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INDEX
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, 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Representative terms from entire chapter:
suppo ting
