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Index
A
Accumulator mechanism, 63, 351
Addition/subtraction
acting out solutions, 157
associative properties of addition, 51
cardinality and, 130, 150, 154
change plus/change minus situations, 23, 32, 33-34, 149, 150, 151, 155, 162-163, 165, 351
commutative properties of addition, 50-51, 160
comparison situations, 149, 151, 161-162, 165
composing/decomposing, 23, 32-33, 45-46, 51, 133, 158-159, 160-161, 164-165, 231
counting on, 25, 51, 111, 130, 145, 148, 151, 152-153, 154, 155, 157-158, 159-160, 163-164, 166
decontextualized problems, 158
derived fact method, 151, 153, 154, 160-161, 165, 166
direct modeling, 151-152, 160
drawing solutions, 157, 159
economically disadvantaged children, 152, 159
equation writing, 162-163
fluency vs. recalled/memorized facts, 166
infants, 65, 67
language differences and, 101, 149, 157
levels in solution methods, 150-154, 160
make-a-10 method, 51, 151, 153, 154, 160, 165, 166
mathematizing real-world situations, 165
measurement and, 208
number word list and, 25, 160, 166-168
oral problems, 152, 154, 159
pattern finding in, 159-160
preschool children, 67-68, 69-70, 71
put together/take apart, 23, 32-33, 34-35, 148, 150, 151, 155, 158, 160, 162, 355, 356
relations and, 31, 149
teaching-learning path, 130
variability in solution methods, 168-169
vocabulary learning, 149, 157
word problems and situations, 32-35, 101, 149, 151-152, 155, 157, 159
Additive comparison situations, 23, 33, 34, 231, 351
Ages 2-3
addition/subtraction teaching-learning path, 130, 154, 155
cardinality, 130, 131, 132-134, 139-140, 150, 154
compositions and decompositions, 177, 185, 186, 187-188, 206
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coordinating components, 131, 138-140
counting correspondences, 131, 134-137, 138-139
measurement, 202, 205-206
number core teaching-learning path, 130, 131-140
number word list, 131, 134, 138
operations teaching-learning path, 150, 155
relations teaching-learning path, 130, 150, 154-155
shapes and objects, 177, 185, 186
spatial relations, 177, 185, 186, 205-206
vocabulary, 154
written number symbols, 131, 138
Age 4 (prekindergarten).
See also Preschool children
cardinality, 141, 142, 150, 155, 156
compositions/ decompositions, 178, 186-187, 189-190, 206
counting correspondences, 136, 141, 144
counting out n things, 141, 145, 156, 157
matching, 155, 156
measurement, 202, 206
number core teaching-learning path, 130, 141-145
number word list, 141, 142-143, 144, 145
operations teaching-learning path, 130, 150-151, 154, 155, 157-159
relations teaching-learning path, 130, 150, 155-157
shapes, 177-178, 186-187, 188
spatial relations, 177-178, 186-187, 188-189
vocabulary, 155, 156-157, 188-189
written number symbols, 138, 141, 144-145
Age 5.
See Kindergarten
Algebraic problem solving, 165
Analog magnitude system, 63, 70, 112, 168, 351
Area, 35, 36, 37, 39, 46, 49, 53, 61-62, 65, 66, 79, 177, 179, 190, 201-204, 205, 207, 209-210, 354, 361
Arithmetic properties, 50-52
Assessment practices.
See also Formative assessment
multiple-choice tests, 100
Attention, 84
Attitudes and beliefs about mathematics
ECE teachers, 8-9, 297-300, 309
parents, 8, 102, 344
preschool children, 12
Attribute blocks, 47, 183-184, 209-210, 248, 351
Automaticity, 128
B
Behavioral and attention regulation, 82-84, 335
Big Math for Little Kids, 256, 263, 271, 309
Bright Beginnings, 269
Building Blocks curriculum, 248, 256, 270, 271, 311, 312
Bureau of Labor Statistics, 291-292
C
Calendar activities, 241-242, 275
California, 230, 233, 290, 293, 310
Cardinality
and addition/subtraction operations, 130, 150, 154
ages 2-3, 130, 131, 132-134, 154
cognitive foundations, 63, 71
coordinations and, 23, 25-26, 130, 131, 139
counting and, 129, 130, 131, 135, 139, 140, 147
defined, 22, 23, 139, 333-334, 336, 351
and equivalence classes, 63, 71
foundational content, 22, 23, 25-26, 31, 32, 34, 36, 53, 336
games, 53
grade 1, 147, 148
infant concepts, 63
instruction practices, 245-246, 300, 333-334
kindergarten, 146
and measurement, 36, 53
number word list and, 25-26, 31, 32, 129, 132, 147, 148
one-to-one correspondence and, 22, 24, 129
parents’ perceptions of children’s abilities, 104
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prekindergarten, 142
principle, 135
socioeconomic variation on tasks of, 103
standards of learning, 230, 231
subitizing and, 132, 134, 141, 142, 146, 150, 151, 155
teaching-learning paths, 129, 130, 131, 132-134, 135, 139, 140, 141, 142, 147, 148, 150, 169
truly operational cardinal number, 166
Charge to committee, 1, 10-13
Child Care Development Fund, 303
Child Development Associate, 292, 303, 318
Child-initiated experiences, 226, 227, 247-248, 250, 298, 352
Chutes and Ladders, 238 n.3, 251
CLASS framework, 235-236
Classroom Assessment Scoring System, 234, 236, 237, 243-244, 299
Classroom context for instruction.
See also Grade 1;
Kindergarten;
Mathematics instruction;
Preschool programs
emotional support, 236
generalized teaching strategies, 241-242
instructional opportunities, 235
instructional supports, 237
kindergarten, 239-242
mathematics content, 235, 237-242
noninstructional activities and time, 235
organization and management, 236-237
prekindergarten, 237-239
teacher-child interactions, 235-237
Classroom Observation of Early Mathematics Environment and Teaching, 270-271, 272
Cognitive flexibility, 83
Cognitive foundations for early learning
behavioral and attentional regulation and, 82-84, 85
cardinality, 63, 71
fraction representations, 61
measurement, 61, 79-82, 84-85
number understanding, 60-71, 84
pattern finding, 69, 83
Piaget’s theories, 60, 70, 72, 79-80, 155-156
preverbal number knowledge, 60-66
proportional relations, 61, 82
spatial thinking and geometry, 61, 71-79, 84, 192-193
universal starting points, 59-60, 85, 95, 96, 104
van Heile’s theories, 77
Cognitively Guided Instruction, 314
Colorado, 229
Comparisons, 30-31, 53, 97
additive, 23, 33, 34, 231, 351
Competence and performance.
See also Cognitive foundations for early learning;
Learning disabilities;
Number understanding or number sense
activities in daily lives and, 100, 102-104, 113
automaticity, 128
committee conclusions, 332-335
English language learners, 100-101
experiences and learning opportunities
and, 100, 103-105, 333-334
family role, 101-105, 113, 333-334
fluency, 7, 25, 111, 125, 128, 138, 141, 145, 153, 154, 156, 157, 158, 159, 160, 166, 169, 250, 352
gender and, 98-99
individual variations in, 127-128, 154, 157, 334-335
international comparisons, 184, 192, 273
language/cultural influences, 98, 106-110
parents’ knowledge and beliefs and, 102
race/ethnicity and, 99-100
socioeconomic status and, 1-2, 70, 96-98, 103, 104-105, 334-335
Composing/decomposing activities, 248
addition/subtraction situations, 23, 32-33, 45-46, 51, 133, 231
decimal system, 45-46
defined, 352
groups of groups, 49
groups of groups of groups, 50
and mathematical connections across content areas, 39-40, 45-46, 48-49, 50, 194
and measurement, 35, 39-40, 46, 202, 206, 207, 360
and multiplication/division, 45, 51-52
numbers, 23, 32-33, 45, 48, 49-50, 154, 157, 161, 164-165, 194, 231
operations, 45
reasoning processes, 35, 45-46, 179
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shapes and objects, 35, 39-40, 46, 48-50, 175, 177-179, 184, 185-188, 189-190, 191, 194, 195, 206, 231, 248
standards of learning, 231, 337
teaching-learning paths, 32, 154, 157, 161, 164-165, 175, 177-179, 184, 185-188, 189-190, 191, 194, 202, 206, 207
triads, 25, 160
unitizing by grouping, 48-49, 207
Computational fluency, 111, 352
Computer manipulatives, 196
Conceptual subidizing, 23, 132-133, 141, 142, 146, 150, 151, 152, 153, 154, 155, 158, 159, 164, 166, 356
Connections across content
arithmetic properties, 50-52
cardinality, 23, 25-26, 130, 131, 139
composing/decomposing and, 39-40, 45-46, 48-52, 194
in data analysis, number, and measurement, 53-55
fractions, 52-53
of number core, 138-140
Core Knowledge, 256, 273
Counting abstraction principle, 135, 136
accumulator mechanism, 63, 351
addition/subtraction operations, 25, 51, 111, 130, 145, 148, 151, 152-153, 154, 155, 157-158, 159, 160, 163, 164, 166
backward counting, 25, 26, 153
cardinality principle, 31, 129, 130, 131, 135, 139, 140, 147
comparing collections, 30-31
competence of preschool children, 61
core content, 145
cultural differences, 133
defined, 25
errors, 136-137, 153
on fingers, 25, 67, 68, 70, 106, 130, 131, 132, 133, 134, 141, 142, 152, 157-158, 162
games, 53, 113
how-to-count principles, 135
importance of fluency, 156
instruction practices, 240
language and, 30, 107-109, 140
learning disabilities and, 111, 112
money, 147
nonverbal, 63
one-to-one correspondence, 22, 24-25, 129, 134-137, 144
order irrelevance principle, 135, 136
pattern finding and, 47, 49, 107, 156, 164
quantizing, 22, 26, 27, 30, 51
skip counting, 25, 48, 55, 147, 148
socioeconomic differences, 97
of solution methods, 25
stable order principle, 135
subidizing and, 164
unitizing, 45, 48, 147, 148
what-to-count principles, 135, 136-137
Counting on, 25, 111, 113, 130, 151, 152-154, 159-160, 163, 164, 166
Creative Curriculum, 256, 268, 269
Credentialing and licensure, 4, 101, 291, 292, 301, 305, 315, 317-319, 320, 342, 347, 352
Curriculum.
See also Mathematics curriculum
definitions, 227
effectiveness research, 265-273
integrated, 149, 227, 353
D
Data analysis, 53-55
Decimal system
groups of groups, 49
groups of groups of groups, 50
language differences and, 106-108, 109-110
number word list and, 26-30, 64
place values, 28, 29-30, 45, 48, 107, 108, 148, 231, 355
unitizing, 45, 48
Developmental dyscalculia, 112
Dynamic Indicators of Basic Early Literacy Skills, 128
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E
Early Childhood education (ECE) system.
See Curriculum;
Early childhood education teachers;
Early childhood workforce;
Grade 1;
Instruction/pedagogy;
Kindergarten;
Preschool;
Standards for mathematics learning
accountability systems, 16
center-based care, 13-14, 15 66, 67, 123, 127, 129, 133, 137, 141, 155, 158, 290, 291, 292, 293, 294, 307, 318, 337, 338, 341
costs of care, 13, 15
demand for care, 15
home-based care, 13-14, 289, 290, 291, 292, 305, 307
legislation and policy initiatives, 15-16
opportunities to learn mathematics, 1-2
state-funded preschool, 15
variety of settings, 13-15
Early childhood education teachers.
See also Professional development
beliefs about early childhood education, 295-297
defined, 289-290, 352
instruction impacts of beliefs and knowledge, 299-300, 309-310
quality and effectiveness, 272, 305-307, 356
values and beliefs about math, 8-9, 297-300, 309
Early Childhood Educator Professional Development Program, 303
Early Childhood Environment Rating Scale, 234, 237, 242, 299
Early Childhood Longitudinal Study
birth cohort, 13
kindergarten cohort, 98, 233-234, 237, 239, 247
Early childhood workforce.
See also Professional development of workforce
compensation, 291-293
defined, 289, 352
demographic characteristics, 290-291
educational experience and background, 291
and mathematics, 297-300
stability and turnover, 293-294
work environment, 294-295
Early Head Start, 9
Early Reading First, 16
Economically disadvantaged children.
See also Head Start
competence in mathematics, 1, 2, 12, 152, 154
equity in early mathematics education, 12-13, 15-16
instruction strategies, 110, 154, 271, 273, 296-298
opportunities to learn mathematics, 1-2, 13, 15
Education of teachers.
See Formal education of workforce;
Professional development of workforce
Education policies
accountability systems, 16
voluntary universal preschool, 15
Effortful control, 83
Emerging Academics Snapshot, 234, 238
English language learners, 100-101, 149, 157
Equation writing, 33, 35, 44, 162-163
Equivalence classes, 63, 64, 71
Equivalence matches, 69
Even Start, 229
Ewing Marion Kauffman Foundation, 10
Executive function, 83-84, 243, 250, 334
F
False belief understanding, 83
Family role.
See also Parents activities, 76, 102-104, 155, 343
committee conclusions, 343-344
math talk, 104
resources, 344
socioeconomic factors, 104-105, 343-344
Feedback
defined, 353
encouragement and affirmation, 238, 244, 245, 352
importance and quality, 125, 169, 182, 237, 243-244, 299
instructional, 137, 353
loops, 244-245, 352
in professional development, 317, 320, 347
prompting thought processes, 245, 340, 355
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providing information, 127, 245, 248, 253, 299, 355
scaffolding, 182, 191, 203, 238, 243, 244, 247, 249-250, 251, 255, 260, 275, 355
sources, 125
training teachers in, 312
Fidelity of Implementation, 270
Finding a pattern. See Pattern finding
Flexible interview, 262-264
organized systems, 263
strengths and weaknesses, 263-264
Florida, 230, 232
Formal education of workforce
academic content, 315
associate degree programs, 303, 304, 342
bachelor’s degree programs, 304-305, 342
characteristics of high-quality experiences, 301-302
committee conclusions, 341-343
context for delivery, 302-305
coursework, 316-317, 341
credentialing and licensure, 4, 101, 291, 292, 301, 305, 315, 317-319, 320, 342, 347, 352
definitions, 300
degree requirements, 315
faculty, 317
field experiences, 317
general landscape of, 303-304
in-service, 3-4, 300, 301-302, 303, 305, 307, 308, 310-314, 319, 320-321, 342, 343, 346-347, 353
and mathematics education for young children, 307-319
outcomes of mathematics preparation, 313-314
pre-service, 300, 301-302, 303, 307, 308, 309, 314-316, 317-318, 319, 320-321, 342, 343, 346, 355
and quality of teaching, 101, 305-307, 313
recommendations, 3-4
Formative assessment
committee conclusions, 340
defined, 255, 353
and effective teaching and, 274, 275, 340, 353
and feedback, 244
flexible interview, 262-264
observation, 257-260
rationale for, 227, 255-256
research needs, 265
tasks, 201, 203, 260-262
Foundational content
cardinality, 22, 23, 25-26, 31, 32, 34, 36, 53, 336
committee conclusions, 335-337
geometry/measurement content, 35-42, 336-337
mathematical connections, 48-55, 337
mathematical process goals, 42-47, 337
number content, 22-35, 336-337
pattern finding, 46-47, 48, 49, 122, 125, 241, 266, 337
Fractions, 7, 35, 36-37, 40, 46, 52-53, 61
Freudenthal, Hans, 175
G
Gender differences
and equity in education, 182
spatial performance, 74, 76, 182
Geometry.
See also Shapes and objects;
Space;
Spatial thinking
achievable and foundational thinking, 184-185
computer use, 196
defined, 353
development of, 71-79
foundational content, 37-42, 184-185
importance, 2, 124-125, 175
instruction practices, 240-241
instructional strategies, 191-196
manipulatives, use of, 195, 196
pictures and diagrams, 195-196
preschool curriculum, 184
standards, 230-232
teacher’s knowledge of, 37
teaching/learning paths, 184-196
vocabulary, 75-76
Georgia, 15, 230, 232-233
Good Start, Grow Smart, 9, 16, 303
Grade 1
cardinality, 147, 148
counting fluency, 163-164
decimal system, 32, 148
number core teaching-learning path, 130, 147-148
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relations/operations teaching-learning path, 32, 130, 151, 152-153, 163-165
Graphs/graphing, 54-55, 161, 168, 208
H
Head Start, 9, 13, 14, 15, 83, 96, 105, 229, 251, 268, 271, 273, 275, 290, 291, 292, 296, 303, 304, 337, 346
Head Start Act, 15
Head Start Family, 347-348
Head Start Impact Study, 271
Heads-to-Toes task, 83-84
High/Scope, 256, 268, 269
I
Individuals with Disabilities Education Act, 229
Infants
accumulator mechanism, 63, 351
addition/subtraction transformation, 60, 65, 66
cardinality concepts, 63
large set size discrimination, 60, 63
measurement, 61-62, 66, 79, 82
number concept, 60, 63-64, 66, 95
numerical order relations, 65-66
small set size discrimination, 60, 61-63, 66
spatial thinking, 72-73, 75, 78
Informal knowledge of young children, 299
Inhibitory control skills, 83, 84
Instruction/pedagogy.
See also Mathematics instruction
defined, 226, 353
direct, 226, 289, 352
explicit, 104, 110, 226, 352
Instructional supports.
See also Feedback
concept development, 244
defined, 353
effectiveness, 147, 243-245
language modeling, 237, 353
preschool and kindergarten, 147, 237
scaffolding, 182, 191, 203, 238, 243, 244, 247, 249-250, 251, 255, 260, 275, 355
Intelligence, 83, 84, 250, 266
Intentional teaching
approaches, 267, 340
benefit to children of low socioeconomic status, 12-13
defined, 226, 353
effectiveness, 194, 227, 237, 273, 340
and formative assessment, 227
mathematizing play experiences, 104, 250
professional development needs, 3-4, 342, 346-347
K
Kindergarten
cardinality, 145, 146, 151
classroom activities, 234-235
classroom organization and management, 236-237
compositions/ decompositions, 160-161, 178-179, 187, 191, 207
counting correspondences, 136, 144, 145, 146-147
emotional support, 236
equation writing, 162-163
instructional supports, 147, 237
math instruction, 239-242
measurement, 161, 202, 207
money understanding, 147
number core teaching-learning path, 130, 136, 145-147
number word list, 146-147, 160
operations teaching-learning path, 130, 151, 154, 159-163
relations teaching-learning path, 130, 151, 161
shapes, 178-179, 187, 190
spatial relations, 178-179, 187, 191
standards for learning, 147, 230-233
teacher-child interactions, 235-237
teen numbers, 146-147
vocabulary, 190
written number symbols, 138, 145, 146-147
L
Language differences
and base-10 structure in problem solving, 106, 109-110
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Chinese, 106-107, 108, 109, 110, 143
counting, 27, 106, 107-109
French, 110
Japanese, 108, 110, 133
Korean, 75, 108, 110, 247
and mathematical learning, 12, 98
morphological markers, 106, 109, 354
number names, 27, 106-107
representations for numbers, 106-107
Spanish, 100, 101, 106, 107
and spatial mathematical skills, 75
Swedish, 110
Language (mathematical) modeling, 237, 353.
See also Math talk
Learning disabilities
characteristics, 111-112
developmental dyscalculia, 112
helping high-risk children, 112-113
prevalence, 110
visual/spatial deficits, 209-210
Learning trajectories. See Teaching-learning paths
Legislation, 15-16
Low-income. See Economically disadvantaged children
M
Manipulatives
classroom practices, 239, 240-241, 248, 251-252
counting, 240, 252, 261
cultural tools, 203
defined, 353
effectiveness, 252
in family settings, 343
geometric, 179, 180, 183, 191, 194, 195, 240, 248, 356, 357
measurement, 199, 203, 207, 208, 209, 240-241
spatial visualization and imagery, 183, 203
virtual, 196, 248, 253, 356-357
Map use, 74-75, 82
Math talk, 43, 44, 104, 157, 245-246, 249, 340, 343
Mathematical connections. See Connections across content
Mathematical process goals
communicating, 42, 43
connecting, 42, 43
math talk, 43, 44
mathematizing, 43-44, 54, 134, 149, 150-151, 165, 181, 196, 250, 256, 257, 274, 334, 337, 354, 355
pattern finding, 43
problem solving, 42, 43-44, 103, 108, 109
reasoning, 42-44
representing, 21, 35, 42-43, 44, 332, 337
specific reasoning processes, 44-47
standards, 42, 232-233
Mathematics curriculum
automatized knowledge, 83, 100
committee conclusions, 338-340
effectiveness research, 2, 265-266, 269-271
embedded (secondary), 2, 227, 238, 248, 269, 275, 316, 339, 355
executive function skills, 83
focused time, 2, 124, 227, 238, 248, 269, 271, 273, 275, 313, 339, 340, 352, 355
guiding principles, 273-274
integrated, 2, 129, 130, 145, 146, 147, 148, 149, 166, 198, 227, 238, 239, 256, 267-268, 269, 339
interventions for economically disadvantaged children, 271-273
NAEYC and NCTM position statement, 122-123
NCTM standards, 122
resources, 267-268
Mathematics instruction.
See also Classroom context for instruction;
Curriculum;
Instructional supports
book reading, 254, 339
committee conclusions, 338-340
computers, 252-254
concrete materials and manipulatives, 196, 251-252
developmentally appropriate practice, 9, 254-255, 268, 270, 296, 298-299, 300
direct, 226, 297
effective strategies, 191-195, 242-254, 272, 339-340
explicit, 104, 110, 169, 208, 226
grouping strategy, 247, 248
guiding principles, 273-274
instructional supports, 243-245
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kindergarten practices, 239-242
math talk, 157, 245-246, 249
movement to engage children, 253-254
opportunities to learn, 1-2
pattern finding, 241, 242, 246, 248, 249, 260, 261
play, 104, 247-251, 339
preschool practices, 237-239, 242, 299-300
teacher-guided, 225-226, 345, 356
teacher-initiated learning experiences, 225-226, 356
teacher pedagogical beliefs and knowledge, 299-300
Mathematizing, 43-44, 54, 134, 149, 150-151, 165, 181, 196, 250, 256, 257, 274, 334, 337, 354, 355
Measurement
achievable and foundational goals, 201, 202, 205
accumulation of distance, 360
addition/subtraction, 80
additivity, 360
area, 35, 36, 37, 39, 46, 49, 53, 61-62, 65, 66, 79, 177, 179, 190, 201-204, 205, 207, 209-210, 354, 361
cardinality and, 36, 53
cognitive foundations, 79-82
compositions and decompositions, 35, 39-40, 46, 202, 206, 207, 360
concepts of, 359-361
conservation of length, 359
conventional, 80-82
counting and, 81, 198, 204
and data analysis, 53-55
defined, 354
development of, 79-82, 196-197
equal partitioning, 46, 198, 201, 204, 206, 359-360, 361
equivalence/nonequivalence, 80-81
foundational content, 36-37, 198
and fractions, 36-37, 52-53
importance, 2, 124-125
infants, 61-62, 66, 79, 82
instructional strategies, 80-82, 199-201, 207-210
language of, 205-206, 207-208
length, 36, 37-38, 45, 46, 50, 52-53, 55, 61-62, 66, 79, 80-81, 197-201, 207, 209, 359
manipulatives, 203-204
and number, 52-63
number lines, 168
number-measurement relation, 360-361
objects and spatial relations, 205-206, 207
origin, 360
Piaget’s theories, 79-80, 199, 200
proportional reasoning, 75, 82
relating parts and wholes, 202, 206, 207
relative coding, 46, 79
ruler use, 53, 80, 81, 167, 168, 197, 198, 199-200, 201, 202, 207, 208, 209, 240-241, 267, 360, 361
scaling, 75, 82, 182
socioeconomic differences, 96
and spatial structuring, 82, 189, 190, 201-204, 209, 361
standards, 230-232
teaching-learning paths, 205-210
thinking about parts, 202, 206, 207
transitivity, 197, 208, 358, 359, 361
units and unit iteration, 36, 37, 45, 46, 53, 80-82, 161, 197, 198, 199, 204, 206, 207, 208-209, 360
visual/holistic thinking, 202, 206
visual/spatial deficits and, 209-210
volume, 35, 36-37, 39, 40, 50, 63, 71, 79, 190, 204-205, 210, 354, 361
zero point, 200, 208
Memory
deficits, 112
faulty, 256
working, 67, 83, 84, 361
Michigan, 230
Mississippi, 15
Multiplication/division, 30, 34, 45, 46, 48, 50, 51-52, 55, 190, 203
Multiplicative comparisons, 34
Multi-State Study of Preschool, 233-237, 239
N
National Assessment of Educational Progress, 100
National Association for the Education of Young Children (NAEYC), 9, 122, 228, 268, 275, 296, 307-308, 309, 318, 319
National Board for Professional Teaching Standards, 318-319
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National Center for Early Development and Learning, 233-237, 239, 268
National Council for Accreditation of Teacher Education, 314, 315, 316, 318
National Council of Teachers of Mathematics (NCTM), 9, 42, 55, 122-123, 228, 230, 239, 241, 307-308, 309, 310, 335
National Education Goals of 1990, 15
National Institute of Child Health and Human Development, 10
National Reporting System, 9
New Jersey, 230, 242
New Mexico, 229
New York, 230, 233
No Child Left Behind (NCLB) Act, 15, 16, 303
North Carolina, 230, 232, 294
Number competencies, 95, 96, 112, 350, 354
Number content.
See also Number core;
Operations core;
Relations core
importance, 2, 124-125
instruction practices, 239-240
standards, 230-232
Number core
cardinality, 22, 23, 25-26, 34, 129, 132-134, 140, 142, 145, 146, 147, 148
composing/decomposing, 23, 32-33, 45, 48, 49-50, 154, 157, 161, 164-165, 194, 231
coordinating components of, 23, 25, 52-53, 131, 138-140
counting correspondences, 22, 23, 24-25, 26, 129, 134-137, 141, 144, 145, 146, 147, 148
counting out n things, 141, 145
and data analysis, 53-55
decimal system, 22, 24, 26-30, 48
foundational content, 22-30
importance, 124-125
integrating components of, 145-148
mathematical perspective, 24-30
number word list, 22, 23, 24, 25-30, 132, 134, 141, 142-143, 144, 145, 146
overview, 22-30
quantity, 22, 24
subitizing, 23, 25, 130, 131, 141, 146
teaching-learning paths, 129-148
teen numbers, 146-147
written number symbols, 22, 23, 24, 26-30, 129, 138, 141, 144-145, 146, 147, 148
Number facts, 25
Number lines, 53, 111, 113, 167-168, 251
Number names. See Number word list
Number representation.
See also Written number symbols
analog magnitude system, 63, 70, 112, 168, 351
approximate, 112
decimal system and, 27, 109-110
fractions, 61
mental, in preschool children, 66-71, 138
object file system, 62, 63, 66, 70, 354
Number sense.
See also Number understanding or number sense
defined, 95, 354
Number triads
composed/decomposed, 25, 32-33, 158-159, 160-161
embedded, 23, 159
Number understanding or number sense, 142, 230, 231, 354.
See also Competence and performance;
Number competencies
addition/subtraction transformations, 67, 96
approximation, 61, 63, 66, 68, 70, 84, 98, 112, 351
conceptual, 63-64
infants, 60, 61-66, 95
kindergarten, 96
learning disabilities and, 112-113
mental number representation, 66-71
order relations, 65-66
preschool children, 66-71, 95
preverbal, 60-66
set comparisons, 96
set size discrimination, 60, 61-66, 67-70, 96, 98
Number word list
activities, 136-137
and addition/subtraction situations, 25, 160, 166-168
and cardinality, 25-26, 31, 32, 129, 132, 147, 148
core content, 24-30, 134, 142-143
decade words, 147
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importance of fluency, 25, 66
language/cultural differences, 27, 70, 106-107, 142-143
number line distinguished from, 167-168
and one-to-one counting correspondence, 134, 138, 144
parental modeling of, 104
pattern finding in, 47, 107, 108, 131, 134, 141, 142-143, 144, 146, 147, 148
as representational tool, 160, 166-168
socioeconomic differences in understanding, 97-98
teaching-learning paths, 129, 132, 134, 138, 141, 142-143, 144, 145, 146
unitizing, 166
verbal knowledge, 140
and written symbols in decimal system, 26-30, 64
Number Worlds, 256, 263, 271
Numerals.
See also Written number symbols
Arabic, 106, 107, 108, 109
defined, 27, 354
Numerosity, 23, 61, 62, 63, 64, 65, 71, 134, 156
defined, 354
matching task, 67-68, 69, 97-98
O
Object file system, 62, 63, 66, 70, 354
Observational assessment, 257-260
organized systems of, 259
strengths and weaknesses, 259-260
Ohio, 230, 233
Oklahoma, 15
One-to-one counting correspondences
accuracy, 135
to cardinality, 22, 24-25, 129, 134-137, 144
defined, 354
number word list and, 134, 138
set size and, 135
teaching-learning paths, 129, 132, 134-137, 138-139, 141, 144, 146, 147, 148
Operations core.
See also Addition/subtraction;
Multiplication/division
change situations, 23, 32-34, 149
comparison situations, 23, 30-31, 33, 151
compose/decompose numbers, 23, 25, 32-33, 158-159, 160-161
computational fluency, 111
connections across core, 34
counting on, 25, 111, 113, 130, 151, 152-154, 159, 160, 163, 164, 166
embedded number triads, 23, 33, 158-159, 166
importance, 124-125
mathematizing real-word situations, 149, 150-151, 165
number list fluency and, 25
overview, 23
put together/take apart situations, 23, 32-33, 34-35, 149, 355
socioeconomic differences, 154, 159
standards, 230-232
subidizing, 130, 150, 151, 152, 154, 155, 158, 159
teaching-learning paths, 130, 149, 150-154, 155, 157-161, 162-165, 166
Order relations, 65-66, 149, 156, 166, 167.
See also Relating and ordering
Ordinal numbers, 167
Ordinal position, 63
Organizing information, 46-47
P
Parents.
See also Family role
engagement in mathematics activities, 102-104
knowledge and beliefs about mathematics, 8, 102, 105, 344
perceptions of children’s abilities, 104
spatial activities and language, 76, 78, 99
Pattern finding.
See also Subitizing activities, 46-47, 102, 156, 189-190, 194, 196, 248
in addition/subtraction, 159-160
attribute blocks, 47, 183-184, 209-210, 248, 351
cognitive foundations, 69, 83
computer manipulatives, 196
conceptual subitizing, 132-133, 142, 153, 155, 158, 356
counting, 47, 49, 107, 156, 164
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defined, 352
foundational content, 46-47, 48, 49, 122, 125, 241, 266, 337
informal knowledge of young children, 299
instruction practices, 241, 242, 246, 248, 249, 260, 261
in multiplication tables, 52
in number words, 47, 107, 108, 131, 134, 141, 142-143, 144, 146, 147, 148
reasoning processes, 43, 46-47
shapes, 47, 176, 177-179, 183-184, 189-190, 191, 194, 196, 351, 355, 357
spatial visualization, 249, 357
standards of learning, 238 n.3
structural relationships, 47, 48, 52
in teaching-learning paths, 122, 125, 131, 132-133, 134, 141, 142-143, 144, 146, 147, 148, 153, 155, 156, 158, 159-160, 164, 176, 177-179, 183-184, 189-190, 191, 194, 196
unitizing, 45, 48
Perceptual subidizing, 131, 132, 164, 166, 356
Performance in mathematics. See Competence and performance
Piaget’s theories and perspectives, 60, 61, 67, 70, 72, 76, 77, 79-80, 155, 156, 166, 183, 199, 200, 255-256, 257, 258, 260, 261-262, 264, 298, 359, 361
Place values, 28, 29-30, 45, 48, 107, 108, 148, 231, 355
Play
block building, 248-249
games, 251
mathematical, 149, 251
practice during, 250
sociodramatic, 249-250
Policies. See Education policies
Pre-K Mathematics, 311
Preschool children.
See also Age 2-3;
Age 4
attitudes about mathematics, 12
calculation tasks, 67-68, 69
map use, 74-75, 82
mathematics curricula, 184
mental number representation, 66-71
numerosity matching task, 67-68, 69, 97-98
set size discrimination, 67-70
shape discrimination, 77
spatial thinking, 73-75
Preschool Curriculum Evaluation Research Study, 269, 270
Preschool Embedded Figures Test, 73-74
Preschool Mathematics Curriculum, 270
Preschool programs
classroom activities, 234-235
classroom organization and management, 236-237
economically disadvantaged children targeted, 15
emotional support, 236
equity in, 124
instructional supports, 237
math instruction, 124-125, 237-239, 242
standards of learning, 228-230
state-funded, 15, 228-230, 233-242
teacher-child interactions, 235-237
voluntary universal, 15
Problem solving, 43, 44, 46, 51, 100, 109-110, 122, 133, 142, 159, 165, 166, 169, 194-195, 228, 229, 239, 247, 251, 252, 255, 273, 274, 297, 299, 337
Process. See Mathematical process goals
Professional development of workforce.
See also Formal education of workforce
access, 303
assessing competence, 315-316
defined, 355
inadequacy in mathematics, 2
preparing teachers to promote mathematics development, 316-319
training, 2, 258-259, 270, 291, 292, 294-295, 298, 300, 301, 302, 303, 305, 306-307, 309, 310, 311, 312, 313, 314, 318, 341, 342, 343, 348, 356
Project IMPACT, 314
Q
Quantizing, 24
auditory, 133
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R
Race/ethnicity
and competence and performance, 99-100
ECE workforce, 14
Reading First, 16
Reasoning processes
“big ideas,” 44-45
composing/decomposing, 35, 40, 45-46, 179
and conceptual development, 244
general, 40, 42-44, 184
geometric and spatial, 36, 76-77, 83, 188, 206, 230, 231, 337
goals, 42-44, 122, 337, 345
NCTM process standards, 42
organizing information, 46, 47
pattern finding, 43, 46-47
prompting for, 245, 256, 355
proportional thinking, 82
relating and ordering, 46, 83
representing, 42-43, 44
specific, 40, 44-47
standards of learning, 229, 230, 231, 233
teachers’ pedagogical beliefs, 297
teaching-learning paths, 184, 197, 199, 206
transitive, 206
unitizing, 45
Recommendations
credentialing of educators (7), 4, 347
curricula and instruction (3), 3, 345
family-community partnerships (8), 4, 347-348
informal learning supports (9), 4, 348
instructional supports (5), 3, 346
math experiences in early childhood (2), 3, 345
national initiative (1), 3, 345
professional development (6), 3-4, 346-347
standards of learning (4), 3, 346
Reggio Emilia, 259, 267
Relating and ordering, 23, 37-38, 46, 61, 97, 337, 355
Relating parts and wholes, 178, 179, 185, 187, 188, 189-190, 191, 193, 202, 206, 207, 355
Relations core
comparison situations, 30-31, 33, 34, 149, 154-155, 161-162
length or density strategies, 23, 130, 149, 150, 154, 155, 156
matching and counting strategies, 155, 156, 161
overview, 23
socioeconomic differences, 97
subidizing, 150, 151, 154, 166
teaching-learning path, 130, 149, 150, 151, 154-157, 161-162, 165, 166-168
vocabulary, 156-157, 161
Representing relationships, 110, 165, 177, 187, 189, 202, 206, 267, 332, 337.
See Number representation;
Numerals;
Written number symbols
data analysis, 54-55
mathematizing, 43, 44
in measurement, 53, 55
as reasoning process, 42-43, 44
Research needs
curriculum evaluation, 349
English language learner needs, 348
instruction practices, 348-349
learning disabilities, 349-350
parental involvement, 349
teacher preparation, 349
Research on mathematics learning
curriculum effectiveness, 265-273
evidence base, 12
NRC reports on early development and learning, 9-10
Rightstart program, 271, 273
Rod-and-Frame Test, 73
S
Scaffolding, 182, 191, 203, 238, 243, 244, 247, 249-250, 251, 255, 260, 275, 355
Self-regulation, 83-84, 250, 264
Set size discrimination, 60-63
Shapes and objects.
See also Geometry
2-D, 37-38, 177-179, 185, 188, 190
3-D, 38-39, 40, 184, 185, 188, 190
as abstractions, 35-36
activities, 193-195
area of, 35, 39-40, 49
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categorization, 180
characteristics, 35, 37-39, 180, 192-193
cognitive foundations, 76-78, 179-180
composing/decomposing, 35, 39-40, 46, 48-50, 52, 175, 177-179, 184, 185-188, 189-190, 191, 194, 195, 206, 231, 248, 336
concept development, 179-180, 192-193
dimensional adjectives, 78
discrimination, 76-79
faces on, 39
manipulatives, 180, 183-184
pattern finding, 47, 176, 177-179, 183-184, 185, 189-190, 191, 194, 196, 351, 355, 357
relating parts and wholes, 176, 178, 179, 185, 187, 188, 189-190, 191, 193, 355
steps in thinking about, 176-179
teaching-learning paths, 185, 186-187, 188, 190, 191-195
thinking about parts, 164, 176, 177, 178, 179, 185, 186, 187, 188, 189, 193, 356
vertices, 38, 39, 178, 188
visual/holistic thinking, 176, 177, 178, 185-189, 357
Socioeconomic status.
See also Economically disadvantaged children
and cardinality tasks, 103
and child care arrangements, 14
and learning opportunities, 104-105
and number understanding, 70
and spatial performance, 74
and teachers’ values and beliefs about instruction, 296-299
Space
2-D, 40-41, 177-179
3-D, 40-41, 50
Cartesian coordinates, 35, 49, 50
models for, 41
movement through, 41-42, 52, 178, 183, 189, 248, 252
Spatial language, 75-76, 77-78, 99, 181, 185
Spatial orientation, 41, 73, 74, 75, 138, 177, 178, 180-182, 188, 189, 192, 194, 336, 355, 357
Spatial relations, 185, 188-189, 191
in 2-D space, 177-179
categorization, 72-73, 75, 77
foundational content, 3, 35, 41-42
vocabulary, 77, 185, 186
Spatial structuring, 35, 40-42, 50, 180-184, 189, 190, 201-204, 209, 361
Spatial thinking.
See also Geometry;
Shapes and objects
activities and tests, 72, 73-74, 76, 77, 99, 183-184
cognitive foundations, 71-79, 180-182
foundational content, 42-43
gender differences, 74, 76, 98-99, 182
importance, 175
infants, 72-73, 75, 78
language (spatial) use and, 75-76, 77-78, 99, 181
manipulatives, 183
and map use, 74-75, 182
and math and science achievement, 71
mental functions in, 71
mental representations, 181-182
mental transformation and superposition of shapes, 42-43, 73-75, 78, 99, 103, 183, 203, 35
modeling, 182
perspective changes, 35, 72, 74, 182
Piaget’s view, 72
preschool, 75-79
socioeconomic differences, 74, 96
symbolic systems, 72
teaching-learning paths, 185, 186-187, 188-189, 191
visualization and imagery, 39, 73-74, 180, 183-184, 230, 231, 249, 336, 355, 357
SRA Real Math Building Blocks, 311
Standards for mathematics learning
committee conclusions, 338-339
Curriculum Focal Points, 55, 123, 125, 228, 229, 230, 335
early learning (prekindergarten), 9, 228-230
kindergarten, 230-233
monitoring implementation, 229
Principles and Standards for School Mathematics, 122, 123, 228, 229, 231, 233
process strands, 232-233
State-wide Early Education Programs (SWEEP) Study, 234
Storytelling Sagas, 254
Structural relationships, 47, 48, 52
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Subitizing
auditory, 133, 164
and cardinality, 132, 134, 141, 142, 146, 150, 151, 155
conceptual, 23, 132-133, 141, 142, 146, 150, 151, 152, 153, 154, 155, 158, 159, 164, 166, 356
for counting on, 164
defined, 25, 132, 356
neurological origins, 112
in number core, 23, 25, 130, 131, 141, 146
in operations core, 130, 150, 151, 152, 154, 155, 158, 159
ordinal numbers, 167
perceptual, 131, 132, 164, 166, 356
range, 63
in relations core, 150, 154, 166
teaching-learning paths, 130, 131, 132-134, 141, 142, 146, 150, 151, 152, 153, 154, 155, 158, 159, 164, 166
Symmetry, 51, 52, 134, 179, 180, 185, 195, 196, 231
T
Tangrams, 179, 183, 191, 194, 248, 356, 357
Tasks
assessment strategy, 260-262
organized systems of, 261
strengths and weaknesses of, 261-262
Teacher-child interactions
classroom organization and management, 236-237
emotional support, 236
preschool and kindergarten, 235-237
Teacher Education Assistance for College and Higher Education Grant Program, 303
Teacher-initiated experiences
defined, 225-226
Teachers. See Early childhood education teachers
Teaching-learning paths.
See also Ages 2-3;
Age 4;
Grade 1
cardinality, 129, 130, 131, 132-134, 135, 139-140, 141, 142, 145, 146, 147, 148, 150, 151, 154, 155, 169
composition/decomposition, 32, 154, 157, 161, 164-165, 175, 177-179, 184, 185-188, 189-190, 191, 194, 202, 206, 207
connections among core component, 23, 25-26, 34, 39-40, 45-46, 48-55, 129, 130, 131, 138-140, 337
context for, 129
counting on to make a solution, 145, 148, 151, 152-153, 154, 155, 156, 157-158, 159-160, 163-165, 166
counting out n things, 141, 145
defined, 121, 353-354
equation writing, 158-159, 162-163
integrating components of, 145, 146-148
number core, 129-148
number word list, 129, 132, 134, 138, 141, 142-143, 144, 145, 146
one-to-one counting correspondences, 129, 132, 134-137, 138-139, 141, 144, 146, 147, 148
operations core, 130, 149, 150-154, 155, 157-161, 162-165, 166
overlearning, 128, 138, 145
pattern finding, 122, 125, 131, 132-133, 134, 141, 142-143, 144, 146, 147, 148, 153, 155, 156, 158, 159-160, 164, 176, 177-179, 183-184, 189-190, 191, 194, 196
problem-solving strategies, 150-154, 168-169
relations core, 130, 149, 150, 151, 154-157, 161-162, 165, 166-168
shapes and objects, 185, 186-187, 188, 190, 191-195
spatial relations, 185, 186-187, 188-189, 191
subitizing, 130, 131, 132-134, 141, 142, 146, 150, 151, 152, 153, 154, 155, 158, 159, 164, 166
teen numbers, 146-147
written number symbols, 53, 70, 112-113, 129, 138, 141, 144-145, 146-147, 148
Teaching to the Big Ideas, 314
Technology-enhanced, Research-based, Instruction, Assessment, and professional Development (TRIAD) model, 311, 312
Test of Early Mathematics Ability, 273
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Texas, 230, 233
Tools of the Mind, 249-250
U
Unitizing, 337
in counting, 23, 45, 48, 147, 148
decimal system, 45, 48
defined, 356
by grouping, 48-49, 207
in measurement, 36, 37, 45, 46, 53, 80-82
number words, 166
and pattern finding, 45, 48
U.S. Department of Education, 269
U.S. Department of Health and Human Services, 10
Office of Head Start, 9, 10
V
Virginia, 230, 232
Visual/spatial deficits and, 209-210
Volume, 35, 36-37, 39, 40, 50, 63, 71, 79, 190, 204-205, 210, 354, 361
Vygotsky’s theories, 200, 203, 249-250, 255, 260
W
Washington, DC, 15
What Works Clearinghouse, 312, 313
Workforce. See Early childhood workforce
WPPSI-3 Block Design subtest, 78
Written number symbols
errors in writing, 147
number words and, 26-30, 64
place value cards, 148
teaching-learning paths, 53, 70, 112-113, 129, 138, 141, 144-145, 146-147, 148
