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INDEX 597 Index This index includes the text of the full version of How Students Learn: History, Mathematics, and Science, which can be found on the CD attached to the back cover. 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, 292296 of sources, 93 Rosemary's Magic Shoes game, 295 Accounts, 5961 296 of Colombian voyages, 192193 Skating Party game, 292295 different ideas about historical, 3839 Analogy to understand the benchmark historical, 5961 experience, 489490 substantiated, 87 Ancient views of the Earth as flat or round, Actions at a distance 196197 exploring similarities and differences the Atlas Farnese, 196 between, 492493 the story of Eratosthenes and the Activity A1 worksheet, 483 Earth's circumference, 196197 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, 79178, 591 Addressing preconceptions, 399403 Assessment-centered, 415 Advantage Assessment-centered classroom selective, 542 environments, 13, 1617, 267, 290, Adventure 292, 555558 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, 130132 consistency between models, 557 working things out for ourselves, Assessment systems 133138 DIAGNOSER, 513 Bridging Assessments. See also Self-assessment from understanding magnetic action formative, 1617, 193 at a distance to understanding preinstruction, 495 gravitational action at a distance, "reflective," 412 508510 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, 8889 Award cards, 293 Building conceptual understanding, Awareness of how you are thinking, 135 procedural fluency, and connected knowledge, 364369 3-slot schema for graphing a line, B 370371 developmental model for learning Bain, Robert B., 23, 179213, 591 functions, 365366 Balzac, Honor de, 236 level 0, 364, 367 Barry, Tr., 578 level 1, 367368 Barton, Keith, 45, 160 level 2, 368 Beakers level 3, 369 a new approach to rational-number Building on children's current learning, 322324 understandings, 267279, 359364 Bede, St., 58 administering and scoring the Bell jar experiment, 484, 489 Number Knowledge Test, 271 Benchmark lessons, 493501, 512n mental counting line structure, 276 weighing in a vacuum, 480483 Number Knowledge Test, 268269 Black box approaches, 519520 understandings of 4-year-olds, 270 "Blastoff!", 298 273 Boorstin, Daniel, 198 understandings of 5-year-olds, 273 Bradford, William, 8488, 96, 108111 274 Bransford, John D., 128, 217256, 397 understandings of 6-year-olds, 274 419, 569592 277 Brendan, St., 71, 8283, 128164, 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, 154155 279 explanation of words in the story, Building resourceful, self-regulating 132133 problem solvers, 371373 finding out what kind of story it is, an integrated understanding of 150164 functions, 372 grid for evidence on, 173174 the question, 128 the shrinking past, 160161 C the story, 128133 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, 335337

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INDEX 599 Carey, Susan, 592 Christian geography, 200 Cartier, Jennifer L., 23, 515565, 592 Circle Land, 286287 Cartoons, 143, 145146, 148, 546549 Claims Peanuts, 309 backing up, 58 sequencing activity, 546547 Classroom environments Case, Robbie, 23 genetic inquiry in, 529534 Causal models to account for patterns principles of learning and, 586588 providing students with opportunities Classroom environments that support to develop, 524 learning with understanding, 555 Causes, 4954 560 exploring the logic of the situation, assessment-centered classroom 5051 environments, 13, 1617, 267, 290, modeling, 562n 292, 555558 as necessary conditions, 53 community-centered classroom "underlying," 35 environments, 13, 1720, 301, Central conceptual structure hypothesis 559560 bidimensional, for number, 279 knowledge-centered classroom dependence of future learning on the environments, 1316, 267, 284, acquisition of this structure, 264 292, 555, 587 265 learner-centered classroom importance of structure to successful environments, 1314, 266, 292, performance on a range of tasks, 555 262263 Clumping information, 69 for whole number, 261262, 275 Codes Change, 4346, 61 cracking, 335 direction of, 44 Cognitive Tutor Algebra, 355, 391 large-scale patterns of, 68 Colombian Exposition, 208 pace of, 44 Columbus' voyages, 189193, 195, 199, as progressive, rational, and limited in 204205, 207208, 587 time, 45 Common preconceptions about Cheese and the Worms, 185 mathematics, 220222 Children as "following rules" to guarantee engaging their emotions and correct answers, 220221 capturing their imagination, as learning to compute, 220 embedding knowledge only some people have the ability to constructed in their hopes, fears, "do math," 221222 and passions, 296298 Community-centered classroom exposing to major forms of number environments, 13, 1720, 301, 415, representation, 283288 559560 as "natural" scientists, 421 learning with understanding, 559560 Children passing the Number Knowledge organizing knowledge around core Test concepts, 1819 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, 6165 China Concepts of History and Teaching teaching of mathematics in, 1516, Approaches (Project CHATA), 38 1819 39, 5153, 56, 62, 82

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600 INDEX Conceptual change, 400403 suggested curricular sequence, 376377 student conceptions of knowledge two different student solutions to an generation and justification in open-ended problem, 385 science, 402403 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, 462467 central, for whole number, 261262, in guided-inquiry science, 427 275 supporting learning through, 460467 Conceptual understanding, 218 of light, 423424 Conceptualization D children's problems with, 137 Connected knowledge, 1516 Dances with Wolves (film), 151 Conquest of Paradise, 208 Darwin, Charles, 542545, 550551, 556, Consistency 573 internal and external, 518 Darwin's model of natural selection in high between models, 557 school evolution, 540554 Constitution, 61 attending to significant disciplinary Context knowledge, 543544 evidence in, 167 attending to student knowledge, 544 Continuity, 44 545 "Controlled experiments," 402 cartoon sequencing activity, 546547 Core concepts, 589 explanation written by students on organizing knowledge around, 1819 the monarch/viceroy case, 553 organizing procedural knowledge and instruction, 545554 skills around, 19 laying the groundwork, 545549 Corne, Michael Felice, 90 understanding, 550552 "Counterintuitive" intuitions Data in history, 33, 42 interpretation of, 403 Counting schema, 272 Data tables from initial recording and with Counting words revisions for analysis, 445 as the crucial link between the world Debugging of quantity and the world of emphasizing, 239240 formal symbols, 280281 Decimals, 332334 order of, 274 magnitude and order in decimal Course outcomes, 181 numbers, 333334 Curriculum and stopwatches, 332333 mandates in, 181 Decisions from Modeling for Understanding in as to what knowledge to teach, 259 Science Education, 555, 559 267, 281282 "openings" in, 245 Deficit past, 154155 Curriculum for moving students through Dependence, 234, 352 the model, 373375 Design of instruction example lessons, 375389 bridging instructional activities, 231 learning slope, 378381 learning environments and, 1220 learning y-intercept, 381384 Development operating on y = x2, 384389 of community knowledge across sample computer screen, 386 cycles of investigation, 460

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INDEX 601 of Darwin's model of natural DNA, 517, 526 selection in high school evolution, "Doing," 32, 48 540554 "Doing math" of physical concepts in infancy, 4 only some people having the ability of understanding through model- for, 221222 based inquiry, 515565 Donovan, M. Suzanne, 128, 397419, Development of conceptual frameworks 569590, 592 for light, 462467 Double-blind procedure, 302 community knowledge from the first Dragon Quest game, 297298 cycle of investigation (first-hand), 463 community knowledge from the E fourth cycle of investigation (first- hand), 467 Earth as flat or round, ancient views of, community knowledge from the 196197 second cycle of investigation Earth's circumference (first-hand), 464 the story of Eratosthenes and, 196197 community knowledge from the third Effects of gravity, 510511 cycle of investigation (second- explaining falling bodies, 510511 hand), 465 explaining motion of projectiles, 511 Development of mathematical proficiency, Egan, Kieran, 592 232236 8-year-olds understandings of, 278 inaccessible algorithms, 236 279 instruction to support mathematical Elementary Science Study proficiency, 233236 Optics unit, 422, 468 a learning path from children's math "Embroidering" stories, 153 worlds for single-digit addition Empathy, 4649, 65, 112 and subtraction, 234235 Encouraging math talk, 228231 Developmental model Encouraging the use of metacognitive for learning functions, 365366 processes to facilitate knowledge DIAGNOSER assessment system, 513 construction, 300302 Diagnosing preconceptions in physics, 404 Engage phase, 428434 Diagnostic assessment, 491492 Engagement of students' preconceptions Diagnostic questions, 478 and building on existing Dialogue knowledge, 45, 223231 internal and external, as support for allowing multiple strategies, 223227 metacognition, 241 designing bridging instructional Direction of change, 44 activities, 231 Disciplinary knowledge, 32 encouraging math talk, 228231 attending to significant, 543544 Engagement of students' problem-solving "second-order," 61 strategies, 225227 Disconfirmation, 415 Equipment Manager, 435 Discrepant events Eratosthenes, 194, 196197 providing students with opportunities European geographic knowledge to experience, 571573 the great interruption in, 200201 Discussion Everyday concepts guided, 579, 582 history and, 3361 DiSessa, Andrea, 5 of scientific methods, argumentation, Distinguishing among kinds of textbook and reasoning, 400 claims of scientific phenomena, 399400 and the nature of sources, 101102

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602 INDEX Evidence, 41, 5458, 61, 65, 112, 120, 165 essential role of factual knowledge in context, 167 and conceptual frameworks in cutting-and-pasting, 167 understanding, 69 finding out about the past from importance of self-monitoring in, 10 received information, 5658 12 historical, 134 "Flat earth," 189199 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 166167 round, 196197 pictures of the past, 166 Formative assessments, 1617, 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, 89 Fourth cycle of investigation Explanations, 156 community knowledge from, 467 of words in the story, 132133 Fourth graders' initial ideas about light, 431 Explanatory power, 518 Fractions and mixed representations of External consistency, 518 rational numbers, 334337 External migration, 68 card games, 335337 External testing, 181 cracking the code, 335 fractions and equivalencies, 334335 Framework of How People Learn F seeking a balanced classroom environment, 242243 Face value Frank, Anne, 109 going beyond, 134 Fundamental physics, 24 Factual knowledge Fundamentalism, 176 manipulating, 7980 Fuson, Karen C., 23, 217256, 593 Falling bodies Future real-world experience, 390 explaining, 510511 Familiarity, 389390 the dangers of what appears to be G 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, 102107 why the Norse colonists didn't stay in Genetics, 516540 America, unit on, 172 attending to students' existing First cycle of investigation knowledge, 517526 community knowledge from, 463 metacognition and engaging students Fish story (Fish Is Fish), 212, 398, 414, 575 in reflective scientific practice, 5-year-olds understandings of, 273 538540 274 simple dominance homework engaging prior understandings in, 45 assignment, 539 student inquiry in, 526538

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INDEX 603 Genetics Construction Kit (GCK), 534537 H homework assignment, example of student work on, 535 "H(ac)", 187188 Genetics content Hall, G. Stanley, 177n learning, 524526 Halsall, William Formsby, 87 Geographic knowledge Help Christian, 200 seeking and giving, 241242 the great interruption in European, Heuristic for teaching and learning science 200201 through guided inquiry, 427455 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, 428434 Gravity and its effects, 477511 fourth graders' initial ideas about activity A1 worksheet, 483 light, 431 analogy to magnetism, 508 investigate phase, 438443 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, 508510 prepare-to-investigate phase, 434438 building an analogy to understand prepare-to-report phase, 443448 the benchmark experience, 489 report phase, 448455 490 "H(ev)", 187 consensus discussion and summary of Higher-order knowledge structure, 276 learning, 490491 Historical accounts, 5961 defining, 477510 different ideas about, 3839 diagnostic assessment, 491492 not copies of the past, 6263 exploring similarities and differences "problematizing," 184188 between actions at a distance, Historical evidence, 134 492493 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, 181199 ideas, 477478 History, 29213 identifying preconceptions, 478480 applying the principles of How People opportunities for students to suggest Learn in teaching high school and test related hypotheses, 484 history, 179213 489 "counterintuitive" intuitions in, 33, 42 twisting a torsion bar, 493501 "doing," 32, 48 weighing in a vacuum, 480483 implications for planning, 164176 Grids, 173175 periods in, 4243 Griffin, Sharon, 23, 257308, 593 putting principles into practice, 79 Group work, 582584 178 Guess My Number, 300 the reality test, 8084 Guidance of student observation and significance in, 45 articulation that "works," 6572 supporting metacognition, 584585 understanding, 3177 Guided inquiry, 495, 579, 582 working with evidence, 84119

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604 INDEX History and everyday ideas, 3361 providing students with opportunities differences in the power of ideas, 36 to make public, 524 37 "second-order," 3233 grounds for caution, 4041 time, 4143 ideas we need to address, 4161 Inaccessible algorithms, 236 the progression of ideas, 3740 Information, 41, 124, 166 understanding the past and "clumping," 69 understanding the discipline of finding, 121 history, 3435 from history, 499 "History-as-account," 187188, 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, 199209 meiotic processes governing, 528 the great interruption in European Initial models geographic knowledge, 200201 providing students with opportunities with tools for historical thinking, 199 to revise in light of anomalous 209 data and in response to critiques History of the Decline and Fall of the of others, 524 Roman Empire, The, 57 Inquiry based information, 470n Hitler, Adolf, 3435, 5960, 586 Instruction, 545554 Holt, John, 218 to support mathematical proficiency, How People Learn: Brain, Mind, Experience, 233236 and School, 1, 25, 3132 Instruction in rational number, 319340 cautions in, 199 alternative instructional approaches, design characteristics described in, 321322 1213, 2022, 257258, 359 children's thinking after instruction, key findings of, 7980, 171173, 176 338340 research summarized in, 241 curriculum overview, 325 violating principles of, 319 fractions and mixed representations How People Learn framework, 411415 of rational numbers, 334337 assessment-centered, 415 introduction of decimals, 332334 community-centered, 415 introduction to percents, 325332 knowledge-centered, 414 knowledge network, 340 learner-centered, 414 pie charts and a part-whole reflective assessment in ThinkerTools, interpretation of rational numbers, 412413 320321 Humor pipes, tubes, and beakers, 322324 enlivening learning and helping build Instruction that supports metacognition, positive relationships with 239242 students, 501 emphasizing debugging, 239240 internal and external dialogue as support for metacognition, 241 I seeking and giving help, 241242 Instructional lines of thinking, 182 Ideas, 4161 Intellectual roles for students to adopt, 436 accounts, 5961 Internal consistency, 518 cause, 4954 Internal migration, 68 change, 4346 Interpretation empathy, 4649 anchoring themes in historical, 186 evidence, 5458 of data, 403 progression of, 3740

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INDEX 605 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, 312316 Invented procedures, 329 new symbols, meanings, and Investigate phase, 438443 representations, 313314 Investigative setup for studying how light reconceptualizing the unit and interacts with solid objects, 437 operations, 315 Irving, Washington, 208 the subconstructs, 314315 Isolation understanding numbers as evidence in, 167 multiplicative relations, 316 Italy "Knowledge packages," 588n instruction about payment for work, Knowledge that should be taught, 259267 6667 central conceptual structure hypothesis, 262265 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., 351393, 593594 Jasper Woodbury series, 391 Kraus, Pamela, 23, 401, 475513, 594 Jefferson, Thomas, 6263 KWL charts, 199, 428430 Johnson, Lyndon, 62 Jonassen, David, 181 Judgments L avoiding expressing, 498 Lamarck, Jean Baptiste de, 550, 573 Larson, Gary, 217 K Learner-centered classroom environments, 1314, 266, 292, 414, 555 Kalchman, Mindy, 23, 217256, 351393, 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, 1220 connected, 1516 assessment-centered classroom disciplinary, 32, 543544 environments, 13, 1617, 267, 290, handed down through generations, 292, 555558 9394 community-centered classroom manipulating factual, 7980 environments, 13, 1720, 301, "metahistorical," 32 559560 organized, 462 knowledge-centered classroom "second-order," 3233 environments, 1316, 267, 284, secret, 72 292, 555, 587 student, 258, 544545 learner-centered classroom of what it means to "do science," environments, 1314, 266, 292, 403407 414, 555 Knowledge-centered classroom perspectives on, 13 environments, 1316, 267, 284, Learning goals for prekindergarten through 292, 414, 555, 587 grade 2, 284285

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606 INDEX Learning paths of knowledge Maps, 86, 140141 building, 258 conceptual, 188 from children's math worlds, for Marfan's syndrome, 533 single-digit addition and Math words, 230 subtraction, 234235 Mathematical proficiency, 218 Learning principles adaptive reasoning, 218 engaging resilient preconceptions, conceptual understanding, 218 569575 procedural fluency, 218 organizing knowledge around core productive disposition, 218 concepts, 575577 strategic competence, 218 principles of learning and classroom Mathematical thinkers environments, 586588 building, 258 pulling threads, 569590 Mathematical understanding, 217256 revisiting the three, 567590 computation without comprehension, supporting metacognition, 577586 218 Learning rational number, 341343 developing mathematical proficiency, metacognition, 342 232236 network of concepts, 341342 learning to use student thinking in prior understandings, 341 teacher video clubs, 244 Learning with understanding, 559560 lesson study cycle, 244 supporting knowledge use in new a metacognitive approach enabling situations, 7 student self-monitoring, 236243 Leather boats, 139141 suggested reading list for teachers, Lee, Peter J., 23, 31178, 576, 594 256 Lesson Study Research Group, 244 teachers as curriculum designers, 245 Life and Voyages of Christopher Columbus, teachers engaging students' The, 208 preconceptions, 219231 "Light catchers," 437. See also Study of light understanding requiring factual Linkage knowledge and conceptual of formal mathematical understanding frameworks, 231236 to informal reasoning, 354355 Mathematics, 215393 Lionni, Lee, 2, 4. See also Fish story as about quantity, not about numbers, Logic of the situation 280 exploring, 5051 as "following rules" to guarantee Lowenthal, David, 185 correct answers, 220221 fostering the development of whole number sense, 257308 M as learning to compute, 220 pipes, tubes, and beakers in, 309349 Ma, Liping, 1516, 1819, 577578 teaching and learning functions, 351 Magic Shoes game, 295296 393 Magnetism Mathematics instruction analogy to gravity, 508 in China, 1516, 1819 Magnitude Mayflower, The in decimal numbers, 333334 arrival of, 84, 87, 90, 9295 of gravitational force, 501508 Medawar, Peter, 406 Magnusson, Shirley J., 421474, 594 Media Management of student activities, 435 technical and passive, 496 Mandates Meiotic processes curricular, 181 governing inheritance, 528 Manipulation of factual knowledge, 7980

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INDEX 607 Mendel, Gregor, 406, 410, 517, 523, 525 Model-based inquiry, 515565 529, 539 classroom environments that support model of simple dominance, 528 learning with understanding, 555 Mental counting line structure, 276 560 Metacognition, 10, 238, 407411, 577586 developing Darwin's model of natural conceptual explanation without selection in high school evolution, conceptual understanding, 578 540554 engaging students in reflective genetics, 516540 scientific practice, 538540 Modeling for Understanding in Science in evaluating the methods used in an Education (MUSE), 516, 548 experiment, 408409 curricula from, 555, 559 guiding student observation and Models, 402403 articulation, 584585 consistency between, 557 of light, 426 of progression in ideas about in Mendel's contribution to genetics, evidence, 166167 410 providing students with opportunities questioning and explaining in high to revise in light of anomalous school science, 582583 data and in response to critiques and rational number, 319, 342 of others, 524 supporting, 577586 Monarch/viceroy case supporting skilled questioning and Darwinian explanation written by explaining in mathematics students on the, 553 problem solving, 580581 Monitoring. See also Self-monitoring Metacognitive approaches to instruction, 2, metacognitive, 10 80 "Monster-free zone," 295 enabling student self-monitoring, Moss, Joan, 23, 309349, 595 236243 Motion of projectiles framework of How People Learn, 242 explaining, 511 243 Multiple strategies, 223227 instruction that supports allowing, 223227 metacognition, 239242 engaging students' problem-solving seeking a balanced classroom strategies, 225227 environment, 242243 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 Minstrell, James, 23, 401, 475513, 594595 N Minus Mouse, 290291 Misconceptions Narrative accounts about momentum, 5 providing students with, 573575 about the scientific method, 414 National Council of Teachers of "Missing-term problem," 317 Mathematics (NCTM), 221, 241, Misunderstandings, 310 259 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 National Science Education Standards, the "world of formal symbols," 455, 561 288292 Native Americans, 41, 8283, 98, 105106 Number Worlds program, 262, 283, 287 NCTM. See National Council of Teachers of 288, 292, 296, 300, 302303 Mathematics Numeric answers, 372 Necessary conditions causes as, 53 Neighborhood Number Line, 295 O Networks of concepts, and rational number, Object Land, 284286, 288 341342 "One world" revolution, 169 of knowledge, building, 258 "Openings" in the curriculum, 245 New conceptualizations Opportunities understanding numbers as to develop causal models to account multiplicative relations, 316 for patterns, 524 New ideas to experience discrepant events that development of, 470n allow them to come to terms with New rules the shortcomings in their everyday discovering, 588 models, 571573 New symbols to make ideas public, 524 meanings, and representations, 313 providing students with, 523524 314 to revise initial models in light of "Nothing" happening, 43 anomalous data and in response Number Knowledge Test, 260, 264, 267 to critiques of others, 524 269, 271, 279, 304305 to search for patterns in data, 524 administering and scoring, 271 to use patterns in data and models to Number worlds, 282302 make predictions, 524 encouraging the use of metacognitive to use prior knowledge to pose processes to facilitate knowledge problems and generate data, 523 construction, 300302 524 engaging children's emotions and Opportunities for children to acquire capturing their imagination, 296 computational fluency as well as 298 conceptual understanding, 298300 exposing children to major forms of Sky Land Blastoff activity, 298299 number representation, 283288 Opportunities for students to suggest and the five forms of representation and test related hypotheses in the lands in which they appear, elaboration activities, 484489 286 inverted cylinder in a cylinder of learning goals for prekindergarten water, 485486 through grade 2, 284285 inverted glass of water, 484485 providing analogs of number leaky bottle, 486 representations that children can water and air in a straw, 486488 actively explore hands-on, 292 weighing" an object in a fluid 296 medium, 488489 providing opportunities for children Opportunities to link the "world of to acquire computational fluency quantity" with the "world of as well as conceptual counting numbers" and the "world understanding, 298300 of formal symbols," 288292 Minus Mouse, 290291

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INDEX 609 Plus Pup, 288290 Periods in history, 4243 Plus Pup meets Minus Mouse, 291292 Physics Optics kit, 422, 468 fundamental, 24 Order instruction in, 1617 of counting words, 274 Picture Land, 285287, 297 in decimal numbers, 333334 Pie charts and a part-whole interpretation Organized knowledge, 462 of rational numbers, 320321 Organizing knowledge around core Pilgrim Fathers and Native Americans, 71, concepts 84119 subtraction with regrouping, 1819 exploring the basis for textbook Origin of Species, 551 claims and the nature of sources, Outcomes of courses, 181 84111 grid for evidence on, 173, 175 ideas, beliefs, and attitudes, 112118 P language of sources, interpretation, and other perspectives, 118119 Pace of change, 44 teacher questions, 112113, 115 Paley, William, 550551, 573 whether people thought like us in the Palincsar, Annemarie Sullivan, 23, 421474, past, 117 595 Pipes Park, Lesley, 455 a new approach to rational-number Part-whole relation, 314 learning, 322324 Pass it on (game), 105 a representation for fullness, 325326 Passive media, 496 Planning, 164176 Passmore, Cynthia M., 23, 515565, 595 of progression in ideas about Past evidence, 166167, 174175 finding out about, 5658 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, 6667 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, 288290 unit on, 170 meeting Minus Mouse, 291292 Percents, 325332, 340 Pocahontas (Disney film), 122 computing with, 329 Pory, John, 8485, 90, 97, 100104, 106 in everyday life, 325 108 "families" of, 331 Positive relationships invented procedures, 329 humor helping to build with students, on number lines, 326329 501 pipes and tubes, as representations Possible Worlds, 406 for fullness, 325326 Power starting from, 322324 explanatory and predictive, 518 string challenges, 329331 Preconceptions, 1, 55, 399403 Percy, George, 122 about people, society, and how the Performance world works, 127128 need to assist, 203 conceptual change, 400403

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610 INDEX drawing on knowledge and "Problematizing" historical accounts, 184188 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 569571 Progress, 4445 engaging resilient, 569575 Progression of ideas, 3740 everyday concepts of scientific different ideas about historical methods, argumentation, and accounts, 3839 reasoning, 400 Progressive change, 45 everyday concepts of scientific Project CHATA. See Concepts of History phenomena, 399400 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, 571573 providing students with narrative accounts of the discovery of Q (targeted) knowledge or the development of (targeted) tools, Quantity, 234 573575 schema for, 272 Preconceptions about how we know about Question Poser, 300301 the past, 121123 Questioning and explaining in high school common student assumptions about science how we know of the past, 123 supporting metacognition, 582583 dangers of what appears to be Questions, 128 familiar, 122 diagnostic, 478 Predictive power, 518 at the heart of using evidence, 124 Preinstruction assessments, 495 many as yet unanswered, 492 Prepare-to-investigate phase, 434438 teachers modeling for students, 477 Prepare-to-report phase, 443448 Quotient interpretation, 314 Principles of How People Learn applied to teaching high school history, 179 213 R designing a "history-considerate" Rational change, 45 learning environment, 199209 Rational number, 341343 transforming topics and objectives metacognition, 342 into historical problems, 181199 network of concepts, 341342 Prior understandings prior understandings, 341 development of physical concepts in Rational-number learning infancy, 4 and the knowledge network, 312316 engaging, 45 metacognition and rational number, 319 of light, 425 new concepts of numbers and new misconceptions about momentum, 5 applications, 312316 providing students with opportunities and the principles of How People to use to pose problems and Learn, 312319 generate data, 523524 students' errors and misconceptions and rational number, 341 based on previous learning, 316 Problem solvers 319 building, 258

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INDEX 611 Real-world experience diagnosing preconceptions in physics, current and future, 390 404 Real-world words, 230 the How People Learn framework, Reality test, 8084 411415 "7-year gap," 82 knowledge of what it means to "do Reciprocal teaching, 11 science," 403407 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, 412413 94 Regrouping Searchers, The (film), 151 subtraction with, 1819 Second cycle of investigation Relative thinking as multiplicative, 311 community knowledge from, 464 Relativism, 176 Second-hand investigation, 455459 Reliability, 126 "Second-order" disciplinary concepts, 61, Religious practices, 113118 73n Reporter, 301 "Second-order" knowledge, 3233, 41 Reporting phase, 427, 448455 acquisition of, 4041 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, 1012 Sagan, Carl, 194, 196197 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, 395565 127 developing understanding through Severin, Tim, 139, 142143 model-based inquiry, 515565 Shemilt, Denis, 23, 56, 79178, 595596 guided inquiry in the science Shrinking past, 160161 classroom, 475513 Significance, 45 information from the history of, 499 historical, 45 leaving many questions as yet Simplicity, 389390 unanswered, 492 6-year-olds understandings of, 274 teaching to promote the development 277 of scientific knowledge and Skating Party game, 292295 reasoning about light at the Skills elementary school level, 421474 defining, 40 unit on the nature of gravity and its Sky Land, 286287 effects, 477511 Blastoff activity, 298299 Science classrooms Smith, John, 122 guided inquiry in, 475513 Sources Scientific inquiry and How People Learn, access to someone who saw for 397419 himself, 93 addressing preconceptions, 399403 briefing sheet, 8889

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612 INDEX distinguishing among kinds of claims, initial GCK population for the final 101102 GCK inquiry, 537 generalizing, 102107 meiotic processes governing getting behind the record to concerns inheritance, 528 of the people who produced Mendel's model of simple dominance, them, 107108 528 interpreting sources in context, 100 Students' errors and misconceptions based maintaining contact with an on previous learning, 316319 eyewitness using knowledge Students' existing knowledge, 517526 handed down through assessing knowledge claims in generations, 9394 genetics, 523 the nature of, 84111 attending to, 544545 scissors-and-paste approach, 94 black box, 520 seeing for yourself, 93 building on and connecting, 258 teacher questions, 92, 9596, 99101 learning genetics content, 524526 trusting the source who was in a providing students with learning position to know, 96 opportunities, 523524 understanding the purpose of the student conceptions of models, 518 source, 9699 Students' preconceptions understanding what is likely to get importance of, 79 recorded and under what Study of light, 422426 circumstances, 108111 conceptual understanding, 423424 working out the facts from other metacognition, 426 sources or available knowledge, prior knowledge, 425 9495 Study of light through inquiry, 426459 Splitting, 323 heuristic for teaching and learning State of affairs science through guided inquiry, changes in, 44 427455 Stearns, Peter, 210 second-hand investigation, 455459 Stewart, James, 23, 515565, 596 Subconstructs "Stop-Start Challenge," 333 the many personalities of rational Stopwatches number, 314315 decimals and, 332333 Subject-specific knowledge in effective Stories science instruction, 467469 "embroidering," 153 Substantiated accounts, 87 Strategic competence, 218 Substantive assumptions String challenges sensitivity to students', 127 guessing mystery objects, 329331 Substantive concepts, 6165 Student assumptions about how we know historical accounts not copies of the of the past, 123 past, 6263 Student conceptions payment for work, 6667 experimentation, 402 Subtraction with regrouping, 1819 inadequacies in arguments, 403 Supporting learning through cycles of interpretation of data, 403 investigation, 460467 of knowledge generation and Supporting skilled questioning and justification in science, 402403 explaining in mathematics models, 402403, 518 problem solving Student inquiry in genetics, 526538 supporting metacognition, 580581 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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INDEX 613 T Teaching mathematics in the primary grades, 257308 Table of values to produce a function, acknowledging teachers' conceptions 353358 and partial understandings, 279 Teacher professional development in 281 Japan, 244 building on children's current Teacher questions, 112113, 115 understandings, 267279 and textbook claims and the nature the case of number worlds, 282302 of sources, 92, 9596, 99101 comparing number worlds and Teachers' conceptions and partial control group outcomes, 304 understandings, 279281 deciding what knowledge to teach, acquiring an understanding of 259267 number as a lengthy, step-by-step defining the knowledge that should process, 280281 be taught, 281282 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, 280281 311 math as not about numbers, but how students learn rational number, about quantity, 280 341343 Teachers engaging students' instruction in rational number, 319 preconceptions, 219231 340 common preconceptions about rational-number learning and the mathematics, 220222 principles of How People Learn, engaging students' preconceptions 312319 and building on existing Teaching to promote the development of knowledge, 223231 scientific knowledge and Teaching reasoning about light at the reciprocal, 11 elementary school level, 421474 Teaching and learning functions in the role of subject-specific knowledge mathematics, 351393 in effective science instruction, addressing the three principles, 359 467469 373 the study of light, 422426 building conceptual understanding, the study of light through inquiry, procedural fluency, and 426459 connected knowledge, 364369 supporting learning through cycles of building on prior knowledge, 359 investigation, 460467 364 Technical media, 496 building resourceful, self-regulating Testimony, 41, 124, 135, 166 problem solvers, 371373 Testing linking formal mathematical external, 181 understanding to informal Textbook claims reasoning, 354355 access to someone who saw for making a table of values to produce a himself, 93 function, 353358 briefing sheet, 8889 teaching functions for understanding, distinguishing among kinds of claims, 373389 101102 teaching to achieve this kind of generalizing, 102107 understanding, 358359 getting behind the record to concerns Teaching as Story Telling, 574 of the people who produced Teaching functions for understanding, 373 them, 107108 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, 9394 and the nature of sources, 84111 scissors-and-paste approach, 94 U seeing for yourself, 93 teacher questions, 92, 9596, 99101 "Underlying" causes, 35 trusting the source who was in a Understanding position to know, 96 essential role of factual knowledge understanding the purpose of the and conceptual frameworks in, source, 9699 69 understanding what is likely to get experts remembering considerably recorded and under what more relevant detail than novices circumstances, 108111 in tasks within their domain, 89 working out the facts from other learning with understanding sources or available knowledge, supporting knowledge use in new 9495 situations, 7 Themes, 44 Understanding of number anchoring in historical representation a lengthy, step-by-step process, 280 and interpretation, 186 281 ThinkerTools, 407, 585 Understanding the purpose of the source Third cycle of investigation and textbook claims and the community knowledge from, 465 nature of sources, 9699 Third International Mathematics and Understanding what is likely to get Science Study, 243 recorded and under what 3-slot schema circumstances for graphing a line, 370371 and textbook claims and the nature Three subtraction methods, 224 of sources, 108111 Time, 4143 Unit-level problem, 189199 change limited in, 45 accounts of Colombian voyages, 192 periods in history, 43 193 Time lines, 129, 159 ancient views of the Earth as flat or Timekeeper, 435 round, 196197 Torsion bar, 493501 Unit on the nature of gravity and its Transforming topics and objectives into effects, 477511 historical problems, 181199 United Kingdom accounting for the "flat earth," 189 adjusting data from, 177n 199 Schools Council History Project, 40, "problematizing" historical accounts, 177n 184188 Units Transmission errors, 123 on filling the world with people, 169 Trusting the source who was in a position on first contacts, whether St. Brendan to know sailed from Ireland to America, and textbook claims and the nature 171 of sources, 96 on first contacts, why the Norse Truth colonists didn't stay in America, twisting, 105, 123 172 Tubes on people going their separate ways, a new approach to rational-number 170 learning, 322324 a representation for fullness, 325326

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INDEX 615 V Work payment for in history, 6667 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, 323324 of sources, 9495 Working things out for ourselves, 133138 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, 139141 Water and air in a straw, 486488 Working through the task, 128164 Website, 562n Working with evidence "Weighing" an object in a fluid medium, Pilgrim Fathers and Native Americans, 488489 84119 Weighing-in-a-vacuum situation, 484, 489 preparing for the task, 121128 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, 264265 Wisdom, 236, 238 Woodbury, Jasper, 391 Word Problems test, 264265 Y Words versus notations, 230 Year-long historical questions, 184188 Words in stories explaining, 132133

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