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How Students Learn: History, Mathematics, and Science in the Classroom
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Index
A
Absolute difference, 311
Absolute thinking
as additive, 311
Access to someone who saw for himself and textbook claims and the nature of sources, 93
Accounts, 59–61
of Colombian voyages, 192–193
different ideas about historical, 38–39
historical, 59–61
substantiated, 87
Actions at a distance
exploring similarities and differences between, 492–493
Activity A1 worksheet, 483
Adams, John, 185
Adaptive reasoning, 218
Adding It Up, 218, 233, 241
Additive reasoning, 311, 321
absolute thinking as, 311
Addressing preconceptions, 399–403
Advantage
selective, 542
Adventure
sense of, 71
Alternative instructional approaches, 321–322
American Association for the Advancement of Science
guidelines of, 398
textbook review by, 16
Analogs of number representations that children can actively explore hands-on, 292–296
Rosemary’s Magic Shoes game, 295–296
Skating Party game, 292–295
Analogy to understand the benchmark experience, 489–490
Ancient views of the Earth as flat or round, 196–197
the Atlas Farnese, 196
the story of Eratosthenes and the Earth’s circumference, 196–197
Anglo-Saxons, 117
Anselm, St., 46
Arguments
inadequacies in, 403
Ashby, Rosalyn, 79–178, 591
Assessment-centered, 415
Assessment-centered classroom environments, 13, 16–17, 267, 290, 292, 555–558
examples of students’ critiques of their own Darwinian explanations, 558
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sample exam question, and consistency between models, 557
Assessment systems DIAGNOSER, 513
Assessments. See also Self-assessment formative, 16–17, 193
preinstruction, 495
“reflective,” 412
Assumptions
substantive, 127
Atlas Farnese, 194, 196
Authority, 135
Award cards, 293
Awareness of how you are thinking, 135
B
Bain, Robert B., 23, 179–213, 591
Balzac, Honoré de, 236
Barry, Tr., 578
Barton, Keith, 45, 160
Beakers
a new approach to rational-number learning, 322–324
Bede, St., 58
Bell jar experiment, 484, 489
Benchmark lessons, 493–501, 512n
weighing in a vacuum, 480–483
Black box approaches, 519–520
“Blastoff!”, 298
Boorstin, Daniel, 198
Bradford, William, 84–88, 96, 108–111
Bransford, John D., 1–28, 217–256, 397–419, 569–592
Brendan, St., 71, 82–83, 128–164, 171
believing historical films when people in them behave as we would, 151
the deficit past, 154–155
explanation of words in the story, 132–133
finding out what kind of story it is, 150–164
grid for evidence on, 173–174
the question, 128
the shrinking past, 160–161
the story, 128–133
thinking from inside the story, 144–150
thinking from outside the story, 138–144
voyage of, 130–132
working things out for ourselves, 133–138
Bridging
from understanding magnetic action at a distance to understanding gravitational action at a distance, 508–510
“Bridging context,” 324, 359
Briefing sheets, 87, 91
and textbook claims and the nature of sources, 88–89
Building conceptual understanding, procedural fluency, and connected knowledge, 364–369
3-slot schema for graphing a line, 370–371
developmental model for learning functions, 365–366
level 0, 364, 367
level 1, 367–368
level 2, 368
level 3, 369
Building on children’s current understandings, 267–279, 359–364
administering and scoring the Number Knowledge Test, 271
mental counting line structure, 276
Number Knowledge Test, 268–269
understandings of 4-year-olds, 270–273
understandings of 5-year-olds, 273–274
understandings of 6-year-olds, 274–277
understandings of 7-year-olds, 277–278
understandings of 8-year-olds, 278–279
Building resourceful, self-regulating problem solvers, 371–373
an integrated understanding of functions, 372
C
Cambridge History Project, 177n
Canada
teaching history in, 151
“Candles” (unit), 456
Card games, 335–337
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Carey, Susan, 592
Cartier, Jennifer L., 23, 515–565, 592
Cartoons, 143, 145–146, 148, 546–549
Peanuts, 309
sequencing activity, 546–547
Case, Robbie, 23
Causal models to account for patterns providing students with opportunities to develop, 524
Causes, 49–54
exploring the logic of the situation, 50–51
modeling, 562n
as necessary conditions, 53
“underlying,” 35
Central conceptual structure hypothesis bidimensional, for number, 279
dependence of future learning on the acquisition of this structure, 264–265
importance of structure to successful performance on a range of tasks, 262–263
for whole number, 261–262, 275
Change, 43–46, 61
direction of, 44
large-scale patterns of, 68
pace of, 44
as progressive, rational, and limited in time, 45
Cheese and the Worms, 185
Children
engaging their emotions and capturing their imagination, embedding knowledge constructed in their hopes, fears, and passions, 296–298
exposing to major forms of number representation, 283–288
as “natural” scientists, 421
Children passing the Number Knowledge Test
and measures of arithmetic learning and achievement, 265
and numerical transfer tests, 263
Children’s Math World project, 219, 223, 227, 229, 231, 236, 241
Children’s thinking after instruction, 338–340
China
teaching of mathematics in, 15–16, 18–19
Christian geography, 200
Circle Land, 286–287
Claims
backing up, 58
Classroom environments
genetic inquiry in, 529–534
principles of learning and, 586–588
Classroom environments that support learning with understanding, 555–560
assessment-centered classroom environments, 13, 16–17, 267, 290, 292, 555–558
community-centered classroom environments, 13, 17–20, 301, 559–560
knowledge-centered classroom environments, 13–16, 267, 284, 292, 555, 587
learner-centered classroom environments, 13–14, 266, 292, 555
Clumping information, 69
Codes
cracking, 335
Cognitive Tutor Algebra, 355, 391
Colombian Exposition, 208
Columbus’ voyages, 189–193, 195, 199, 204–205, 207–208, 587
Common preconceptions about mathematics, 220–222
as “following rules” to guarantee correct answers, 220–221
as learning to compute, 220
only some people have the ability to “do math,” 221–222
Community-centered classroom environments, 13, 17–20, 301, 415, 559–560
learning with understanding, 559–560
organizing knowledge around core concepts, 18–19
Comparing number worlds and control group outcomes, 304
Competence developed by students, 1
Comprehensive Test of Basic Skills, 412
Computing with percent, 329
Concepts
substantive, 61–65
Concepts of History and Teaching Approaches (Project CHATA), 38–39, 51–53, 56, 62, 82
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Conceptual change, 400–403
student conceptions of knowledge generation and justification in science, 402–403
Conceptual explanations
without conceptual understanding, 578
Conceptual structure
bidimensional central, for number, 279
central, for whole number, 261–262, 275
Conceptual understanding, 218
of light, 423–424
Conceptualization
children’s problems with, 137
Connected knowledge, 15–16
Conquest of Paradise, 208
Consistency
internal and external, 518
between models, 557
Constitution, 61
Context
evidence in, 167
Continuity, 44
“Controlled experiments,” 402
Core concepts, 589
organizing knowledge around, 18–19
organizing procedural knowledge and skills around, 19
Corne, Michael Felice, 90
“Counterintuitive” intuitions
in history, 33, 42
Counting schema, 272
Counting words
as the crucial link between the world of quantity and the world of formal symbols, 280–281
order of, 274
Course outcomes, 181
Curriculum
mandates in, 181
from Modeling for Understanding in Science Education, 555, 559
“openings” in, 245
Curriculum for moving students through the model, 373–375
example lessons, 375–389
learning slope, 378–381
learning y-intercept, 381–384
operating on y = x2, 384–389
sample computer screen, 386
suggested curricular sequence, 376–377
two different student solutions to an open-ended problem, 385
Cut-and-paste, 167
Cycles of investigation
development of community knowledge across cycles of investigation, 460
development of conceptual frameworks for light, 462–467
in guided-inquiry science, 427
supporting learning through, 460–467
D
Dances with Wolves (film), 151
Darwin, Charles, 542–545, 550–551, 556, 573
Darwin’s model of natural selection in high school evolution, 540–554
attending to significant disciplinary knowledge, 543–544
attending to student knowledge, 544–545
cartoon sequencing activity, 546–547
explanation written by students on the monarch/viceroy case, 553
instruction, 545–554
laying the groundwork, 545–549
understanding, 550–552
Data
interpretation of, 403
Data tables from initial recording and with revisions for analysis, 445
Debugging
emphasizing, 239–240
Decimals, 332–334
magnitude and order in decimal numbers, 333–334
and stopwatches, 332–333
Decisions
as to what knowledge to teach, 259–267, 281–282
Deficit past, 154–155
Dependence, 234, 352
Design of instruction
bridging instructional activities, 231
learning environments and, 12–20
Development
of community knowledge across cycles of investigation, 460
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of Darwin’s model of natural selection in high school evolution, 540–554
of physical concepts in infancy, 4
of understanding through model-based inquiry, 515–565
Development of conceptual frameworks for light, 462–467
community knowledge from the first cycle of investigation (first-hand), 463
community knowledge from the fourth cycle of investigation (first-hand), 467
community knowledge from the second cycle of investigation (first-hand), 464
community knowledge from the third cycle of investigation (second-hand), 465
Development of mathematical proficiency, 232–236
inaccessible algorithms, 236
instruction to support mathematical proficiency, 233–236
a learning path from children’s math worlds for single-digit addition and subtraction, 234–235
Developmental model
for learning functions, 365–366
DIAGNOSER assessment system, 513
Diagnosing preconceptions in physics, 404
Diagnostic assessment, 491–492
Diagnostic questions, 478
Dialogue
internal and external, as support for metacognition, 241
Direction of change, 44
Disciplinary knowledge, 32
attending to significant, 543–544
“second-order,” 61
Disconfirmation, 415
Discrepant events
providing students with opportunities to experience, 571–573
Discussion
guided, 579, 582
DiSessa, Andrea, 5
Distinguishing among kinds of textbook claims
and the nature of sources, 101–102
DNA, 517, 526
“Doing,” 32, 48
“Doing math”
only some people having the ability for, 221–222
Donovan, M. Suzanne, 1–28, 397–419, 569–590, 592
Double-blind procedure, 302
Dragon Quest game, 297–298
E
Earth as flat or round, ancient views of, 196–197
Earth’s circumference
the story of Eratosthenes and, 196–197
Effects of gravity, 510–511
explaining falling bodies, 510–511
explaining motion of projectiles, 511
Egan, Kieran, 592
8-year-olds understandings of, 278–279
Elementary Science Study
Optics unit, 422, 468
“Embroidering” stories, 153
Empathy, 46–49, 65, 112
Encouraging math talk, 228–231
Encouraging the use of metacognitive processes to facilitate knowledge construction, 300–302
Engage phase, 428–434
Engagement of students’ preconceptions and building on existing knowledge, 4–5, 223–231
allowing multiple strategies, 223–227
designing bridging instructional activities, 231
encouraging math talk, 228–231
Engagement of students’ problem-solving strategies, 225–227
Equipment Manager, 435
Eratosthenes, 194, 196–197
European geographic knowledge
the great interruption in, 200–201
Everyday concepts
history and, 33–61
of scientific methods, argumentation, and reasoning, 400
of scientific phenomena, 399–400
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Evidence, 41, 54–58, 61, 65, 112, 120, 165
in context, 167
cutting-and-pasting, 167
finding out about the past from received information, 56–58
historical, 134
information as, 166
in isolation, 167
model of progression in ideas about, 166–167
pictures of the past, 166
questions at the heart of using, 124
testimony as, 166
Experiments on Plant Hybridization, 529
Experts remembering considerably more relevant detail than novices in tasks within their domain, 8–9
Explanations, 156
of words in the story, 132–133
Explanatory power, 518
External consistency, 518
External migration, 68
External testing, 181
F
Face value
going beyond, 134
Factual knowledge
manipulating, 79–80
Falling bodies
explaining, 510–511
Familiarity, 389–390
the dangers of what appears to be familiar, 122
Feynman, Richard, 24, 403
Filling the world with people
unit on, 169
First contacts
whether St. Brendan sailed from Ireland to America, unit on, 171
why the Norse colonists didn’t stay in America, unit on, 172
First cycle of investigation
community knowledge from, 463
Fish story (Fish Is Fish), 2–12, 398, 414, 575
5-year-olds understandings of, 273–274
engaging prior understandings in, 4–5
essential role of factual knowledge and conceptual frameworks in understanding, 6–9
importance of self-monitoring in, 10–12
“Flat earth,” 189–199
accounts of Colombian voyages, 192–193
ancient views of the Earth as flat or round, 196–197
Formative assessments, 16–17, 193
Forms of representation
4-year-olds understandings of, 270–273
and the lands in which they appear, 286
Fourth cycle of investigation
community knowledge from, 467
Fourth graders’ initial ideas about light, 431
Fractions and mixed representations of rational numbers, 334–337
card games, 335–337
cracking the code, 335
fractions and equivalencies, 334–335
Framework of How People Learn
seeking a balanced classroom environment, 242–243
Frank, Anne, 109
Fundamental physics, 24
Fundamentalism, 176
Fuson, Karen C., 23, 217–256, 593
Future real-world experience, 390
G
Galapagos tortoises, 558
GCK. See Genetics Construction Kit
General ideas, 162
General meaning of slope, 363
Generalizing and textbook claims and the nature of sources, 102–107
Genetics, 516–540
attending to students’ existing knowledge, 517–526
metacognition and engaging students in reflective scientific practice, 538–540
simple dominance homework assignment, 539
student inquiry in, 526–538
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Genetics Construction Kit (GCK), 534–537
homework assignment, example of student work on, 535
Genetics content
learning, 524–526
Geographic knowledge
Christian, 200
the great interruption in European, 200–201
Gibbon, Edward, 57
GIsML Community of Practice, 470n
“Globalization,” 169
Gould, Stephen Jay, 198
Gragg, Charles, 236
Gravity and its effects, 477–511
activity A1 worksheet, 483
analogy to magnetism, 508
bridging from understanding magnetic action at a distance to understanding gravitational action at a distance, 508–510
building an analogy to understand the benchmark experience, 489–490
consensus discussion and summary of learning, 490–491
defining, 477–510
diagnostic assessment, 491–492
exploring similarities and differences between actions at a distance, 492–493
factors on which the magnitude of gravitational force depends, 501–508
finding out about students’ initial ideas, 477–478
identifying preconceptions, 478–480
opportunities for students to suggest and test related hypotheses, 484–489
twisting a torsion bar, 493–501
weighing in a vacuum, 480–483
Grids, 173–175
Griffin, Sharon, 23, 257–308, 593
Group work, 582–584
Guess My Number, 300
Guidance of student observation and articulation
supporting metacognition, 584–585
Guided inquiry, 495, 579, 582
H
“H(ac)”, 187–188
Hall, G. Stanley, 177n
Halsall, William Formsby, 87
Help
seeking and giving, 241–242
Heuristic for teaching and learning science through guided inquiry, 427–455
cycle of investigation in guided-inquiry science, 427
data tables from initial recording and with revisions for analysis, 445
engage phase, 428–434
fourth graders’ initial ideas about light, 431
investigate phase, 438–443
investigative setup for studying how light interacts with solid objects, 437
prepare-to-investigate phase, 434–438
prepare-to-report phase, 443–448
report phase, 448–455
“H(ev)”, 187
Higher-order knowledge structure, 276
Historical accounts, 59–61
different ideas about, 38–39
not copies of the past, 62–63
“problematizing,” 184–188
Historical evidence, 134
Historical films, 151
Historical lines of thinking, 182
Historical problems
transforming topics and objectives into, 181–199
History, 29–213
applying the principles of How People Learn in teaching high school history, 179–213
“counterintuitive” intuitions in, 33, 42
“doing,” 32, 48
implications for planning, 164–176
periods in, 42–43
putting principles into practice, 79–178
the reality test, 80–84
significance in, 45
that “works,” 65–72
understanding, 31–77
working with evidence, 84–119
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History and everyday ideas, 33–61
differences in the power of ideas, 36–37
grounds for caution, 40–41
ideas we need to address, 41–61
the progression of ideas, 37–40
understanding the past and understanding the discipline of history, 34–35
“History-as-account,” 187–188, 203
“History-as-event,” 187, 203
“History-considerate” learning environments
designing, 199–209
the great interruption in European geographic knowledge, 200–201
with tools for historical thinking, 199–209
History of the Decline and Fall of the Roman Empire, The, 57
Hitler, Adolf, 34–35, 59–60, 586
Holt, John, 218
How People Learn: Brain, Mind, Experience, and School, 1, 25, 31–32
cautions in, 199
design characteristics described in, 12–13, 20–22, 257–258, 359
key findings of, 79–80, 171–173, 176
research summarized in, 241
violating principles of, 319
How People Learn framework, 411–415
assessment-centered, 415
community-centered, 415
knowledge-centered, 414
learner-centered, 414
reflective assessment in ThinkerTools, 412–413
Humor
enlivening learning and helping build positive relationships with students, 501
I
Ideas, 41–61
accounts, 59–61
cause, 49–54
change, 43–46
empathy, 46–49
evidence, 54–58
progression of, 37–40
providing students with opportunities to make public, 524
“second-order,” 32–33
time, 41–43
Inaccessible algorithms, 236
Information, 41, 124, 166
“clumping,” 69
finding, 121
from history, 499
from the history of science, 499
inquiry based, 470n
storing in memory, 180
Inheritance
meiotic processes governing, 528
Initial models
providing students with opportunities to revise in light of anomalous data and in response to critiques of others, 524
Inquiry based information, 470n
Instruction, 545–554
to support mathematical proficiency, 233–236
Instruction in rational number, 319–340
alternative instructional approaches, 321–322
children’s thinking after instruction, 338–340
curriculum overview, 325
fractions and mixed representations of rational numbers, 334–337
introduction of decimals, 332–334
introduction to percents, 325–332
knowledge network, 340
pie charts and a part-whole interpretation of rational numbers, 320–321
pipes, tubes, and beakers, 322–324
Instruction that supports metacognition, 239–242
emphasizing debugging, 239–240
internal and external dialogue as support for metacognition, 241
seeking and giving help, 241–242
Instructional lines of thinking, 182
Intellectual roles for students to adopt, 436
Internal consistency, 518
Internal migration, 68
Interpretation
anchoring themes in historical, 186
of data, 403
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Interpreting sources in context and textbook claims and the nature of sources, 100
Intuitions in history
“counterintuitive,” 33, 42
Invented procedures, 329
Investigate phase, 438–443
Investigative setup for studying how light interacts with solid objects, 437
Irving, Washington, 208
Isolation
evidence in, 167
Italy
instruction about payment for work, 66–67
J
Japan
teacher professional development in, 244
Jasper Woodbury series, 391
Jefferson, Thomas, 62–63
Johnson, Lyndon, 62
Jonassen, David, 181
Judgments
avoiding expressing, 498
K
Kalchman, Mindy, 23, 217–256, 351–393, 593
Knowledge. See also Prior understandings building learning paths and networks of, 258
connected, 15–16
disciplinary, 32, 543–544
handed down through generations, 93–94
manipulating factual, 79–80
“metahistorical,” 32
organized, 462
“second-order,” 32–33
secret, 72
student, 258, 544–545
of what it means to “do science,” 403–407
Knowledge-centered classroom environments, 13–16, 267, 284, 292, 414, 555, 587
Knowledge claims
in genetics, assessing, 523
Knowledge networks, 340
new concepts of numbers and new applications, 312–316
new symbols, meanings, and representations, 313–314
reconceptualizing the unit and operations, 315
the subconstructs, 314–315
understanding numbers as multiplicative relations, 316
“Knowledge packages,” 588n
Knowledge that should be taught, 259–267
central conceptual structure hypothesis, 262–265
children passing the Number Knowledge Test, 263, 265
measures of arithmetic learning and achievement, 265
numerical transfer tests, 263
Koedinger, Kenneth R., 351–393, 593–594
Kraus, Pamela, 23, 401, 475–513, 594
KWL charts, 199, 428–430
L
Lamarck, Jean Baptiste de, 550, 573
Larson, Gary, 217
Learner-centered classroom environments, 13–14, 266, 292, 414, 555
Learning
an active process, 476
humor enlivening, 501
Learning environments and the design of instruction, 12–20
assessment-centered classroom environments, 13, 16–17, 267, 290, 292, 555–558
community-centered classroom environments, 13, 17–20, 301, 559–560
knowledge-centered classroom environments, 13–16, 267, 284, 292, 555, 587
learner-centered classroom environments, 13–14, 266, 292, 414, 555
perspectives on, 13
Learning goals for prekindergarten through grade 2, 284–285
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Learning paths of knowledge
building, 258
from children’s math worlds, for single-digit addition and subtraction, 234–235
Learning principles
engaging resilient preconceptions, 569–575
organizing knowledge around core concepts, 575–577
principles of learning and classroom environments, 586–588
pulling threads, 569–590
revisiting the three, 567–590
supporting metacognition, 577–586
Learning rational number, 341–343
metacognition, 342
network of concepts, 341–342
prior understandings, 341
Learning with understanding, 559–560
supporting knowledge use in new situations, 7
Leather boats, 139–141
Lee, Peter J., 23, 31–178, 576, 594
Lesson Study Research Group, 244
Life and Voyages of Christopher Columbus, The, 208
“Light catchers,” 437.
See also Study of light
Linkage
of formal mathematical understanding to informal reasoning, 354–355
Lionni, Lee, 2, 4.
See also Fish story
Logic of the situation
exploring, 50–51
Lowenthal, David, 185
M
Ma, Liping, 15–16, 18–19, 577–578
Magic Shoes game, 295–296
Magnetism
analogy to gravity, 508
Magnitude
in decimal numbers, 333–334
of gravitational force, 501–508
Magnusson, Shirley J., 421–474, 594
Management of student activities, 435
Mandates
curricular, 181
Manipulation of factual knowledge, 79–80
Maps, 86, 140–141
conceptual, 188
Marfan’s syndrome, 533
Math words, 230
Mathematical proficiency, 218
adaptive reasoning, 218
conceptual understanding, 218
procedural fluency, 218
productive disposition, 218
strategic competence, 218
Mathematical thinkers
building, 258
Mathematical understanding, 217–256
computation without comprehension, 218
developing mathematical proficiency, 232–236
learning to use student thinking in teacher video clubs, 244
lesson study cycle, 244
a metacognitive approach enabling student self-monitoring, 236–243
suggested reading list for teachers, 256
teachers as curriculum designers, 245
teachers engaging students’ preconceptions, 219–231
understanding requiring factual knowledge and conceptual frameworks, 231–236
Mathematics, 215–393
as about quantity, not about numbers, 280
as “following rules” to guarantee correct answers, 220–221
fostering the development of whole number sense, 257–308
as learning to compute, 220
pipes, tubes, and beakers in, 309–349
teaching and learning functions, 351–393
Mathematics instruction
in China, 15–16, 18–19
Mayflower, The
arrival of, 84, 87, 90, 92–95
Medawar, Peter, 406
Media
technical and passive, 496
Meiotic processes
governing inheritance, 528
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Mendel, Gregor, 406, 410, 517, 523, 525–529, 539
model of simple dominance, 528
Mental counting line structure, 276
Metacognition, 10, 238, 407–411, 577–586
conceptual explanation without conceptual understanding, 578
engaging students in reflective scientific practice, 538–540
in evaluating the methods used in an experiment, 408–409
guiding student observation and articulation, 584–585
of light, 426
in Mendel’s contribution to genetics, 410
questioning and explaining in high school science, 582–583
and rational number, 319, 342
supporting, 577–586
supporting skilled questioning and explaining in mathematics problem solving, 580–581
Metacognitive approaches to instruction, 2, 80
enabling student self-monitoring, 236–243
framework of How People Learn, 242–243
instruction that supports metacognition, 239–242
seeking a balanced classroom environment, 242–243
supporting student and teacher learning through a classroom discourse community, 237
Metacognitive monitoring, 10
“Metahistorical” knowledge, 32
“Metamemory,” 11
Migration
internal and external, 68
Miller Analogies Test, 404
“Mindtools,” 181
Minstrell, James, 23, 401, 475–513, 594–595
Minus Mouse, 290–291
Misconceptions
about momentum, 5
about the scientific method, 414
“Missing-term problem,” 317
Misunderstandings, 310
Model-based inquiry, 515–565
classroom environments that support learning with understanding, 555–560
developing Darwin’s model of natural selection in high school evolution, 540–554
genetics, 516–540
Modeling for Understanding in Science Education (MUSE), 516, 548
curricula from, 555, 559
Models, 402–403
consistency between, 557
of progression in ideas about evidence, 166–167
providing students with opportunities to revise in light of anomalous data and in response to critiques of others, 524
Monarch/viceroy case
Darwinian explanation written by students on the, 553
Monitoring.
See also Self-monitoring metacognitive, 10
“Monster-free zone,” 295
Moss, Joan, 23, 309–349, 595
Motion of projectiles
explaining, 511
Multiple strategies, 223–227
allowing, 223–227
engaging students’ problem-solving strategies, 225–227
three subtraction methods, 224
Multiplicative operators, 315
Multiplicative reasoning
relative thinking as, 311
MUSE. See Modeling for Understanding in Science Education
Mystery
sense of, 71
“Mystery Object Challenge,” 329
N
Narrative accounts
providing students with, 573–575
National Council of Teachers of Mathematics (NCTM), 221, 241, 259
standards from, 305
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How Students Learn: History, Mathematics, and Science in the Classroom
National Curriculum for History, 177n
National Research Council, 1, 218, 221, 233
guidelines of, 398
National Science Education Standards, 455, 561
Native Americans, 41, 82–83, 98, 105–106
NCTM. See National Council of Teachers of Mathematics
Necessary conditions
causes as, 53
Neighborhood Number Line, 295
Networks
of concepts, and rational number, 341–342
of knowledge, building, 258
New conceptualizations
understanding numbers as multiplicative relations, 316
New ideas
development of, 470n
New rules
discovering, 588
New symbols
meanings, and representations, 313–314
“Nothing” happening, 43
Number Knowledge Test, 260, 264, 267–269, 271, 279, 304–305
administering and scoring, 271
Number worlds, 282–302
encouraging the use of metacognitive processes to facilitate knowledge construction, 300–302
engaging children’s emotions and capturing their imagination, 296–298
exposing children to major forms of number representation, 283–288
the five forms of representation and the lands in which they appear, 286
learning goals for prekindergarten through grade 2, 284–285
providing analogs of number representations that children can actively explore hands-on, 292–296
providing opportunities for children to acquire computational fluency as well as conceptual understanding, 298–300
providing opportunities to link the “world of quantity” with the “world of counting numbers” and the “world of formal symbols,” 288–292
Number Worlds program, 262, 283, 287–288, 292, 296, 300, 302–303
Numeric answers, 372
O
Object Land, 284–286, 288
“One world” revolution, 169
“Openings” in the curriculum, 245
Opportunities
to develop causal models to account for patterns, 524
to experience discrepant events that allow them to come to terms with the shortcomings in their everyday models, 571–573
to make ideas public, 524
providing students with, 523–524
to revise initial models in light of anomalous data and in response to critiques of others, 524
to search for patterns in data, 524
to use patterns in data and models to make predictions, 524
to use prior knowledge to pose problems and generate data, 523–524
Opportunities for children to acquire computational fluency as well as conceptual understanding, 298–300
Sky Land Blastoff activity, 298–299
Opportunities for students to suggest and test related hypotheses in elaboration activities, 484–489
inverted cylinder in a cylinder of water, 485–486
inverted glass of water, 484–485
leaky bottle, 486
water and air in a straw, 486–488
weighing” an object in a fluid medium, 488–489
Opportunities to link the “world of quantity” with the “world of counting numbers” and the “world of formal symbols,” 288–292
Minus Mouse, 290–291
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Plus Pup, 288–290
Plus Pup meets Minus Mouse, 291–292
Optics kit, 422, 468
Order
of counting words, 274
in decimal numbers, 333–334
Organized knowledge, 462
Organizing knowledge around core concepts
subtraction with regrouping, 18–19
Origin of Species, 551
Outcomes of courses, 181
P
Pace of change, 44
Paley, William, 550–551, 573
Palincsar, Annemarie Sullivan, 23, 421–474, 595
Park, Lesley, 455
Part-whole relation, 314
Pass it on (game), 105
Passive media, 496
Passmore, Cynthia M., 23, 515–565, 595
Past
finding out about, 56–58
pictures of, 166
Patterns in data
providing students with opportunities to search for, 524
providing students with opportunities to use to make predictions, 524
Payment for work in history, 66–67
Peanuts cartoon, 309
Pedagogical words
meaningful, 230
People going their separate ways
unit on, 170
Percents, 325–332, 340
computing with, 329
in everyday life, 325
“families” of, 331
invented procedures, 329
on number lines, 326–329
pipes and tubes, as representations for fullness, 325–326
starting from, 322–324
string challenges, 329–331
Percy, George, 122
Performance
need to assist, 203
Periods in history, 42–43
Physics
fundamental, 24
instruction in, 16–17
Picture Land, 285–287, 297
Pie charts and a part-whole interpretation of rational numbers, 320–321
Pilgrim Fathers and Native Americans, 71, 84–119
exploring the basis for textbook claims and the nature of sources, 84–111
grid for evidence on, 173, 175
ideas, beliefs, and attitudes, 112–118
language of sources, interpretation, and other perspectives, 118–119
teacher questions, 112–113, 115
whether people thought like us in the past, 117
Pipes
a new approach to rational-number learning, 322–324
a representation for fullness, 325–326
Planning, 164–176
of progression in ideas about evidence, 166–167, 174–175
unit on filling the world with people, 169
unit on first contacts, whether St. Brendan sailed from Ireland to America, 171
unit on first contacts, why the Norse colonists didn’t stay in America, 172
unit on people going their separate ways, 170
Plausibility, 138
Plus Pup, 288–290
meeting Minus Mouse, 291–292
Pocahontas (Disney film), 122
Pory, John, 84–85, 90, 97, 100–104, 106–108
Positive relationships
humor helping to build with students, 501
Possible Worlds, 406
Power
explanatory and predictive, 518
Preconceptions, 1, 55, 399–403
about people, society, and how the world works, 127–128
conceptual change, 400–403
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drawing on knowledge and experiences that students commonly bring to the classroom but are generally not activated with regard to the topic of study, 569–571
engaging resilient, 569–575
everyday concepts of scientific methods, argumentation, and reasoning, 400
everyday concepts of scientific phenomena, 399–400
importance of students’, 79
providing opportunities for students to experience discrepant events that allow them to come to terms with the shortcomings in their everyday models, 571–573
providing students with narrative accounts of the discovery of (targeted) knowledge or the development of (targeted) tools, 573–575
Preconceptions about how we know about the past, 121–123
common student assumptions about how we know of the past, 123
dangers of what appears to be familiar, 122
Predictive power, 518
Preinstruction assessments, 495
Prepare-to-investigate phase, 434–438
Prepare-to-report phase, 443–448
Principles of How People Learn applied to teaching high school history, 179–213
designing a “history-considerate” learning environment, 199–209
transforming topics and objectives into historical problems, 181–199
Prior understandings
development of physical concepts in infancy, 4
engaging, 4–5
of light, 425
misconceptions about momentum, 5
providing students with opportunities to use to pose problems and generate data, 523–524
and rational number, 341
Problem solvers
building, 258
“Problematizing” historical accounts, 184–188
Procedural fluency, 218
Productive disposition, 218
Proficiency
mathematical, 218
Progress, 44–45
Progression of ideas, 37–40
different ideas about historical accounts, 38–39
Progressive change, 45
Project CHATA. See Concepts of History and Teaching Approaches
Projectiles
explaining motion of, 511
Proportion, 234, 340
Pump Algebra Tutor. See Cognitive Tutor Algebra
Q
Quantity, 234
schema for, 272
Question Poser, 300–301
Questioning and explaining in high school science
supporting metacognition, 582–583
Questions, 128
diagnostic, 478
at the heart of using evidence, 124
many as yet unanswered, 492
teachers modeling for students, 477
Quotient interpretation, 314
R
Rational change, 45
Rational number, 341–343
metacognition, 342
network of concepts, 341–342
prior understandings, 341
Rational-number learning
and the knowledge network, 312–316
metacognition and rational number, 319
new concepts of numbers and new applications, 312–316
and the principles of How People Learn, 312–319
students’ errors and misconceptions based on previous learning, 316–319
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Real-world experience
current and future, 390
Real-world words, 230
Reality test, 80–84
“7-year gap,” 82
Reciprocal teaching, 11
Reconceptualizing the unit and operations, 315
Recorder, 435
Reflective assessments, 412
in ThinkerTools, 412–413
Regrouping
subtraction with, 18–19
Relative thinking as multiplicative, 311
Relativism, 176
Reliability, 126
Religious practices, 113–118
Reporter, 301
Reporting phase, 427, 448–455
Representations, 372
anchoring themes in historical, 186
Reproductive success, 542
Revolution, 61
S
Sagan, Carl, 194, 196–197
Sales, Kirkpatrick, 208
Schemas
2-slot and 3-slot, 370
counting and quantity, 272
Schools Council History Project, 40, 177n
Science, 395–565
developing understanding through model-based inquiry, 515–565
guided inquiry in the science classroom, 475–513
information from the history of, 499
leaving many questions as yet unanswered, 492
teaching to promote the development of scientific knowledge and reasoning about light at the elementary school level, 421–474
unit on the nature of gravity and its effects, 477–511
Science classrooms
guided inquiry in, 475–513
Scientific inquiry and How People Learn, 397–419
addressing preconceptions, 399–403
diagnosing preconceptions in physics, 404
the How People Learn framework, 411–415
knowledge of what it means to “do science,” 403–407
Scientific method
misconceptions about, 414
Scissors-and-paste approach and textbook claims and the nature of sources, 94
Searchers, The (film), 151
Second cycle of investigation
community knowledge from, 464
Second-hand investigation, 455–459
“Second-order” disciplinary concepts, 61, 73n
“Second-order” knowledge, 32–33, 41
acquisition of, 40–41
Secret knowledge, 72
Seeing for yourself and textbook claims and the nature of sources, 93
Seixas, Peter, 151
Selective advantage, 542
Self-assessment, 12
Self-monitoring
importance of, 10–12
metacognitive monitoring, 10
Sensitivity
“7-year gap,” 82
7-year-olds understandings of, 277–278
to students’ substantive assumptions, 127
Severin, Tim, 139, 142–143
Shemilt, Denis, 23, 56, 79–178, 595–596
Shrinking past, 160–161
Significance, 45
historical, 45
Simplicity, 389–390
6-year-olds understandings of, 274–277
Skating Party game, 292–295
Skills
defining, 40
Sky Land, 286–287
Blastoff activity, 298–299
Smith, John, 122
Sources
access to someone who saw for himself, 93
briefing sheet, 88–89
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distinguishing among kinds of claims, 101–102
generalizing, 102–107
getting behind the record to concerns of the people who produced them, 107–108
interpreting sources in context, 100
maintaining contact with an eyewitness using knowledge handed down through generations, 93–94
the nature of, 84–111
scissors-and-paste approach, 94
seeing for yourself, 93
teacher questions, 92, 95–96, 99–101
trusting the source who was in a position to know, 96
understanding the purpose of the source, 96–99
understanding what is likely to get recorded and under what circumstances, 108–111
working out the facts from other sources or available knowledge, 94–95
Splitting, 323
State of affairs
changes in, 44
Stearns, Peter, 210
Stewart, James, 23, 515–565, 596
“Stop-Start Challenge,” 333
Stopwatches
decimals and, 332–333
Stories
“embroidering,” 153
Strategic competence, 218
String challenges
guessing mystery objects, 329–331
Student assumptions about how we know of the past, 123
Student conceptions
experimentation, 402
inadequacies in arguments, 403
interpretation of data, 403
of knowledge generation and justification in science, 402–403
models, 402–403, 518
Student inquiry in genetics, 526–538
example of student work on a GCK homework assignment, 535
genetic inquiry in the classroom, 529–534
initial GCK population for the final GCK inquiry, 537
meiotic processes governing inheritance, 528
Mendel’s model of simple dominance, 528
Students’ errors and misconceptions based on previous learning, 316–319
Students’ existing knowledge, 517–526
assessing knowledge claims in genetics, 523
attending to, 544–545
black box, 520
building on and connecting, 258
learning genetics content, 524–526
providing students with learning opportunities, 523–524
student conceptions of models, 518
Students’ preconceptions
importance of, 79
Study of light, 422–426
conceptual understanding, 423–424
metacognition, 426
prior knowledge, 425
Study of light through inquiry, 426–459
heuristic for teaching and learning science through guided inquiry, 427–455
second-hand investigation, 455–459
Subconstructs
the many personalities of rational number, 314–315
Subject-specific knowledge in effective science instruction, 467–469
Substantiated accounts, 87
Substantive assumptions
sensitivity to students’, 127
Substantive concepts, 61–65
historical accounts not copies of the past, 62–63
payment for work, 66–67
Subtraction with regrouping, 18–19
Supporting learning through cycles of investigation, 460–467
Supporting skilled questioning and explaining in mathematics problem solving
supporting metacognition, 580–581
Supporting student and teacher learning through a classroom discourse community, 237
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T
Table of values to produce a function, 353–358
Teacher professional development in Japan, 244
Teacher questions, 112–113, 115
and textbook claims and the nature of sources, 92, 95–96, 99–101
Teachers’ conceptions and partial understandings, 279–281
acquiring an understanding of number as a lengthy, step-by-step process, 280–281
counting words as the crucial link between the world of quantity and the world of formal symbols, 280–281
math as not about numbers, but about quantity, 280
Teachers engaging students’ preconceptions, 219–231
common preconceptions about mathematics, 220–222
engaging students’ preconceptions and building on existing knowledge, 223–231
Teaching
reciprocal, 11
Teaching and learning functions in mathematics, 351–393
addressing the three principles, 359–373
building conceptual understanding, procedural fluency, and connected knowledge, 364–369
building on prior knowledge, 359–364
building resourceful, self-regulating problem solvers, 371–373
linking formal mathematical understanding to informal reasoning, 354–355
making a table of values to produce a function, 353–358
teaching functions for understanding, 373–389
teaching to achieve this kind of understanding, 358–359
Teaching as Story Telling, 574
Teaching functions for understanding, 373–389
Teaching mathematics in the primary grades, 257–308
acknowledging teachers’ conceptions and partial understandings, 279–281
building on children’s current understandings, 267–279
the case of number worlds, 282–302
comparing number worlds and control group outcomes, 304
deciding what knowledge to teach, 259–267
defining the knowledge that should be taught, 281–282
Teaching the rational number system, 309–349
additive and multiplicative reasoning, 311
how students learn rational number, 341–343
instruction in rational number, 319–340
rational-number learning and the principles of How People Learn, 312–319
Teaching to promote the development of scientific knowledge and reasoning about light at the elementary school level, 421–474
the role of subject-specific knowledge in effective science instruction, 467–469
the study of light, 422–426
the study of light through inquiry, 426–459
supporting learning through cycles of investigation, 460–467
Technical media, 496
Testimony, 41, 124, 135, 166
Testing
external, 181
Textbook claims
access to someone who saw for himself, 93
briefing sheet, 88–89
distinguishing among kinds of claims, 101–102
generalizing, 102–107
getting behind the record to concerns of the people who produced them, 107–108
interpreting sources in context, 100
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maintaining contact with an eyewitness using knowledge handed down through generations, 93–94
and the nature of sources, 84–111
scissors-and-paste approach, 94
seeing for yourself, 93
teacher questions, 92, 95–96, 99–101
trusting the source who was in a position to know, 96
understanding the purpose of the source, 96–99
understanding what is likely to get recorded and under what circumstances, 108–111
working out the facts from other sources or available knowledge, 94–95
Themes, 44
anchoring in historical representation and interpretation, 186
ThinkerTools, 407, 585
Third cycle of investigation
community knowledge from, 465
Third International Mathematics and Science Study, 243
3-slot schema
for graphing a line, 370–371
Three subtraction methods, 224
Time, 41–43
change limited in, 45
periods in history, 43
Time lines, 129, 159
Timekeeper, 435
Torsion bar, 493–501
Transforming topics and objectives into historical problems, 181–199
accounting for the “flat earth,” 189–199
“problematizing” historical accounts, 184–188
Transmission errors, 123
Trusting the source who was in a position to know
and textbook claims and the nature of sources, 96
Truth
twisting, 105, 123
Tubes
a new approach to rational-number learning, 322–324
a representation for fullness, 325–326
Turner, Frederick Jackson, 58
Twisting the truth, 105, 123
2-slot schemas, 370
U
“Underlying” causes, 35
Understanding
essential role of factual knowledge and conceptual frameworks in, 6–9
experts remembering considerably more relevant detail than novices in tasks within their domain, 8–9
learning with understanding supporting knowledge use in new situations, 7
Understanding of number
a lengthy, step-by-step process, 280–281
Understanding the purpose of the source and textbook claims and the nature of sources, 96–99
Understanding what is likely to get recorded and under what circumstances
and textbook claims and the nature of sources, 108–111
Unit-level problem, 189–199
accounts of Colombian voyages, 192–193
ancient views of the Earth as flat or round, 196–197
Unit on the nature of gravity and its effects, 477–511
United Kingdom
adjusting data from, 177n
Schools Council History Project, 40, 177n
Units
on filling the world with people, 169
on first contacts, whether St. Brendan sailed from Ireland to America, 171
on first contacts, why the Norse colonists didn’t stay in America, 172
on people going their separate ways, 170
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V
Verbal interpretations, 372
Visual proportional estimation starting from, and halving and doubling, 323–324
W
War (card game), 336
Warm-Up period, 298, 300
Water and air in a straw, 486–488
Website, 562n
“Weighing” an object in a fluid medium, 488–489
Weighing-in-a-vacuum situation, 484, 489
Whole number
central conceptual structure for, 261–262, 275
Wilson, Suzanne M., 596
Wineburg, Samuel S., 100
Wisdom, 236, 238
Woodbury, Jasper, 391
Word Problems test, 264–265
Words
versus notations, 230
Words in stories
explaining, 132–133
Work
payment for in history, 66–67
Working out the facts from other sources or available knowledge
and textbook claims and the nature of sources, 94–95
Working things out for ourselves, 133–138
being aware of how we are thinking, 135
going beyond face value, 134
how far a leather boat could have managed to sail, 139–141
Working through the task, 128–164
Working with evidence
Pilgrim Fathers and Native Americans, 84–119
preparing for the task, 121–128
the St. Brendan’s voyage task, 128–164
World’s Fair of 1892, 208
Wrap-Up period, 301
Written Arithmetic test, 264–265
Y
Year-long historical questions, 184–188
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