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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Index A language differences and, 101, 149, 157 levels in solution methods, 150-154, 160 Accumulator mechanism, 63, 351 make-a-10 method, 51, 151, 153, 154, Addition/subtraction 160, 165, 166 acting out solutions, 157 mathematizing real-world situations, associative properties of addition, 51 165 cardinality and, 130, 150, 154 measurement and, 208 change plus/change minus situations, 23, number word list and, 25, 160, 166-168 32, 33-34, 149, 150, 151, 155, 162- oral problems, 152, 154, 159 163, 165, 351 pattern finding in, 159-160 commutative properties of addition, 50- preschool children, 67-68, 69-70, 71 51, 160 put together/take apart, 23, 32-33, 34- comparison situations, 149, 151, 161- 35, 148, 150, 151, 155, 158, 160, 162, 165 162, 355, 356 composing/decomposing, 23, 32-33, relations and, 31, 149 45-46, 51, 133, 158-159, 160-161, teaching-learning path, 130 164-165, 231 variability in solution methods, 168-169 counting on, 25, 51, 111, 130, 145, 148, vocabulary learning, 149, 157 151, 152-153, 154, 155, 157-158, word problems and situations, 32-35, 159-160, 163-164, 166 101, 149, 151-152, 155, 157, 159 decontextualized problems, 158 Additive comparison situations, 23, 33, 34, derived fact method, 151, 153, 154, 231, 351 160-161, 165, 166 Ages 2-3 direct modeling, 151-152, 160 addition/subtraction teaching-learning drawing solutions, 157, 159 path, 130, 154, 155 economically disadvantaged children, cardinality, 130, 131, 132-134, 139-140, 152, 159 150, 154 equation writing, 162-163 compositions and decompositions, 177, fluency vs. recalled/memorized facts, 166 185, 186, 187-188, 206 infants, 65, 67 1

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2 INDEX coordinating components, 131, 138-140 Attention, 84 counting correspondences, 131, 134- Attitudes and beliefs about mathematics 137, 138-139 ECE teachers, 8-9, 297-300, 309 measurement, 202, 205-206 parents, 8, 102, 344 number core teaching-learning path, preschool children, 12 130, 131-140 Attribute blocks, 47, 183-184, 209-210, number word list, 131, 134, 138 248, 351 operations teaching-learning path, 150, Automaticity, 128 155 relations teaching-learning path, 130, B 150, 154-155 shapes and objects, 177, 185, 186 Behavioral and attention regulation, 82-84, spatial relations, 177, 185, 186, 205-206 335 vocabulary, 154 Big Math for Little Kids, 256, 263, 271, written number symbols, 131, 138 309 Age 4 (prekindergarten). See also Preschool Bright Beginnings, 269 children Building Blocks curriculum, 248, 256, 270, cardinality, 141, 142, 150, 155, 156 271, 311, 312 compositions/ decompositions, 178, 186- Bureau of Labor Statistics, 291-292 187, 189-190, 206 counting correspondences, 136, 141, 144 C counting out n things, 141, 145, 156, 157 Calendar activities, 241-242, 275 matching, 155, 156 California, 230, 233, 290, 293, 310 measurement, 202, 206 Cardinality number core teaching-learning path, and addition/subtraction operations, 130, 141-145 130, 150, 154 number word list, 141, 142-143, 144, ages 2-3, 130, 131, 132-134, 154 145 cognitive foundations, 63, 71 operations teaching-learning path, 130, coordinations and, 23, 25-26, 130, 131, 150-151, 154, 155, 157-159 139 relations teaching-learning path, 130, counting and, 129, 130, 131, 135, 139, 150, 155-157 140, 147 shapes, 177-178, 186-187, 188 defined, 22, 23, 139, 333-334, 336, 351 spatial relations, 177-178, 186-187, and equivalence classes, 63, 71 188-189 foundational content, 22, 23, 25-26, 31, vocabulary, 155, 156-157, 188-189 32, 34, 36, 53, 336 written number symbols, 138, 141, games, 53 144-145 grade 1, 147, 148 Age 5. See Kindergarten infant concepts, 63 Algebraic problem solving, 165 instruction practices, 245-246, 300, Analog magnitude system, 63, 70, 112, 168, 333-334 351 kindergarten, 146 Area, 35, 36, 37, 39, 46, 49, 53, 61-62, 65, and measurement, 36, 53 66, 79, 177, 179, 190, 201-204, 205, number word list and, 25-26, 31, 32, 207, 209-210, 354, 361 129, 132, 147, 148 Arithmetic properties, 50-52 one-to-one correspondence and, 22, 24, Assessment practices. See also Formative 129 assessment parents’ perceptions of children’s multiple-choice tests, 100 abilities, 104

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 INDEX prekindergarten, 142 universal starting points, 59-60, 85, 95, principle, 135 96, 104 socioeconomic variation on tasks of, 103 van Heile’s theories, 77 standards of learning, 230, 231 Cognitively Guided Instruction, 314 subitizing and, 132, 134, 141, 142, 146, Colorado, 229 150, 151, 155 Comparisons, 30-31, 53, 97 teaching-learning paths, 129, 130, 131, additive, 23, 33, 34, 231, 351 132-134, 135, 139, 140, 141, 142, Competence and performance. See also 147, 148, 150, 169 Cognitive foundations for early truly operational cardinal number, 166 learning; Learning disabilities; Charge to committee, 1, 10-13 Number understanding or number Child Care Development Fund, 303 sense Child Development Associate, 292, 303, activities in daily lives and, 100, 102- 318 104, 113 Child-initiated experiences, 226, 227, 247- automaticity, 128 248, 250, 298, 352 committee conclusions, 332-335 Chutes and Ladders, 238 n.3, 251 English language learners, 100-101 CLASS framework, 235-236 experiences and learning opportunities Classroom Assessment Scoring System, 234, and, 100, 103-105, 333-334 236, 237, 243-244, 299 family role, 101-105, 113, 333-334 Classroom context for instruction. See also fluency, 7, 25, 111, 125, 128, 138, 141, Grade 1; Kindergarten; Mathematics 145, 153, 154, 156, 157, 158, 159, instruction; Preschool programs 160, 166, 169, 250, 352 emotional support, 236 gender and, 98-99 generalized teaching strategies, 241-242 individual variations in, 127-128, 154, instructional opportunities, 235 157, 334-335 instructional supports, 237 international comparisons, 184, 192, kindergarten, 239-242 273 mathematics content, 235, 237-242 language/cultural influences, 98, 106-110 noninstructional activities and time, 235 parents’ knowledge and beliefs and, 102 organization and management, 236-237 race/ethnicity and, 99-100 prekindergarten, 237-239 socioeconomic status and, 1-2, 70, 96- teacher-child interactions, 235-237 98, 103, 104-105, 334-335 Classroom Observation of Early Composing/decomposing Mathematics Environment and activities, 248 Teaching, 270-271, 272 addition/subtraction situations, 23, 32- Cognitive flexibility, 83 33, 45-46, 51, 133, 231 Cognitive foundations for early learning decimal system, 45-46 behavioral and attentional regulation defined, 352 and, 82-84, 85 groups of groups, 49 cardinality, 63, 71 groups of groups of groups, 50 fraction representations, 61 and mathematical connections across measurement, 61, 79-82, 84-85 content areas, 39-40, 45-46, 48-49, number understanding, 60-71, 84 50, 194 pattern finding, 69, 83 and measurement, 35, 39-40, 46, 202, Piaget’s theories, 60, 70, 72, 79-80, 206, 207, 360 155-156 and multiplication/division, 45, 51-52 preverbal number knowledge, 60-66 numbers, 23, 32-33, 45, 48, 49-50, 154, proportional relations, 61, 82 157, 161, 164-165, 194, 231 spatial thinking and geometry, 61, 71- operations, 45 79, 84, 192-193 reasoning processes, 35, 45-46, 179

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 INDEX shapes and objects, 35, 39-40, 46, 48- importance of fluency, 156 50, 175, 177-179, 184, 185-188, instruction practices, 240 189-190, 191, 194, 195, 206, 231, language and, 30, 107-109, 140 248 learning disabilities and, 111, 112 standards of learning, 231, 337 money, 147 teaching-learning paths, 32, 154, 157, nonverbal, 63 161, 164-165, 175, 177-179, 184, one-to-one correspondence, 22, 24-25, 185-188, 189-190, 191, 194, 202, 129, 134-137, 144 206, 207 order irrelevance principle, 135, 136 triads, 25, 160 pattern finding and, 47, 49, 107, 156, unitizing by grouping, 48-49, 207 164 Computational fluency, 111, 352 quantizing, 22, 26, 27, 30, 51 Computer manipulatives, 196 skip counting, 25, 48, 55, 147, 148 Conceptual subidizing, 23, 132-133, 141, socioeconomic differences, 97 142, 146, 150, 151, 152, 153, 154, of solution methods, 25 155, 158, 159, 164, 166, 356 stable order principle, 135 Connections across content subidizing and, 164 arithmetic properties, 50-52 unitizing, 45, 48, 147, 148 cardinality, 23, 25-26, 130, 131, 139 what-to-count principles, 135, 136-137 composing/decomposing and, 39-40, 45- Counting on, 25, 111, 113, 130, 151, 152- 46, 48-52, 194 154, 159-160, 163, 164, 166 in data analysis, number, and Creative Curriculum, 256, 268, 269 measurement, 53-55 Credentialing and licensure, 4, 101, 291, fractions, 52-53 292, 301, 305, 315, 317-319, 320, of number core, 138-140 342, 347, 352 Core Knowledge, 256, 273 Curriculum. See also Mathematics Counting curriculum abstraction principle, 135, 136 definitions, 227 accumulator mechanism, 63, 351 effectiveness research, 265-273 addition/subtraction operations, 25, 51, integrated, 149, 227, 353 111, 130, 145, 148, 151, 152-153, 154, 155, 157-158, 159, 160, 163, D 164, 166 backward counting, 25, 26, 153 Data analysis, 53-55 cardinality principle, 31, 129, 130, 131, Decimal system 135, 139, 140, 147 groups of groups, 49 comparing collections, 30-31 groups of groups of groups, 50 competence of preschool children, 61 language differences and, 106-108, core content, 145 109-110 cultural differences, 133 number word list and, 26-30, 64 defined, 25 place values, 28, 29-30, 45, 48, 107, errors, 136-137, 153 108, 148, 231, 355 on fingers, 25, 67, 68, 70, 106, 130, unitizing, 45, 48 131, 132, 133, 134, 141, 142, 152, Developmental dyscalculia, 112 157-158, 162 Dynamic Indicators of Basic Early Literacy games, 53, 113 Skills, 128 how-to-count principles, 135

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5 INDEX E Early Head Start, 9 Early Reading First, 16 Early Childhood education (ECE) system. Economically disadvantaged children. See See Curriculum; Early childhood also Head Start education teachers; Early childhood competence in mathematics, 1, 2, 12, workforce; Grade 1; Instruction/ 152, 154 pedagogy; Kindergarten; Preschool; equity in early mathematics education, Standards for mathematics learning 12-13, 15-16 accountability systems, 16 instruction strategies, 110, 154, 271, center-based care, 13-14, 15, 66, 67, 273, 296-298 123, 127, 129, 133, 137, 141, 155, opportunities to learn mathematics, 1-2, 158, 290, 291, 292, 293, 294, 307, 13, 15 318, 337, 338, 341 Education of teachers. See Formal costs of care, 13, 15 education of workforce; Professional demand for care, 15 development of workforce home-based care, 13-14, 289, 290, 291, Education policies 292, 305, 307 accountability systems, 16 legislation and policy initiatives, 15-16 voluntary universal preschool, 15 opportunities to learn mathematics, 1-2 Effortful control, 83 state-funded preschool, 15 Emerging Academics Snapshot, 234, 238 variety of settings, 13-15 English language learners, 100-101, 149, Early childhood education teachers. See also 157 Professional development Equation writing, 33, 35, 44, 162-163 beliefs about early childhood education, Equivalence classes, 63, 64, 71 295-297 Equivalence matches, 69 defined, 289-290, 352 Even Start, 229 instruction impacts of beliefs and Ewing Marion Kauffman Foundation, 10 knowledge, 299-300, 309-310 Executive function, 83-84, 243, 250, 334 quality and effectiveness, 272, 305-307, 356 values and beliefs about math, 8-9, 297- F 300, 309 False belief understanding, 83 Early Childhood Educator Professional Family role. See also Parents Development Program, 303 activities, 76, 102-104, 155, 343 Early Childhood Environment Rating Scale, committee conclusions, 343-344 234, 237, 242, 299 math talk, 104 Early Childhood Longitudinal Study resources, 344 birth cohort, 13 socioeconomic factors, 104-105, kindergarten cohort, 98, 233-234, 237, 343-344 239, 247 Feedback Early childhood workforce. See also defined, 353 Professional development of encouragement and affirmation, 238, workforce 244, 245, 352 compensation, 291-293 importance and quality, 125, 169, 182, defined, 289, 352 237, 243-244, 299 demographic characteristics, 290-291 instructional, 137, 353 educational experience and background, loops, 244-245, 352 291 in professional development, 317, 320, and mathematics, 297-300 347 stability and turnover, 293-294 prompting thought processes, 245, 340, work environment, 294-295 355

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6 INDEX providing information, 127, 245, 248, flexible interview, 262-264 253, 299, 355 observation, 257-260 scaffolding, 182, 191, 203, 238, 243, rationale for, 227, 255-256 244, 247, 249-250, 251, 255, 260, research needs, 265 275, 355 tasks, 201, 203, 260-262 sources, 125 Foundational content training teachers in, 312 cardinality, 22, 23, 25-26, 31, 32, 34, Fidelity of Implementation, 270 36, 53, 336 Finding a pattern. See Pattern finding committee conclusions, 335-337 Flexible interview, 262-264 geometry/measurement content, 35-42, organized systems, 263 336-337 strengths and weaknesses, 263-264 mathematical connections, 48-55, 337 Florida, 230, 232 mathematical process goals, 42-47, 337 Formal education of workforce number content, 22-35, 336-337 academic content, 315 pattern finding, 46-47, 48, 49, 122, 125, associate degree programs, 303, 304, 241, 266, 337 342 Fractions, 7, 35, 36-37, 40, 46, 52-53, 61 bachelor’s degree programs, 304-305, Freudenthal, Hans, 175 342 characteristics of high-quality G experiences, 301-302 committee conclusions, 341-343 Gender differences context for delivery, 302-305 and equity in education, 182 coursework, 316-317, 341 spatial performance, 74, 76, 182 credentialing and licensure, 4, 101, 291, Geometry. See also Shapes and objects; 292, 301, 305, 315, 317-319, 320, Space; Spatial thinking 342, 347, 352 achievable and foundational thinking, definitions, 300 184-185 degree requirements, 315 computer use, 196 faculty, 317 defined, 353 field experiences, 317 development of, 71-79 general landscape of, 303-304 foundational content, 37-42, 184-185 in-service, 3-4, 300, 301-302, 303, 305, importance, 2, 124-125, 175 307, 308, 310-314, 319, 320-321, instruction practices, 240-241 342, 343, 346-347, 353 instructional strategies, 191-196 and mathematics education for young manipulatives, use of, 195, 196 children, 307-319 pictures and diagrams, 195-196 outcomes of mathematics preparation, preschool curriculum, 184 313-314 standards, 230-232 pre-service, 300, 301-302, 303, 307, teacher’s knowledge of, 37 308, 309, 314-316, 317-318, 319, teaching/learning paths, 184-196 320-321, 342, 343, 346, 355 vocabulary, 75-76 and quality of teaching, 101, 305-307, Georgia, 15, 230, 232-233 313 Good Start, Grow Smart, 9, 16, 303 recommendations, 3-4 Grade 1 Formative assessment cardinality, 147, 148 committee conclusions, 340 counting fluency, 163-164 defined, 255, 353 decimal system, 32, 148 and effective teaching and, 274, 275, number core teaching-learning path, 340, 353 130, 147-148 and feedback, 244

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 INDEX relations/operations teaching-learning Intentional teaching path, 32, 130, 151, 152-153, approaches, 267, 340 163-165 benefit to children of low socioeconomic Graphs/graphing, 54-55, 161, 168, 208 status, 12-13 defined, 226, 353 effectiveness, 194, 227, 237, 273, 340 H and formative assessment, 227 mathematizing play experiences, 104, Head Start, 9, 13, 14, 15, 83, 96, 105, 229, 250 251, 268, 271, 273, 275, 290, 291, professional development needs, 3-4, 292, 296, 303, 304, 337, 346 342, 346-347 Head Start Act, 15 Head Start Family, 347-348 Head Start Impact Study, 271 K Heads-to-Toes task, 83-84 High/Scope, 256, 268, 269 Kindergarten cardinality, 145, 146, 151 classroom activities, 234-235 I classroom organization and management, 236-237 Individuals with Disabilities Education Act, compositions/ decompositions, 160-161, 229 178-179, 187, 191, 207 Infants counting correspondences, 136, 144, accumulator mechanism, 63, 351 145, 146-147 addition/subtraction transformation, 60, emotional support, 236 65, 66 equation writing, 162-163 cardinality concepts, 63 instructional supports, 147, 237 large set size discrimination, 60, 63 math instruction, 239-242 measurement, 61-62, 66, 79, 82 measurement, 161, 202, 207 number concept, 60, 63-64, 66, 95 money understanding, 147 numerical order relations, 65-66 number core teaching-learning path, small set size discrimination, 60, 61-63, 130, 136, 145-147 66 number word list, 146-147, 160 spatial thinking, 72-73, 75, 78 operations teaching-learning path, 130, Informal knowledge of young children, 299 151, 154, 159-163 Inhibitory control skills, 83, 84 relations teaching-learning path, 130, Instruction/pedagogy. See also Mathematics 151, 161 instruction shapes, 178-179, 187, 190 defined, 226, 353 spatial relations, 178-179, 187, 191 direct, 226, 289, 352 standards for learning, 147, 230-233 explicit, 104, 110, 226, 352 teacher-child interactions, 235-237 Instructional supports. See also Feedback teen numbers, 146-147 concept development, 244 vocabulary, 190 defined, 353 written number symbols, 138, 145, effectiveness, 147, 243-245 146-147 language modeling, 237, 353 preschool and kindergarten, 147, 237 scaffolding, 182, 191, 203, 238, 243, L 244, 247, 249-250, 251, 255, 260, 275, 355 Language differences Intelligence, 83, 84, 250, 266 and base-10 structure in problem solving, 106, 109-110

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 INDEX Chinese, 106-107, 108, 109, 110, 143 math talk, 43, 44 counting, 27, 106, 107-109 mathematizing, 43-44, 54, 134, 149, French, 110 150-151, 165, 181, 196, 250, 256, Japanese, 108, 110, 133 257, 274, 334, 337, 354, 355 Korean, 75, 108, 110, 247 pattern finding, 43 and mathematical learning, 12, 98 problem solving, 42, 43-44, 103, 108, morphological markers, 106, 109, 354 109 number names, 27, 106-107 reasoning, 42-44 representations for numbers, 106-107 representing, 21, 35, 42-43, 44, 332, Spanish, 100, 101, 106, 107 337 and spatial mathematical skills, 75 specific reasoning processes, 44-47 Swedish, 110 standards, 42, 232-233 Language (mathematical) modeling, 237, Mathematics curriculum 353. See also Math talk automatized knowledge, 83, 100 Learning disabilities committee conclusions, 338-340 characteristics, 111-112 effectiveness research, 2, 265-266, developmental dyscalculia, 112 269-271 helping high-risk children, 112-113 embedded (secondary), 2, 227, 238, 248, prevalence, 110 269, 275, 316, 339, 355 visual/spatial deficits, 209-210 executive function skills, 83 Learning trajectories. See Teaching-learning focused time, 2, 124, 227, 238, 248, paths 269, 271, 273, 275, 313, 339, 340, Legislation, 15-16 352, 355 Low-income. See Economically guiding principles, 273-274 disadvantaged children integrated, 2, 129, 130, 145, 146, 147, 148, 149, 166, 198, 227, 238, 239, 256, 267-268, 269, 339 M interventions for economically disadvantaged children, 271-273 Manipulatives NAEYC and NCTM position statement, classroom practices, 239, 240-241, 248, 122-123 251-252 NCTM standards, 122 counting, 240, 252, 261 resources, 267-268 cultural tools, 203 Mathematics instruction. See also defined, 353 Classroom context for instruction; effectiveness, 252 Curriculum; Instructional supports in family settings, 343 book reading, 254, 339 geometric, 179, 180, 183, 191, 194, committee conclusions, 338-340 195, 240, 248, 356, 357 computers, 252-254 measurement, 199, 203, 207, 208, 209, concrete materials and manipulatives, 240-241 196, 251-252 spatial visualization and imagery, 183, developmentally appropriate practice, 9, 203 254-255, 268, 270, 296, 298-299, virtual, 196, 248, 253, 356-357 300 Map use, 74-75, 82 direct, 226, 297 Math talk, 43, 44, 104, 157, 245-246, 249, effective strategies, 191-195, 242-254, 340, 343 272, 339-340 Mathematical connections. See Connections explicit, 104, 110, 169, 208, 226 across content grouping strategy, 247, 248 Mathematical process goals guiding principles, 273-274 communicating, 42, 43 instructional supports, 243-245 connecting, 42, 43

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 INDEX kindergarten practices, 239-242 and number, 52-63 math talk, 157, 245-246, 249 number lines, 168 movement to engage children, 253-254 number-measurement relation, 360-361 opportunities to learn, 1-2 objects and spatial relations, 205-206, pattern finding, 241, 242, 246, 248, 207 249, 260, 261 origin, 360 play, 104, 247-251, 339 Piaget’s theories, 79-80, 199, 200 preschool practices, 237-239, 242, proportional reasoning, 75, 82 299-300 relating parts and wholes, 202, 206, 207 teacher-guided, 225-226, 345, 356 relative coding, 46, 79 teacher-initiated learning experiences, ruler use, 53, 80, 81, 167, 168, 197, 225-226, 356 198, 199-200, 201, 202, 207, 208, teacher pedagogical beliefs and 209, 240-241, 267, 360, 361 knowledge, 299-300 scaling, 75, 82, 182 Mathematizing, 43-44, 54, 134, 149, 150- socioeconomic differences, 96 151, 165, 181, 196, 250, 256, 257, and spatial structuring, 82, 189, 190, 274, 334, 337, 354, 355 201-204, 209, 361 Measurement standards, 230-232 achievable and foundational goals, 201, teaching-learning paths, 205-210 202, 205 thinking about parts, 202, 206, 207 accumulation of distance, 360 transitivity, 197, 208, 358, 359, 361 addition/subtraction, 80 units and unit iteration, 36, 37, 45, 46, additivity, 360 53, 80-82, 161, 197, 198, 199, 204, area, 35, 36, 37, 39, 46, 49, 53, 61-62, 206, 207, 208-209, 360 65, 66, 79, 177, 179, 190, 201-204, visual/holistic thinking, 202, 206 205, 207, 209-210, 354, 361 visual/spatial deficits and, 209-210 cardinality and, 36, 53 volume, 35, 36-37, 39, 40, 50, 63, 71, cognitive foundations, 79-82 79, 190, 204-205, 210, 354, 361 compositions and decompositions, 35, zero point, 200, 208 39-40, 46, 202, 206, 207, 360 Memory concepts of, 359-361 deficits, 112 conservation of length, 359 faulty, 256 conventional, 80-82 working, 67, 83, 84, 361 counting and, 81, 198, 204 Michigan, 230 and data analysis, 53-55 Mississippi, 15 defined, 354 Multiplication/division, 30, 34, 45, 46, 48, development of, 79-82, 196-197 50, 51-52, 55, 190, 203 equal partitioning, 46, 198, 201, 204, Multiplicative comparisons, 34 206, 359-360, 361 Multi-State Study of Preschool, 233-237, equivalence/nonequivalence, 80-81 239 foundational content, 36-37, 198 and fractions, 36-37, 52-53 N importance, 2, 124-125 infants, 61-62, 66, 79, 82 National Assessment of Educational instructional strategies, 80-82, 199-201, Progress, 100 207-210 National Association for the Education of language of, 205-206, 207-208 Young Children (NAEYC), 9, 122, length, 36, 37-38, 45, 46, 50, 52-53, 55, 228, 268, 275, 296, 307-308, 309, 61-62, 66, 79, 80-81, 197-201, 207- 318, 319 209, 359 National Board for Professional Teaching manipulatives, 203-204 Standards, 318-319

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0 INDEX National Center for Early Development and teen numbers, 146-147 Learning, 233-237, 239, 268 written number symbols, 22, 23, 24, National Council for Accreditation of 26-30, 129, 138, 141, 144-145, 146, Teacher Education, 314, 315, 316, 147, 148 318 Number facts, 25 National Council of Teachers of Number lines, 53, 111, 113, 167-168, 251 Mathematics (NCTM), 9, 42, 55, Number names. See Number word list 122-123, 228, 230, 239, 241, 307- Number representation. See also Written 308, 309, 310, 335 number symbols National Education Goals of 1990, 15 analog magnitude system, 63, 70, 112, National Institute of Child Health and 168, 351 Human Development, 10 approximate, 112 National Reporting System, 9 decimal system and, 27, 109-110 New Jersey, 230, 242 fractions, 61 New Mexico, 229 mental, in preschool children, 66-71, New York, 230, 233 138 No Child Left Behind (NCLB) Act, 15, 16, object file system, 62, 63, 66, 70, 354 303 Number sense. See also Number North Carolina, 230, 232, 294 understanding or number sense Number competencies, 95, 96, 112, 350, defined, 95, 354 354 Number triads Number content. See also Number core; composed/decomposed, 25, 32-33, 158- Operations core; Relations core 159, 160-161 importance, 2, 124-125 embedded, 23, 159 instruction practices, 239-240 Number understanding or number sense, standards, 230-232 142, 230, 231, 354. See also Number core Competence and performance; cardinality, 22, 23, 25-26, 34, 129, 132- Number competencies 134, 140, 142, 145, 146, 147, 148 addition/subtraction transformations, composing/decomposing, 23, 32-33, 45, 67, 96 48, 49-50, 154, 157, 161, 164-165, approximation, 61, 63, 66, 68, 70, 84, 194, 231 98, 112, 351 coordinating components of, 23, 25, 52- conceptual, 63-64 53, 131, 138-140 infants, 60, 61-66, 95 counting correspondences, 22, 23, 24- kindergarten, 96 25, 26, 129, 134-137, 141, 144, 145, learning disabilities and, 112-113 146, 147, 148 mental number representation, 66-71 counting out n things, 141, 145 order relations, 65-66 and data analysis, 53-55 preschool children, 66-71, 95 decimal system, 22, 24, 26-30, 48 preverbal, 60-66 foundational content, 22-30 set comparisons, 96 importance, 124-125 set size discrimination, 60, 61-66, 67-70, integrating components of, 145-148 96, 98 mathematical perspective, 24-30 Number word list number word list, 22, 23, 24, 25-30, activities, 136-137 132, 134, 141, 142-143, 144, 145, and addition/subtraction situations, 25, 146 160, 166-168 overview, 22-30 and cardinality, 25-26, 31, 32, 129, 132, quantity, 22, 24 147, 148 subitizing, 23, 25, 130, 131, 141, 146 core content, 24-30, 134, 142-143 teaching-learning paths, 129-148 decade words, 147

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1 INDEX importance of fluency, 25, 66 comparison situations, 23, 30-31, 33, language/cultural differences, 27, 70, 151 106-107, 142-143 compose/decompose numbers, 23, 25, number line distinguished from, 167-168 32-33, 158-159, 160-161 and one-to-one counting computational fluency, 111 correspondence, 134, 138, 144 connections across core, 34 parental modeling of, 104 counting on, 25, 111, 113, 130, 151, pattern finding in, 47, 107, 108, 131, 152-154, 159, 160, 163, 164, 166 134, 141, 142-143, 144, 146, 147, embedded number triads, 23, 33, 158- 148 159, 166 as representational tool, 160, 166-168 importance, 124-125 socioeconomic differences in mathematizing real-word situations, understanding, 97-98 149, 150-151, 165 teaching-learning paths, 129, 132, 134, number list fluency and, 25 138, 141, 142-143, 144, 145, 146 overview, 23 unitizing, 166 put together/take apart situations, 23, verbal knowledge, 140 32-33, 34-35, 149, 355 and written symbols in decimal system, socioeconomic differences, 154, 159 26-30, 64 standards, 230-232 Number Worlds, 256, 263, 271 subidizing, 130, 150, 151, 152, 154, Numerals. See also Written number symbols 155, 158, 159 Arabic, 106, 107, 108, 109 teaching-learning paths, 130, 149, 150- defined, 27, 354 154, 155, 157-161, 162-165, 166 Numerosity, 23, 61, 62, 63, 64, 65, 71, Order relations, 65-66, 149, 156, 166, 167. 134, 156 See also Relating and ordering defined, 354 Ordinal numbers, 167 matching task, 67-68, 69, 97-98 Ordinal position, 63 Organizing information, 46-47 O P Object file system, 62, 63, 66, 70, 354 Observational assessment, 257-260 Parents. See also Family role organized systems of, 259 engagement in mathematics activities, strengths and weaknesses, 259-260 102-104 Ohio, 230, 233 knowledge and beliefs about Oklahoma, 15 mathematics, 8, 102, 105, 344 One-to-one counting correspondences perceptions of children’s abilities, 104 accuracy, 135 spatial activities and language, 76, 78, to cardinality, 22, 24-25, 129, 134-137, 99 144 Pattern finding. See also Subitizing defined, 354 activities, 46-47, 102, 156, 189-190, number word list and, 134, 138 194, 196, 248 set size and, 135 in addition/subtraction, 159-160 teaching-learning paths, 129, 132, 134- attribute blocks, 47, 183-184, 209-210, 137, 138-139, 141, 144, 146, 147, 248, 351 148 cognitive foundations, 69, 83 Operations core. See also Addition/ computer manipulatives, 196 subtraction; Multiplication/division conceptual subitizing, 132-133, 142, change situations, 23, 32-34, 149 153, 155, 158, 356 counting, 47, 49, 107, 156, 164

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2 INDEX defined, 352 set size discrimination, 67-70 foundational content, 46-47, 48, 49, shape discrimination, 77 122, 125, 241, 266, 337 spatial thinking, 73-75 informal knowledge of young children, Preschool Curriculum Evaluation Research 299 Study, 269, 270 instruction practices, 241, 242, 246, Preschool Embedded Figures Test, 73-74 248, 249, 260, 261 Preschool Mathematics Curriculum, 270 in multiplication tables, 52 Preschool programs in number words, 47, 107, 108, 131, classroom activities, 234-235 134, 141, 142-143, 144, 146, 147, classroom organization and 148 management, 236-237 reasoning processes, 43, 46-47 economically disadvantaged children shapes, 47, 176, 177-179, 183-184, 189- targeted, 15 190, 191, 194, 196, 351, 355, 357 emotional support, 236 spatial visualization, 249, 357 equity in, 124 standards of learning, 238 n.3 instructional supports, 237 structural relationships, 47, 48, 52 math instruction, 124-125, 237-239, in teaching-learning paths, 122, 125, 242 131, 132-133, 134, 141, 142-143, standards of learning, 228-230 144, 146, 147, 148, 153, 155, 156, state-funded, 15, 228-230, 233-242 158, 159-160, 164, 176, 177-179, teacher-child interactions, 235-237 183-184, 189-190, 191, 194, 196 voluntary universal, 15 unitizing, 45, 48 Problem solving, 43, 44, 46, 51, 100, 109- Perceptual subidizing, 131, 132, 164, 166, 110, 122, 133, 142, 159, 165, 166, 356 169, 194-195, 228, 229, 239, 247, Performance in mathematics. See 251, 252, 255, 273, 274, 297, 299, Competence and performance 337 Piaget’s theories and perspectives, 60, 61, Process. See Mathematical process goals 67, 70, 72, 76, 77, 79-80, 155, 156, Professional development of workforce. See 166, 183, 199, 200, 255-256, 257, also Formal education of workforce 258, 260, 261-262, 264, 298, 359, access, 303 361 assessing competence, 315-316 Place values, 28, 29-30, 45, 48, 107, 108, defined, 355 148, 231, 355 inadequacy in mathematics, 2 Play preparing teachers to promote block building, 248-249 mathematics development, 316-319 games, 251 training, 2, 258-259, 270, 291, 292, mathematical, 149, 251 294-295, 298, 300, 301, 302, 303, practice during, 250 305, 306-307, 309, 310, 311, 312, sociodramatic, 249-250 313, 314, 318, 341, 342, 343, 348, Policies. See Education policies 356 Pre-K Mathematics, 311 Project IMPACT, 314 Preschool children. See also Age 2-3; Age 4 attitudes about mathematics, 12 Q calculation tasks, 67-68, 69 map use, 74-75, 82 Quantizing, 24 mathematics curricula, 184 auditory, 133 mental number representation, 66-71 numerosity matching task, 67-68, 69, 97-98

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 INDEX R Relations core comparison situations, 30-31, 33, 34, Race/ethnicity 149, 154-155, 161-162 and competence and performance, length or density strategies, 23, 130, 99-100 149, 150, 154, 155, 156 ECE workforce, 14 matching and counting strategies, 155, Reading First, 16 156, 161 Reasoning processes overview, 23 “big ideas,” 44-45 socioeconomic differences, 97 composing/decomposing, 35, 40, 45-46, subidizing, 150, 151, 154, 166 179 teaching-learning path, 130, 149, and conceptual development, 244 150, 151, 154-157, 161-162, 165, general, 40, 42-44, 184 166-168 geometric and spatial, 36, 76-77, 83, vocabulary, 156-157, 161 188, 206, 230, 231, 337 Representing relationships, 110, 165, goals, 42-44, 122, 337, 345 177, 187, 189, 202, 206, 267, 332, NCTM process standards, 42 337. See Number representation; organizing information, 46, 47 Numerals; Written number symbols pattern finding, 43, 46-47 data analysis, 54-55 prompting for, 245, 256, 355 mathematizing, 43, 44 proportional thinking, 82 in measurement, 53, 55 relating and ordering, 46, 83 as reasoning process, 42-43, 44 representing, 42-43, 44 Research needs specific, 40, 44-47 curriculum evaluation, 349 standards of learning, 229, 230, 231, English language learner needs, 348 233 instruction practices, 348-349 teachers’ pedagogical beliefs, 297 learning disabilities, 349-350 teaching-learning paths, 184, 197, 199, parental involvement, 349 206 teacher preparation, 349 transitive, 206 Research on mathematics learning unitizing, 45 curriculum effectiveness, 265-273 Recommendations evidence base, 12 credentialing of educators (7), 4, 347 NRC reports on early development and curricula and instruction (3), 3, 345 learning, 9-10 family-community partnerships (8), 4, Rightstart program, 271, 273 347-348 Rod-and-Frame Test, 73 informal learning supports (9), 4, 348 instructional supports (5), 3, 346 math experiences in early childhood (2), S 3, 345 Scaffolding, 182, 191, 203, 238, 243, 244, national initiative (1), 3, 345 247, 249-250, 251, 255, 260, 275, professional development (6), 3-4, 355 346-347 Self-regulation, 83-84, 250, 264 standards of learning (4), 3, 346 Set size discrimination, 60-63 Reggio Emilia, 259, 267 Shapes and objects. See also Geometry Relating and ordering, 23, 37-38, 46, 61, 2-D, 37-38, 177-179, 185, 188, 190 97, 337, 355 3-D, 38-39, 40, 184, 185, 188, 190 Relating parts and wholes, 178, 179, 185, as abstractions, 35-36 187, 188, 189-190, 191, 193, 202, activities, 193-195 206, 207, 355 area of, 35, 39-40, 49

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 INDEX categorization, 180 foundational content, 3, 35, 41-42 characteristics, 35, 37-39, 180, 192-193 vocabulary, 77, 185, 186 cognitive foundations, 76-78, 179-180 Spatial structuring, 35, 40-42, 50, 180-184, composing/decomposing, 35, 39-40, 46, 189, 190, 201-204, 209, 361 48-50, 52, 175, 177-179, 184, 185- Spatial thinking. See also Geometry; Shapes 188, 189-190, 191, 194, 195, 206, and objects 231, 248, 336 activities and tests, 72, 73-74, 76, 77, concept development, 179-180, 192-193 99, 183-184 dimensional adjectives, 78 cognitive foundations, 71-79, 180-182 discrimination, 76-79 foundational content, 42-43 faces on, 39 gender differences, 74, 76, 98-99, 182 manipulatives, 180, 183-184 importance, 175 pattern finding, 47, 176, 177-179, 183- infants, 72-73, 75, 78 184, 185, 189-190, 191, 194, 196, language (spatial) use and, 75-76, 77-78, 351, 355, 357 99, 181 relating parts and wholes, 176, 178, manipulatives, 183 179, 185, 187, 188, 189-190, 191, and map use, 74-75, 182 193, 355 and math and science achievement, 71 steps in thinking about, 176-179 mental functions in, 71 teaching-learning paths, 185, 186-187, mental representations, 181-182 188, 190, 191-195 mental transformation and superposition thinking about parts, 164, 176, 177, of shapes, 42-43, 73-75, 78, 99, 103, 178, 179, 185, 186, 187, 188, 189, 183, 203, 356 193, 356 modeling, 182 vertices, 38, 39, 178, 188 perspective changes, 35, 72, 74, 182 visual/holistic thinking, 176, 177, 178, Piaget’s view, 72 185-189, 357 preschool, 75-79 Socioeconomic status. See also Economically socioeconomic differences, 74, 96 disadvantaged children symbolic systems, 72 and cardinality tasks, 103 teaching-learning paths, 185, 186-187, and child care arrangements, 14 188-189, 191 and learning opportunities, 104-105 visualization and imagery, 39, 73-74, and number understanding, 70 180, 183-184, 230, 231, 249, 336, and spatial performance, 74 355, 357 and teachers’ values and beliefs about SRA Real Math Building Blocks, 311 instruction, 296-299 Standards for mathematics learning Space committee conclusions, 338-339 2-D, 40-41, 177-179 Curriculum Focal Points, 55, 123, 125, 3-D, 40-41, 50 228, 229, 230, 335 Cartesian coordinates, 35, 49, 50 early learning (prekindergarten), 9, models for, 41 228-230 movement through, 41-42, 52, 178, 183, kindergarten, 230-233 189, 248, 252 monitoring implementation, 229 Spatial language, 75-76, 77-78, 99, 181, Principles and Standards for School 185 Mathematics, 122, 123, 228, 229, Spatial orientation, 41, 73, 74, 75, 138, 231, 233 177, 178, 180-182, 188, 189, 192, process strands, 232-233 194, 336, 355, 357 State-wide Early Education Programs Spatial relations, 185, 188-189, 191 (SWEEP) Study, 234 in 2-D space, 177-179 Storytelling Sagas, 254 categorization, 72-73, 75, 77 Structural relationships, 47, 48, 52

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5 INDEX Subitizing composition/decomposition, 32, 154, auditory, 133, 164 157, 161, 164-165, 175, 177-179, and cardinality, 132, 134, 141, 142, 184, 185-188, 189-190, 191, 194, 146, 150, 151, 155 202, 206, 207 conceptual, 23, 132-133, 141, 142, 146, connections among core component, 23, 150, 151, 152, 153, 154, 155, 158, 25-26, 34, 39-40, 45-46, 48-55, 129, 159, 164, 166, 356 130, 131, 138-140, 337 for counting on, 164 context for, 129 defined, 25, 132, 356 counting on to make a solution, 145, neurological origins, 112 148, 151, 152-153, 154, 155, 156, in number core, 23, 25, 130, 131, 141, 157-158, 159-160, 163-165, 166 146 counting out n things, 141, 145 in operations core, 130, 150, 151, 152, defined, 121, 353-354 154, 155, 158, 159 equation writing, 158-159, 162-163 ordinal numbers, 167 integrating components of, 145, 146-148 perceptual, 131, 132, 164, 166, 356 number core, 129-148 range, 63 number word list, 129, 132, 134, 138, in relations core, 150, 154, 166 141, 142-143, 144, 145, 146 teaching-learning paths, 130, 131, 132- one-to-one counting correspondences, 134, 141, 142, 146, 150, 151, 152, 129, 132, 134-137, 138-139, 141, 153, 154, 155, 158, 159, 164, 166 144, 146, 147, 148 Symmetry, 51, 52, 134, 179, 180, 185, 195, operations core, 130, 149, 150-154, 196, 231 155, 157-161, 162-165, 166 overlearning, 128, 138, 145 pattern finding, 122, 125, 131, 132-133, T 134, 141, 142-143, 144, 146, 147, 148, 153, 155, 156, 158, 159-160, Tangrams, 179, 183, 191, 194, 248, 356, 164, 176, 177-179, 183-184, 189- 357 190, 191, 194, 196 Tasks problem-solving strategies, 150-154, assessment strategy, 260-262 168-169 organized systems of, 261 relations core, 130, 149, 150, 151, 154- strengths and weaknesses of, 261-262 157, 161-162, 165, 166-168 Teacher-child interactions shapes and objects, 185, 186-187, 188, classroom organization and 190, 191-195 management, 236-237 spatial relations, 185, 186-187, 188-189, emotional support, 236 191 preschool and kindergarten, 235-237 subitizing, 130, 131, 132-134, 141, 142, Teacher Education Assistance for College 146, 150, 151, 152, 153, 154, 155, and Higher Education Grant 158, 159, 164, 166 Program, 303 teen numbers, 146-147 Teacher-initiated experiences written number symbols, 53, 70, 112- defined, 225-226 113, 129, 138, 141, 144-145, 146- Teachers. See Early childhood education 147, 148 teachers Teaching to the Big Ideas, 314 Teaching-learning paths. See also Ages 2-3; Technology-enhanced, Research-based, Age 4; Grade 1 Instruction, Assessment, and cardinality, 129, 130, 131, 132-134, professional Development (TRIAD) 135, 139-140, 141, 142, 145, 146, model, 311, 312 147, 148, 150, 151, 154, 155, 169 Test of Early Mathematics Ability, 273

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6 INDEX V Texas, 230, 233 Tools of the Mind, 249-250 Virginia, 230, 232 Visual/spatial deficits and, 209-210 Volume, 35, 36-37, 39, 40, 50, 63, 71, 79, U 190, 204-205, 210, 354, 361 Unitizing, 337 Vygotsky’s theories, 200, 203, 249-250, in counting, 23, 45, 48, 147, 148 255, 260 decimal system, 45, 48 defined, 356 W by grouping, 48-49, 207 in measurement, 36, 37, 45, 46, 53, Washington, DC, 15 80-82 What Works Clearinghouse, 312, 313 number words, 166 Workforce. See Early childhood workforce and pattern finding, 45, 48 WPPSI-3 Block Design subtest, 78 U.S. Department of Education, 269 Written number symbols U.S. Department of Health and Human errors in writing, 147 Services, 10 number words and, 26-30, 64 Office of Head Start, 9, 10 place value cards, 148 teaching-learning paths, 53, 70, 112- 113, 129, 138, 141, 144-145, 146- 147, 148