INDEX

A

Abacus, use of in Japan, 218n

Abstraction, 111n, 160

recommendation for linking experience to, 426

Accessible generalizable methods, for multidigit subtraction, 205

Accuracy

of calculators, 247

of decimal approximations, 88–90

improvement with practice, 121

of subtraction, 191

of teacher self-reports, 47

Achievement, in school mathematics in the U.S., 55–57

Adaptive reasoning, 5, 10, 116, 138–139, 170, 380, 383–384

and mathematical proficiency, 129–131

in the teaching of mathematics, 383–384

Addition. See also Multidigit addition

algorithms for, 199–204

associativity of, 77

carrying in, 203

commutativity of, 75, 77

distributivity of multiplication over, 78

of fractions, 86, 320–322

problem types, 185

properties of, 82

single-digit, 187–190

Additive concepts

identity, 82

inverse, 82–83

Address, of real number, 90

Algebra, 256–279.

See also Generalizing activities;

Transformational activities of algebra

beginning, 255–256

characterizations of, 294–295n

developing meaning, 272–274

as generalized arithmetic, 256

mentally graphing to solve an equation, 275

representational activities of, 257

role of technology, 274–276

as symbol transformation, 256

table completion task from NAEP, 260

two methods for solving equations, 273

using technology to learn, 420

what the number-proficient child brings, 270– 272

Algebra for all, 279–280

promoting, 420

Algebraic thinking, developing, 419

Algorithms, 102–106, 195–196

for addition, 199–204

children devising their own, 197

common, for multidigit division, 211

for division, 210–212

efficiency of, 103

examples of, 104–106

generality of, 103

learning numerical, 414

for multiplication, 104–106, 195, 206–210

precision of, 103

simplicity of, 103

for subtraction, 204–206



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Adding + It Up: Helping Children Learn Mathematics INDEX A Abacus, use of in Japan, 218n Abstraction, 111n, 160 recommendation for linking experience to, 426 Accessible generalizable methods, for multidigit subtraction, 205 Accuracy of calculators, 247 of decimal approximations, 88–90 improvement with practice, 121 of subtraction, 191 of teacher self-reports, 47 Achievement, in school mathematics in the U.S., 55–57 Adaptive reasoning, 5, 10, 116, 138–139, 170, 380, 383–384 and mathematical proficiency, 129–131 in the teaching of mathematics, 383–384 Addition. See also Multidigit addition algorithms for, 199–204 associativity of, 77 carrying in, 203 commutativity of, 75, 77 distributivity of multiplication over, 78 of fractions, 86, 320–322 problem types, 185 properties of, 82 single-digit, 187–190 Additive concepts identity, 82 inverse, 82–83 Address, of real number, 90 Algebra, 256–279. See also Generalizing activities; Transformational activities of algebra beginning, 255–256 characterizations of, 294–295n developing meaning, 272–274 as generalized arithmetic, 256 mentally graphing to solve an equation, 275 representational activities of, 257 role of technology, 274–276 as symbol transformation, 256 table completion task from NAEP, 260 two methods for solving equations, 273 using technology to learn, 420 what the number-proficient child brings, 270– 272 Algebra for all, 279–280 promoting, 420 Algebraic thinking, developing, 419 Algorithms, 102–106, 195–196 for addition, 199–204 children devising their own, 197 common, for multidigit division, 211 for division, 210–212 efficiency of, 103 examples of, 104–106 generality of, 103 learning numerical, 414 for multiplication, 104–106, 195, 206–210 precision of, 103 simplicity of, 103 for subtraction, 204–206

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Adding + It Up: Helping Children Learn Mathematics America 2000, 34 American Federation of Teachers, 34–35 Arabic numerals, 18, 163, 166, 175n Archimedes, 112–113n Area interpretation of multiplication, 107 Area measure, 283 space-filling concept for, 283 Arithmetic. See also Mental arithmetic; Single-digit arithmetic abstract nature of, 74 and algebra, 256, 261 and geometry, 87–93 and mathematics, 293–294 multidigit, 121–122 number systems of, 72 paper-and-pencil, 20 preschool, 169 rules of, 73–75, 274, 280 Arithmetic operations, properties of, 73, 75–78 Arizona, exam passing rates in, 42 Arrays, rectangular, interpretation of multiplication, 77, 207–208 Assessment, 349–350 cut scores for, 42 high stakes, 41–42 internal and external, 39–40 of mathematics knowledge in the U.S., 35–36 of school mathematics in the U.S., 31, 39–44 of students, 349–350 Assessment Standards for School Mathematics, 34 Associative law, 75 Associativity of addition, 77 Automaticity, attaining, 351 Averages, 290 B “Back to basics” movement, 115 Base-10 blocks, 96, 203, 221n place-value system, 198 Beginning algebra, 255–256 Bird and worm problem, 129 Blocks base-10, 96, 203, 221n building, 106–110 Book purchase problem, 261 Borrowing, in subtraction, 204–205 Building blocks, number concepts as, 106–110 C CAD. See Computer-assisted-drawing tools Calculators, 45–46, 354–356 different types of, 100 four function, 100n graphing, 269 instructional recommendations for using, 427 order-of-magnitude accuracy of, 247 recommendation for using, 427 scientific, 100n use in Sweden, 355 Calculus, comprehending, 295n California standards for knowledge of mathematics, 34 textbook system in, 36 California State Board of Education, 50 Call for Change, A, 51 Canada, levels of mathematics achievement in, 56 Cardinal numbers, 160 Carrying, 203 Cases. See also Vignettes programs focusing on, 392–395 Central tendency, measures of, 289 CGI. See Cognitively Guided Instruction Chance, learning about, 291–293 Children. See also Preschoolers’ mathematical proficiency devising their own algorithms, 197 China addition method in, 188 decimal system in, 175n fractions in, 236 learning number names in, 164–168, 175n Chocolate distribution problem, 266 Classroom discourse, 345–346 recommendation for managing, 425–426 Classroom vignettes. See Vignettes Cognitive science, 117–118, 145n, 218n Cognitively Guided Instruction (CGI), 389, 391– 392, 400n Combinations. See Number combinations Combinatorics, 109 Committee on Mathematics Learning, 2, 26 Communities of learners, 344–345 of mathematics specialists, 397–398 of practice, 397–398 Commutativity of addition, 75, 77 of multiplication, 77 Compare, problem types, 185 Comparing prices, teaching about, 326–327

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Adding + It Up: Helping Children Learn Mathematics Competence. See Mathematical proficiency; Strategic competence “Complementary number-to-10” strategy, 218n Composite units, 249n Compound units, 249n Computation, with rational numbers, 238 Computer-assisted-drawing (CAD) tools, 287 Computers graphics on, 16, 269 instructional recommendations for using, 427 Conceptual understanding, 5, 10, 116, 136–137, 158–159, 380–382 and mathematical proficiency, 118–120 Concrete materials. See also Manipulatives not the same as physical, 426 Conditional probability, 292–293 Conditions, as aids to understanding, 127 Conference Board of the Mathematical Sciences, 397 Connections, supporting, 235–236 Content, 333–338, 350–356 and calculators, 354–356 and homework, 352–353 and manipulatives, 353–354 opportunities to learn, 333–335 planning, 337–338 and practice, 351–352 task selection and use, 335–336 Contexts for instruction, 314 for learning, solving problems as providing, 420–421 meaningful, for word problems, 183–187 Conventional instruction, what can be learned from, 240–241 Cookie distribution problem, 376–377 Cooperative learning, 50, 344–349 Coordinating improvement efforts, in teaching mathematics in the U.S., 58–59 Council for Basic Education, 35 Council of Chief State School Officers, 52–53 Counting, 181 and the origins of the number concept, 159– 160 understanding and mastering, 161–162 Curriculum. See also Curriculum recommendations decisions, 10–11, 410–424 guides and frameworks for, 34 mathematics, in U.S., 33–35 standards for, 34 Curriculum and Evaluation Standards for School Mathematics, 33–34, 36 Curriculum recommendations, 10–11, 410–424 building on informal knowledge, 410–411 developing algebraic thinking, 419 developing proportional reasoning, 417 expanding the number domain, 418 extending the place-value system, 416–417 giving students time to practice, 422–423 giving time to instruction, 422 improving materials for instruction, 421–422 learning about numbers, 412–413 learning number names, 411–412 learning numerical algorithms, 414 operating with single-digit numbers, 413 promoting algebra for all, 420 representing and operating with rational numbers, 415–416 solving problems as a context for learning, 420–421 using estimation and mental arithmetic, 415 using technology to learn algebra, 420 using the number line, 418 Cycle shop problem, 126 Czech Republic, levels of mathematics achievement in, 56 D Data analyzing, 290–291 describing, 289 learning to use, 288–291 organizing, 289–290 reading, 289–290 representations of, 290 Decimal system, 96. See also Base-10 Derived number combinations, 188 Developing algebraic thinking, recommendation for, 419 Developing geometric reasoning, 284–288 reasoning about more advanced concepts, 287–288 reasoning about shape and form, 284–287 Developing mathematical proficiency, 8, 13–14, 246–247, 255–312, 432 acquiring measure concepts, 281–284 algebra for all, 279–280 from arithmetic to mathematics, 293–294 beginning algebra, 255–256 concept of negative numbers, 245 developing geometric reasoning, 284–288 developmental themes, 216–218 discontinuities in, 233–234

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Adding + It Up: Helping Children Learn Mathematics early, and language, 163 estimation, 215–216 generalizing and justifying activities of algebra, 276–279 with integers, 244–246 main activities of algebra, 256–279 measurement and geometry, 281 mental arithmetic, 214–215 multidigit whole number calculations, 195– 214 operations with single-digit whole numbers, 182–195 overtime, 135 with rational numbers, 7–8, 231–241 representational activities of algebra, 261–270 representing rational numbers, 236–244 statistics and probability, 288–293 supporting connections, 235–236 transformational activities of algebra, 270–276 using informal knowledge, 232–233 with whole numbers, 6–7, 181–229 Developing meaning, 263–270 in algebra, 272–274 Developing proficiency in teaching mathematics, 10, 369–405. See also Professional development attaining a profound understanding of fundamental mathematics, 370 communities of practice, 397–398 effective professional development, 398–399 knowledge base for teaching mathematics, 370–380 patterns in predicting student proficiency, 217 proficient teaching of mathematics, 380–385 programs to develop proficient teaching, 385– 397 what it takes to teach for mathematical proficiency, 369–370 Developing proportional reasoning, recommendation for, 417 Developing specialized knowledge, recommendation for, 428–429 Developmental themes, 216–218 Discontinuities in proficiency, 233–234 Discourse, managing, 345–346, 425–426 Disparities in mathematical proficiency, 148n addressing, 344 gender, 148n racial, 55, 148n socioeconomic, 143 Disposition, 146–147n. See also Productive disposition Distributivity, of multiplication over addition, 78 Division. See also Multidigit division algorithms for, 210–212 of fractions, 83–86, 386–388 single-digit, 192–193 subtraction and, 78–80 Division of Elementary, Secondary, and Informal Education, 2, 26 Domain. See Number domain Dynamic geometry software, 298n E Early development of mathematical proficiency, and language, 163 Effectiveness, of professional development, 398– 399 Efficiency of algorithms, 103 of representations, 99 Elementary and Secondary Education Act, 34 Elements, Euclid’s, 82 Enactment, as instructional interaction, 9 England investigations of mathematics competence in, 39 teaching experiments in, 265 English language, number names in, 164–168 Equal sign, announcing a result, 270, 390 Equality, 86 in a professional development group, 390 statement of, 75 Equity. See also Disparities in mathematical proficiency and remediation, 172–174 Errors. See Students’ errors Estimation, 215–216, 221n recommendation for using, 415 Euclid, 82 European-Latino subtraction algorithm, 219n Evidence from research, 23–26 Expanded algorithms for multidigit division, 211–212 for multidigit multiplication, 209 Expanding the number domain, recommendation for, 418 Expectations low, 343 of success, maintaining, 339–340 of teachers, 338–339 Experience, linking to abstraction, 426 Experiments in teaching, 265 what can be learned from, 240–241

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Adding + It Up: Helping Children Learn Mathematics Exponential functions, 295n Exponents, 98 F Finite decimals, number system of, 88–90 First International Mathematics Study, 360n Florida special funding provided for mathematics teaching, 41 standards of analyzed, 35 Fluency. See also Procedural fluency relation to understanding, 196 Fordham Foundation, 35 Form, reasoning about, 284–287 Formulas for the arithmetic of fractions, 86 for the arithmetic of negation, 83 Fractions addition of, 86, 320–322 division of, 83–86 equality in, 86 formulas for the arithmetic of, 86 multiplication of, 86 notation for, 86, 112n reciprocals of reciprocals, 86 France, investigations of mathematics competence in, 39 Functions, 274 graphs of, 274 G Gas price problem, 125 Gauss, Carl Friedrich, 108 Gender disparities, 148n Generality of algorithms, 103 of representations, 100 Generalized arithmetic algebra as, 256 Generalizing activities, 258, 276–279 justifying generalizations, 276–278 predicting patterns, 278–279 problems that involve, 277 Geometry, 107, 281 and arithmetic, 87–93 Germany, video studies of mathematics teaching in, 49–50, 280 Givens, as aids to understanding, 127 Goals 2000, 34 Gradualness, 217 Graphing, using calculators, 269 Grouping. See also Regrouping of quantities, 96–99 of students, 50, 112n, 265, 346–349 H Handshake problem, 107–109 Hawaii, requirements for professional development, 54 High-stakes assessments, 41–42 Hindu-Arabic numerals, 18, 163, 166, 175n Holmes Group/Partnership, 52 Homework, 352–353. See also Independent work Hong Kong investigations of mathematics competence in, 39 levels of mathematics achievement in, 56 I Illinois, requirements for professional development, 54 Independence of events, 293 Independent work, recommendation for assigning, 426–427 Informal knowledge, 232–233 building on, 410–411 Instruction as interaction, 313–315 issues in improving, 356–359 in multidigit procedures, importance of, 197 recommendation for giving time to, 422 varied approaches to, 197–198, 382 Instructional materials. See also Manipulatives; Textbooks recommendation for improving, 421–422 use of, 7, 39, 282 Instructional programs, for school mathematics in the U.S., 31, 36–39 Instructional recommendations, 11, 424–427 assigning independent work, 426–427 linking experience to abstraction, 426 managing classroom discourse, 425–426 planning for instruction, 424–425 using calculators and computers, 427 Instructional routines, 11, 382 Instructional triangle, 314 Integers, 72, 244–246 subtraction and, 80–83

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Adding + It Up: Helping Children Learn Mathematics Interaction instruction as, 313–315 with students, 343–344 Interactive perspective, on teaching and learning, 359n Interstate New Teacher Assessment and Standards Consortium, 61n Intertwined strands of mathematical proficiency, 5, 116–117, 217 Inverse operations, 270 relationships, 79 variation functions, 295n Ireland, levels of mathematics achievement in, 56 Irrelevance, of counting order, 160 J Japan assessment of mathematics knowledge in, 35– 36 levels of mathematics achievement in, 56 study of development of proficient teaching in, 396 use of abacus, 218n video studies of mathematics teaching in, 49– 50, 280 Join, problem types, 185 Justification, 130, 273, 276–279. See also Generalizing activities of generalizations, 276–278 problems that involve, 277 of procedures, 130 and proof, 138–139, 170 K Key ideas about number, 110–111 Knowledge of classroom practice, 379–380 clusters of, 120 of instructional practice, 372 of mathematics, 372–378 of students, 371–372, 378–379 Knowledge base building on informal, 410–411 for teaching mathematics, 370–380 Korea levels of mathematics achievement in, 56 subtraction algorithm from, 219n L Language, and early mathematical development, 163 Learners, communities of, 344–345 Learning about chance, 291–293 about numbers, 412–413 and conditional probability, 292–293 current patterns of, 246 independence in, 293 number names, 411–412 numerical algorithms, 414 opportunities for teachers, 333–335 probability comparisons across sample spaces, 292 probability of an event, 291–292 and sample space, 291 single-digit arithmetic, 194–195 solving problems as a context for, 420–421 symbolic, 198 Learning difficulties, 342 Learning goals, for school mathematics in the U.S., 31, 33–36 Learning orientation, versus performance orientation, 171 Learning progression for single-digit addition, 187 for single-digit subtraction, 190 Length measure, 281–282 Lesson study in Japan, 396 Lessons, 337 observed, 48–51 programs focusing on, 395–397 Letters representing unknowns, 270 Licensing requirements, by state, 53 Limitations, on preschoolers’ mathematical proficiency, 172 Line. See Number line Linguistic structure, of number names, 163–166 Linking experience to abstraction, recommendation for, 426 Longitudinal Study of American Youth (LSAY), 374, 399n Louisiana, licensing requirements in, 53 Low expectations, self-fulfilling prophecies of, 343 LSAY. See Longitudinal Study of American Youth

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Adding + It Up: Helping Children Learn Mathematics M Managing discourse, 345–346 recommendation for, 425–426 Manipulatives, 45, 198, 353–354 Massachusetts, exam passing rates in, 42 Mastering counting, 161–162 Materials. See Instructional materials Mathematical knowledge, 372–378 children bringing to school, 5–6, 157–180 of preschoolers, 158–174 state standards for, 34 “street mathematics,” 146n of teachers, 371, 373–378 Mathematical proficiency, 5, 115–154 adaptive reasoning and, 129–131 conceptual understanding and, 118–120 developing over time, 135 discontinuities in, 233–234 intertwined strands of, 5, 116–117, 217 monitoring progress toward, 431–432 need for all students to possess, 142–144 not all or nothing, 135 procedural fluency, 121–124 productive disposition and, 131–133 properties of, 133–135 recommendations concerning, 10–11, 13–14, 408–410, 432 strategic competence and, 124–129 unique position of, 59n in various domains of mathematics, 141–142 Mathematical proficiency of U.S. students today, 136–141 in adaptive reasoning, 138–139 in conceptual understanding, 136–137 and population growth in two towns, 140 in procedural fluency, 137–138 in productive disposition, 139–141 in strategic competence, 138 Mathematical tasks, programs investigating, using cases from real practice, 393–394 Mathematics. See also Teaching mathematics looking at, 20–21 of number, 71–114 power of, 115 programs focusing on, 385–389 Mathematics and learning, 15–29 looking at mathematics, 20–21 mathematics and reading, 17–20 nature of the evidence, 21–24 quality of research studies, 23 role of research in improving school mathematics, 24–26 Mathematics specialists, 397–398 Measure concepts acquiring, 281–284 area, 283 central tendency, 289 and geometry, 281 length, 281–282 volume, 284 Measurement, 281–284 Memory techniques, mnemonic, 119 Mental arithmetic, 214–215, 356 recommendation for using, 415 Mental graphing, in algebra, 275 Metacognition, 117–118 Mexico, teaching experiments in, 265 Michigan, standards of analyzed, 35 Missing-value problems, 243 Mnemonic techniques, 119 Models for multidigit addition, 204 for multidigit multiplication, 208 for multidigit subtraction, 206 Monitoring progress toward mathematical proficiency, recommendation for, 12–13, 431–432 Motivation, 339–341, 360n maintaining an expectation of success, 339–340 valuing learning activities, 340–341 Multidigit addition algorithms for, 199–204 model for, 204 Multidigit division algorithms for, 210–212 common algorithm for, 211 expanded algorithm and model for, 211 expanded algorithm for, with fewer steps, 212 Multidigit multiplication algorithms for, 206–210 beginning algorithm for, 195 common U.S. algorithm for, 207 expanded algorithm for, 209 model for, 208 Multidigit numbers, 195–214 importance of instruction in gaining proficiency with, 197 summary of findings on, 212–214 third-grade class finding 54+48, 200 Multidigit subtraction accessible generalizable methods for, 205 algorithms for, 204–206 common error in, 123 common U.S. algorithm for, 205 model for, 206

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Adding + It Up: Helping Children Learn Mathematics Multiple representations, 95 Multiplication. See also Multidigit multiplication algorithms for, 206–210 beginning an algorithm for, 195 commutativity of, 77 distributivity of over addition, 78 of fractions, 86 and negation, 83 by powers of 10, teaching about, 316–318 properties of, 85 single-digit, 191–192 Multiplicative concepts, 248n identity, 85 inverse, 85 N NAEP. See National Assessment of Educational Progress Names. See Number names Nation at Risk, A, 34 National Advisory Committee on Mathematical Education (NACOME), 48 National Assessment of Educational Progress (NAEP), 36–37, 40, 42, 45–47, 53–57, 117, 234, 285, 356, 374, 432 scores for long-term trend assessment, 136– 138, 141, 143, 147n, 242, 259–260 National Board for Professional Teaching Standards, 54 National Center for Improving Student Learning and Achievement in Mathematics and Science, 60n National centers for research and development in school mathematics, 39 National Council of Teachers of Mathematics (NCTM), 33–34, 36 standards promulgated by, 33–35, 47 National Educational Longitudinal Study (NELS), 347 National Longitudinal Study of Mathematical Abilities (NLSMA), 374 National Research Council (NRC), 2, 17, 44, 132 Strategic Education Research Program, 62n National Science Foundation, 3, 34, 38, 48 Directorate for Education and Human Resources, 3, 26 Natural numbers, 111n NCTM. See National Council of Teachers of Mathematics Needs. See Special needs Negation, 83 multiplication and, 83 opposites of opposites, 83 subtraction and, 83 Negative numbers, 111n concept of, 245 NELS. See National Educational Longitudinal Study Netherlands, investigations of mathematics competence in, 39 New Jersey, requirements for professional development, 54 New Mexico, requirements for professional development, 54 New York exam passing rates in, 42 requirements for professional development, 54 NLSMA. See National Longitudinal Study of Mathematical Abilities North Carolina requirements for professional development, 54 standards for knowledge of mathematics, 34 NRC. See National Research Council Number combinations, 6, 182 derived, 188 Number domain, recommendation for expanding, 418 Number line, 245, 282, 418 linking arithmetic and geometry, 87–93 Number names in Chinese, English, and Spanish, 164–166, 175n learning, 411–412 linguistic structure of, 163–166 psychological consequences of, 166–168 Number-proficient students, 261–263, 270–272 Numbers, 20, 71–114. See also Multidigit numbers; Negative numbers; Rational numbers; Real number system; Representations of numbers; Single-digit numbers building blocks, 106–110 cardinal, 160 choosing and translating among representations, 99–102 decimal, 88–90 grouping and place value, 96–99 key ideas about, 110–111 learning about, 412–413 meanings of, 71, 158

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Adding + It Up: Helping Children Learn Mathematics natural, 111n nested systems of, 93–94 operations on, 75–78, 86 origins of concept of, 159–160 rational, 85–87 representations of, 94–96 systems of, 72, 88–90, 93–94 triangular, 108 whole, 73–75 Numeration system, Hindu-Arabic, 18, 163, 166, 175n Numerical algorithms, learning, 414 O Observed lessons, in teaching mathematics in the U.S. , 48–51 Octagon problem, 109 Office of Educational Research and Improvement, 3, 26 One-to-one relationships, 160 Operating with integers, 245–246 with rational numbers, 415–416 with single-digit whole numbers, 182–195, 413 Opportunity to learn, 333–335 Opposites of opposites, 83 Order irrelevance, 160 Order-of-magnitude accuracy, of calculators, 247 Oregon, requirements for professional development, 54 Organizing data, 289–290 Orientation, 91, 111n Origins of the number concept, counting and, 159–160 P Part-part-whole, problem types, 185 Partial products, 207 Passing rates on exams, by state, 42 Patterns, 217, 278–279 Performance orientation, versus learning orientation, 171 Piaget, Jean, 158, 174n Pile of marbles problem, 184, 186 Pizza sharing problem, 237 Place-value system base-10, 198 extending, 416–417 grouping and, 96–99 Plane geometry, 82 Planning for instruction, 337–338 recommendations concerning, 424–425 Polydrons, 298n Population growth illustration, 140 Positive rational numbers, 84 Powers, 98 of 10, multiplying by, 316–318 Practice for students, 351–352 communities of, 397–398 kinds of, 351–352 recommendation for providing time for, 422– 423 role of, 351 in single-digit calculations, 193 Precision of algorithms, 103 of representations, 101–102 Predicting patterns, 278–279 in the development of student proficiency, 217 Preschool arithmetic, 169 Preschoolers’ mathematical proficiency, 158–174 adaptive reasoning, 170 conceptual understanding, 158–159 counting and the origins of the number concept, 159–160 equity and remediation, 172–174 limitations of preschoolers’ mathematical proficiency, 172 procedural fluency, 162–168 productive disposition, 171 strategic competence, 168–170 understanding and mastering counting, 159– 160 Prices, teaching about comparing, 326–327 Principles and Standards for School Mathematics, 34 Probability of an event, 291–292 comparisons across sample spaces, 292 conditional, 292–293 statistics and, 288–293 Problem model, 125, 268 Problem solving as a context for learning, 420–421 focusing on, 219n, 383 Problem types, 185 addition, 185 compare, 185 involving generalizing and justifying activities, 277 join, 185

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Adding + It Up: Helping Children Learn Mathematics part-part-whole, 185 “routine,” 146n separate, 185 subtraction, 185 word, 169–170, 183–187 Problems bird and worm, 129 book purchase, 261 chocolate distribution, 266 concepts, 125–126 cookie distribution, 376–377 cycle shop, 126 gas price, 125 handshake, 107–109 octagon, 109 pile of marbles, 184, 186 pizza sharing, 237 routine and nonroutine, 125–126 toy cars, 183 water business, 269 weather balloon, 268 Procedural fluency, 5, 10, 116, 121–124, 137–138, 162–168, 380, 382 and a common error in multidigit subtraction, 123 and counting, 162–163 language and early mathematical development, 163 linguistic structure of number names, 163–166 and mathematical proficiency, 121–124 psychological consequences of number names, 166–168 Procedures, 187 benefit of getting students to explain, 221n justifying, 130 Productive disposition, 5, 10, 116, 139–141, 171, 380, 384–385 and mathematical proficiency, 131–133 in the teaching of mathematics, 384–385 Professional development, 31, 51–54 capitalizing on professional meetings, 430 developing specialized knowledge, 428–429 effectiveness of, 398–399 recommendations concerning, 12, 428–431 state requirements for, 54 sustaining, 430–431 for teaching mathematics in the U.S., 51–54 working together, 430 Professional development groups, investigating the concept of equality in, 390 Professional meetings, capitalizing on, 430 Professional Standards for Teaching Mathematics, 34, 51 Proficient teaching of mathematics, 8–9, 380–385. See also Mathematical proficiency; Student proficiency; Teaching for mathematical proficiency and adaptive reasoning, 383–384 and instructional routines, 382 Japanese lesson study, 396 and productive disposition, 384–385 programs to develop, 385–397 and strategic competence, 382–383 and understanding of core knowledge, 381– 382 Programs to develop proficient teaching of mathematics, 385–397 focusing on cases, 392–395 focusing on lesson study, 395–397 focusing on mathematics, 385–389 focusing on student thinking, 389–392 investigating division of fractions, 386–388 investigating equality in a professional development group, 390 investigating mathematical tasks using cases, 393–394 Japanese lesson study, 396 Promoting algebra for all, recommendation concerning, 420 Proof, 130. See also Generalizing activities; Justification Properties of addition, 82 additive identity, 82 additive inverse, 82–83 associativity of, 77 commutativity of addition, 77 Properties of arithmetic operations, 77–78, 82 Properties of mathematical proficiency, 133–135 developing over time, 135 interwoven strands, 133–134 not all or nothing, 135 Properties of multiplication, 85 commutativity of, 77 distributivity of, over addition, 78 multiplicative identity, 85 multiplicative inverse, 85 reciprocals, 85 Proportional reasoning, 8, 241–244 developing, 417 Psychological consequences, of number names, 166–168 Q Quadrilaterals, children’s understanding of, 284

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Adding + It Up: Helping Children Learn Mathematics Quality issues, 23 generalizability of research, 23 relevance of research, 23 soundness of research, 23 R Racial disparities, 55, 148n addressing, 344 Rational numbers, 72, 85–87 computation with, 238 positive, 84 representing and operating with, 415–416 teaching about representations of, 324–325 Reading, mathematics and, 17–20 Reading data, 289–290 Reading Recovery program, 28n Real number system, 90 and its subsystems, 94 Reasoning. See also Adaptive reasoning; Proportional reasoning about advanced concepts, 287–288 about shape and form, 284–287 Reciprocals, 85, 98 Reciprocals of reciprocals, 86 Recitation, 48 Recommendations, 10–14, 407–432 curricular, 10–11, 410–424 for further research, 357–359 instructional, 11, 424–427 for mathematical proficiency, 10–11, 408–410 for monitoring progress, 12–13, 431–432 for supporting the development of mathematical proficiency, 13–14, 432 for teacher preparation and professional development, 12, 428–431 Regrouping, 203, 205 Relevance of research, 23 Remediation, and equity, 172–174 Reported practices, in teaching mathematics in the U.S., 45–47 Representational activities of algebra, 256–257, 261–270 building on spreadsheet experiences, 266 developing meaning, 263–270 water business problem, 269 weather balloon problem, 268 what the number-proficient student brings, 261–263 Representations of numbers, 94–96, 99–102 choosing and translating among, 99–102 clarity of, 100 of data, 290 efficiency of, 99 generality of, 100 multiple, 95 precision of, 101–102 symbol-based, 234, 399n transparency of, 99 Representations of rational numbers, 236–244 computing with, 238 conventional and experimental instruction in, 240–241 learning from students’ errors, 238–240 recommendation for, 415–416 teaching about, 324–325 Representativeness issues, 293 Research. See also Research on teaching convergent, 25 determinants of quality, 22–23 generalizable, 23 relevant, 23 role in improving school mathematics, 24–25 sound, 23 Research on teaching about students and content, 350–356 about teachers and content, 333–338 about teachers and students, 338–350 findings from, 333–356 Research recommendations. See Recommendations S Sample spaces, 291 probability comparisons across, 292 Scaffolding, 336 School mathematics in the U.S., 4, 31–70 assessments of, 39–44 instructional programs and materials goals, 36–39 learning goals, 33–36 and teaching, 45–54 Schools and Staffing Surveys, 54 Second International Mathematics Study (SIMS), 59n Self-fulfilling prophecies, 343 Separate, problem types, 185 Set-combination, 74 Shape, reasoning about, 284–287 Simplicity, of algorithms, 103 SIMS. See Second International Mathematics Study

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Adding + It Up: Helping Children Learn Mathematics Singapore investigations of mathematics competence in, 39 levels of mathematics achievement in, 56–57 Single-digit arithmetic, 6 addition, 185, 187–190 division, 192–193 multiplication, 191–192 practice for students in, 193 subtraction, 185, 190–191 Single-digit numbers, 187–193. See also Multidigit numbers findings on learning about, 194–195 learning progression for, 187, 190 making ten, 189 operating with, 413 thinking strategies for working with, 192 word problems using, 183–187 Socioeconomic disparities, 143 Solutions, as aids to understanding fundamental mathematics, 127 Solving problems focusing on, 219n, 383 as providing contexts for learning, 420–421 Solving word problems, 169–170 Soundness of research, 23 Space-filling, concept for measuring area, 283 Spanish language, number names in, 164–165 Special needs, teaching students with, 341–343 Special Study on Essential Skills in Mathematics (Japan), 40 Specialists, in mathematics, 397–398 Specialized knowledge, developing, 428–429 Spreadsheets, 16 building on experiences with, 266 Stable order of counting words, 160 Stakes, assessments with high, 41–42 Standardized tests, 42–43 defined, 60n State licensing requirements, 53 State passing rates on exams, 42 State requirements for professional development, 54 State standards for knowledge of mathematics, 34 Statistics, probability and, 288–293 Stereotype threat, 133 Sticks, subtraction using, 128 Strands of proficiency. See Intertwined strands of mathematical proficiency Strategic competence, 5, 10, 116, 138, 168–170, 380, 382–383 and mathematical proficiency, 124–129 preschool arithmetic, 169 solving word problems, 169–170 subtraction using sticks, 128 in teaching mathematics, 382–383 Strategic Education Research Program, 62n “Street mathematics,” 146n Student proficiency, patterns in predicting development of, 217 Student thinking, programs focusing on, 389–392 Students, 338–356 assessment of, 349–350 communities of learners, 344–345 cooperative groups, 348–349 doing homework, 352–353 given time to practice, 422–423 giving directives to, 162 grouping, 346–348 interacting with other students, 343–344 managing discourse among, 345–346 motivating, 339–341 practicing, 351–352 with special needs, 341–343 and tasks, 350 teacher expectations of, 338–339 using calculators, 354–356 using manipulatives, 353–354 Students’ errors learning from, 238–240 systematic patterns of, 196 Subtraction. See also Multidigit subtraction algorithms for, 204–206, 219n borrowing in, 204–205 and division, 78–80 and the integers, 80–83 and negation, 83 problem types, 185 single-digit, 190–191 using sticks, 128 Supporting connections, in meaning of rational numbers, 235–236 Supporting the development of mathematical proficiency, recommendation for, 13–14, 432 Sustaining professional development, recommendation for, 430–431 Sweden, use of calculators in, 355 Symbol-based representation, 234, 399n Symbolic learning, 198 Systems of numbers, 72, 88–90 nested, 93–94

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Adding + It Up: Helping Children Learn Mathematics T Table completion task, from NAEP, 260 Task selection and use, teachers’, 335–336 Teacher certification, in teaching mathematics in the U.S., 51–54 Teacher preparation, 31, 51–54 capitalizing on professional meetings, 430 developing specialized knowledge, 428–429 recommendations for, 12, 428–431 sustaining professional development, 430–431 for teaching mathematics in the U.S., 51–54 working together, 430 Teachers, 333–350 assessing students, 349–350 certification of, 31, 51–54 creating communities of learners, 344–345 creating cooperative groups, 348–349 expectations of, 338–339 grouping students, 346–348 interacting with different students, 343–344 managing discourse, 345–346 motivating students, 339–341 opportunities for learning, 333–335 planning content, 337–338 providing opportunities to learn, 333–335 task selection and use, 335–336 teaching students with special needs, 341–343 Teachers’ mathematical knowledge and student achievement, 373–377 and their teaching practice, 377–378 Teaching for mathematical proficiency, 8–9, 313– 368 findings from research on, 333–356 four classroom vignettes, 315–328 instruction as interaction, 313–315 the instructional triangle, 314 issues in improving, 356–359 Teaching mathematics achievement in, 55–57 adaptive reasoning in, 383–384 adding fractions, 320–322 comparing prices, 326–327 experiments in, 265 findings from research on, 333–356 multiplying by powers of 10, 316–318 representations of rational numbers, 324–325 understanding of fundamental mathematics in, 381–382 Teaching mathematics in the U.S., 45–54 achievement, 55–57 coordinating improvement efforts, 58–59 observed lessons, 48–51 reported practices, 45–47 teacher preparation, certification, and professional development, 51–54 Teaching students with special needs, 341–343 Techniques, mnemonic, 119 Technology, using to learn algebra, 274–276, 420 Ten, making, 189 Tendencies. See Central tendency Testing, Teaching, and Learning, 44 Texas standards for knowledge of mathematics, 34 textbook system in, 36 Textbook system, in the U.S., 36–37 Textbooks addition algorithm, in U.S., 203 division algorithm, in U.S., 212 make-a-ten procedure, in U.S., 188 mathematics, in U.S., 27 multiplication algorithm, in U.S., 208 for prospective teachers, 71 word problem solution method, in traditional, 186 Thinking strategies, 192–193. See also Student thinking Third International Mathematics and Science Study (TIMSS), 32–33, 41, 56–57, 356 Video Study, 49–50, 280, 359n Tiling, concept of measuring area, 283 Time for instruction, 422 to practice, 422–423 TIMSS. See Third International Mathematics and Science Study “Top from bottom” error, 204–205 Toy cars problem, 183 Tracking, 346 Trading, 203–204 Transformational activities of algebra, 257–259, 270–276 developing meaning, 272–274 mentally graphing to solve an equation, 275 role of technology, 274–276 two methods for solving equations, 273 what the number-proficient child brings, 270– 272 Translating among representations, 99–102 clarity, 100 efficiency, 99 example, 101–102 generality, 100 precision, 101–102 transparency, 99

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Adding + It Up: Helping Children Learn Mathematics Transparency of algorithms, 103 of representations, 99 Tremont Hotel analogy, 58 Triangle. See Instructional triangle Triangular numbers, 108 U Understanding counting, 161–162 Understanding fundamental mathematics. See also Conceptual understanding attaining a profound, 370 relation to fluency, 196 solutions as aids to, 127 in the teaching of mathematics, 381–382 unknowns as aids to, 127 Undoing operations, 270, 273 United Kingdom, teaching experiments in, 265 United States assessments of school mathematics in, 39–44 instructional programs and materials goals in, 36–39 learning goals in, 33–36 levels of mathematics achievement in, 56–57 state of school mathematics in, 4, 31–70 teaching of school mathematics in, 45–54 U.S. Constitution, 33 U.S. Department of Education, 38 Office of Educational Research and Improvement, 3, 26 Unknowns, as aids to understanding fundamental mathematics, 127 V Valuing learning activities, 340–341 Video Study (TIMSS), 49–50, 280, 359n Vignettes, 315–333 about adding fractions, 320–322 about comparing prices, 326–327 about multiplying by powers of 10, 316–318 about representations of rational numbers, 324–325 comparing the lessons, 328–333 Virginia exam passing rates in, 42 standards for knowledge of mathematics, 34 Volume measure, 284 W Water business problem, 269 Weather balloon problem, 268 Whole numbers, 72–75 multidigit, 195–214 Wisconsin, licensing requirements in, 53 Wise educational environments, 133, 145n Word problems meaningful context for, 183–187 solving, 169–170 Working together, 430 recommendation for, 430 Z Zero, 111n