GUIDING CHILDREN'S LEARNING OF MATHEMATICS

578 Index Calculators, 80–83 mastery checklists, 105–106
addition, practicing, 207 recording information as, 97
Averages. See Mean appropriate use of, 80–81 Children’s literature, 77
Aztec calendar, 479 benefits of, 81–82 act-it-out strategy and, 119–120
with common fractions, 296, 304, 326–327 counting concepts, 146
Balance scales, 368–369, 473 for decade multiplication, 241 deductive reasoning and, 397
Ball, D., 263 with decimal fractions, 307, 325–327 and measurement, 438
Bar graphs, 492–493, 495–498 and dividing common fractions, 318 multicultural activities, 33
MathMaster calculator, 212–213 plane figures, identifying, 401
mean, determining, 512 number sentences and, 125–126 similarity examples in, 408
median, determining, 512 for percent problems, 350 Chinese counting language, 169
See also Histograms repeated subtraction with, 250 Chip trading materials, 164
Barnette, R., 29 research on, 82–83 Christ, G., 304
Baroody, Arthur, 208 subtraction, practicing, 207 Circle graphs, 477, 504–506
Barry, David, 32, 363 See also Casio FX-55 calculator; TI-15 clean numbers for, 504
Base-10 blocks calculator misconceptions about, 506
for addition, 233–234 Circles, 402
division, modeling, 245–246 Calendars area of, 467–468
Base-10 numeration system, 163–167 Aztec calendar, 479 circumference of, 76, 468
assessing understanding, 165–167, 169 Gregorian calendar, 479 similarity of, 408
materials, 163–164 Mayan calendar, 479 Circumference, 468
See also Place-value materials months, number of days in, 479 manipulatives for, 76
Battista, M., 84 multicultural activities, 33 Classifying, 141–142
Battleship, 89 temperature measurement and, 454 large numbers, chart for, 172
Bead cards, 194 time measurement and, 458 manipulatives for, 76
Bean sticks, regrouping with, 232 polygons, 417–420
Beglau, M., 83 Campbell, P. B., 29 separation, understanding of, 398
Behaviorism, 48 Canavale, D., 83 shape relationships and, 20
cognitivism compared, 49 Canterbury Tales, The (Chaucer), 255 sorting boxes, 141–142
Bell curve, 107 Carbo, M., 43 triangles, 419
Benchmarks, 94 Card games, 139 Classroom action research, 56
estimation and, 175 Classroom arrangements, 73
fractions, ordering, 276–277 for numbers and numerals, 153 Clements, D., 83, 88
for measurement, 449 speed of operations and, 206 Clocks
ordering fractions with, 275 Cardinal numbers, 145–146 angles in, 453
Bias, test, 109 counting and, 147 and measurement, 439, 440, 456–457
Bibby, P., 177 Cardinal principal, 147 reading, 456–457
Big numbers. See Large numbers Carpenter, Thomas P., 53 Closed questions, 72
Bilger, B., 40 Carraher, D., 364 Clue, 89
Bimodal values, 499 Carrying, 231 Coerr, Eleanor, 32
Bite-and-push method, 427–428 Carter, G., 104 Cognitive theories, 48
Black Snowman, The (Mendez), 32 Cartesian cross-product, 127, 208, 210 behaviorism compared, 49
Bley, N. S., 39 Cartesian geometry, 412–413 developmental stages, 49–50
Blocks, 74 Casey, B., 392 thinking to learn, 52
Dienes blocks, 50 Casio FX-55 calculator, 304 Coins, 74
number patterns with, 18 with common fractions, 326–327 Collaboration and communication, 21
pattern blocks, 362 Cassidy, J., 85 Colloquialisms, avoidance of, 35
Body/kinesthetic intelligence, 43 Caulfield, R., 157 Colors
Boiling point of water, 479 CDs with textbooks, 84–85 manipulatives, 76
Book of al-Khowarazmi on Hindu Number, Cebulla, K., 54–55, 82 United States maps, coloring, 400
Celsius, 443 Combinations
The, 144 Celsius temperature, 478 and Cartesian cross-product, 127, 208,
Books. See Children’s literature; Textbooks Centimeters
Borja, R., 83 area measurements, 465 210
Box-and-whisker plots, 508–510 converting, 450 and probability, 532–533
Boys. See Gender issues cubic centimeters, 452 Common fractions, 253–254
Brain working with, 463 addition, 292–293
Central tendency measures, 491, 499
pattern making and, 51–52 differences between, 513–514 algorithms for computing, 294–297
thinking to learn, 52 misconceptions about, 514 calculator, using, 296, 304
Brainteasers, 41 research on, 511 with different denominators, 297–299
Breaking old perceptions, 131 See also Mean; Median; Mode with like denominators, 294
Bright, G., 258 Chacon, P., 29 mixed numerals, adding with, 299–302
Brown, E., 383 Challenge problems, 42 algorithms for computing, 294–297
Brownell, William, 48–49 Charts assessment for, 260–265, 280
Bruchac, Joseph, 32 large numbers, classifying, 172 benchmarks for ordering, 275
Bruer, J. T., 122 and mathematical representation, 22 calculators for adding, 296, 304, 326–327
Bruner, Jerome, 48, 49, 50, 55 See also Hundreds chart circles, 261
Burns, Marilyn, 49, 70, 114, 132, 258, 363, Chaucer, Geoffrey, 255 comparing, 272
Checklists comparison model, 257
454, 463 at end of instruction, 105–106

Cabri Geometry, 84
Calculations, 4

Index 579

converting forms, 323–324, 326 Compensation algorithm Counting charts, 147
worksheet for, 349 with addition, 234–235 Counting Jennie (Pittman), 148
with subtraction, 238 Counting on, 199, 201–202
division, 315 Count on Your Fingers African Style
algorithms, development of, 315–318 Compensation principle, 129
whole number, dividing fraction by, in measurement, 448 (Zaslavsky), 32
318 Count Your Way Through Africa (Haskins), 32
Completion subtraction, 191–192, 198 Craig, Michelle, 460
greater than one, 265–266 with common fractions, 293, 296 Crazy Eights, 139
greatest common factor (GCF), 302–303 Creative thinking, 131
introduction of, 259–267 Composite numbers, 178–179 Crews, Donald, 149, 197
least common multiple (LCM), 302–303 manipulatives for, 76 Criterion-referenced tests, 108
misunderstanding, development of, Sieve of Eratosthenes, 179, 181 Cross-product algorithm

265 Comprehensive mathematics education, equivalent fractions and, 340
mixed numerals, 266 27–28 proportion, solving for, 33–344
Cubes, 406–407, 420–421
adding and subtracting with, 299–302 Computational fluency, 14, 208, 229 nets of, 416–417
lesson plan for, 311 “Computer Delusion” (Oppenheimer), 83 See also Unifix cubes
misconceptions about, 314 Computers Cubic centimeters, 452
multiplying, 312–315 Cubic inches/feet, 452
multiplication, 305–315 equity issues and, 37 and volume measurement, 472
fractions by fractions, 308–309, 312 games, 89–90 Cubic numbers, 178
misconceptions, 307 research on use of, 83 manipulatives for, 76
with mixed numerals, 312–315 speed of operations and, 206 Cuevas, G., 132
between whole numbers and fractions, See also Internet; Software Cuisenaire rods, 63
Computing Technology for Math Excellence, for addition, 196
307–308, 310–311 for base-10 numeration, 164
number lines, 263–264 84 decimal fractions with, 268
Concrete experiences, 50 perimeter measurement with, 450
comparisons with, 272–274 Concrete operations, 49–50 Cullinan, B., 31
number sense and operations with, Culture. See Ethnicity
number constancy and, 151 Cups, 452–453
318–319 Cones, volume of, 472 Curriculum
probability expressed as, 528 Congruence, 407–408 assessment and, 94
ratios, expression of, 257–258 content standards, 12
relating to decimal fractions, 323–324 extending concept of, 421–422 mathematics curriculum principle, 6
set in equal-size groups, 256–257 manipulatives, 76 multiculturalism in, 32–34
set of objects, partitioning, 265 Connected mathematics, 21–22 problem-solving curriculum, 132
simplifying, 319–320, 326 Conservation Customary measurement system, 442
single things, partitioning, 259–260 and measurement, 445 Cylinders, 406
stencils, 264 of number task, 148–151 volume, measuring, 471
subtraction, 292–293 Constructions, manipulatives for, 76
Constructivism, 48 D’Ambrosio, Uribitan, 31–32, 36
with different denominators, 297–299 social constructivism, 50 Dassart, D., 82
illustration of work with, 296 themes for teaching mathematics, 55–56 Data analysis
mixed numerals, subtracting with, Consumer Reports, 525
Content pedagogy, 60 content standards, 12, 16–17
299–302 Content standards, 7, 12 highs of children and, 24
teaching, 255–256 integrating process and, 23–25 Data collection, 490–491
for units in equal-size parts, 256 Continuous data, 440 central tendency measures, 491
unlike denominators, adding and subtract- Continuous properties, 438–439 line plots, 492
Converting. See Common fractions; Decimal as third-grade objective, 59
ing with, 297–299 See also Tables
use of term, 255–256 fractions; Metric system Data concepts
See also Equivalent fractions; Whole Cooperative learning, 70–72 correlation standards, 499–500
Coordinate geometry, 395, 412–414 extending, 499–502
numbers National Council of Teachers of Mathemat-
Communication, 21 manipulatives for, 76
reasoning, 413 ics (NCTM) standards, 490, 499–500
all possibilities strategies, 128 Coordination and spatial sense, 391 representing data, 491
common fractions, 274 Copeland, R., 150 sources of data, 491
diagram-drawing strategy, 124 Cordes, C., 83 statistics, 498–499
process standards, 13 Counting, 146–151 See also Histograms; Tables
understanding of, 60 Anno’s Counting Book, 146–147 Davenport, L. F., 29
Community involvement, 60 assessment, 102 Decade multiplication, 240–242
Commutative operations, 200 counting on, 199, 201–202 division, estimation of, 249
Commutative property of addition, 201 discrete objects, 438 for larger numbers, 242
fact families, 204–205 English counting words, 163 Decagons, 403
Commutative property of multiplication, Hindu-Arabic numeration system and, Decimal fractions, 253–254
addition, 292–293, 304–305
219 144–145
learning strategies, 220 language table, 169 with calculator, 307
Compact numeral forms, 172 manipulatives for, 76
Comparing/contrasting, 138–141 multicultural activities, 33
box-and-whisker plots for, 509 overemphasis on, 208
card games and, 139 problems with, 154
common fractions and, 257 research, 157
Comparison subtraction, 191, 198 sequencing and, 143
with common fractions, 293, 296 See also Skip counting
Compatible numbers, 201, 238

580 Index

Decimal fractions (continued) Disconnected mathematics, 21 English counting words, 163
calculators with, 307, 325–327 Discrete data, 440 English system, 441–442
converting forms, 323–324, 326 Discrete objects, 438
worksheet for, 349 Discriminating skills, 138–141 converting
division of, 324 measurements, 463
equivalent fractions, 276 tasks, 139 units to, 450
expanded notation for, 292 Disequilibrium, 50
extending understanding of, 290–292 Distributive property of multiplication, 220 scales, 473
hundredths, introducing, 268–269, 270 standard units in, 449
introducing, 267–272 algorithms and, 239, 242 volume measurement, 452
and length measurement, 464–465 facts, learning, 221 weight units, 473
mixed numerals with, 272 mixed numerals, multiplying, 314–315 Enneagons, 403
multiplying with, 320 Diversity in learning, 60 Equal-addition algorithm, 236–238
newspaper examples, 272 Dividends, 210, 218 Equal participation, 70
and number density, 290, 292 Division, 188 Equal signs, 194
number lines, comparisons with, 272–274 accuracy, building, 224 meaning of, 368
relating common fractions to, 323–324 algorithms, 242–247 Equations
rounding, 277–279 concepts, 13–14 balance aspect of, 368
smaller decimal fractions, 271–272 decade division, 242 extending understanding of, 378–379
subtraction with, 292–293, 304–305 of decimal fractions, 324 introducing, 368
teaching, 255–256 developing concepts, 208 manipulatives for, 76
tenths, introducing, 268, 269 estimation and, 176 See also Balance scales
terminating decimals, 324 facts strategies, 221–224 Equilateral triangles, 401, 419
thousandths, introducing, 271–272 inspection and, 245 perimeters of, 451
use of term, 255–256 introducing, 216–217 similarity of, 407
inverse operations, 210, 218, 221 Equity issues, 28
Decimal points ladder algorithm, 245–246 gifted and talented students, 40–42
addition and subtraction with, 304–305 meanings of, 210–211 Internet and, 85
misconceptions about, 271 mental computation with, 247 learning difficulties, 37–40
number sentences and, 125 limited English proficiency (LEP) stu-
Decimals with regrouping, 246
Hindu-Arabic numeration system and, repeated subtraction, 210–211 dents, 34–37
144–145 sharing/partitioning, 211–212 multiculturalism, 31
manipulatives for, 76 skills development, 209 technological equity, 37
See also Decimal fractions speed, building, 224 See also Gender issues
stories, 217–218 Equity principle, 6, 7–8
Decomposition, 235–236 zero in, 221, 223 grouping and, 70
Deduction stage of geometric understand- See also Common fractions; Remainders Equivalent fractions, 276
Divisor, 210, 218 proportions and, 339–340
ing, 396–397 zero as, 221 Eratosthenes, 179, 181
Dee, Ruby, 32 D’Nealian number forms, 154 Escher, M. C., 428
Degrees of angles, 476 Dobrynina, G., 364 Estimation, 175–177
Del Grande, J., 391 Dodecagons, 375 of area, 465–466
Delivering instruction, 68–72 Dodecahedrons, 420–421 defined, 173
Delpit, L., 34 Domain, 372 expanded algorithms, 234
Denominators, 256 Dominoes and addition, 194 factorials, 248–249
Doorbell Rang, The (Hutchins), 119, 217–218 front-end estimation, 234
derivation of term, 258 Double-classification board, 141–142 with large numbers, 247–249
different denominators, adding and sub- Double facts, 202 of length measurements, 464–465
Dunham, P. H., 82 manipulatives for, 76
tracting with, 297–299 Dürer, Albrecht, 416 mental computation, 238–239
like denominators, adding and subtracting DVDs, use of, 90 procedural learning, skills in, 51
of quotients, 249
with, 294 EBSCO, 52 Snack Stand Supply Problem, 177
and ordering fractions, 277 Education Week, 37 of weight, 473
use of term, 260 Effective instruction, 58–60 Estimation jars, 175–176
Density, 474, 475 Egypt Ethnicity, 30–34
De Paola, T., 32 cooperative learning and, 72
Descartes, René, 412 multiplication, 244 equity principle and, 7–8
Developmental stages, 49–50 rope stretchers, 465 ethnomathematics, 31–34
Diagnostic assessment, 99 Einstein, Albert, 391 test bias, 109
Diagrams Electronic measurement, 440 See also Multicultural activities
mathematical representation and, 22 Elementary School Journal, 54 Ethnomathematics, 31–34
and problem-solving, 120, 122, 124 Ellington, A., 82 Euclid, 396
See also Venn diagrams Ellipses, 402 Euclidean geometry, 395, 396
Dick, T. P., 82 Emotions and memory, 51 angles, 403–405, 406
Dienes, Z. P., 50 Empirical probability, 528–529 congruence, 407–408
Dienes blocks, 50 in geometric setting, 537–538 lines and line segments, 403–405
Digital clocks, 456 Enactive stage of representation, 50 plane figures, 400–403
Dilations, 425 Enclosure concept, 399 points, 403–405
Directed teaching/thinking lessons, 63–66 English, L. D., 54 rays, 403–405
writing plans for, 65–66
Direct measurement, 439–440
iteration, 440
Dirr, P., 83

Index 581

similarity, 407–408 Fraction bars, 258 research on gender advantage, 392
symmetry, 409–411 Fraction circles, 261 stereotypes and, 30
teaching, 400–411 Geoboards, 402
Euler, Leonhard, 304 unlike denominators, adding fractions coordinate geometry activities, 412–413
Euler’s formula, 421 with, 298 perimeters, measurement of, 451
Even numbers. See Odd and even numbers Geofix Shapes, 407
Exchanging place-value materials, 165 Fraction Pie mathlet, 283–284 nets of a cube, 416–417
Expanded notation Fractions, 253–254 for Platonic solids, 420
for addition, 233 Geometer’s Sketchpad, 84
for decimal fractions, 292 algorithms for computing, 294–297 Geometric interpretation of multiplication,
decompensation and, 235–236 assessment for, 280
mixed numerals, multiplying, 314–315 bar symbol, 258 208, 210, 214
Expanded numeral forms, 172 manipulatives for, 76 area measurement, 215
Expected value, 536–539 National Council of Teachers of Mathe- Geometric probability, 533–536
Explorations: Integrating Hand-Held Technol- Geometry, 389–390
matics (NCTM) standards, 254, 288 content standards, 12, 14–15
ogy into the Elementary Mathematics ordering fractions, benchmarks for, extending concepts, 414–430
Classroom (Olson et al.), 304 learning centers for, 74
Exponential notation, 172 276–277 manipulatives for, 76
Extralinguistic clues, 35 ratios compared, 336 National Council of Teachers of Mathemat-
Eye-motor coordination, 391 research
ics (NCTM) standards, 390, 414–415
Face of polyhedrons, 407 abstract, 53 probability, geometric, 533–536
Fact families, 204–205 common fraction representation, 259 proportional reasoning in, 342–343
Factorials, 248–249 on misconceptions, 53 spatial sense development, 390–392
Factoring, manipulatives for, 76 operations with fractions, 294 stages of understanding, 396–397
Factor trees simpler terms, renaming in, 319–320, 326 in three dimensions, 406–407
teaching, 255–256 See also Coordinate geometry; Euclidean
for division, 221 unit fractions, 260
for least common multiple (LCM), See also Common fractions; Decimal frac- geometry; Shapes; Topological geom-
etry; Transformational geometry; van
302–303 tions; Equivalent fractions Hiele model
Fact strategies, manipulatives for, 76 Fraction stencils, 264 Gershaw, D., 29
Fahrenheit temperature, 478 Fraction strips, 276 Gifted and talented students, 40–42
Falkner, K., 56 Frames of the Mind: The Theory of Multiple Giganti, Paul, 61, 197, 198
Feedback Girls. See Gender issues
Intelligences (Gardner), 43 Girl Who Loved Wild Horses, The (Gobel), 32
and computer games, 89 Freezing point of water, 479 Gnadinger, C., 31
in formative assessment, 102–104 French National Assembly, 442 GO!, 33
Feeling box, 139 Front-end estimation, 234 Goals. See Objectives
Feet Gobel, Paul, 32
area measurements, 465 with large numbers, 247–248 Go Fish, 139
square feet, 472 Frustration and addition/subtraction, Goldin, Gerald A., 55
and volume measurement, 472 Gombard, Antoine, 532
See also Cubic inches/feet 206–207 Goral, M., 31
Fennell, S., 199 Functions Grading
Fermat, Pierre de, 532 alternative assessment in, 107
Fibonacci, 258 clock, working with, 381, 383 of homework, 69–70
Fibonacci sequence, 178 extending meaning of, 379–383 Graeber, A., 299
Figure-ground perception, 391 formal definitions of, 372 Graham, A. O., 247
Finger plays, 146 graphing, 381 Grams, 444, 473
Finite numbers, 272 input and output values, 372–373 problem card for weighing in, 474
Flash cards introducing, 369–373 Grandfather Tang’s Story (Tompert), 32
for addition/subtraction facts, 206, 207 Fundamental theorem of arithmetic, 180 Graphs and graphing
for division facts, 224 Fun with Numbers (Maslin), 32 box-and-whisker plots, 508–510
for multiplication facts, 224 Futures Channel, 90 as first-grade objective, 58
triangle flash cards, 204–205, 224 Fuys, D., 396 functions, 381
Flexibility mathematical representation and, 22
number sense and, 162 Gagne, Robert, 66 patterns, 376–378
in problem-solving, 131, 132 Galda, L., 31 picture graphs, 494–495
Flips, 411–412, 425 Galileo, 54–55 as problem-solving strategy, 122–125
Flournoy, Valerie, 32 Gallagher, A., 29 recording form for assessing, 98
Folk stories, multicultural, 32 Gallistel, C. R., 147, 154 software, 497
Ford, Gerald, 442 Gallons, 452 stem-and-leaf plots, 506–508
Formal operational stage, 50 Gambler’s fallacy, 536 use of graphs, 492–493
Formative assessment, 102–104 Games See also Bar graphs; Circle graphs; Line
Formulas graphs
area formulas, 466–470 act-it-out strategy and, 119 Graunt, John, 499
for circle area, 468 computer games, 89–90 Greatest common factor (GCF), 302–303
Euler’s formula, 421 deductive reasoning games, 397 Greenes, C., 30, 132, 358, 360
for trapezoids, 467 matching/discriminating skills and, 139 Greenwich time, 440
volume, measuring, 471 multicultural activities, 33
spatial directions, 398
See also Card games
Gardner, Howard, 43, 391
Garger, S., 28
Gelman, Rachel, 147, 154
Gelosia multiplication, 244
Gender issues, 29–30
equity principle and, 7–8

582 Index

Gregorian calendar, 479 Hundred Penny Box, The (Mathis), 32 Isometric drawings, 416
Grifalconi, Ann, 32 Hundreds chart, 163 Isosceles triangles, 419
Groups Iteration, 440
addition, illustration of, 232–233
cooperative learning in, 70–72 patterns in, 363 Jefferson, Thomas, 442
for gifted and talented students, 42 Hundredths, introducing, 268–269, 270 Jensen, Eric, 52
for instruction, 70 Hunting, R., 259 Johnson, A., 32, 33, 83, 144, 300, 443
for limited English proficiency (LEP) Hutchins, Pat, 119, 217–218 Jones, E., 383
Hypotenuse, 428 Jones, G., 157
students, 36 Journal of Educational Research, 54
place-value materials, grouping, 165 Icosahedrons, 420–421 Journal of Research in Mathematics Educa-
See also Regrouping Identity element, 201, 219
Grouws, D., 54–55, 82 IEPs (individual education plans), 38 tion, 54
Groves, S., 82 If You Made a Million (Schwartz & Kellogg), Journals
Guess and check strategy, 126–127
Guided practice, 64, 69 170, 171 assessment, writing as, 104–105
Guild, P., 28 Ikonic stage of representation, 50 collecting assessment information with, 97
Improper fractions, 265–266 research summaries, 54
Halves Inches Julian calendar, 479
introducing, 261
representation of half, 22 area measurements, 465 Kagan, M., 43
converting, 450 Kagan, Spencer, 43, 70, 71
Handbook of International Research in Math- and volume measurement, 472 Kain, Carolyn, 32
ematics Education (English), 54 See also Cubic inches/feet Kamii, C., 165, 169, 446
Inclusion and learning disabilities, 38 Kaput, J., 344
Handbook of Research on Mathematics Independent practice, 64–65, 69 Karlin, Bernie, 146
Teaching and Learning (Grouws), 54 Indirect measurement, 439–440 Kaufman, J., 29
Individual accountability, 70 Keller, B., 88, 90
Handwriting numerals, 153–154 Individual learning styles, 42–44 Kellogg, Steven, 170
Hart, E., 88, 90 Individuals with Disabilities Education Act Kempe, C. H., 481
Hart, Leslie, 51–52, 117, 359 Kilograms, 443, 473
Haskins, Jim, 32 (IDEA), 37–38
Health studies, 23 Infinite numbers, 272 converting, 450
Hectares, 466 problem card for weighing in, 474
Henry I, England, 442 pi, 467 Kilometers, estimation of, 464
Henry VII, England, 442 teacher-student interaction on, 273, Known-wanted problems, 129–130
Heptagons, 403 Knuckle months, 479
Herman, J. L., 94 275–276 Konold, C., 511
Heuristic, 115 Informal activities, 63 Krueger, A., 43, 116, 307
Hexagons, 402–403 Informal deduction stage of geometric Krutetskii, Igor, 41

tessellations of, 425 understanding, 396–397 Ladder algorithm, 245–246
Hexahedrons. See Cubes Information processing theory, 51 La Disme (Stevin), 268
Hide-and-Seek Cards, subtracting with, 205 InfoTrac College Edition, 52 Ladson-Billings, Gloria, 30–31
Hiebert, James, 51 Instructional environment, 72–75 Lai, Meng-Lung, 208
High, defined, 510 Instruction and assessment, 94, 102–104 Language
Hillman, S., 80 Integers, 179–181
Hindu-Arabic numeration system, 144–145, counting language table, 169
manipulatives for, 76 limited English proficiency (LEP) stu-
161 signs for, 180–181
for fractions, 290 Interdisciplinary teaching, 22, 23 dents, 34–37
and metric system, 443 International Bureau of Weights and Mea- mathematics and, 23
Histograms, 502–510 palindromes, 398–399
box-and-whisker plots, 508–510 sures, 443 patterns in, 359
line graphs, 502–504 Internet, 85–89 Large numbers, 167–172
stem-and-leaf plots, 506–508 addition strategies, 231–239
See also Circle graphs applets, 86–88 estimation with, 247–249
Historical probability, 524 data resources, list of, 86 National Council of Teachers of Mathemat-
History of mathematics, 33 interactive Internet sites, 87
Holistic rubrics, 99 ics (NCTM) standards, 230–231
Homework, 69–70 list of, 88 number sense and, 173, 247–249
collecting assessment information with, 97 learning disabilities, resources for, 40 rounding with, 247–249
for gifted and talented students, 42 useful sites, list of, 86 subtraction strategies, 231–239
self-assessment of, 104 virtual manipulatives, 86–89 vertical notation, 199–200
Homophones, use of, 36 See also Computers See also Algorithms
Hoppe, C., 19 Interpersonal intelligence, 43 La Thiende (Stevin), 268
Horizontal notation, 199 Interpersonal skills, 71 Latino Americans. See Ethnicity
How Many Snails: A Counting Book (Giganti Interquartile range, 509 Lattice multiplication, 244
Interstate New Teacher Assessment and Sup- Learning adventures, 20
& Crew), 61, 197, 198 Learning centers, 73–74
How Much Is a Million? (Schwartz & Kel- port Consortium (INTASC) standards, for measurement activities, 450
60 Learning disabilities, 37–40
logg), 170, 171 Intrapersonal intelligence, 43 cooperative learning and, 72
How to Solve It (Polya), 115 Inverse operations, 196–197 Internet resources, 40
Hoyt, R., 90 division, 210, 218, 221
Huber, B., 29 fact families, 204–205
Humbree, R., 82 multiplication, 210, 218
Invert-and-multiply algorithm, 317
Investigations, problem-based, 66–68
Irregular polygons, 403

Index 583

Learning in the Real World, 83 concepts and, 76 compensation principle, 448
Learning modalities, 43 model-building, 120–121 concept standards, 15–16
Learning principle, 6, 7–8 and number constancy, 151 content standards, 12
Learning styles, 42–44 probability simulations, 539 of continuous properties, 438–439
Learning theories subtraction, modeling, 198 direct measurement, 439
virtual manipulatives, 86–89 English measurement system, 441–442
behaviorism, 48 See also Place-value materials; specific extending concepts, 460–480
cognitive theories, 48 indirect measurement, 439–440
Least common multiple (LCM), 302–303 types manipulatives for, 76
Lechner, G., 131 Many-to-many correspondence, 155–156 of money, 439, 458–460
Lefevre, Patricia, 51 Many-to-one correspondence, 155–156 National Council of Teachers of Mathemat-
Lefrancois, G. R., 49, 51 MapQuest, 425
Legend of the Indian Paintbrush (de Paola), Maps, 14 ics (NCTM) standard, 438, 460–461
nonstandard units of, 446–449
32 similarity concept and, 421–422 of perimeters, 450–451
Leiva, M. A., 114, 360, 362 Markers for multiplication sentence, 213–214 precision in, 441
Length measurement, 450–451 Marlowe, Christopher, 500–502 problem-solving, 480
Marshall, J., 83 processes, 439–440
estimation and, 464–465 Martian Climate Orbiter, 442 qualitative comparison, 446
extending concepts, 461–465 Martine, S., 89 research on concepts, 446
rope stretchers, 465 Marzano, R. J., 72, 138 standard units of, 449–450
Leonardo of Pisa, 258 Mass, 473. See also Weight task card for, 75
Lesson plans Masslin, Jan, 32 teaching/learning model for, 444–446
blank form for, 66 Master Ruler, 461–462 of temperature, 454
daily lesson plans, 62–63 Mastery checklists, 105–106 of time, 454–458
for directed teaching/thinking lessons, Matching skills, 138–141 See also Area; Central tendency measures;

64–66 tasks, 139 Data collection; English system;
for mixed numerals, 311 Materials Length measurement; Metric system;
subtraction, sample plan for, 67 Volume/capacity
Lester, F., Jr., 19, 115 for addition, 193–194 Measurement division, 216–217
Levine, S. C., 30, 392 for base-10 numeration, 163–165 with common fractions, 315, 316
Library, technological equity and, 37 for money measurement, 459–460 for decimal fractions, 324
Likert-type scales, 98 multiplication algorithms and, 239 Measurements-at-Home Hunt, 481
Limited English proficiency (LEP) students, and number constancy, 151 Median, 498, 499
printed materials, 75, 77 bar graphs for, 512
34–37 See also specific types box-and-whisker plots, 508–510
grouping and, 70–71 Math cards for learning disabled students, extreme data cluster and, 514
test bias and, 109 meaning of, 511–512
Linear measures 38–39 misconceptions about, 511
English system, 442 Mathematical probability, 524 representation of, 22
metric system, 443 Mathematical sentences research on, 511
Linear relationships, 504 stem-and-leaf plots for, 512
Line graphs, 502–504 horizontal notation, 199 Memory
data points for, 503 vertical notation, 199–200 information processing theory and, 51
Line plots, 492–493 writing, 125–126 visual memory, 391
Lines, 403–405 Mathematics curriculum principle, 6, 7 Mendez, Phil, 32
symmetry lines, 410 Mathematics Teaching in the Middle School, Mental computation, 238–239
Line segments, 403–405 with division, 247
Line symmetry, 410 24 with multiplication, 247
Linguistic intelligence, 43 Mathis, Sharon Bell, 32 Mental math manipulatives, 76
Linn, M. C., 30, 392 Mathlets for fractions, 283–284 Mental models and length measurement,
Literature. See Children’s literature MathMaster calculator, 212–213 464–465
Liters, 444, 452 Mayan culture Mental rotation, 392
converting, 450 Mercer, M. D., 37, 38
Logical/mathematical intelligence, 43 calendar, 479 Merriam, Eve, 146
Long-range planning, 61 zero, use of, 145 Mestre, Jose, 53
Long-term memory, 51 Mazes Metaphor examples, 140
Low, defined, 510 spatial directions, 398 Meters, 444
Low-stress algorithms topological geometry and, 400 area measurements, 465
for addition, 233–234 McCook, A., 29 converting, 450
expanded algorithms, 234 McDermott, Gerald, 32 Metersticks, 268, 441, 461
for subtraction, 238 McKissack, Patricia, 32 Metric Conversion Act, 442
Lyons, Mary, 32 McMillan, S., 88 Metric system, 268, 442–444
Mean, 498, 499 advantages of, 443–444
Make 10, 202–203 bar graphs for, 512 converting
Malloy, Carol, 72 box-and-whisker plots for, 510 measurements, 463
Malotka, C., 80 extreme data cluster and, 514 units to, 450
Mancala, 33, 89 representation of, 22 equivalents, 444
Manhattan Project, 539 research on, 511 scales, 473
Manipulatives, 74–75 Meaning theory, 48–49
Measurement, 437–439
for addition, 232 accuracy in, 441
algebra tiles, 379 of angles, 453, 475–478
appropriate units for, 441
approximation, 440–441

584 Index

Metric system (continued) Multiple intelligences theory, 43 number standards, 188–190
standard units in, 449 Multiples table, proportions with, 340–343 operations standards, 188–190
volume measurement, 452 Multiple teaching strategies, 60 probability standard, 526
weight units, 473 Multiplicands, 214 proportional reasoning standards, 334
Multiplication, 188 See also Principles and Standards for
Metric tons, 473
Meyer, J., 50 accuracy, building, 224 School Mathematics
Miles algorithms, 239–247 National Library of Virtual Manipulatives, 87
alternative algorithms for, 240 Native Americans. See Ethnicity
estimation of, 464 arrays, 214–215 Naturalistic intelligence, 43
square miles, 466 Cartesian cross-product, 127, 208, 210 Navigation series (NCTM), 84, 132
Miller, E., 83 concepts, 13–14 Near doubles, 202–203
Milligrams, 473 decade multiplication, 240–242 Near squares, 221
Milliliters, 452 with decimal fractions, 320 Negative integers, 179–180
A Million Fish . . . More or Less (McKissack), developing concepts, 208
estimation and, 176 signs for, 180–181
32 factorials, 248–249 Nets, 416–417
Minority students. See Ethnicity introducing, 212–216
Minutes, 454 inverse operations, 210, 218 for Platonic solids, 420
Mira manipulative, 422 learning strategies, 220–221 Neural networks, 51
Missing-addend form, 192 meanings of, 208–210 Newsletters on problem-solving activities,
Mix, Kelly, 208 mental computation with, 247
Mixed numerals, 266 with nine, 221 132
with one, 221 Newspapers
with decimal fractions, 272 rating scales for assessment, 100
See also Common fractions with regrouping, 242–243 decimal fractions and, 272
Mobius strip, 415 rubric scales for assessment, 100 large number searches, 171–172
Mode, 498 skills development, 209 as problem-based project source, 68
defined, 499 speed, building, 224 Nightingale, Florence, 506
research on, 511 squared facts, 221 No Child Left Behind Act of 2001, 3, 5
uses of, 513 by two, 221 fractions, understanding of, 289
Models with zero, 220–221 and gifted and talented students, 40
area models for common fractions, 265 See also Common fractions; Commutative See also Assessment; Standardized tests
building, 120–121 Nominal numbers, 145–146
for geometric figures, 401 property of multiplication; Distribu- Nonlinear relationships, 504
length measurement, mental model for, tive property of multiplication; Nonproportional materials, 163–165
Geometric interpretation of Nonstandard measures, 446–449
464–465 multiplication area, measuring, 465
mathematical representation and, 22 Multiplication operators, 214 Normal distribution, 107
Molina, D., 368 Multiplication sentence, 213–214 Norm-referenced tests, 107–108
Money Multipliers, 214 Norris, K., 443
coin values, 459 Murphy, C., 177 Norwood, K. S., 104
manipulatives for, 76 Muschla, G., 305 Notation
measurement of, 439, 458–460 Muschla, J., 305 for algorithms, 240
multicultural instruction and, 268 Music horizontal notation, 199
Montague, M., 177 intelligence, musical, 43 multicultural activities, 33
Monte Carlo method, 539 repetition in, 359 scientific notation, 172
Months, number of days in, 479 See also Expanded notation; Vertical
Moore, V., 88 Napier, John, 244
Moritz, J. B., 511, 538 Napier’s rods, 244 notation
Morrow, L., 391 National Assessment of Educational Prog- Number concepts
Motion geometry. See Transformational
ress (NAEP), 219 beginning concepts, 143–145
geometry National Center for Improving Student Learn- composite numbers, 178–179
Motivation, understanding of, 60 infinite numbers, discussion of, 273,
Moyer, P., 87 ing and Achievement in Mathematics
Multicultural activities, 31 and Science, 54 275–276
National Council of Teachers of Mathematics manipulatives for, 76
for calendars, 458, 479 (NCTM), 3 National Council of Teachers of Mathemat-
classifying activities, 141 algebra standards, 138, 358, 373
for data collection, 490 on comprehensive mathematics educa- ics (NCTM) standards, 162
fractional problems and, 315 tion, 27–28 patterns, 178
for measurement, 446 correction standard, 499–500 prime numbers, 178–179
patterns, 376 data standards, 490 research on, 177
place-value materials and, 165 extending number/operations standards, Number constancy, 148–151
for probability, 528 230–231 Number lines, 14, 155
on Pythagorean theorem, 428 fractions, standards for, 254, 288 and equivalent fractions, 276
recreational mathematics, 33 geometry standards, 390, 414–415 for percent, 351
shapes, 402 measurement standards, 438, 460–461 rounding, 174
storytelling, 31, 32 Navigation series, 84, 132
for symmetry, 411, 422 number and operations standards, 138 decimal fractions, 277–278
for tessellations, 425–426 number concept standards, 162 See also Common fractions; Decimal
topological geometry, 400
Multicultural Game Book, The (Orlando), 528 fractions
Multiple instructional approaches, 68 Number operations. See Operations
Number puzzles, 126–127
Numbers, 13–14

card games for, 153
compatible numbers, 201

Index 585

content standards, 12 Ordering, 142–143 on problem-solving, 106
large numbers, working with, 167–172 common fractions, benchmarks for, 275 on volume/capacity, 106
linking numerals and, 151–155 counting and, 143 Perimeters
mental computation, 238–239 fractions, 276–277 area compared, 468
National Council of Teachers of Mathemat- in subtraction sentences, 200 manipulatives for, 76
topological geometry and, 398–399 measuring, 450–451
ics (NCTM) standards, 138, 188–190 Petersen, A. C., 30, 392
research, 157 Order-irrelevance principle, 147 Pew Internet and American Life Project, 85
subitizing, 151–152 Ordinal numbers, 145–146 Physical education and mathematics, 23
thinking with, 172–177 Pi, 335
types of, 145–146 introducing, 154–155 and circle area, 467–468
See also Large numbers; Odd and even Orlando, Louise, 528 Piaget, Jean, 48, 49–50, 55, 148–151, 340,
Outliers, 510
numbers; Prime numbers Overlearning, 38 395, 397
Number sense, 162–163 Owens, D., 302 Picture graphs, 494–495
Pictures
common fractions, operations with, Palindromes, 398–399
318–319 Pan balances, 368–369 mathematical representation and, 22
Parallelograms, 417–418 and problem-solving, 120, 122
and large numbers, 173 Park, S., 40 Pie graphs. See Circle graphs
with large numbers, 247–249 Partitioning, 211–212 Pints, 452
research on, 208 Partitive division. See Sharing/partitive Pittman, Helena Clare, 148
Number sentences, 125–126 Place-value charts, 172
Number systems, 33 division Place-value materials, 76, 163–165
Numeral cards, 153–154 Parts, breaking problem into, 129 for addition, 196
Numerals, 145–146 Pascal, Blaise, 130, 530–532, 533 exchanging, trading and regrouping, 165
card games for, 153 Pascal’s Triangle, 530–532, 533 representation, 22–23
compact forms, 172 Patchwork Quilt, The (Flournoy), 32 Place-value system, 163
expanded forms, 172 Pattern blocks, 362 Plane figures, 400–403
linking numbers and, 151–155 Patterns, 51–52, 178 perimeters, measurement of, 451
set cards for matching, 152 Planning
writing, 153–154 auditory patterns, 360 for assessment, 94–101
Numerators, 256 extending understanding of, 373–378 long-range planning, 61
derivation of term, 258 formative assessment of, 102–104 objectives, use of, 60–63
and ordering fractions, 277 functions and, 369–370 understanding of, 60
use of term, 260 graphing, 376–378 unit planning, 61–62
in language, 359 See also Lesson plans
Obelus, 258 manipulatives for, 76 Plato, 407, 420
Object graphs, 494–495 in math learning, 359–363 Platonic solids, 420–421
Objectives misconceptions about, 360 Playing cards. See Card games
people patterns, 118 Point of view, changing, 131
as guide for instruction, 58–60 problem-solving and, 116–118 Points, 403–405
performance objectives, 94–95, 108 in series, 375–376 symmetry, 410–411
for planning instruction, 60–63 task cards and, 360 Pollatsek, A., 511
and rubrics, 99 in thinking, 358–359 Polya, George, 115
Object permanence, 49 See also Algebra Polygons, 402–403
Obtuse angles, 404–405, 453 Pavlov, I., 48, 49 area measurement, 466–467
Obtuse triangles, 419 Pedagogy, 60 classifying, 417–420
Octagons, 403 Pencils, 79 congruence, 407–408
tessellations of, 425 Pentagons, 402–403 irregular polygons, 403
Octahedrons, 420–421 tessellations of, 425 regular polygons, 403
Odd and even numbers, 156 Pentominoes, 416 similarity, 407–408
manipulatives for, 76 Percent Polyhedrons, 407, 420–421
patterns and, 178 calculators for solving, 350 face of, 407
Oliveres, Antonio y, 259 common uses of, 345 Polynomials, manipulatives for, 76
Olson, J., 304 developing concepts of, 344–345 Popcorn Book (de Paola), 32
One-to-many correspondence, 155–156 drawings illustrating, 350 Population
One-to-one correspondence, 147, 155–156 greater than 100, 351–352 bar graphs, 495
Open-ended questions, 72 manipulatives for, 76 circle graphs for, 506
Open sentences, 364 meaning of, 345 large numbers and, 171–172
Operations, 13–14 proportional reasoning and, 351 ranking of, 500
content standards, 12 sales, percent off, 345 statistics and, 499
National Council of Teachers of Mathemat- working with, 346–352 Portfolios as summative assessment strategy,
Percentile scores, 107–108
ics (NCTM) standards, 188–190 Perception of spatial relationships, 391 106
research on, 208 Perceptual constancy, 391 Position-in-space perception, 391
vertical notation, 199–200 Performance indicators, 95–96 Positive integers, 179–180
See also Addition; Common fractions; assessment tasks and, 96
Performance objectives, 94–95 signs for, 180–181
Decimal fractions; Division; Inverse in criterion-referenced tests, 108 Positive interdependence, 70
operations; Large numbers; Multipli- Performance tasks Pounds, 473
cation; Subtraction angle measurement, 478
Operators, ratios as, 258, 336 converting, 450
Oppenheimer, Ted, 83
Oral reports, assessing, 98

586 Index

Practice, 69 research on, 53 Quizzes, 105–107
independent practice, 64–65, 69 rubric for, 99 collecting assessment information with,
simpler problem-solving strategy, 129 97
Preassessment, 101–102 sorting boxes, 142 subtraction quiz, 105
Precision in measurement, 441 tasks, 106
Prensky, M., 89 tools for, 116–131 Quotient, 210, 218
Preoperational stage, 49 working backward strategy, 129–130 estimation of, 249
Prime numbers, 178–179 Procedural learning, 51
Process-oriented problems, 81 Race. See Ethnicity
arrays, 214 Process standards, 7, 13 Radius of circle, 468
manipulatives for, 76 integrating content and, 23–25 Railroads and time zones, 480
Sieve of Eratosthenes, 179, 181 learning processes, 19–23 Rajah’s Rice, The (Barry), 32, 363
Principles and Standards for School Math- Product-oriented problems, 81 Range, 372, 498
Professional development
ematics, 5–6, 55 multicultural diversity and, 34 interquartile range, 509
vision of, 11 understanding of, 60 statistical range, 510
Printed materials, 75, 77 Professional organizations Rate pairs, 338–339
Prism volume, measuring, 472 National Assessment of Educational Prog- Rates
Probability, 523–524 proportions and, 338–340
combinations, 532–533 ress (NAEP), 219 unit rates, 342
common fractions, expression as, 528 National Center for Improving Student Rating scales
content standards, 12, 16–17 at end of instruction, 106
expected value, 536–539 Learning and Achievement in Math- for multiplication assessment, 100
experiments, 529–536 ematics and Science, 54 recording information as, 98
extending understanding, 528–529 research summaries, 54 scoring rubrics, 99
geometric probability, 533–536 See also National Council of Teachers of Rational counting, 146–147
investigations, 529–536 Mathematics (NCTM Ratios, 333–334
learning center assignment, 74 Proofs, 20 common fractions expressing, 257–258
manipulatives for, 76 Proportional reasoning, 333–334 independent sets, relationship between,
misconception about, 528 difficulties with, 336–338
National Council of Teachers of Mathemat- in geometry, 342–343 335
grades 6–8 expectations, 334–335 manipulatives for, 76
ics (NCTM) standard, 526 and percent, 351 meaning of, 335–336
relationships, 336 qualitative problems, 337 misconceptions about, 336
research, 536 and rate pairs, 338–339 as operators, 258, 336
sampling, 528–529, 533 research on, 344 part-part sense, 335
simulations, 539 See also Ratios part-whole sense, 335
spinners, 535–536 Proportions, 333–334 percent in, 345
teaching, 525–528 base-10 numeration, materials for, rate, ratio as, 335
types of, 524 163–165 relationship examples, 257–258
Problem-based learning, 20 equivalent fractions, 339–340 similarity concept and, 421–422
multicultural activities, 33 manipulatives for, 76 Rays, 403–405
projects/investigations, 66–68 multiples table for, 340–343 Reading
Problem cards unitizing, 342 mathematics and, 23
for area measurements, 465 Protractors, 475, 477 multicultural stories, 31, 32
on density, 475 circle graphs with, 477 See also Children’s literature
for measurement, 450 correct alignment of, 478 Ready recall, 205–206
volume/capacity, measuring, 470–471 Proximal development zone. See Zone of Reasoning, 20
for weighing activities, 474 proximal development act-it-out strategies, 119, 120
Problem-solving, 4–5, 113–115 Proximity, 397–398 all possibilities strategies, 128
act-it-out strategy, 118–120 Public opinion surveys, 533 diagram-drawing strategy, 124
with addition, 193 Puzzles, 74 manipulatives for, 76
all possibilities, accounting for, 127–129 number puzzles, 126–127 model-building activities, 121
approaches to teaching, 115 Sudoku puzzle, 33, 131 process standards, 13
breaking old perceptions, 131 topological geometry and, 400 Sieve of Eratosthenes, 181
creative thinking and, 131 Pyramids, volume of, 472 sorting boxes, 142
criteria for, 101 Pythagoras, 428 See also Proportional reasoning
diagram-drawing and, 120, 122, 124 Pythagorean theorem, 428–430 Recording information, 96–99
ethnomathematics and, 34 manipulatives for, 76 anecdotal records, 97
four-step process, 115 checklists, 97
guess and check strategy, 126–127 Quadrilaterals, 402–403 Recreational mathematics, 33
implementing curriculum, 132 classifying, 417–419 Rectangles, 417–418
mathematical sentence-writing, 125–126 similarity of, 407–408 measuring area of, 15
measurement problem-solving, 480 perimeters of, 451
model-building for, 120–121 Qualitative comparison, 446 Rectangular prisms, 406
parts, breaking problem into, 129 Quarts, 452 Reece, C. S., 446
patterns, using, 116–118 Reflections, 411–412, 425
performance levels for, 101 converting, 450 manipulatives for, 76
picture-drawing and, 120, 122 Questions Regrouping, 231
point of view, changing, 131 division with, 246
process standards, 13, 19–20 assessment questions, 95 multiplication with, 242–243
and real-life skills, 114–115 encouraging questions, 72
for journals, 104–105

Index 587

Regular polygons, 403 Rounding, 174–175 Shapes
similarity of, 407 decimal fractions, 277–279 classification and, 20
defined, 173 models for, 401
Regular solids, 420–421 estimation, 247 three-dimensional shapes, 406–407
Rekenweb, 87 with large numbers, 247–249 transformational geometry and, 410–411
Relationships mental computation, 238–239
See also Number lines Sharing/partitive division, 211–212, 217
matching and, 138 with common fractions, 315, 316
patterns, looking for, 178 Rubrics for decimal fractions, 324
ratios and, 257–258 analytic rubrics, 99
Relative position, 397–398 at end of instruction, 106 Sharp EL-500L calculator, 326–327
Remainders, 191, 212 guide for developing, 100 Shaughnessy, M., 524, 538
working with, 218–219 holistic rubrics, 99 Shermo, D., 40
Renaming, 231 for multiplication assessment, 100 Shi-Chieh, Chu, 533
Renzulli, Joseph, 40, 41 recording information as, 97 Short-term memory, 51
Repeated addition, 208, 209–210 scoring rubrics, 99 Sides
Repeated subtraction, 210–211 for unit fractions, 262
with calculator, 250 of polyhedrons, 407
with common fractions, 315 Ruf, D., 40 triangles, classifying, 419
Repeating decimals, 324 Rulers, 74 Sieve of Eratosthenes, 179, 181
Representation, 22–23 Russian peasant multiplication, 244 Signal amid noise, 511
A-B-C pattern representations, 18 Signs, for positive/negative integers, 180–181.
act-it-out strategies, 119, 120 Sadako and the Thousand Paper Cranes
associative properties, 202 (Coerr), 32 See also Equal signs
commutative properties, 202 Similarity
levels of, 50, 62 Sadker, D., 29
model-building activities, 121 Sadker, M., 29 extending concept of, 421–422
process standards, 13 Sage, S., 20 manipulatives for, 76
Research Sales taxes, 345 Simile examples, 140
action research, 56 Sampling, 528–529, 533 Simpler problem-solving strategy, 129
on calculators, 82–83 Sarama, J., 83 Simplifying common fractions, 319–320, 326
on central tendency measures, 511 Scaffolding, 50 Simulations, 539
on computer use, 83 Simultaneous interaction, 70
on cooperative learning, 72 preassessment and, 101 Size manipulatives, 76
on counting, 157 Scalene triangles, 419 Skemp, Richard, 48, 50, 55
gender advantage in mathematics, 392 Scales, 74, 473–474. See also Balance Sketchpad, 84
on graphs and tables, 511 Skinner, B. F., 48, 49
on kindergartner’s problem-solving, 53 scales Skip counting, 155
on learning, 52–56 Scanlon, R., 29 and multiplication, 212, 220
on measurement concepts, 446 Schacter, J., 83 and squared facts, 221
on number concepts, 177 Schema Skolnick, Joan, 122
on numbers, 157 Slang, avoidance of, 35
on number sense, 208 for classifying, 141 Slavin, R., 72
on operations, 208 and concrete experiences, 50 Slide rules, 79
on probability, 536 in development theory, 50 Slides, 411–412, 425
proportional reasoning approaches, 344 Schenk de Regniers, Beatrice, 193 Smaller parts, breaking problem into, 129
recommendations from, 54–55 School involvement, 60 Smoot, Oliver R., Jr., 449
reviews of, 53–55 Schroeder, T., 19, 115 Smoot measurement unit, 449
on software programs, 84 Schwartz, David, 170, 171 Snack Stand Supply Problem, 177
spatial skills, 122 Science and mathematics, 23 Social constructivism, 50
on tables and graphs, 511 Scientific method, 113 Social interactions, 50
variables, use of, 364 Scientific notation, 172 Social studies and mathematics, 23
Reversal errors, 154 Scoring rubrics, 99 Software, 83–85
Reversibility, sense of, 398–399 Secada, W. G., 30 for box-and-whisker plots, 510
Rhombus, 417–418 Self-assessment graphing software, 497
Ridge, Laurence, 40, 41 for addition/subtraction facts, 206 list of, 85
Right angles, 404–405, 453 angle measurement, 478 for multiplication/division accuracy, 224
similarity and, 407 of homework, 70, 104 Soldier, Lee Little, 72
Right triangles, 419 during instruction, 104 Solidus, 258, 259
Pythagorean theorem, 428–430 Sensorimotor stage, 49 So Many Cats! (Schenk de Regniers), 193
Rigor stage of geometric understanding, Separation, understanding of, 398 Sorting, manipulatives for, 76
Sequencing, 142–143 Sorting boxes, 141–142
396–397 counting and, 143 Soto-Johnson, H., 29
Rivers, C., 29 Fibonacci sequence, 178 Space for instruction, 73
Roman (Julian) calendar, 479 for time measurement, 454–455 Spatial intelligence, 43, 391
Roman numerals, 144 topological geometry and, 398–399 Spatial sense, 14–15
Rope stretchers, 465 Seriation, 142–143 development of, 390–392, 415
Rotational symmetry, 410–411 counting and, 143 gender and, 30, 392
Rotations, 411–412, 425 patterns in series, 375–376 manipulatives for, 76
Sesame Street, 139 Mobius strip, 415
manipulatives for, 76 Set cards for matching numerals, 152 topological geometry and, 399
Rote counting, 146–147 Shadows and indirect measurement, 439 Speed
Shakespeare, William, 500–502 in addition, 205–207
Shape cards, 261

588 Index properties of, 200–201 expanding concepts, 478
quiz, 105 measurement of, 440, 454
Speed (continued) sequence of skills for, 190 Templates for rubrics, 99
in division, 224 speed in, 205–207 Ten Black Dots (Crews), 149
in multiplication, 224 stories, 198–199 Ten minus 1, 202–203
in subtraction, 205–207 zero in, 236 Ten Plus, 203
See also Common fractions; Takeaway Tenths, introducing, 268, 269
Spinners Terminating decimals, 324
expected value and, 536–537 subtraction; Whole-part-part Tessellations, 425–428
geometric probability and, 535–536 subtraction bite-and-push method, 427–428
Sudoku puzzle, 33, 131 manipulatives for, 76
Sports Illustrated/Sports Illustrated for Kids, Suggate, J., 157 Tests
272 Sums, 190–191 collecting assessment information with,
introduction of term, 194
Squared facts, 221 Sundials, 457 97
Square feet, 472 Super, D., 302 timed tests, 206
Square miles, 466 Sutton, J., 43, 116, 307 See also Quizzes; Standardized tests
Square neighbors, 221 Swensson, A., 122 Tetrahedrons, 420–421
Square numbers, 178 Symbolic stage of representation, 50 Texas Assessment of Academic Skills Test,
Symbols
manipulatives for, 76 and limited English proficiency (LEP) 122, 123, 125
Squares students, 36 Texas Instruments
mathematical representation and, 22
fractions and, 259–260 in preoperational stage, 49 TI-10 Explorer, 307
misconceptions about, 415 in procedural learning, 51 See also TI-15 calculator
perimeters of, 451 Symmetry, 409–411 Textbooks, 75, 77
similarity of, 407 extending concept of, 422–424 CDs with, 84–85
tessellations of, 425 in Islam, 422 Theoretical probability, 528–529
as trapezoids, 417–418 manipulatives for, 76 Thermometers, 478
Stable-order principle, 147 Synchronicity and clocks, 440 Thinking-learning skills, 52, 138–143
Standard deviation, box-and-whisker plots Thirteen Moons on the Turtle’s Back
Tables, 491–494
for, 510 extending concepts, 500 (Bruchac), 32
Standardized tests, 94 mathematical representation and, 22 Thompson, Francis, 205, 236
as problem-solving strategy, 122–125 Thorndike, E. L., 48
bias, 109 stem-and-leaf plots from, 507–508 Thornton, C., 39
concerns about, 108–109 Thorp, L., 20
criterion-referenced tests, 108 Tagboard ruler, 463 Thousandths, introducing, 271–272
interpreting, 107–109 Takeaway subtraction, 191, 197 Three-dimensional geometry
norm-referenced tests, 107–108
Standard units of measurement, 449–450 with common fractions, 293, 296 extending meaning of, 420–421
Statistics, 498–499 Tanenhaus, E., 247, 299 introducing, 406–407
extending concepts, 510–516 Tangram pieces, 411–412 TI-15 calculator, 81, 304
Steeh, J., 29 Tapasak, R. C., 29 with common fractions, 326–327
Stem-and-leaf plots, 506–508 Tape measures, 450–451 TI-10 Explorer, 307
median, determining, 512 Tarr, J., 526 Tiling patterns, 425–428
Stereotypes, gender, 30 Task cards Time
Stevin, Simon, 268, 442 duration, 454
Stimulus-response theory, 48 in learning centers, 74 extending concepts, 479–480
Story of Money, The (Kain), 32 for measurement, 75 for instruction, 72–73
Story Quilts of Harriat Powers (Lyons), 32 and pattern work, 360 measurement of, 440, 454–458
Storytelling and multiculturalism, 31, 32 for rounding, 174–175 sundials, 457
Student development, 60 Tasks for assessment, 96 telling time, 16
Subitizing, 151, 153 Taxes, percent in, 345 See also Calendars; Clocks
Subtraction, 188, 190–192 Taylor, L., 396 Timed tests, 206
accuracy in, 205–207 Teaching algorithms, 233 Time zones, 479
addition, inverse relationship with, Teaching Children Mathematics, 24, 54, 56, Tischler, R., 396
Tompert, Ann, 32
196–197 132 Tons, 473
algorithms for, 235–240 Teaching principle, 6, 8 Topological geometry, 395
alternative algorithms for, 236–237 Teaching to the test issue, 108 enclosure concept, 399
calculators, practicing with, 207 Team building, 71 mazes and, 400
compensation algorithm, 238 Technology order, understanding of, 398–399
concepts, 13–14 proximity, 397–398
decimal fractions, 292–293, 304–305 calculators, 80–83 puzzles and, 400
decomposition, 235–236 equity, technological, 37 relative position, 397–398
developing concepts of, 192 understanding of, 60 separation, understanding of, 398
equal-addition algorithm, 236–238 video technology, 90 teaching, 397–400
estimation and, 176 See also Computers; Internet Tracking, grouping and, 70
flash cards for, 206, 207 Technology principle, 6, 8–9 Trading, 231
identity element for, 201 Temperature place-value materials, 165
introducing, 196–199 Celsius, 443 Traffic signs, shapes of, 401
large numbers, strategies for, 231–239 converting from Celsius to Fahrenheit, Transference, 439
learning strategies for, 204–205
lesson sample lesson plan for, 67 479
low-stress algorithm, 238
mental computation, 238–239
preassessment for, 102

Index 589

Transformational geometry, 395 van Garderen, D., 177 metric system, 443
extending concepts of, 424–430 van Hiele, Dina, 396–397 teaching measurement of, 453
manipulatives for, 76 van Hiele-Geldof, Pierre Marie, 396–397 Wenglinsky, H., 83
Pythagorean theorem, 428–430 van Hiele model, 392, 396–397 West, M., 344
teaching, 411–412 What Are You Teaching My Child? (Burns),
tessellations, 425–428 plane figures in, 400
three-dimensional geometry, 406 132
Transitional algorithms, 233 triangles and, 401 Wheatley, G. H., 427
Transitivity concept, 439 Variables, 363–367 Whitin, D., 492
Translations, 411–412, 425 misconceptions about, 364, 366 Whole-class instruction, 70
research on use of, 364 Whole numbers, 179–180
manipulatives for, 76 use of, 364
Trapezoids, 417–418 Venn, John, 304 common fractions, multiplication with,
Venn diagrams, 122, 123, 304 307–308, 310–311
area measurement, 466–467 least common multiple (LCM) and, 303
formulas for, 467 for quadrilaterals, 417, 419 dividing fraction by, 318
Trial and error strategy, 126–127 for trapezoids, 417, 419 manipulatives for, 76
Trials, 529–530 Vertical notation, 199–200 rounding decimals to, 277–278
Triangle flash cards, 204–205, 224 common fractions, adding and subtract- Whole-part-part subtraction, 191, 197
Triangles, 402–403 with common fractions, 293, 296
area measurement, 466 ing, 297 Who Wants to Be a Millionaire?, 119–120
classification of, 419 Videos Williams, S., 368
misconceptions about, 401 Wipeout, 325
similarity of, 422–423 technology, 90 Wood, D., 177
tessellations of, 425 What Are You Teaching My Child? (Burns), Working backward strategy, 129–130
See also Equilateral triangles; Pythago- A World of Mathematics: Activities for Grades
132
rean theorem; Right triangles Village of Round and Square Houses, The 4, 5, and 6 Using the TI-15 (Christ),
Trigonometry, 439 304
Trundle wheels, 450–451, 461 (Grifalconi), 32 Wright, Frank Lloyd, 395
Tsankova, E., 364 Virginia State Board of Education model, Wright, R. J., 157
Tsankova, J., 30 Writing
Tukey, John, 508 34–36 as assessment, 104–105
Turns, 411–412, 425 Virtual manipulatives, 88 numerals, 153–154
Tutoring by gifted and talented students, 42 Vision of mathematics, 4–5 reversal errors, 154
12 Ways to Get to 11 (Merriam & Karlin), 146 Visualization stage of geometric understand-
Two Ways to Count to Ten: A Liberian Folk- Yards
ing, 396–397 converting, 450
tale (Dee), 32 Visually impaired students, 39–40 length of, 442
Visual memory, 391
Unifix cubes, 127 Volume/capacity, 443 Yardsticks, 441, 461
for bar graphs, 404 Yeatts, K., 132
for common fractions, 265 area, conversion to, 472
for commutative law for addition, 201 English system, 442 Zahorik, J. A., 56
for linking numbers and numerals, 151 extending concepts, 470–472 Zanger, V., 34
perimeter, measurement of, 450 formulas for, 471 Zaslavsky, Claudia, 32
place-value materials, 163–164 manipulatives for, 76 Zero
performance task, 106
United States maps, coloring, 400 problem cards for measuring, 470–471 decade multiplication and, 241–242
United States National Institute of Standards teaching measurement of, 452–453 in division, 221, 223
Vygotsky, Lev, 50, 55, 101 Hindu-Arabic numeration system and, 145
and Technology, 440 as identity element, 201
United States Naval Observatory time, 440 Watches. See Clocks multiplying with zero, 220–221
Unit fractions, 260 Water, boiling/freezing point, 479 in subtraction, 236
Watson, J. B., 49, 511 Zone of proximal development, 50
scoring rubric for, 262 Watson, J. M., 536 preassessment and, 101
Unitizing, 342 Weather data, 525
Unit rates, 342 Weight
Units
defined, 473
directed teaching/thinking lessons, 64 density, 474, 475
planning for, 61–62 English system measures, 442
Uno, 139 estimation of, 473
Usiskin, Z., 81 extending concepts, 472–474
Uyeda, Dee, 265 measurement of, 440