**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

Adult learners, 11

Advanced Technological Education program, 3, 165

Aesthetic value of mathematics, 4, 35, 36

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

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