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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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