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From page 371...
... 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, 67 371
From page 372...
... . 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 counting out n things, 141, 145, 156, C 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.
From page 373...
... 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, 32Cognitive 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
From page 374...
... 374 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, 164, 166 D 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
From page 375...
... 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 Early Childhood Educator Professional False belief understanding, 83 Development Program, 303 Family role.
From page 376...
... 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
From page 377...
... 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
From page 378...
... 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
From page 379...
... INDEX 379 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 importance, 2, 124-125 N 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
From page 380...
... 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
From page 381...
... 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
From page 382...
... See Education policies 356 Pre-K Mathematics, 311 Project IMPACT, 314 Preschool children. See also Age 2-3; Age 4 attitudes about mathematics, 12 calculation tasks, 67-68, 69 Q 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
From page 383...
... , 3, 346 Self-regulation, 83-84, 250, 264 Reggio Emilia, 259, 267 Set size discrimination, 60-63 Relating and ordering, 23, 37-38, 46, 61, Shapes and objects. See also Geometry 97, 337, 355 2-D, 37-38, 177-179, 185, 188, 190 Relating parts and wholes, 178, 179, 185, 3-D, 38-39, 40, 184, 185, 188, 190 187, 188, 189-190, 191, 193, 202, as abstractions, 35-36 206, 207, 355 activities, 193-195 area of, 35, 39-40, 49
From page 384...
... 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)
From page 385...
... 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
From page 386...
... Department of Education, 269 WPPSI-3 Block Design subtest, 78 U.S. Department of Health and Human Written number symbols Services, 10 errors in writing, 147 Office of Head Start, 9, 10 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


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