Index

A

Absolute difference, 311

Absolute thinking

as additive, 311

Access to someone who saw for himself and textbook claims and the nature of sources, 93

Accounts, 59–61

of Colombian voyages, 192–193

different ideas about historical, 38–39

historical, 59–61

substantiated, 87

Actions at a distance

exploring similarities and differences between, 492–493

Activity A1 worksheet, 483

Adams, John, 185

Adaptive reasoning, 218

Adding It Up, 218, 233, 241

Additive reasoning, 311, 321

absolute thinking as, 311

Addressing preconceptions, 399–403

Advantage

selective, 542

Adventure

sense of, 71

Alternative instructional approaches, 321–322

American Association for the Advancement of Science

guidelines of, 398

textbook review by, 16

Analogs of number representations that children can actively explore hands-on, 292–296

Rosemary’s Magic Shoes game, 295–296

Skating Party game, 292–295

Analogy to understand the benchmark experience, 489–490

Ancient views of the Earth as flat or round, 196–197

the Atlas Farnese, 196

the story of Eratosthenes and the Earth’s circumference, 196–197

Anglo-Saxons, 117

Anselm, St., 46

Arguments

inadequacies in, 403

Ashby, Rosalyn, 79–178, 591

Assessment-centered, 415

Assessment-centered classroom environments, 13, 16–17, 267, 290, 292, 555–558

examples of students’ critiques of their own Darwinian explanations, 558



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597 INDEX Index A American Association for the Advancement of Science Absolute difference, 311 guidelines of, 398 Absolute thinking textbook review by, 16 as additive, 311 Analogs of number representations that Access to someone who saw for himself children can actively explore and textbook claims and the nature hands-on, 292–296 of sources, 93 Rosemary’s Magic Shoes game, 295– Accounts, 59–61 296 of Colombian voyages, 192–193 Skating Party game, 292–295 different ideas about historical, 38–39 Analogy to understand the benchmark historical, 59–61 experience, 489–490 substantiated, 87 Ancient views of the Earth as flat or round, Actions at a distance 196–197 exploring similarities and differences the Atlas Farnese, 196 between, 492–493 the story of Eratosthenes and the Activity A1 worksheet, 483 Earth’s circumference, 196–197 Adams, John, 185 Anglo-Saxons, 117 Adaptive reasoning, 218 Anselm, St., 46 Adding It Up, 218, 233, 241 Arguments Additive reasoning, 311, 321 inadequacies in, 403 absolute thinking as, 311 Ashby, Rosalyn, 79–178, 591 Addressing preconceptions, 399–403 Assessment-centered, 415 Advantage Assessment-centered classroom selective, 542 environments, 13, 16–17, 267, 290, Adventure 292, 555–558 sense of, 71 examples of students’ critiques of Alternative instructional approaches, 321– their own Darwinian explanations, 322 558

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598 INDEX sample exam question, and voyage of, 130–132 consistency between models, 557 working things out for ourselves, Assessment systems 133–138 DIAGNOSER, 513 Bridging Assessments. See also Self-assessment from understanding magnetic action formative, 16–17, 193 at a distance to understanding preinstruction, 495 gravitational action at a distance, “reflective,” 412 508–510 Assumptions “Bridging context,” 324, 359 substantive, 127 Briefing sheets, 87, 91 Atlas Farnese, 194, 196 and textbook claims and the nature Authority, 135 of sources, 88–89 Award cards, 293 Building conceptual understanding, Awareness of how you are thinking, 135 procedural fluency, and connected knowledge, 364–369 3-slot schema for graphing a line, B 370–371 developmental model for learning Bain, Robert B., 23, 179–213, 591 functions, 365–366 Balzac, Honoré de, 236 level 0, 364, 367 Barry, Tr., 578 level 1, 367–368 Barton, Keith, 45, 160 level 2, 368 Beakers level 3, 369 a new approach to rational-number Building on children’s current learning, 322–324 understandings, 267–279, 359–364 Bede, St., 58 administering and scoring the Bell jar experiment, 484, 489 Number Knowledge Test, 271 Benchmark lessons, 493–501, 512n mental counting line structure, 276 weighing in a vacuum, 480–483 Number Knowledge Test, 268–269 Black box approaches, 519–520 understandings of 4-year-olds, 270– “Blastoff!”, 298 273 Boorstin, Daniel, 198 understandings of 5-year-olds, 273– Bradford, William, 84–88, 96, 108–111 274 Bransford, John D., 1–28, 217–256, 397– understandings of 6-year-olds, 274– 419, 569–592 277 Brendan, St., 71, 82–83, 128–164, 171 understandings of 7-year-olds, 277– believing historical films when people 278 in them behave as we would, 151 understandings of 8-year-olds, 278– the deficit past, 154–155 279 explanation of words in the story, Building resourceful, self-regulating 132–133 problem solvers, 371–373 finding out what kind of story it is, an integrated understanding of 150–164 functions, 372 grid for evidence on, 173–174 the question, 128 C the shrinking past, 160–161 the story, 128–133 thinking from inside the story, 144– Cambridge History Project, 177n 150 Canada thinking from outside the story, 138– teaching history in, 151 144 “Candles” (unit), 456 Card games, 335–337

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599 INDEX Carey, Susan, 592 Christian geography, 200 Cartier, Jennifer L., 23, 515–565, 592 Circle Land, 286–287 Cartoons, 143, 145–146, 148, 546–549 Claims Peanuts, 309 backing up, 58 sequencing activity, 546–547 Classroom environments Case, Robbie, 23 genetic inquiry in, 529–534 Causal models to account for patterns principles of learning and, 586–588 providing students with opportunities Classroom environments that support to develop, 524 learning with understanding, 555– Causes, 49–54 560 exploring the logic of the situation, assessment-centered classroom 50–51 environments, 13, 16–17, 267, 290, modeling, 562n 292, 555–558 as necessary conditions, 53 community-centered classroom “underlying,” 35 environments, 13, 17–20, 301, Central conceptual structure hypothesis 559–560 bidimensional, for number, 279 knowledge-centered classroom dependence of future learning on the environments, 13–16, 267, 284, acquisition of this structure, 264– 292, 555, 587 265 learner-centered classroom importance of structure to successful environments, 13–14, 266, 292, performance on a range of tasks, 555 262–263 Clumping information, 69 for whole number, 261–262, 275 Codes Change, 43–46, 61 cracking, 335 direction of, 44 Cognitive Tutor Algebra, 355, 391 large-scale patterns of, 68 Colombian Exposition, 208 pace of, 44 Columbus’ voyages, 189–193, 195, 199, as progressive, rational, and limited in 204–205, 207–208, 587 time, 45 Common preconceptions about Cheese and the Worms, 185 mathematics, 220–222 Children as “following rules” to guarantee engaging their emotions and correct answers, 220–221 capturing their imagination, as learning to compute, 220 embedding knowledge only some people have the ability to constructed in their hopes, fears, “do math,” 221–222 and passions, 296–298 Community-centered classroom exposing to major forms of number environments, 13, 17–20, 301, 415, representation, 283–288 559–560 as “natural” scientists, 421 learning with understanding, 559–560 Children passing the Number Knowledge organizing knowledge around core Test concepts, 18–19 and measures of arithmetic learning Comparing number worlds and control and achievement, 265 group outcomes, 304 and numerical transfer tests, 263 Competence developed by students, 1 Children’s Math World project, 219, 223, Comprehensive Test of Basic Skills, 412 227, 229, 231, 236, 241 Computing with percent, 329 Children’s thinking after instruction, 338– Concepts 340 substantive, 61–65 China Concepts of History and Teaching teaching of mathematics in, 15–16, Approaches (Project CHATA), 38– 18–19 39, 51–53, 56, 62, 82

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600 INDEX Conceptual change, 400–403 suggested curricular sequence, 376–377 student conceptions of knowledge two different student solutions to an generation and justification in open-ended problem, 385 science, 402–403 Cut-and-paste, 167 Conceptual explanations Cycles of investigation without conceptual understanding, development of community 578 knowledge across cycles of Conceptual structure investigation, 460 bidimensional central, for number, development of conceptual 279 frameworks for light, 462–467 central, for whole number, 261–262, in guided-inquiry science, 427 275 supporting learning through, 460–467 Conceptual understanding, 218 of light, 423–424 D Conceptualization children’s problems with, 137 Dances with Wolves (film), 151 Connected knowledge, 15–16 Darwin, Charles, 542–545, 550–551, 556, Conquest of Paradise, 208 573 Consistency Darwin’s model of natural selection in high internal and external, 518 school evolution, 540–554 between models, 557 attending to significant disciplinary Constitution, 61 knowledge, 543–544 Context attending to student knowledge, 544– evidence in, 167 545 Continuity, 44 cartoon sequencing activity, 546–547 “Controlled experiments,” 402 explanation written by students on Core concepts, 589 the monarch/viceroy case, 553 organizing knowledge around, 18–19 instruction, 545–554 organizing procedural knowledge and laying the groundwork, 545–549 skills around, 19 understanding, 550–552 Corne, Michael Felice, 90 Data “Counterintuitive” intuitions interpretation of, 403 in history, 33, 42 Data tables from initial recording and with Counting schema, 272 revisions for analysis, 445 Counting words Debugging as the crucial link between the world emphasizing, 239–240 of quantity and the world of Decimals, 332–334 formal symbols, 280–281 magnitude and order in decimal order of, 274 numbers, 333–334 Course outcomes, 181 and stopwatches, 332–333 Curriculum Decisions mandates in, 181 as to what knowledge to teach, 259– from Modeling for Understanding in 267, 281–282 Science Education, 555, 559 Deficit past, 154–155 “openings” in, 245 Dependence, 234, 352 Curriculum for moving students through Design of instruction the model, 373–375 bridging instructional activities, 231 example lessons, 375–389 learning environments and, 12–20 learning slope, 378–381 Development learning y-intercept, 381–384 of community knowledge across operating on y = x2, 384–389 cycles of investigation, 460 sample computer screen, 386

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601 INDEX of Darwin’s model of natural DNA, 517, 526 selection in high school evolution, “Doing,” 32, 48 540–554 “Doing math” of physical concepts in infancy, 4 only some people having the ability of understanding through model- for, 221–222 based inquiry, 515–565 Donovan, M. Suzanne, 1–28, 397–419, Development of conceptual frameworks 569–590, 592 for light, 462–467 Double-blind procedure, 302 community knowledge from the first Dragon Quest game, 297–298 cycle of investigation (first-hand), 463 E community knowledge from the fourth cycle of investigation (first- Earth as flat or round, ancient views of, hand), 467 196–197 community knowledge from the Earth’s circumference second cycle of investigation the story of Eratosthenes and, 196–197 (first-hand), 464 Effects of gravity, 510–511 community knowledge from the third explaining falling bodies, 510–511 cycle of investigation (second- explaining motion of projectiles, 511 hand), 465 Egan, Kieran, 592 Development of mathematical proficiency, 8-year-olds understandings of, 278– 232–236 279 inaccessible algorithms, 236 Elementary Science Study instruction to support mathematical Optics unit, 422, 468 proficiency, 233–236 “Embroidering” stories, 153 a learning path from children’s math Empathy, 46–49, 65, 112 worlds for single-digit addition Encouraging math talk, 228–231 and subtraction, 234–235 Encouraging the use of metacognitive Developmental model processes to facilitate knowledge for learning functions, 365–366 construction, 300–302 DIAGNOSER assessment system, 513 Engage phase, 428–434 Diagnosing preconceptions in physics, 404 Engagement of students’ preconceptions Diagnostic assessment, 491–492 and building on existing Diagnostic questions, 478 knowledge, 4–5, 223–231 Dialogue allowing multiple strategies, 223–227 internal and external, as support for designing bridging instructional metacognition, 241 activities, 231 Direction of change, 44 encouraging math talk, 228–231 Disciplinary knowledge, 32 Engagement of students’ problem-solving attending to significant, 543–544 strategies, 225–227 “second-order,” 61 Equipment Manager, 435 Disconfirmation, 415 Eratosthenes, 194, 196–197 Discrepant events European geographic knowledge providing students with opportunities the great interruption in, 200–201 to experience, 571–573 Everyday concepts Discussion history and, 33–61 guided, 579, 582 of scientific methods, argumentation, DiSessa, Andrea, 5 and reasoning, 400 Distinguishing among kinds of textbook of scientific phenomena, 399–400 claims and the nature of sources, 101–102

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602 INDEX Evidence, 41, 54–58, 61, 65, 112, 120, 165 essential role of factual knowledge in context, 167 and conceptual frameworks in cutting-and-pasting, 167 understanding, 6–9 finding out about the past from importance of self-monitoring in, 10– received information, 56–58 12 historical, 134 “Flat earth,” 189–199 information as, 166 accounts of Colombian voyages, 192– in isolation, 167 193 model of progression in ideas about, ancient views of the Earth as flat or 166–167 round, 196–197 pictures of the past, 166 Formative assessments, 16–17, 193 questions at the heart of using, 124 Forms of representation testimony as, 166 4-year-olds understandings of, 270– Experiments on Plant Hybridization, 529 273 Experts remembering considerably more and the lands in which they appear, relevant detail than novices in 286 tasks within their domain, 8–9 Fourth cycle of investigation Explanations, 156 community knowledge from, 467 of words in the story, 132–133 Fourth graders’ initial ideas about light, 431 Explanatory power, 518 Fractions and mixed representations of External consistency, 518 rational numbers, 334–337 External migration, 68 card games, 335–337 External testing, 181 cracking the code, 335 fractions and equivalencies, 334–335 Framework of How People Learn F seeking a balanced classroom environment, 242–243 Face value Frank, Anne, 109 going beyond, 134 Fundamental physics, 24 Factual knowledge Fundamentalism, 176 manipulating, 79–80 Fuson, Karen C., 23, 217–256, 593 Falling bodies Future real-world experience, 390 explaining, 510–511 Familiarity, 389–390 G the dangers of what appears to be familiar, 122 Feynman, Richard, 24, 403 Galapagos tortoises, 558 Filling the world with people GCK. See Genetics Construction Kit unit on, 169 General ideas, 162 First contacts General meaning of slope, 363 whether St. Brendan sailed from Generalizing and textbook claims and the Ireland to America, unit on, 171 nature of sources, 102–107 why the Norse colonists didn’t stay in Genetics, 516–540 America, unit on, 172 attending to students’ existing First cycle of investigation knowledge, 517–526 community knowledge from, 463 metacognition and engaging students Fish story (Fish Is Fish), 2–12, 398, 414, 575 in reflective scientific practice, 5-year-olds understandings of, 273– 538–540 274 simple dominance homework engaging prior understandings in, 4–5 assignment, 539 student inquiry in, 526–538

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603 INDEX H Genetics Construction Kit (GCK), 534–537 homework assignment, example of student work on, 535 “H(ac)”, 187–188 Genetics content Hall, G. Stanley, 177n learning, 524–526 Halsall, William Formsby, 87 Geographic knowledge Help Christian, 200 seeking and giving, 241–242 the great interruption in European, Heuristic for teaching and learning science 200–201 through guided inquiry, 427–455 Gibbon, Edward, 57 cycle of investigation in guided- GIsML Community of Practice, 470n inquiry science, 427 “Globalization,” 169 data tables from initial recording and Gould, Stephen Jay, 198 with revisions for analysis, 445 Gragg, Charles, 236 engage phase, 428–434 Gravity and its effects, 477–511 fourth graders’ initial ideas about activity A1 worksheet, 483 light, 431 analogy to magnetism, 508 investigate phase, 438–443 bridging from understanding investigative setup for studying how magnetic action at a distance to light interacts with solid objects, understanding gravitational action 437 at a distance, 508–510 prepare-to-investigate phase, 434–438 building an analogy to understand prepare-to-report phase, 443–448 the benchmark experience, 489– report phase, 448–455 490 “H(ev)”, 187 consensus discussion and summary of Higher-order knowledge structure, 276 learning, 490–491 Historical accounts, 59–61 defining, 477–510 different ideas about, 38–39 diagnostic assessment, 491–492 not copies of the past, 62–63 exploring similarities and differences “problematizing,” 184–188 between actions at a distance, Historical evidence, 134 492–493 Historical films, 151 factors on which the magnitude of Historical lines of thinking, 182 gravitational force depends, 501– Historical problems 508 transforming topics and objectives finding out about students’ initial into, 181–199 ideas, 477–478 History, 29–213 identifying preconceptions, 478–480 applying the principles of How People opportunities for students to suggest Learn in teaching high school and test related hypotheses, 484– history, 179–213 489 “counterintuitive” intuitions in, 33, 42 twisting a torsion bar, 493–501 “doing,” 32, 48 weighing in a vacuum, 480–483 implications for planning, 164–176 Grids, 173–175 periods in, 42–43 Griffin, Sharon, 23, 257–308, 593 putting principles into practice, 79– Group work, 582–584 178 Guess My Number, 300 the reality test, 80–84 Guidance of student observation and significance in, 45 articulation that “works,” 65–72 supporting metacognition, 584–585 understanding, 31–77 Guided inquiry, 495, 579, 582 working with evidence, 84–119

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604 INDEX History and everyday ideas, 33–61 providing students with opportunities differences in the power of ideas, 36– to make public, 524 37 “second-order,” 32–33 grounds for caution, 40–41 time, 41–43 ideas we need to address, 41–61 Inaccessible algorithms, 236 the progression of ideas, 37–40 Information, 41, 124, 166 understanding the past and “clumping,” 69 understanding the discipline of finding, 121 history, 34–35 from history, 499 “History-as-account,” 187–188, 203 from the history of science, 499 “History-as-event,” 187, 203 inquiry based, 470n “History-considerate” learning storing in memory, 180 environments Inheritance designing, 199–209 meiotic processes governing, 528 the great interruption in European Initial models geographic knowledge, 200–201 providing students with opportunities with tools for historical thinking, 199– to revise in light of anomalous 209 data and in response to critiques of others, 524 History of the Decline and Fall of the Roman Empire, The, 57 Inquiry based information, 470n Hitler, Adolf, 34–35, 59–60, 586 Instruction, 545–554 Holt, John, 218 to support mathematical proficiency, 233–236 How People Learn: Brain, Mind, Experience, and School, 1, 25, 31–32 Instruction in rational number, 319–340 cautions in, 199 alternative instructional approaches, design characteristics described in, 321–322 12–13, 20–22, 257–258, 359 children’s thinking after instruction, key findings of, 79–80, 171–173, 176 338–340 research summarized in, 241 curriculum overview, 325 violating principles of, 319 fractions and mixed representations How People Learn framework, 411–415 of rational numbers, 334–337 assessment-centered, 415 introduction of decimals, 332–334 community-centered, 415 introduction to percents, 325–332 knowledge-centered, 414 knowledge network, 340 learner-centered, 414 pie charts and a part-whole reflective assessment in ThinkerTools, interpretation of rational numbers, 412–413 320–321 Humor pipes, tubes, and beakers, 322–324 enlivening learning and helping build Instruction that supports metacognition, positive relationships with 239–242 students, 501 emphasizing debugging, 239–240 internal and external dialogue as support for metacognition, 241 I seeking and giving help, 241–242 Instructional lines of thinking, 182 Ideas, 41–61 Intellectual roles for students to adopt, 436 accounts, 59–61 Internal consistency, 518 cause, 49–54 Internal migration, 68 change, 43–46 Interpretation empathy, 46–49 anchoring themes in historical, 186 evidence, 54–58 of data, 403 progression of, 37–40

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605 INDEX Interpreting sources in context and Knowledge claims textbook claims and the nature of in genetics, assessing, 523 sources, 100 Knowledge networks, 340 Intuitions in history new concepts of numbers and new “counterintuitive,” 33, 42 applications, 312–316 Invented procedures, 329 new symbols, meanings, and Investigate phase, 438–443 representations, 313–314 Investigative setup for studying how light reconceptualizing the unit and interacts with solid objects, 437 operations, 315 Irving, Washington, 208 the subconstructs, 314–315 Isolation understanding numbers as evidence in, 167 multiplicative relations, 316 Italy “Knowledge packages,” 588n instruction about payment for work, Knowledge that should be taught, 259–267 66–67 central conceptual structure hypothesis, 262–265 children passing the Number J Knowledge Test, 263, 265 measures of arithmetic learning and Japan achievement, 265 teacher professional development in, numerical transfer tests, 263 244 Koedinger, Kenneth R., 351–393, 593–594 Jasper Woodbury series, 391 Kraus, Pamela, 23, 401, 475–513, 594 Jefferson, Thomas, 62–63 KWL charts, 199, 428–430 Johnson, Lyndon, 62 Jonassen, David, 181 L Judgments avoiding expressing, 498 Lamarck, Jean Baptiste de, 550, 573 Larson, Gary, 217 K Learner-centered classroom environments, 13–14, 266, 292, 414, 555 Kalchman, Mindy, 23, 217–256, 351–393, Learning 593 an active process, 476 Knowledge. See also Prior understandings humor enlivening, 501 building learning paths and networks Learning environments and the design of of, 258 instruction, 12–20 connected, 15–16 assessment-centered classroom disciplinary, 32, 543–544 environments, 13, 16–17, 267, 290, handed down through generations, 292, 555–558 93–94 community-centered classroom manipulating factual, 79–80 environments, 13, 17–20, 301, “metahistorical,” 32 559–560 organized, 462 knowledge-centered classroom “second-order,” 32–33 environments, 13–16, 267, 284, secret, 72 292, 555, 587 student, 258, 544–545 learner-centered classroom of what it means to “do science,” environments, 13–14, 266, 292, 403–407 414, 555 Knowledge-centered classroom perspectives on, 13 environments, 13–16, 267, 284, Learning goals for prekindergarten through 292, 414, 555, 587 grade 2, 284–285

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606 INDEX Learning paths of knowledge Maps, 86, 140–141 building, 258 conceptual, 188 from children’s math worlds, for Marfan’s syndrome, 533 single-digit addition and Math words, 230 subtraction, 234–235 Mathematical proficiency, 218 Learning principles adaptive reasoning, 218 engaging resilient preconceptions, conceptual understanding, 218 569–575 procedural fluency, 218 organizing knowledge around core productive disposition, 218 concepts, 575–577 strategic competence, 218 principles of learning and classroom Mathematical thinkers environments, 586–588 building, 258 pulling threads, 569–590 Mathematical understanding, 217–256 revisiting the three, 567–590 computation without comprehension, supporting metacognition, 577–586 218 Learning rational number, 341–343 developing mathematical proficiency, metacognition, 342 232–236 network of concepts, 341–342 learning to use student thinking in prior understandings, 341 teacher video clubs, 244 Learning with understanding, 559–560 lesson study cycle, 244 supporting knowledge use in new a metacognitive approach enabling situations, 7 student self-monitoring, 236–243 Leather boats, 139–141 suggested reading list for teachers, Lee, Peter J., 23, 31–178, 576, 594 256 Lesson Study Research Group, 244 teachers as curriculum designers, 245 teachers engaging students’ Life and Voyages of Christopher Columbus, The, 208 preconceptions, 219–231 “Light catchers,” 437. See also Study of light understanding requiring factual Linkage knowledge and conceptual of formal mathematical understanding frameworks, 231–236 to informal reasoning, 354–355 Mathematics, 215–393 Lionni, Lee, 2, 4. See also Fish story as about quantity, not about numbers, Logic of the situation 280 exploring, 50–51 as “following rules” to guarantee Lowenthal, David, 185 correct answers, 220–221 fostering the development of whole number sense, 257–308 M as learning to compute, 220 pipes, tubes, and beakers in, 309–349 Ma, Liping, 15–16, 18–19, 577–578 teaching and learning functions, 351– Magic Shoes game, 295–296 393 Magnetism Mathematics instruction analogy to gravity, 508 in China, 15–16, 18–19 Magnitude Mayflower, The in decimal numbers, 333–334 arrival of, 84, 87, 90, 92–95 of gravitational force, 501–508 Medawar, Peter, 406 Magnusson, Shirley J., 421–474, 594 Media Management of student activities, 435 technical and passive, 496 Mandates Meiotic processes curricular, 181 governing inheritance, 528 Manipulation of factual knowledge, 79–80

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607 INDEX Mendel, Gregor, 406, 410, 517, 523, 525– Model-based inquiry, 515–565 529, 539 classroom environments that support model of simple dominance, 528 learning with understanding, 555– Mental counting line structure, 276 560 Metacognition, 10, 238, 407–411, 577–586 developing Darwin’s model of natural conceptual explanation without selection in high school evolution, conceptual understanding, 578 540–554 engaging students in reflective genetics, 516–540 scientific practice, 538–540 Modeling for Understanding in Science in evaluating the methods used in an Education (MUSE), 516, 548 experiment, 408–409 curricula from, 555, 559 guiding student observation and Models, 402–403 articulation, 584–585 consistency between, 557 of light, 426 of progression in ideas about in Mendel’s contribution to genetics, evidence, 166–167 410 providing students with opportunities questioning and explaining in high to revise in light of anomalous school science, 582–583 data and in response to critiques and rational number, 319, 342 of others, 524 supporting, 577–586 Monarch/viceroy case supporting skilled questioning and Darwinian explanation written by explaining in mathematics students on the, 553 problem solving, 580–581 Monitoring. See also Self-monitoring Metacognitive approaches to instruction, 2, metacognitive, 10 80 “Monster-free zone,” 295 enabling student self-monitoring, Moss, Joan, 23, 309–349, 595 236–243 Motion of projectiles framework of How People Learn, 242– explaining, 511 243 Multiple strategies, 223–227 instruction that supports allowing, 223–227 metacognition, 239–242 engaging students’ problem-solving seeking a balanced classroom strategies, 225–227 environment, 242–243 three subtraction methods, 224 supporting student and teacher Multiplicative operators, 315 learning through a classroom Multiplicative reasoning discourse community, 237 relative thinking as, 311 Metacognitive monitoring, 10 MUSE. See Modeling for Understanding in “Metahistorical” knowledge, 32 Science Education “Metamemory,” 11 Mystery Migration sense of, 71 internal and external, 68 “Mystery Object Challenge,” 329 Miller Analogies Test, 404 “Mindtools,” 181 N Minstrell, James, 23, 401, 475–513, 594–595 Minus Mouse, 290–291 Narrative accounts Misconceptions providing students with, 573–575 about momentum, 5 National Council of Teachers of about the scientific method, 414 Mathematics (NCTM), 221, 241, “Missing-term problem,” 317 259 Misunderstandings, 310 standards from, 305

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608 INDEX National Curriculum for History, 177n providing opportunities to link the National Research Council, 1, 218, 221, 233 “world of quantity” with the guidelines of, 398 “world of counting numbers” and the “world of formal symbols,” National Science Education Standards, 455, 561 288–292 Native Americans, 41, 82–83, 98, 105–106 Number Worlds program, 262, 283, 287– NCTM. See National Council of Teachers of 288, 292, 296, 300, 302–303 Mathematics Numeric answers, 372 Necessary conditions causes as, 53 O Neighborhood Number Line, 295 Networks Object Land, 284–286, 288 of concepts, and rational number, “One world” revolution, 169 341–342 “Openings” in the curriculum, 245 of knowledge, building, 258 Opportunities New conceptualizations to develop causal models to account understanding numbers as for patterns, 524 multiplicative relations, 316 to experience discrepant events that New ideas allow them to come to terms with development of, 470n the shortcomings in their everyday New rules models, 571–573 discovering, 588 to make ideas public, 524 New symbols providing students with, 523–524 meanings, and representations, 313– to revise initial models in light of 314 anomalous data and in response “Nothing” happening, 43 to critiques of others, 524 Number Knowledge Test, 260, 264, 267– to search for patterns in data, 524 269, 271, 279, 304–305 to use patterns in data and models to administering and scoring, 271 make predictions, 524 Number worlds, 282–302 to use prior knowledge to pose encouraging the use of metacognitive problems and generate data, 523– processes to facilitate knowledge 524 construction, 300–302 Opportunities for children to acquire engaging children’s emotions and computational fluency as well as capturing their imagination, 296– conceptual understanding, 298–300 298 Sky Land Blastoff activity, 298–299 exposing children to major forms of Opportunities for students to suggest and number representation, 283–288 test related hypotheses in the five forms of representation and elaboration activities, 484–489 the lands in which they appear, inverted cylinder in a cylinder of 286 water, 485–486 learning goals for prekindergarten inverted glass of water, 484–485 through grade 2, 284–285 leaky bottle, 486 providing analogs of number water and air in a straw, 486–488 representations that children can weighing” an object in a fluid actively explore hands-on, 292– medium, 488–489 296 Opportunities to link the “world of providing opportunities for children quantity” with the “world of to acquire computational fluency counting numbers” and the “world as well as conceptual of formal symbols,” 288–292 understanding, 298–300 Minus Mouse, 290–291

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609 INDEX Plus Pup, 288–290 Periods in history, 42–43 Plus Pup meets Minus Mouse, 291–292 Physics Optics kit, 422, 468 fundamental, 24 Order instruction in, 16–17 of counting words, 274 Picture Land, 285–287, 297 in decimal numbers, 333–334 Pie charts and a part-whole interpretation Organized knowledge, 462 of rational numbers, 320–321 Organizing knowledge around core Pilgrim Fathers and Native Americans, 71, concepts 84–119 subtraction with regrouping, 18–19 exploring the basis for textbook Origin of Species, 551 claims and the nature of sources, Outcomes of courses, 181 84–111 grid for evidence on, 173, 175 ideas, beliefs, and attitudes, 112–118 P language of sources, interpretation, and other perspectives, 118–119 Pace of change, 44 teacher questions, 112–113, 115 Paley, William, 550–551, 573 whether people thought like us in the Palincsar, Annemarie Sullivan, 23, 421–474, past, 117 595 Pipes Park, Lesley, 455 a new approach to rational-number Part-whole relation, 314 learning, 322–324 Pass it on (game), 105 a representation for fullness, 325–326 Passive media, 496 Planning, 164–176 Passmore, Cynthia M., 23, 515–565, 595 of progression in ideas about Past evidence, 166–167, 174–175 finding out about, 56–58 unit on filling the world with people, pictures of, 166 169 Patterns in data unit on first contacts, whether St. providing students with opportunities Brendan sailed from Ireland to to search for, 524 America, 171 providing students with opportunities unit on first contacts, why the Norse to use to make predictions, 524 colonists didn’t stay in America, Payment for work in history, 66–67 172 Peanuts cartoon, 309 unit on people going their separate Pedagogical words ways, 170 meaningful, 230 Plausibility, 138 People going their separate ways Plus Pup, 288–290 unit on, 170 meeting Minus Mouse, 291–292 Percents, 325–332, 340 Pocahontas (Disney film), 122 computing with, 329 Pory, John, 84–85, 90, 97, 100–104, 106– in everyday life, 325 108 “families” of, 331 Positive relationships invented procedures, 329 humor helping to build with students, on number lines, 326–329 501 pipes and tubes, as representations Possible Worlds, 406 for fullness, 325–326 Power starting from, 322–324 explanatory and predictive, 518 string challenges, 329–331 Preconceptions, 1, 55, 399–403 Percy, George, 122 about people, society, and how the Performance world works, 127–128 need to assist, 203 conceptual change, 400–403

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610 INDEX drawing on knowledge and “Problematizing” historical accounts, 184–188 experiences that students Procedural fluency, 218 commonly bring to the classroom Productive disposition, 218 but are generally not activated Proficiency with regard to the topic of study, mathematical, 218 569–571 Progress, 44–45 engaging resilient, 569–575 Progression of ideas, 37–40 everyday concepts of scientific different ideas about historical methods, argumentation, and accounts, 38–39 reasoning, 400 Progressive change, 45 everyday concepts of scientific Project CHATA. See Concepts of History phenomena, 399–400 and Teaching Approaches importance of students’, 79 Projectiles providing opportunities for students explaining motion of, 511 to experience discrepant events Proportion, 234, 340 that allow them to come to terms Pump Algebra Tutor. See Cognitive Tutor with the shortcomings in their Algebra everyday models, 571–573 providing students with narrative Q accounts of the discovery of (targeted) knowledge or the Quantity, 234 development of (targeted) tools, schema for, 272 573–575 Question Poser, 300–301 Preconceptions about how we know about Questioning and explaining in high school the past, 121–123 science common student assumptions about supporting metacognition, 582–583 how we know of the past, 123 Questions, 128 dangers of what appears to be diagnostic, 478 familiar, 122 at the heart of using evidence, 124 Predictive power, 518 many as yet unanswered, 492 Preinstruction assessments, 495 teachers modeling for students, 477 Prepare-to-investigate phase, 434–438 Quotient interpretation, 314 Prepare-to-report phase, 443–448 Principles of How People Learn applied to teaching high school history, 179– R 213 designing a “history-considerate” Rational change, 45 learning environment, 199–209 Rational number, 341–343 transforming topics and objectives metacognition, 342 into historical problems, 181–199 network of concepts, 341–342 Prior understandings prior understandings, 341 development of physical concepts in Rational-number learning infancy, 4 and the knowledge network, 312–316 engaging, 4–5 metacognition and rational number, 319 of light, 425 new concepts of numbers and new misconceptions about momentum, 5 applications, 312–316 providing students with opportunities and the principles of How People to use to pose problems and Learn, 312–319 generate data, 523–524 students’ errors and misconceptions and rational number, 341 based on previous learning, 316– Problem solvers 319 building, 258

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611 INDEX Real-world experience diagnosing preconceptions in physics, current and future, 390 404 Real-world words, 230 the How People Learn framework, Reality test, 80–84 411–415 “7-year gap,” 82 knowledge of what it means to “do Reciprocal teaching, 11 science,” 403–407 Reconceptualizing the unit and operations, Scientific method 315 misconceptions about, 414 Recorder, 435 Scissors-and-paste approach and textbook Reflective assessments, 412 claims and the nature of sources, in ThinkerTools, 412–413 94 Regrouping Searchers, The (film), 151 subtraction with, 18–19 Second cycle of investigation Relative thinking as multiplicative, 311 community knowledge from, 464 Relativism, 176 Second-hand investigation, 455–459 Reliability, 126 “Second-order” disciplinary concepts, 61, Religious practices, 113–118 73n Reporter, 301 “Second-order” knowledge, 32–33, 41 Reporting phase, 427, 448–455 acquisition of, 40–41 Representations, 372 Secret knowledge, 72 anchoring themes in historical, 186 Seeing for yourself and textbook claims Reproductive success, 542 and the nature of sources, 93 Revolution, 61 Seixas, Peter, 151 Selective advantage, 542 Self-assessment, 12 S Self-monitoring importance of, 10–12 Sagan, Carl, 194, 196–197 metacognitive monitoring, 10 Sales, Kirkpatrick, 208 Sensitivity Schemas “7-year gap,” 82 2-slot and 3-slot, 370 7-year-olds understandings of, 277– counting and quantity, 272 278 Schools Council History Project, 40, 177n to students’ substantive assumptions, Science, 395–565 127 developing understanding through Severin, Tim, 139, 142–143 model-based inquiry, 515–565 Shemilt, Denis, 23, 56, 79–178, 595–596 guided inquiry in the science Shrinking past, 160–161 classroom, 475–513 Significance, 45 information from the history of, 499 historical, 45 leaving many questions as yet Simplicity, 389–390 unanswered, 492 6-year-olds understandings of, 274– teaching to promote the development 277 of scientific knowledge and Skating Party game, 292–295 reasoning about light at the Skills elementary school level, 421–474 defining, 40 unit on the nature of gravity and its Sky Land, 286–287 effects, 477–511 Blastoff activity, 298–299 Science classrooms Smith, John, 122 guided inquiry in, 475–513 Sources Scientific inquiry and How People Learn, access to someone who saw for 397–419 himself, 93 addressing preconceptions, 399–403 briefing sheet, 88–89

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612 INDEX distinguishing among kinds of claims, initial GCK population for the final 101–102 GCK inquiry, 537 generalizing, 102–107 meiotic processes governing getting behind the record to concerns inheritance, 528 of the people who produced Mendel’s model of simple dominance, them, 107–108 528 interpreting sources in context, 100 Students’ errors and misconceptions based maintaining contact with an on previous learning, 316–319 eyewitness using knowledge Students’ existing knowledge, 517–526 handed down through assessing knowledge claims in generations, 93–94 genetics, 523 the nature of, 84–111 attending to, 544–545 scissors-and-paste approach, 94 black box, 520 seeing for yourself, 93 building on and connecting, 258 teacher questions, 92, 95–96, 99–101 learning genetics content, 524–526 trusting the source who was in a providing students with learning position to know, 96 opportunities, 523–524 understanding the purpose of the student conceptions of models, 518 source, 96–99 Students’ preconceptions understanding what is likely to get importance of, 79 recorded and under what Study of light, 422–426 circumstances, 108–111 conceptual understanding, 423–424 working out the facts from other metacognition, 426 sources or available knowledge, prior knowledge, 425 94–95 Study of light through inquiry, 426–459 Splitting, 323 heuristic for teaching and learning State of affairs science through guided inquiry, changes in, 44 427–455 Stearns, Peter, 210 second-hand investigation, 455–459 Stewart, James, 23, 515–565, 596 Subconstructs “Stop-Start Challenge,” 333 the many personalities of rational Stopwatches number, 314–315 decimals and, 332–333 Subject-specific knowledge in effective Stories science instruction, 467–469 “embroidering,” 153 Substantiated accounts, 87 Strategic competence, 218 Substantive assumptions String challenges sensitivity to students’, 127 guessing mystery objects, 329–331 Substantive concepts, 61–65 Student assumptions about how we know historical accounts not copies of the of the past, 123 past, 62–63 Student conceptions payment for work, 66–67 experimentation, 402 Subtraction with regrouping, 18–19 inadequacies in arguments, 403 Supporting learning through cycles of interpretation of data, 403 investigation, 460–467 of knowledge generation and Supporting skilled questioning and justification in science, 402–403 explaining in mathematics models, 402–403, 518 problem solving Student inquiry in genetics, 526–538 supporting metacognition, 580–581 example of student work on a GCK Supporting student and teacher learning homework assignment, 535 through a classroom discourse genetic inquiry in the classroom, 529– community, 237 534

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613 INDEX T Teaching mathematics in the primary grades, 257–308 Table of values to produce a function, acknowledging teachers’ conceptions 353–358 and partial understandings, 279– Teacher professional development in 281 Japan, 244 building on children’s current Teacher questions, 112–113, 115 understandings, 267–279 and textbook claims and the nature the case of number worlds, 282–302 of sources, 92, 95–96, 99–101 comparing number worlds and Teachers’ conceptions and partial control group outcomes, 304 understandings, 279–281 deciding what knowledge to teach, acquiring an understanding of 259–267 number as a lengthy, step-by-step defining the knowledge that should process, 280–281 be taught, 281–282 counting words as the crucial link Teaching the rational number system, 309– between the world of quantity 349 and the world of formal symbols, additive and multiplicative reasoning, 280–281 311 math as not about numbers, but how students learn rational number, about quantity, 280 341–343 Teachers engaging students’ instruction in rational number, 319– preconceptions, 219–231 340 common preconceptions about rational-number learning and the mathematics, 220–222 principles of How People Learn, engaging students’ preconceptions 312–319 and building on existing Teaching to promote the development of knowledge, 223–231 scientific knowledge and Teaching reasoning about light at the reciprocal, 11 elementary school level, 421–474 Teaching and learning functions in the role of subject-specific knowledge mathematics, 351–393 in effective science instruction, addressing the three principles, 359– 467–469 373 the study of light, 422–426 building conceptual understanding, the study of light through inquiry, procedural fluency, and 426–459 connected knowledge, 364–369 supporting learning through cycles of building on prior knowledge, 359– investigation, 460–467 364 Technical media, 496 building resourceful, self-regulating Testimony, 41, 124, 135, 166 problem solvers, 371–373 Testing linking formal mathematical external, 181 understanding to informal Textbook claims reasoning, 354–355 access to someone who saw for making a table of values to produce a himself, 93 function, 353–358 briefing sheet, 88–89 teaching functions for understanding, distinguishing among kinds of claims, 373–389 101–102 teaching to achieve this kind of generalizing, 102–107 understanding, 358–359 getting behind the record to concerns Teaching as Story Telling, 574 of the people who produced Teaching functions for understanding, 373– them, 107–108 389 interpreting sources in context, 100

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614 INDEX maintaining contact with an Turner, Frederick Jackson, 58 eyewitness using knowledge Twisting the truth, 105, 123 handed down through 2-slot schemas, 370 generations, 93–94 and the nature of sources, 84–111 U scissors-and-paste approach, 94 seeing for yourself, 93 “Underlying” causes, 35 teacher questions, 92, 95–96, 99–101 Understanding trusting the source who was in a essential role of factual knowledge position to know, 96 and conceptual frameworks in, understanding the purpose of the 6–9 source, 96–99 experts remembering considerably understanding what is likely to get more relevant detail than novices recorded and under what in tasks within their domain, 8–9 circumstances, 108–111 learning with understanding working out the facts from other supporting knowledge use in new sources or available knowledge, situations, 7 94–95 Understanding of number Themes, 44 a lengthy, step-by-step process, 280– anchoring in historical representation 281 and interpretation, 186 Understanding the purpose of the source ThinkerTools, 407, 585 and textbook claims and the Third cycle of investigation nature of sources, 96–99 community knowledge from, 465 Understanding what is likely to get Third International Mathematics and recorded and under what Science Study, 243 circumstances 3-slot schema and textbook claims and the nature for graphing a line, 370–371 of sources, 108–111 Three subtraction methods, 224 Unit-level problem, 189–199 Time, 41–43 accounts of Colombian voyages, 192– change limited in, 45 193 periods in history, 43 ancient views of the Earth as flat or Time lines, 129, 159 round, 196–197 Timekeeper, 435 Unit on the nature of gravity and its Torsion bar, 493–501 effects, 477–511 Transforming topics and objectives into United Kingdom historical problems, 181–199 adjusting data from, 177n accounting for the “flat earth,” 189– Schools Council History Project, 40, 199 177n “problematizing” historical accounts, Units 184–188 on filling the world with people, 169 Transmission errors, 123 on first contacts, whether St. Brendan Trusting the source who was in a position sailed from Ireland to America, to know 171 and textbook claims and the nature on first contacts, why the Norse of sources, 96 colonists didn’t stay in America, Truth 172 twisting, 105, 123 on people going their separate ways, Tubes 170 a new approach to rational-number learning, 322–324 a representation for fullness, 325–326

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615 INDEX V Work payment for in history, 66–67 Verbal interpretations, 372 Working out the facts from other sources Visual proportional estimation or available knowledge starting from, and halving and and textbook claims and the nature doubling, 323–324 of sources, 94–95 Working things out for ourselves, 133–138 being aware of how we are thinking, W 135 going beyond face value, 134 War (card game), 336 how far a leather boat could have Warm-Up period, 298, 300 managed to sail, 139–141 Water and air in a straw, 486–488 Working through the task, 128–164 Website, 562n Working with evidence “Weighing” an object in a fluid medium, Pilgrim Fathers and Native Americans, 488–489 84–119 Weighing-in-a-vacuum situation, 484, 489 preparing for the task, 121–128 Whole number the St. Brendan’s voyage task, 128– central conceptual structure for, 261– 164 262, 275 World’s Fair of 1892, 208 Wilson, Suzanne M., 596 Wrap-Up period, 301 Wineburg, Samuel S., 100 Written Arithmetic test, 264–265 Wisdom, 236, 238 Woodbury, Jasper, 391 Y Word Problems test, 264–265 Words versus notations, 230 Year-long historical questions, 184–188 Words in stories explaining, 132–133

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