11, 12, 15, 24­29, 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, 18­23, 27, 42­44, 56, 64, 71­74, 115­116, 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, 54­56

Engineering, 3, 37­38, 47, 77

Environmental science, 27, 31, 32, 47, 49

Equity issues, 11, 73, 116

Error of measurement, 22, 119, 145, 146

Estimation, 12­13, 21, 30, 31, 76, 131, 145
      tasks, 37­38, 45­48, 119­122, 123­126, 139, 145­146

Evaluation, see Assessment; Standards

Everybody Counts, ix, 9, 103

Exponential functions, 4, 76, 99, 109­110, 111

F

Federal government
      assessments, 59, 62, 69, 73, 77­79
      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, 87­90, 98, 99, 105, 109­110, 111­114, 124­125, 133

Foreign countries, 18­23, 76­78

Formulae, 9, 16, 69, 81, 84­86, 87­88, 90, 112, 113, 117, 121­122, 146, 151
      rules of thumb, 39, 95, 123­126

Fractals, 146

Functions, 22, 39, 40, 56, 75­76, 98­99, 110, 112­113, 158
      concept of, 37
      exponential, 4, 76, 99, 109­110, 111
      linear, 33­34, 54­56, 78, 95, 98, 99, 105, 108, 113, 123­126, 133, 147­152, 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, 145­146
      curricular design, 95, 100, 104, 109
      fractals, 146
      representation, 65, 77, 95, 117­118, 119­122, 147­152
      mapmaking, 110, 146
      ordered pairs, 95, 120
      tasks, 39, 64­65, 76, 77, 109, 110, 117­118, 119­122, 145­146, 147­152, 158
      topology, 104
      vectors, 95, 117­118

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, 133­134
      algebra, 39­40, 55, 80­82, 87­90, 105, 109­110, 111­114, 124, 133
      data analysis, 12, 21, 42­44, 115­116, 153
      geometry, 65, 77, 95, 117­118, 119­122, 147­152
      modeling, 49­53, 55

Group learning, 1, 12, 25, 32, 49, 68, 69
      assessment of, 61
      business planning, 18­23
      teacher involvement, 135­136, 139­140

H

Habits of mind, 94, 102­106

Health care, 12, 26, 30, 62
      tasks, 39­40, 42­44, 80­82, 115­118, 145­146

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, 18­23, 76­78

L

Legislation, Federal, 3, 26, 29, 60

Life by the Numbers, 30

Linear functions, 33­34, 54­56, 78, 95, 98, 99, 105, 108, 113, 123­126, 133, 147­152, 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, 83­86
      see also Conceptual understanding; Habits of mind; Rules of thumb

Minority groups, see Equity issues

Models and modeling, 11, 13, 22, 64
      curricular design, 93­94, 99­101, 104
      energy conservation, 13, 33, 54­56
      linear, 54, 133­134
      optimization, 49­53
      problem solving, 98­99
      solar system, 64­65
      trigonometry, 31
      see also headings beginning "Tasks . . ."

Motivation
      aesthetic value of mathematics, 4, 35, 36
      science, connection with, 25, 26, 64, 66
      students, 10­11, 36­37, 77, 79, 98, 108, 113
       negative, 12, 15, 24, 36
      teachers, 134­135, 136, 140, 141­144

N

NAEP, see National Assessment of Educational Progress

National Assessment of Educational Progress, 62, 73, 77­79

National Center for Restructuring Education, Schools, and Teaching, 25­26

National Council of Teachers of Mathematics, 28, 137
      standards, 28, 61, 63­64, 137

National Science Education Standards, 60, 64­66

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, 25­26
      assessment using, 61, 70­71, 78
      standards, 63­64
      teacher involvement in assessment, 130­131, 141­144

Optimization problems, 49­53

Ordered pairs, 95, 120

Order of magnitude, 45

P

Partitioning, 145

Post-secondary education, 2, 3, 16, 107­110, 164
      admissions tests, see ACT; SAT
      preparation for, 93, 107­110
      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, 115­118, 119­122, 158

Problem-centered teaching, 9­11, 12, 33, 130, 132­140
      research support, 11, 73

Problem solving, 1, 3, 10­11, 12, 16, 27­28, 31, 79, 132
      assessment of, 61
      business/marketing application, 18­23
      classroom environment, 139
      curricular design, 97, 98, 100, 103
      difficulty level, 138, 139
      mental mathematics, 83­84
      research support for problem-centered teaching, 11, 73
      skills, relationship with, 59, 95, 97, 103
      standards, 60, 61, 64, 68
      teacher involvement, 130, 132­140
      theory, relationship with, 103
      types of problems, 98­99
      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, 54­56, 78, 94, 104, 115­118

Psychological factors, see Motivation

Q

Quadratic functions, 32, 69, 95, 99, 121, 124

R

Rates, 62, 78, 80­82, 87­90, 95, 100, 110, 111­114, 115­118, 147­152

Reasoning, see Conceptual understanding; Habits of mind; Mental mathematics; Problem solving

Recursion, 62, 75, 76, 80­82, 87­90, 105, 111­114

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, 119­122

Rules of thumb, 39, 95, 123­126

S

Salaries, see Wages and salaries

Sampling, 19­20, 22, 44

SAT, 61, 71­74

SCANS, see Secretary's Commission on Achieving Necessary Skills

Scholastic Aptitude Test, see SAT

School-to-work experiences, 3, 10, 12­13, 26­27
      algebra, role in, 2, 12, 35­41, 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, 25­27, 31, 75­77
      standards, 60, 63­66
      tasks, 64­65, 80­82
      TIMSS, 4, 95

Secretary's Commission on Achieving Necessary Skills, 60­61, 67­69

Simpson's paradox, 95

Skills, see Algorithms and procedures; Communication skills; Conceptual understandings; Memorization

Spreadsheets, 32, 68, 80­82, 87­90, 94­95, 103, 105, 111­112, 133, 153, 158

Standards, 3, 59­90
      academic/vocational education integrated, 28
      assessments, relation to, 59, 61, 141­144
      calculus, 28
      communication skills, 60, 62, 64, 66, 68, 69, 78­79, 137
      curricular design, 60­62, 63­66, 95, 104, 107­108, 137
      definitional issues, 60
      national, 59, 61
      National Skills Standards Board, 29(n.5), 60
      NCTM, 28, 61, 63­64, 137
      NSES, 60, 64­66
      parents' role in standardized testing, 61, 70, 74
      problem solving, 60, 61, 63­64, 68, 130, 137­138
      SCANS, 60­61, 67­69
      State government role, 59, 60
      teaching methods, 130, 137­140, 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, 71­74

Statistics, 4, 74, 75, 100, 115­116, 158
      sampling, 19­20, 22, 44
      tasks, 18­23, 42­44
      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, 94­95, 97­101, 104, 108­110, 138, 159
      source of, 163­166
      see also Open-ended problems; Problem solving; Tasks, mathematical, by content; Tasks, mathematical, by context

Tasks, mathematical, by content, 4, 9
      algebra, 32, 33­34, 54­56, 76, 80­82, 83­86, 87­90, 98, 99, 100, 109­110, 111­114, 117, 118, 121­122, 123­124, 133, 158
      calculus, 54­56, 121­122, 145­146
      data collection and analysis, 16, 18­23, 27, 42­44, 56, 71­74, 115­116, 153
      estimation, 37­38, 45­48, 119­122, 123­126, 139, 145­146
      fractals, 146
      geometry, 39, 64­65, 76, 77, 109, 110, 117­118, 119­122, 145­146, 147­152, 158
      optimization, 49­53
      probability, 21, 76, 115­118, 119­122, 158
      proportional reasoning, 38, 54­56, 78, 115­118
      rates, 78, 80­82, 87­90, 100, 110, 111­114, 115­118, 147­152
      recursion, 80­82, 87­90, 105, 111­114
      representation, 39­40, 42­44, 49­53, 109­110, 115­118, 119­122, 133, 147­152
      statistics, 18­23, 42­44
      vectors, 117­118
      weighted averages, 115­118
      see also Functions; Open-ended problems; School-to-work experiences; Spreadsheets

Tasks, mathematical, by context
      agriculture, 39
      automobiles, 105, 123­124, 125­126, 140, 147­152, 153
      baseball, 116, 158
      body proportions, 125, 145­146
      business, 18­23, 45, 46­47, 49­53, 76, 83, 84­86, 119­121
      construction, 37­38, 39, 109
      cooking, 125
      energy conservation, 33, 54­56
      environmental science, 27, 31, 32, 47, 49
      finance, 87­90, 98, 99, 105, 109­110, 111­114, 124­125, 133
      health care, 39­40, 42­44, 80­82, 115­118, 145­146
      hypothetical, 139
      mapmaking, 110, 146
      mathematics, 76, 77, 83, 84, 105, 121­122, 158
      piano tuning, 45
      police protection, 47
      salaries, 45, 115, 116
      school and education, 16, 33­34, 45, 46, 71­74, 116­117
      science, 64­65, 80­82
      tax, 47­48, 78, 99, 125
      wages and salaries, 45, 115, 116

Tax calculations, 47­48, 78, 99, 125

Taxonomy of Educational Objectives, 97

Teacher education, 3, 4, 127­153
      inservice teachers, 141­144
      preservice teachers, 130, 134­135, 137, 139

Teachers, 3, 4
      assessment, involvement in, 130, 141­144
      attitudes, 134, 140
      communication skills, 134, 139­140
      local support for, 130
      motivation, 134­135, 136, 140, 142­143
      problem solving, 132­140
      student interactions, 10, 129, 135­136, 139­140

Teaching, 3, 14­15, 129­153
      curricular design, 129­130, 132­133
      feedback, 129, 139
      interdisciplinary collaboration, 26, 27
      memorization, 15, 95, 134
      standards, 63­64, 130, 137­140, 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, 24­29, 73

Trigonometry, 12, 31, 75, 109

Two-year colleges, 2, 3, 15, 26, 27, 29, 60, 140, 164

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, 117­118

Visual aids, see Graphical representations and tables

Vocational education, 2, 109, 164
      academic education and, 11, 12, 15, 24­29, 164
      see also Technical education

W

Wages and salaries, 26
      tasks, 45, 115, 116

Weighted averages, 115­118

What Work Requires of Schools, 60, 67, 68­69

Word problems, 15, 32­33, 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