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Adding + It Up: Helping Children Learn Mathematics
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INDEX
A
Abacus, use of in Japan, 218n
Abstraction, 111n, 160
recommendation for linking experience to, 426
Accessible generalizable methods, for multidigit subtraction, 205
Accuracy
of calculators, 247
of decimal approximations, 88–90
improvement with practice, 121
of subtraction, 191
of teacher self-reports, 47
Achievement, in school mathematics in the U.S., 55–57
Adaptive reasoning, 5, 10, 116, 138–139, 170, 380, 383–384
and mathematical proficiency, 129–131
in the teaching of mathematics, 383–384
Addition. See also Multidigit addition
algorithms for, 199–204
associativity of, 77
carrying in, 203
commutativity of, 75, 77
distributivity of multiplication over, 78
of fractions, 86, 320–322
problem types, 185
properties of, 82
single-digit, 187–190
Additive concepts
identity, 82
inverse, 82–83
Address, of real number, 90
Algebra, 256–279.
See also Generalizing activities;
Transformational activities of algebra
beginning, 255–256
characterizations of, 294–295n
developing meaning, 272–274
as generalized arithmetic, 256
mentally graphing to solve an equation, 275
representational activities of, 257
role of technology, 274–276
as symbol transformation, 256
table completion task from NAEP, 260
two methods for solving equations, 273
using technology to learn, 420
what the number-proficient child brings, 270– 272
Algebra for all, 279–280
promoting, 420
Algebraic thinking, developing, 419
Algorithms, 102–106, 195–196
for addition, 199–204
children devising their own, 197
common, for multidigit division, 211
for division, 210–212
efficiency of, 103
examples of, 104–106
generality of, 103
learning numerical, 414
for multiplication, 104–106, 195, 206–210
precision of, 103
simplicity of, 103
for subtraction, 204–206
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America 2000, 34
American Federation of Teachers, 34–35
Arabic numerals, 18, 163, 166, 175n
Archimedes, 112–113n
Area interpretation of multiplication, 107
Area measure, 283
space-filling concept for, 283
Arithmetic. See also Mental arithmetic;
Single-digit arithmetic
abstract nature of, 74
and algebra, 256, 261
and geometry, 87–93
and mathematics, 293–294
multidigit, 121–122
number systems of, 72
paper-and-pencil, 20
preschool, 169
rules of, 73–75, 274, 280
Arithmetic operations, properties of, 73, 75–78
Arizona, exam passing rates in, 42
Arrays, rectangular, interpretation of multiplication, 77, 207–208
Assessment, 349–350
cut scores for, 42
high stakes, 41–42
internal and external, 39–40
of mathematics knowledge in the U.S., 35–36
of school mathematics in the U.S., 31, 39–44
of students, 349–350
Assessment Standards for School Mathematics, 34
Associative law, 75
Associativity of addition, 77
Automaticity, attaining, 351
Averages, 290
B
“Back to basics” movement, 115
Base-10
blocks, 96, 203, 221n
place-value system, 198
Beginning algebra, 255–256
Bird and worm problem, 129
Blocks
base-10, 96, 203, 221n
building, 106–110
Book purchase problem, 261
Borrowing, in subtraction, 204–205
Building blocks, number concepts as, 106–110
C
CAD. See Computer-assisted-drawing tools
Calculators, 45–46, 354–356
different types of, 100
four function, 100n
graphing, 269
instructional recommendations for using, 427
order-of-magnitude accuracy of, 247
recommendation for using, 427
scientific, 100n
use in Sweden, 355
Calculus, comprehending, 295n
California
standards for knowledge of mathematics, 34
textbook system in, 36
California State Board of Education, 50
Call for Change, A, 51
Canada, levels of mathematics achievement in, 56
Cardinal numbers, 160
Carrying, 203
Cases. See also Vignettes programs focusing on, 392–395
Central tendency, measures of, 289
CGI. See Cognitively Guided Instruction
Chance, learning about, 291–293
Children. See also Preschoolers’ mathematical proficiency
devising their own algorithms, 197
China
addition method in, 188
decimal system in, 175n
fractions in, 236
learning number names in, 164–168, 175n
Chocolate distribution problem, 266
Classroom discourse, 345–346
recommendation for managing, 425–426
Classroom vignettes. See Vignettes
Cognitive science, 117–118, 145n, 218n
Cognitively Guided Instruction (CGI), 389, 391– 392, 400n
Combinations. See Number combinations
Combinatorics, 109
Committee on Mathematics Learning, 2, 26
Communities
of learners, 344–345
of mathematics specialists, 397–398
of practice, 397–398
Commutativity
of addition, 75, 77
of multiplication, 77
Compare, problem types, 185
Comparing prices, teaching about, 326–327
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Competence. See Mathematical proficiency;
Strategic competence
“Complementary number-to-10” strategy, 218n
Composite units, 249n
Compound units, 249n
Computation, with rational numbers, 238
Computer-assisted-drawing (CAD) tools, 287
Computers
graphics on, 16, 269
instructional recommendations for using, 427
Conceptual understanding, 5, 10, 116, 136–137, 158–159, 380–382
and mathematical proficiency, 118–120
Concrete materials. See also Manipulatives not the same as physical, 426
Conditional probability, 292–293
Conditions, as aids to understanding, 127
Conference Board of the Mathematical Sciences, 397
Connections, supporting, 235–236
Content, 333–338, 350–356
and calculators, 354–356
and homework, 352–353
and manipulatives, 353–354
opportunities to learn, 333–335
planning, 337–338
and practice, 351–352
task selection and use, 335–336
Contexts
for instruction, 314
for learning, solving problems as providing, 420–421
meaningful, for word problems, 183–187
Conventional instruction, what can be learned from, 240–241
Cookie distribution problem, 376–377
Cooperative learning, 50, 344–349
Coordinating improvement efforts, in teaching mathematics in the U.S., 58–59
Council for Basic Education, 35
Council of Chief State School Officers, 52–53
Counting, 181
and the origins of the number concept, 159– 160
understanding and mastering, 161–162
Curriculum. See also Curriculum recommendations
decisions, 10–11, 410–424
guides and frameworks for, 34
mathematics, in U.S., 33–35
standards for, 34
Curriculum and Evaluation Standards for School Mathematics, 33–34, 36
Curriculum recommendations, 10–11, 410–424
building on informal knowledge, 410–411
developing algebraic thinking, 419
developing proportional reasoning, 417
expanding the number domain, 418
extending the place-value system, 416–417
giving students time to practice, 422–423
giving time to instruction, 422
improving materials for instruction, 421–422
learning about numbers, 412–413
learning number names, 411–412
learning numerical algorithms, 414
operating with single-digit numbers, 413
promoting algebra for all, 420
representing and operating with rational numbers, 415–416
solving problems as a context for learning, 420–421
using estimation and mental arithmetic, 415
using technology to learn algebra, 420
using the number line, 418
Cycle shop problem, 126
Czech Republic, levels of mathematics achievement in, 56
D
Data
analyzing, 290–291
describing, 289
learning to use, 288–291
organizing, 289–290
reading, 289–290
representations of, 290
Decimal system, 96.
See also Base-10
Derived number combinations, 188
Developing algebraic thinking, recommendation for, 419
Developing geometric reasoning, 284–288
reasoning about more advanced concepts, 287–288
reasoning about shape and form, 284–287
Developing mathematical proficiency, 8, 13–14, 246–247, 255–312, 432
acquiring measure concepts, 281–284
algebra for all, 279–280
from arithmetic to mathematics, 293–294
beginning algebra, 255–256
concept of negative numbers, 245
developing geometric reasoning, 284–288
developmental themes, 216–218
discontinuities in, 233–234
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early, and language, 163
estimation, 215–216
generalizing and justifying activities of algebra, 276–279
with integers, 244–246
main activities of algebra, 256–279
measurement and geometry, 281
mental arithmetic, 214–215
multidigit whole number calculations, 195– 214
operations with single-digit whole numbers, 182–195
overtime, 135
with rational numbers, 7–8, 231–241
representational activities of algebra, 261–270
representing rational numbers, 236–244
statistics and probability, 288–293
supporting connections, 235–236
transformational activities of algebra, 270–276
using informal knowledge, 232–233
with whole numbers, 6–7, 181–229
Developing meaning, 263–270
in algebra, 272–274
Developing proficiency in teaching mathematics, 10, 369–405.
See also Professional development
attaining a profound understanding of fundamental mathematics, 370
communities of practice, 397–398
effective professional development, 398–399
knowledge base for teaching mathematics, 370–380
patterns in predicting student proficiency, 217
proficient teaching of mathematics, 380–385
programs to develop proficient teaching, 385– 397
what it takes to teach for mathematical proficiency, 369–370
Developing proportional reasoning, recommendation for, 417
Developing specialized knowledge, recommendation for, 428–429
Developmental themes, 216–218
Discontinuities in proficiency, 233–234
Discourse, managing, 345–346, 425–426
Disparities in mathematical proficiency, 148n
addressing, 344
gender, 148n
racial, 55, 148n
socioeconomic, 143
Disposition, 146–147n.
See also Productive disposition
Distributivity, of multiplication over addition, 78
Division. See also Multidigit division
algorithms for, 210–212
of fractions, 83–86, 386–388
single-digit, 192–193
subtraction and, 78–80
Division of Elementary, Secondary, and Informal Education, 2, 26
Domain. See Number domain
Dynamic geometry software, 298n
E
Early development of mathematical proficiency, and language, 163
Effectiveness, of professional development, 398– 399
Efficiency
of algorithms, 103
of representations, 99
Elementary and Secondary Education Act, 34
Elements, Euclid’s, 82
Enactment, as instructional interaction, 9
England
investigations of mathematics competence in, 39
teaching experiments in, 265
English language, number names in, 164–168
Equal sign, announcing a result, 270, 390
Equality, 86
in a professional development group, 390
statement of, 75
Equity. See also Disparities in mathematical proficiency
and remediation, 172–174
Errors. See Students’ errors
Estimation, 215–216, 221n
recommendation for using, 415
Euclid, 82
European-Latino subtraction algorithm, 219n
Evidence from research, 23–26
Expanded algorithms
for multidigit division, 211–212
for multidigit multiplication, 209
Expanding the number domain, recommendation for, 418
Expectations
low, 343
of success, maintaining, 339–340
of teachers, 338–339
Experience, linking to abstraction, 426
Experiments in teaching, 265
what can be learned from, 240–241
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Exponential functions, 295n
Exponents, 98
F
Finite decimals, number system of, 88–90
First International Mathematics Study, 360n
Florida
special funding provided for mathematics teaching, 41
standards of analyzed, 35
Fluency. See also Procedural fluency relation to understanding, 196
Fordham Foundation, 35
Form, reasoning about, 284–287
Formulas
for the arithmetic of fractions, 86
for the arithmetic of negation, 83
Fractions
addition of, 86, 320–322
division of, 83–86
equality in, 86
formulas for the arithmetic of, 86
multiplication of, 86
notation for, 86, 112n
reciprocals of reciprocals, 86
France, investigations of mathematics competence in, 39
Functions, 274
graphs of, 274
G
Gas price problem, 125
Gauss, Carl Friedrich, 108
Gender disparities, 148n
Generality
of algorithms, 103
of representations, 100
Generalized arithmetic
algebra as, 256
Generalizing activities, 258, 276–279
justifying generalizations, 276–278
predicting patterns, 278–279
problems that involve, 277
Geometry, 107, 281
and arithmetic, 87–93
Germany, video studies of mathematics teaching in, 49–50, 280
Givens, as aids to understanding, 127
Goals 2000, 34
Gradualness, 217
Graphing, using calculators, 269
Grouping. See also Regrouping
of quantities, 96–99
of students, 50, 112n, 265, 346–349
H
Handshake problem, 107–109
Hawaii, requirements for professional development, 54
High-stakes assessments, 41–42
Hindu-Arabic numerals, 18, 163, 166, 175n
Holmes Group/Partnership, 52
Homework, 352–353.
See also Independent work
Hong Kong
investigations of mathematics competence in, 39
levels of mathematics achievement in, 56
I
Illinois, requirements for professional development, 54
Independence of events, 293
Independent work, recommendation for assigning, 426–427
Informal knowledge, 232–233
building on, 410–411
Instruction
as interaction, 313–315
issues in improving, 356–359
in multidigit procedures, importance of, 197
recommendation for giving time to, 422
varied approaches to, 197–198, 382
Instructional materials. See also Manipulatives;
Textbooks
recommendation for improving, 421–422
use of, 7, 39, 282
Instructional programs, for school mathematics in the U.S., 31, 36–39
Instructional recommendations, 11, 424–427
assigning independent work, 426–427
linking experience to abstraction, 426
managing classroom discourse, 425–426
planning for instruction, 424–425
using calculators and computers, 427
Instructional routines, 11, 382
Instructional triangle, 314
Integers, 72, 244–246
subtraction and, 80–83
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Interaction
instruction as, 313–315
with students, 343–344
Interactive perspective, on teaching and learning, 359n
Interstate New Teacher Assessment and Standards Consortium, 61n
Intertwined strands of mathematical proficiency, 5, 116–117, 217
Inverse
operations, 270
relationships, 79
variation functions, 295n
Ireland, levels of mathematics achievement in, 56
Irrelevance, of counting order, 160
J
Japan
assessment of mathematics knowledge in, 35– 36
levels of mathematics achievement in, 56
study of development of proficient teaching in, 396
use of abacus, 218n
video studies of mathematics teaching in, 49– 50, 280
Join, problem types, 185
Justification, 130, 273, 276–279.
See also Generalizing activities
of generalizations, 276–278
problems that involve, 277
of procedures, 130
and proof, 138–139, 170
K
Key ideas about number, 110–111
Knowledge
of classroom practice, 379–380
clusters of, 120
of instructional practice, 372
of mathematics, 372–378
of students, 371–372, 378–379
Knowledge base
building on informal, 410–411
for teaching mathematics, 370–380
Korea
levels of mathematics achievement in, 56
subtraction algorithm from, 219n
L
Language, and early mathematical development, 163
Learners, communities of, 344–345
Learning
about chance, 291–293
about numbers, 412–413
and conditional probability, 292–293
current patterns of, 246
independence in, 293
number names, 411–412
numerical algorithms, 414
opportunities for teachers, 333–335
probability comparisons across sample spaces, 292
probability of an event, 291–292
and sample space, 291
single-digit arithmetic, 194–195
solving problems as a context for, 420–421
symbolic, 198
Learning difficulties, 342
Learning goals, for school mathematics in the U.S., 31, 33–36
Learning orientation, versus performance orientation, 171
Learning progression
for single-digit addition, 187
for single-digit subtraction, 190
Length measure, 281–282
Lesson study in Japan, 396
Lessons, 337
observed, 48–51
programs focusing on, 395–397
Letters representing unknowns, 270
Licensing requirements, by state, 53
Limitations, on preschoolers’ mathematical proficiency, 172
Line. See Number line
Linguistic structure, of number names, 163–166
Linking experience to abstraction, recommendation for, 426
Longitudinal Study of American Youth (LSAY), 374, 399n
Louisiana, licensing requirements in, 53
Low expectations, self-fulfilling prophecies of, 343
LSAY. See Longitudinal Study of American Youth
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M
Managing discourse, 345–346
recommendation for, 425–426
Manipulatives, 45, 198, 353–354
Massachusetts, exam passing rates in, 42
Mastering counting, 161–162
Materials. See Instructional materials
Mathematical knowledge, 372–378
children bringing to school, 5–6, 157–180
of preschoolers, 158–174
state standards for, 34
“street mathematics,” 146n
of teachers, 371, 373–378
Mathematical proficiency, 5, 115–154
adaptive reasoning and, 129–131
conceptual understanding and, 118–120
developing over time, 135
discontinuities in, 233–234
intertwined strands of, 5, 116–117, 217
monitoring progress toward, 431–432
need for all students to possess, 142–144
not all or nothing, 135
procedural fluency, 121–124
productive disposition and, 131–133
properties of, 133–135
recommendations concerning, 10–11, 13–14, 408–410, 432
strategic competence and, 124–129
unique position of, 59n
in various domains of mathematics, 141–142
Mathematical proficiency of U.S. students today, 136–141
in adaptive reasoning, 138–139
in conceptual understanding, 136–137
and population growth in two towns, 140
in procedural fluency, 137–138
in productive disposition, 139–141
in strategic competence, 138
Mathematical tasks, programs investigating, using cases from real practice, 393–394
Mathematics. See also Teaching mathematics
looking at, 20–21
of number, 71–114
power of, 115
programs focusing on, 385–389
Mathematics and learning, 15–29
looking at mathematics, 20–21
mathematics and reading, 17–20
nature of the evidence, 21–24
quality of research studies, 23
role of research in improving school mathematics, 24–26
Mathematics specialists, 397–398
Measure concepts
acquiring, 281–284
area, 283
central tendency, 289
and geometry, 281
length, 281–282
volume, 284
Measurement, 281–284
Memory techniques, mnemonic, 119
Mental arithmetic, 214–215, 356
recommendation for using, 415
Mental graphing, in algebra, 275
Metacognition, 117–118
Mexico, teaching experiments in, 265
Michigan, standards of analyzed, 35
Missing-value problems, 243
Mnemonic techniques, 119
Models
for multidigit addition, 204
for multidigit multiplication, 208
for multidigit subtraction, 206
Monitoring progress toward mathematical proficiency, recommendation for, 12–13, 431–432
Motivation, 339–341, 360n
maintaining an expectation of success, 339–340
valuing learning activities, 340–341
Multidigit addition
algorithms for, 199–204
model for, 204
Multidigit division
algorithms for, 210–212
common algorithm for, 211
expanded algorithm and model for, 211
expanded algorithm for, with fewer steps, 212
Multidigit multiplication
algorithms for, 206–210
beginning algorithm for, 195
common U.S. algorithm for, 207
expanded algorithm for, 209
model for, 208
Multidigit numbers, 195–214
importance of instruction in gaining proficiency with, 197
summary of findings on, 212–214
third-grade class finding 54+48, 200
Multidigit subtraction
accessible generalizable methods for, 205
algorithms for, 204–206
common error in, 123
common U.S. algorithm for, 205
model for, 206
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Multiple representations, 95
Multiplication. See also Multidigit multiplication
algorithms for, 206–210
beginning an algorithm for, 195
commutativity of, 77
distributivity of over addition, 78
of fractions, 86
and negation, 83
by powers of 10, teaching about, 316–318
properties of, 85
single-digit, 191–192
Multiplicative concepts, 248n
identity, 85
inverse, 85
N
NAEP. See National Assessment of Educational Progress
Names. See Number names
Nation at Risk, A, 34
National Advisory Committee on Mathematical Education (NACOME), 48
National Assessment of Educational Progress (NAEP), 36–37, 40, 42, 45–47, 53–57, 117, 234, 285, 356, 374, 432
scores for long-term trend assessment, 136– 138, 141, 143, 147n, 242, 259–260
National Board for Professional Teaching Standards, 54
National Center for Improving Student Learning and Achievement in Mathematics and Science, 60n
National centers for research and development in school mathematics, 39
National Council of Teachers of Mathematics (NCTM), 33–34, 36
standards promulgated by, 33–35, 47
National Educational Longitudinal Study (NELS), 347
National Longitudinal Study of Mathematical Abilities (NLSMA), 374
National Research Council (NRC), 2, 17, 44, 132
Strategic Education Research Program, 62n
National Science Foundation, 3, 34, 38, 48
Directorate for Education and Human Resources, 3, 26
Natural numbers, 111n
NCTM. See National Council of Teachers of Mathematics
Needs. See Special needs
Negation, 83
multiplication and, 83
opposites of opposites, 83
subtraction and, 83
Negative numbers, 111n
concept of, 245
NELS. See National Educational Longitudinal Study
Netherlands, investigations of mathematics competence in, 39
New Jersey, requirements for professional development, 54
New Mexico, requirements for professional development, 54
New York
exam passing rates in, 42
requirements for professional development, 54
NLSMA. See National Longitudinal Study of Mathematical Abilities
North Carolina
requirements for professional development, 54
standards for knowledge of mathematics, 34
NRC. See National Research Council
Number combinations, 6, 182
derived, 188
Number domain, recommendation for expanding, 418
Number line, 245, 282, 418
linking arithmetic and geometry, 87–93
Number names
in Chinese, English, and Spanish, 164–166, 175n
learning, 411–412
linguistic structure of, 163–166
psychological consequences of, 166–168
Number-proficient students, 261–263, 270–272
Numbers, 20, 71–114.
See also Multidigit numbers;
Negative numbers;
Rational numbers;
Real number system;
Representations of numbers;
Single-digit numbers
building blocks, 106–110
cardinal, 160
choosing and translating among representations, 99–102
decimal, 88–90
grouping and place value, 96–99
key ideas about, 110–111
learning about, 412–413
meanings of, 71, 158
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natural, 111n
nested systems of, 93–94
operations on, 75–78, 86
origins of concept of, 159–160
rational, 85–87
representations of, 94–96
systems of, 72, 88–90, 93–94
triangular, 108
whole, 73–75
Numeration system, Hindu-Arabic, 18, 163, 166, 175n
Numerical algorithms, learning, 414
O
Observed lessons, in teaching mathematics in the U.S. , 48–51
Octagon problem, 109
Office of Educational Research and Improvement, 3, 26
One-to-one relationships, 160
Operating
with integers, 245–246
with rational numbers, 415–416
with single-digit whole numbers, 182–195, 413
Opportunity to learn, 333–335
Opposites of opposites, 83
Order irrelevance, 160
Order-of-magnitude accuracy, of calculators, 247
Oregon, requirements for professional development, 54
Organizing data, 289–290
Orientation, 91, 111n
Origins of the number concept, counting and, 159–160
P
Part-part-whole, problem types, 185
Partial products, 207
Passing rates on exams, by state, 42
Patterns, 217, 278–279
Performance orientation, versus learning orientation, 171
Piaget, Jean, 158, 174n
Pile of marbles problem, 184, 186
Pizza sharing problem, 237
Place-value system
base-10, 198
extending, 416–417
grouping and, 96–99
Plane geometry, 82
Planning for instruction, 337–338
recommendations concerning, 424–425
Polydrons, 298n
Population growth illustration, 140
Positive rational numbers, 84
Powers, 98
of 10, multiplying by, 316–318
Practice for students, 351–352
communities of, 397–398
kinds of, 351–352
recommendation for providing time for, 422– 423
role of, 351
in single-digit calculations, 193
Precision
of algorithms, 103
of representations, 101–102
Predicting patterns, 278–279
in the development of student proficiency, 217
Preschool arithmetic, 169
Preschoolers’ mathematical proficiency, 158–174
adaptive reasoning, 170
conceptual understanding, 158–159
counting and the origins of the number concept, 159–160
equity and remediation, 172–174
limitations of preschoolers’ mathematical proficiency, 172
procedural fluency, 162–168
productive disposition, 171
strategic competence, 168–170
understanding and mastering counting, 159– 160
Prices, teaching about comparing, 326–327
Principles and Standards for School Mathematics, 34
Probability
of an event, 291–292
comparisons across sample spaces, 292
conditional, 292–293
statistics and, 288–293
Problem model, 125, 268
Problem solving
as a context for learning, 420–421
focusing on, 219n, 383
Problem types, 185
addition, 185
compare, 185
involving generalizing and justifying activities, 277
join, 185
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part-part-whole, 185
“routine,” 146n
separate, 185
subtraction, 185
word, 169–170, 183–187
Problems
bird and worm, 129
book purchase, 261
chocolate distribution, 266
concepts, 125–126
cookie distribution, 376–377
cycle shop, 126
gas price, 125
handshake, 107–109
octagon, 109
pile of marbles, 184, 186
pizza sharing, 237
routine and nonroutine, 125–126
toy cars, 183
water business, 269
weather balloon, 268
Procedural fluency, 5, 10, 116, 121–124, 137–138, 162–168, 380, 382
and a common error in multidigit subtraction, 123
and counting, 162–163
language and early mathematical development, 163
linguistic structure of number names, 163–166
and mathematical proficiency, 121–124
psychological consequences of number names, 166–168
Procedures, 187
benefit of getting students to explain, 221n
justifying, 130
Productive disposition, 5, 10, 116, 139–141, 171, 380, 384–385
and mathematical proficiency, 131–133
in the teaching of mathematics, 384–385
Professional development, 31, 51–54
capitalizing on professional meetings, 430
developing specialized knowledge, 428–429
effectiveness of, 398–399
recommendations concerning, 12, 428–431
state requirements for, 54
sustaining, 430–431
for teaching mathematics in the U.S., 51–54
working together, 430
Professional development groups, investigating the concept of equality in, 390
Professional meetings, capitalizing on, 430
Professional Standards for Teaching Mathematics, 34, 51
Proficient teaching of mathematics, 8–9, 380–385.
See also Mathematical proficiency;
Student proficiency;
Teaching for mathematical proficiency
and adaptive reasoning, 383–384
and instructional routines, 382
Japanese lesson study, 396
and productive disposition, 384–385
programs to develop, 385–397
and strategic competence, 382–383
and understanding of core knowledge, 381– 382
Programs to develop proficient teaching of mathematics, 385–397
focusing on cases, 392–395
focusing on lesson study, 395–397
focusing on mathematics, 385–389
focusing on student thinking, 389–392
investigating division of fractions, 386–388
investigating equality in a professional development group, 390
investigating mathematical tasks using cases, 393–394
Japanese lesson study, 396
Promoting algebra for all, recommendation concerning, 420
Proof, 130.
See also Generalizing activities;
Justification
Properties of addition, 82
additive identity, 82
additive inverse, 82–83
associativity of, 77
commutativity of addition, 77
Properties of arithmetic operations, 77–78, 82
Properties of mathematical proficiency, 133–135
developing over time, 135
interwoven strands, 133–134
not all or nothing, 135
Properties of multiplication, 85
commutativity of, 77
distributivity of, over addition, 78
multiplicative identity, 85
multiplicative inverse, 85
reciprocals, 85
Proportional reasoning, 8, 241–244
developing, 417
Psychological consequences, of number names, 166–168
Q
Quadrilaterals, children’s understanding of, 284
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Quality issues, 23
generalizability of research, 23
relevance of research, 23
soundness of research, 23
R
Racial disparities, 55, 148n
addressing, 344
Rational numbers, 72, 85–87
computation with, 238
positive, 84
representing and operating with, 415–416
teaching about representations of, 324–325
Reading, mathematics and, 17–20
Reading data, 289–290
Reading Recovery program, 28n
Real number system, 90
and its subsystems, 94
Reasoning. See also Adaptive reasoning;
Proportional reasoning
about advanced concepts, 287–288
about shape and form, 284–287
Reciprocals, 85, 98
Reciprocals of reciprocals, 86
Recitation, 48
Recommendations, 10–14, 407–432
curricular, 10–11, 410–424
for further research, 357–359
instructional, 11, 424–427
for mathematical proficiency, 10–11, 408–410
for monitoring progress, 12–13, 431–432
for supporting the development of mathematical proficiency, 13–14, 432
for teacher preparation and professional development, 12, 428–431
Regrouping, 203, 205
Relevance of research, 23
Remediation, and equity, 172–174
Reported practices, in teaching mathematics in the U.S., 45–47
Representational activities of algebra, 256–257, 261–270
building on spreadsheet experiences, 266
developing meaning, 263–270
water business problem, 269
weather balloon problem, 268
what the number-proficient student brings, 261–263
Representations of numbers, 94–96, 99–102
choosing and translating among, 99–102
clarity of, 100
of data, 290
efficiency of, 99
generality of, 100
multiple, 95
precision of, 101–102
symbol-based, 234, 399n
transparency of, 99
Representations of rational numbers, 236–244
computing with, 238
conventional and experimental instruction in, 240–241
learning from students’ errors, 238–240
recommendation for, 415–416
teaching about, 324–325
Representativeness issues, 293
Research. See also Research on teaching
convergent, 25
determinants of quality, 22–23
generalizable, 23
relevant, 23
role in improving school mathematics, 24–25
sound, 23
Research on teaching
about students and content, 350–356
about teachers and content, 333–338
about teachers and students, 338–350
findings from, 333–356
Research recommendations. See Recommendations
S
Sample spaces, 291
probability comparisons across, 292
Scaffolding, 336
School mathematics in the U.S., 4, 31–70
assessments of, 39–44
instructional programs and materials goals, 36–39
learning goals, 33–36
and teaching, 45–54
Schools and Staffing Surveys, 54
Second International Mathematics Study (SIMS), 59n
Self-fulfilling prophecies, 343
Separate, problem types, 185
Set-combination, 74
Shape, reasoning about, 284–287
Simplicity, of algorithms, 103
SIMS. See Second International Mathematics Study
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Singapore
investigations of mathematics competence in, 39
levels of mathematics achievement in, 56–57
Single-digit arithmetic, 6
addition, 185, 187–190
division, 192–193
multiplication, 191–192
practice for students in, 193
subtraction, 185, 190–191
Single-digit numbers, 187–193.
See also Multidigit numbers
findings on learning about, 194–195
learning progression for, 187, 190
making ten, 189
operating with, 413
thinking strategies for working with, 192
word problems using, 183–187
Socioeconomic disparities, 143
Solutions, as aids to understanding fundamental mathematics, 127
Solving problems
focusing on, 219n, 383
as providing contexts for learning, 420–421
Solving word problems, 169–170
Soundness of research, 23
Space-filling, concept for measuring area, 283
Spanish language, number names in, 164–165
Special needs, teaching students with, 341–343
Special Study on Essential Skills in Mathematics (Japan), 40
Specialists, in mathematics, 397–398
Specialized knowledge, developing, 428–429
Spreadsheets, 16
building on experiences with, 266
Stable order of counting words, 160
Stakes, assessments with high, 41–42
Standardized tests, 42–43
defined, 60n
State licensing requirements, 53
State passing rates on exams, 42
State requirements for professional development, 54
State standards for knowledge of mathematics, 34
Statistics, probability and, 288–293
Stereotype threat, 133
Sticks, subtraction using, 128
Strands of proficiency. See Intertwined strands of mathematical proficiency
Strategic competence, 5, 10, 116, 138, 168–170, 380, 382–383
and mathematical proficiency, 124–129
preschool arithmetic, 169
solving word problems, 169–170
subtraction using sticks, 128
in teaching mathematics, 382–383
Strategic Education Research Program, 62n
“Street mathematics,” 146n
Student proficiency, patterns in predicting development of, 217
Student thinking, programs focusing on, 389–392
Students, 338–356
assessment of, 349–350
communities of learners, 344–345
cooperative groups, 348–349
doing homework, 352–353
given time to practice, 422–423
giving directives to, 162
grouping, 346–348
interacting with other students, 343–344
managing discourse among, 345–346
motivating, 339–341
practicing, 351–352
with special needs, 341–343
and tasks, 350
teacher expectations of, 338–339
using calculators, 354–356
using manipulatives, 353–354
Students’ errors
learning from, 238–240
systematic patterns of, 196
Subtraction. See also Multidigit subtraction
algorithms for, 204–206, 219n
borrowing in, 204–205
and division, 78–80
and the integers, 80–83
and negation, 83
problem types, 185
single-digit, 190–191
using sticks, 128
Supporting connections, in meaning of rational numbers, 235–236
Supporting the development of mathematical proficiency, recommendation for, 13–14, 432
Sustaining professional development, recommendation for, 430–431
Sweden, use of calculators in, 355
Symbol-based representation, 234, 399n
Symbolic learning, 198
Systems of numbers, 72, 88–90
nested, 93–94
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T
Table completion task, from NAEP, 260
Task selection and use, teachers’, 335–336
Teacher certification, in teaching mathematics in the U.S., 51–54
Teacher preparation, 31, 51–54
capitalizing on professional meetings, 430
developing specialized knowledge, 428–429
recommendations for, 12, 428–431
sustaining professional development, 430–431
for teaching mathematics in the U.S., 51–54
working together, 430
Teachers, 333–350
assessing students, 349–350
certification of, 31, 51–54
creating communities of learners, 344–345
creating cooperative groups, 348–349
expectations of, 338–339
grouping students, 346–348
interacting with different students, 343–344
managing discourse, 345–346
motivating students, 339–341
opportunities for learning, 333–335
planning content, 337–338
providing opportunities to learn, 333–335
task selection and use, 335–336
teaching students with special needs, 341–343
Teachers’ mathematical knowledge
and student achievement, 373–377
and their teaching practice, 377–378
Teaching for mathematical proficiency, 8–9, 313– 368
findings from research on, 333–356
four classroom vignettes, 315–328
instruction as interaction, 313–315
the instructional triangle, 314
issues in improving, 356–359
Teaching mathematics
achievement in, 55–57
adaptive reasoning in, 383–384
adding fractions, 320–322
comparing prices, 326–327
experiments in, 265
findings from research on, 333–356
multiplying by powers of 10, 316–318
representations of rational numbers, 324–325
understanding of fundamental mathematics in, 381–382
Teaching mathematics in the U.S., 45–54
achievement, 55–57
coordinating improvement efforts, 58–59
observed lessons, 48–51
reported practices, 45–47
teacher preparation, certification, and professional development, 51–54
Teaching students with special needs, 341–343
Techniques, mnemonic, 119
Technology, using to learn algebra, 274–276, 420
Ten, making, 189
Tendencies. See Central tendency
Testing, Teaching, and Learning, 44
Texas
standards for knowledge of mathematics, 34
textbook system in, 36
Textbook system, in the U.S., 36–37
Textbooks
addition algorithm, in U.S., 203
division algorithm, in U.S., 212
make-a-ten procedure, in U.S., 188
mathematics, in U.S., 27
multiplication algorithm, in U.S., 208
for prospective teachers, 71
word problem solution method, in traditional, 186
Thinking strategies, 192–193.
See also Student thinking
Third International Mathematics and Science Study (TIMSS), 32–33, 41, 56–57, 356
Video Study, 49–50, 280, 359n
Tiling, concept of measuring area, 283
Time
for instruction, 422
to practice, 422–423
TIMSS. See Third International Mathematics and Science Study
“Top from bottom” error, 204–205
Toy cars problem, 183
Tracking, 346
Trading, 203–204
Transformational activities of algebra, 257–259, 270–276
developing meaning, 272–274
mentally graphing to solve an equation, 275
role of technology, 274–276
two methods for solving equations, 273
what the number-proficient child brings, 270– 272
Translating among representations, 99–102
clarity, 100
efficiency, 99
example, 101–102
generality, 100
precision, 101–102
transparency, 99
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Transparency
of algorithms, 103
of representations, 99
Tremont Hotel analogy, 58
Triangle. See Instructional triangle
Triangular numbers, 108
U
Understanding counting, 161–162
Understanding fundamental mathematics. See also Conceptual understanding
attaining a profound, 370
relation to fluency, 196
solutions as aids to, 127
in the teaching of mathematics, 381–382
unknowns as aids to, 127
Undoing operations, 270, 273
United Kingdom, teaching experiments in, 265
United States
assessments of school mathematics in, 39–44
instructional programs and materials goals in, 36–39
learning goals in, 33–36
levels of mathematics achievement in, 56–57
state of school mathematics in, 4, 31–70
teaching of school mathematics in, 45–54
U.S. Constitution, 33
U.S. Department of Education, 38
Office of Educational Research and Improvement, 3, 26
Unknowns, as aids to understanding fundamental mathematics, 127
V
Valuing learning activities, 340–341
Video Study (TIMSS), 49–50, 280, 359n
Vignettes, 315–333
about adding fractions, 320–322
about comparing prices, 326–327
about multiplying by powers of 10, 316–318
about representations of rational numbers, 324–325
comparing the lessons, 328–333
Virginia
exam passing rates in, 42
standards for knowledge of mathematics, 34
Volume measure, 284
W
Water business problem, 269
Weather balloon problem, 268
Whole numbers, 72–75
multidigit, 195–214
Wisconsin, licensing requirements in, 53
Wise educational environments, 133, 145n
Word problems
meaningful context for, 183–187
solving, 169–170
Working together, 430
recommendation for, 430
Z
Zero, 111n
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