Index

• A-ha moments, 137
• Ability grouping, 154–155, 226–228
• Accountable talk, 144–146, 145 (figure)
  • language frames and, 146, 147 (figure)
  • restating strategy and, 147–148
  • revoicing strategy and, 146
  • small group collaboration and, 157–159
  • See also Deep mathematics
• Algorithms, 3, 29, 139–140
• Alternate ranking, 155–156, 156 (figure)
• Applications, 75–76
• Assessment, 200, 231–232
  • advanced students and, 225
  • curriculum-embedded assessments and, 223
  • formative evaluation and, 200–208, 223
  • group collaborative assessments, 62, 63 (figure)
  • learning intention and, 39
  • preassessments, role of, 66–67
  • progress monitoring and, 223
  • summative evaluation and, 208–211, 223
  • See also Feedback; Success criteria
• Australia’s Mathematics Curriculum Key Ideas, 54
• Automaticity, 77–78

• Barometer of influence, 6–7 (figures), 21–22, 22 (figure)
• Bartolot, E., 194
• Bay-Williams, J. M., 122, 123, 124
• Benbow, C. P., 225
• Bennett, N., 156
• Bereiter, C., 184
• Biddle, M., 124
• Biggs, J., 27, 28
• Boaler, J., 2
• Booth, J., 114
• Breyfogle, M. L., 89, 90, 91
• Brookhart, S. M., 129, 203, 204
• Butler, R., 203

• Career readiness, 1–2
• Carmody, G., 21
• Cass, A., 156
• Center on Disability & Community Inclusion, 148
• Clarke, D., 116
• Clarke, S., 43
• Classroom discussion, 15, 85
  • accountable talk and, 144–148, 145 (figure), 147 (figure)
  • checking for understanding and, 88–89, 92–93
  • cues and, 94, 96, 96 (figure)
  • deepened understanding, questions for, 86–93
  • direct instruction vs. dialogic approach and, 23–26, 25–26 (figure)
  • effective questions, formulation of, 93
  • focusing questions and, 89, 90–92, 92 (figure)
  • funneling questions and, 89–90, 92 (figure)
  • norms/rules of, 148–150
  • prior knowledge, activation of, 93–94
  • prompts and, 93–94, 95 (figure)
  • purposeful questions and, 86–88, 93
  • rich discourse, characteristics of, 85
  • scaffolded learning and, 90, 93
  • surface mathematics and, 104
  • transfer learning and, 188–189
  • See also Deep mathematics; Guided learning; Surface mathematics
• Cobb, P., 150
• Cognitive demand, 80
  • mathematical tasks, characteristics of, 80–83, 81–82 (figure)
  • mathematical tasks, examples of, 83, 84 (figure)
  • See also Guided learning
• Cognitively Guided Instruction, 75–76
• Cole, M. W., 120
• Collaboration. See Deep mathematics
• Collis, K., 27, 28
• Common Core State Standards (CCSS), 3
  • international mathematical practice/process standards and, 242–243 (appendix)
  • Standards for Mathematical Practice, 7, 53, 136, 137, 240–241 (appendix)
  • See also Mathematics instruction; Teaching practices
• Complex problems, 76–80, 77 (figure)
• Contribution checklists, 159, 160 (figure)
• Conversation roundtable, 164–165, 164 (figure)
• Corno, L., 203
• Council of Chief State School Officers (CCSSO), 3, 53, 117, 219
• Cues, 94, 96, 96 (figure)
• Cunningham, P. M., 123

• Deep learning, 23, 27, 30–31, 34–35, 34 (figure)
  • See also Deep mathematics
• Deep mathematics, 134
  • accountable talk and, 144–148, 145 (figure), 147 (figure), 157–159, 166
  • algorithms, exploration of, 139–140
  • classroom-based physical activity and, 160–161
  • classroom discussion, norms of, 148–150
  • collaboration, appropriate use of, 153–154
  • collaborative learning, supports for, 159–161
  • connections/generalizations and, 138–140, 139–140 (figure)
  • contribution checklists and, 159, 160 (figure)
  • conversation roundtables and, 164–165, 164 (figure)
  • deep learning, nature of, 136–141
  • discourse, role of, 136–137, 148–150, 166
  • discourse strategies, selection of, 167–168
  • effective conversational teaching practices and, 151, 152 (figure)
  • exercises vs. problem solving and, 142, 143 (figure)
  • fluency, development of, 137–139, 138 (figure)
  • group size and, 156–157
  • grouping strategies and, 154–157, 156 (figure)
  • individual accountability, supports for, 162–165
  • individual exploration, time for, 165
  • Initiate-Respond-Evaluate questioning cycle and, 150
  • inside-outside circles and, 167
  • language frames and, 146, 147 (figure)
  • manipulatives, strategic use of, 170–171
  • mathematical discourse, sociomathematical norms and, 150–151
  • mathematical representations, connections among, 169–170, 169 (figure)
  • mathematical talk and, 142–150
  • mathematical tasks for, 141–142
  • metacognition and, 141, 164
  • peer support/tutoring and, 161
  • physical layout, effective discourse and, 167
  • restating strategy and, 147–148
  • revoicing strategy and, 146
  • small group collaboration/discussion strategies and, 151–165
  • teacher talk, limited use of, 142
  • Think-Pair-Square activity and, 162–163
  • volume lesson, high-cognitive demand task and, 134–136, 135 (figure)
  • whole class collaboration/discourse strategies and, 165–168
  • writing rules and, 163–164, 164 (figure)
  • See also Deep learning; Mathematics instruction; Surface mathematics; Teaching practices; Transfer learning; Visible learning
• DeLashmutt, K., 130
• Dialogic approach, 23–26, 25–26 (figure)
  • See also Classroom discussion
• Differentiated instruction, 27, 199, 211
  • advanced students, interventions for, 224–226
  • classroom structures for, 211–212
  • content, adjustment of, 212–213
  • definition of, 211
  • grouping strategies and, 154–157, 156 (figure), 212, 226–228
  • ineffective strategies and, 226–231
  • instructional adjustments for, 212–214
  • process, adjustment of, 213
  • product, adjustment of, 213–214
  • progress monitoring and, 223
  • zone of proximal development and, 212
  • See also Feedback; Learning; Mathematics instruction; Response to Intervention (RTI); Teaching practices
• Difficulty, 76–80, 77 (figure)
• Direct instruction, 3, 23
  • definition of, 23, 24
  • dialogic approach, comparison with, 23–26, 25–26 (figure)
  • surface mathematics and, 116–119
• Discourse, 136–137
  • See also Classroom discussion; Deep mathematics; Dialogic approach; Guided learning; Visible learning
• Distributed practice, 73, 128–130
• Drill-and-kill instruction, 3

• Effect sizes, 5–6, 235–239 (appendix)
  • ability grouping and, 154, 227, 228
  • aggregated information and, 5–6, 22
  • barometer of influence and, 6–7 (figures), 21–22, 22 (figure)
  • classroom cohesion and, 51, 161
  • classroom discussion and, 142, 153, 189
  • compare/contrast new with old problems and, 183
  • concentration/persistence/engagement and, 154
  • cooperative vs. individualistic learning and, 153
  • definition of, 19–20
  • dialogic approach and, 24
  • direct instruction and, 24
  • examples of, 6–7, 6–7 (figures)
  • expectations and, 61
  • feedback and, 129, 203
  • goal setting and, 61
  • homework and, 230
  • influence and, 20–21
  • labeling students and, 229
  • learning styles-instruction match and, 228
  • meta-analyses and, 19
  • metacognitive strategies and, 39, 120, 185
  • micro-teaching and, 113
  • mnemonics and, 130
  • organized conceptual knowledge and, 189
  • peer tutoring and, 161, 190
  • problem-solving teaching and, 192
  • questioning and, 165
  • Response to Intervention and, 226
  • retention and, 226
  • self-reported grades/student expectations and, 35, 57
  • self-verbalization/self-questioning and, 109, 119, 148, 185
  • simulations and, 22–23
  • small group learning and, 227
  • spaced vs. massed practice and, 129
  • statistical significance and, 19
  • student self-monitoring and, 35
  • study skills and, 230
  • tactile stimulation programs and, 161
  • teacher clarity and, 40
  • teacher-student relationships and, 48
  • test prep instruction and, 229
  • visible learning practices and, 19–20
  • vocabulary instruction and, 120
  • vocabulary programs and, 119
  • worked examples and, 113
  • zone of desired effects and, 21, 22 (figure)
  • See also Visible learning evidence
• Elawar, M. C., 203
• Ellis, A. B., 116
• Evidence-based approach, 18–20, 105
  • See also Response to Intervention (RTI); Visible learning; Visible learning evidence
• Exercises, 73, 142, 143 (figure)
• Exit Ticket strategy, 64, 101, 102 (figure), 103, 198–199, 199 (figure), 200, 202, 227–228
• Expert blind spot, 40–41

• Far transfer, 177, 178–179
• Feedback, 129–130, 184, 231
  • characteristics of, variations in, 204 (figure)
  • exit tickets and, 198–199, 199 (figure), 200, 202
  • formative evaluation and, 200–208, 223
  • instructional adjustments and, 201–203, 202 (figure), 210–211
  • just-in-time/just-for-me information and, 129–130, 203
  • levels of feedback and, 205–207
  • mistakes, opportunity of, 207–208
  • Numbered Heads Together strategy and, 202
  • Purposeful Sampling strategy and, 202
  • Response Cards strategy and, 202
  • spaced practice and, 128–130
  • student-focused feedback and, 203–208, 204 (figure)
  • summative evaluation and, 208–211, 223
  • understanding, checks for, 201, 202 (figure)
  • See also Assessment; Differentiated instruction; Response to Intervention (RTI)
• Fendick, F., 40
• Fisher, D., 28, 50, 63, 65, 95, 96, 121, 145, 147, 160, 164, 186, 188
• Flexible grouping, 155, 227
• Focusing questions, 89, 90–92, 92 (figure), 166
• Formative evaluation, 200–208, 223, 231
• Frey, N., 28, 50, 63, 65, 95, 96, 121, 145, 147, 160, 164, 186, 188, 226
• Funneling questions, 89–90, 92 (figure), 166

• Garofalo, J., 185
• Genius Hour, 191
• Gojak, L. M., 28, 50, 63, 65, 121, 145, 147, 160, 164, 186, 188
• Graphic organizers, 125, 126 (figure)
• Gresham, G., 215
• Grouping strategies, 154–157, 156 (figure), 212, 226–228
• Guided learning, 72
  • applications and, 75–76
  • appropriate mathematical tasks and, 72–84
  • automaticity and, 77–78
  • checking for understanding and, 88–89, 92–93
  • classroom discussion and, 85–96, 92 (figure), 95–96 (figures)
  • cognitive demand, taxonomy of tasks and, 80–83, 81–82 (figure), 84 (figure)
  • difficulty vs. complexity and, 76–80, 77 (figure)
  • exercises vs. problems and, 73–76
  • false positives, avoidance of, 93
  • feedback and, 129–130
  • fluency vs. speed and, 73–74
  • mathematical modeling, opportunities for, 74
  • mathematical tasks, requirements for, 80
  • non-routine/complex problems and, 76
  • prior knowledge, connection with, 80, 93–94
  • procedural skills vs. conceptual understanding and, 74–75
  • scaffolded learning and, 42–43, 90, 93, 116
  • spaced/distributed practice and, 73, 128–130
  • stamina, development of, 78
  • strategic thinking and, 78–79
  • See also Assessment; Deep mathematics; Feedback; Learning; Surface mathematics; Teaching practices; Visible learning
• Guided questions, 109–113, 110–111 (figures)
• Gutiérrez, K., 191

• Habits of mind, 53, 119–120, 233, 233 (figure)
• Hastie, S., 185, 186
• Hattie, J. A. C., 4, 5, 6, 7, 18, 22, 24, 28, 35, 39, 43, 50, 52, 63, 65, 78, 121, 145, 147, 160, 164, 175, 186, 188, 192, 201, 205, 230, 233, 239
• Herbel-Eisenmann, B. A., 89, 90, 91
• Hinge point, 21
• Homework, 230

• “I can” statements, 59, 60 (figure), 61
• Influence, 20–21
  • barometer of influence and, 6–7 (figures), 21–22, 22 (figure)
  • See also Effect size; Visible learning
• Initiate-Respond-Evaluate (I-R-E) questioning cycle, 150
• Inquiry-based instruction, 3
• Inside-outside circles, 167
• International mathematical practice/process standards, 242–243 (appendix)
• Ito, M., 191

• Jackson, R., 12
• Jacobs, H. R., 225
• Jeon, K., 131
• Just-in-time/just-for-me information, 129–130, 203

• Kolb, D. A., 41

• Lange, K., 114
• Language frames, 146, 147 (figure)
• Language learning intentions, 48–51, 50 (figure)
• Learning, 0
  • barriers to, 38
  • deep learning and, 23, 27, 30–31, 34–35, 34 (figure)
  • definition of, 35
  • learning cycle, elements in, 41
  • prior knowledge and, 34 (figure), 44
  • recursive nature of, 16
  • rote learning and, 29, 31, 82, 103
  • surface learning and, 23, 27, 29–30, 34–35, 34 (figure)
  • See also Assessment; Feedback; Guided learning; Learning intentions; Mathematics instruction; Success criteria; Teaching practices; Transfer learning; Visible learning
• Learning intentions, 27, 38, 39–40
  • establishment of, 53–54
  • hallmarks of, 42–43
  • inviting/disinviting lessons and, 45–48, 47 (figure)
  • language learning intentions and, 48–51, 50 (figure)
  • learning cycle and, 41
  • lesson-based application of, 55–56
  • mathematics instruction and, 40–43
  • prior knowledge, activation of, 44
  • recursive element in, 42
  • scaffolded process and, 42–43
  • social learning intentions and, 51–55, 55 (figure)
  • standards and, 41
  • student exploration/inquiry and, 42, 43
  • student ownership of, 43
  • Tier 2/Tier 3 vocabulary and, 49, 50 (figure)
  • See also Learning; Success criteria; Surface mathematics; Teaching practices
• Leinwand, S., 129
• Lester, F. K., Jr., 185
• Little, M., 215
• Livers, S. D., 122, 123, 124
• Livingstone, S., 191
• Lobato, J., 116
• Lorson, K., 160
• Lubinski, D., 225
• Lyon, A., 160

• Manipulatives, 125–128, 127 (figure), 170–171
• Marzano, R. J., 185
• Mathematical practices. See Standards for Mathematical Practice (MP)
• Mathematical talk, 0
  • deep mathematics and, 142–150
  • surface mathematics and, 104, 106–120
  • transfer learning and, 188–189
  • See also Classroom discussion
• Mathematics Curriculum Key Ideas, 54
• Mathematics education, 1–2
  • career success and, 2
  • higher-level mathematics courses and, 2
  • natural mathematics skill and, 2
  • persistence and, 2
  • See also Mathematics instruction; Teaching practices; Visible learning
• Mathematics instruction, 2–3
  • balanced approach to, 3
  • classroom discussion, influence of, 7, 7 (figure)
  • dialogic approach and, 23–26, 25–26 (figure)
  • direct instruction and, 3, 23–26, 25–26 (figure)
  • fluency, repeated practice and, 3
  • inquiry-/problem-based instruction and, 3
  • mistakes, opportunities of, 16–17, 18
  • precision teaching and, 26
  • problem-based instruction and, 3
  • procedures/algorithms, explicit teaching of, 3
  • project-based learning and, 24
  • quality instruction, effective practices of, 3–4, 17–18, 24
  • real-life applications and, 3
  • rigor, high-impact instruction and, 3
  • scripted lessons and, 16
  • teaching test-taking, influence of, 6, 6 (figure)
  • textbooks, use of, 17
  • traditional approach to, 3
  • visible learning evidence, mobilization of, 4–7, 6–7 (figures)
  • visible learning, structure of, 8–10
  • See also Deep mathematics; Differentiated instruction; Guided learning; Learning intentions; Mathematics education; Success criteria; Surface mathematics; Teaching practices; Visible learning
• McGinn, K., 114
• Mellman, W., 28, 50, 63, 65, 121, 145, 147, 160, 164, 186, 188
• Meta-analyses, 5, 18–19
  • mathematical talk and, 119–120
  • See also Effect size; Visible learning
• Metacognitive skills, 29, 33, 39, 93, 141, 164, 176, 185
  • mathematical talk and, 119–120, 121 (figure)
  • self-questioning and, 185–187, 186 (figure)
  • self-reflection, 187–188, 188 (figure)
• Micro-teaching, 113
• Minoughan, S., 160
• Mirror neuron systems, 106–107
• Mistakes, 16–17, 18, 113, 115, 130, 207–208
• Mnemonics, 130–131
• Moore, S. D., 28, 50, 63, 65, 121, 145, 147, 150, 164, 186, 188
• Munter, C., 23, 24, 26, 116

• National Association for Gifted Children, 225
• National Center on Student Progress Monitoring, 223
• National Council of Teachers of Mathematics (NCTM), 3, 4, 7, 24, 43, 51, 80, 86, 89, 169
• National Governors Association Center for Best Practices, 3, 53, 117, 219
• National Library for Virtual Manipulatives, 125
• National Mathematics Advisory Panel, 217
• Near transfer, 177–178
• Non-routine problems, 76
• Number talk, 107–109
• Numbered Heads Together strategy, 202

• Online resources, 246–247 (appendix)
• Orlowski, M., 160

• Palincsar, A. S., 185
• Parsons, J., 194
• payscale.com, 1
• Peer tutoring, 161, 190
• Penuel, B., 191
• Piaget, J., 83
• Plato, 2
• Practice opportunities, 73, 128–130
• Precision teaching, 26
• Prior knowledge, 34 (figure), 44, 80, 93–94, 174
• Problem-based instruction, 3, 33, 73, 142, 143 (figure)
  • applications and, 75–76
  • difficulty vs. complexity and, 76–80, 77 (figure)
  • mathematical modeling and, 74
  • non-routine/complex problems and, 76
  • transfer learning and, 192–193
  • See also Deep mathematics; Differentiated instruction; Guided learning; SOLO (structure of observed learning outcomes) model; Transfer learning
• Problem-based learning (PBL), 192
• Procedures, 3, 74
• Project-based learning, 24
• Prompts, 93–94, 95 (figure), 114 (figure), 115, 187
• Purkey, W. W., 45
• Purposeful Sampling strategy, 202

• Questioning strategies. See Classroom discussion; Surface mathematics; Transfer learning

• Reciprocal teaching, 193–194
• Reilly, Y., 194
• Relevancy, 183
• Response Cards strategy, 202
• Response to Intervention (RTI), 199, 214–215, 226
  • addition/subtraction situations and, 221 (figure)
  • explicit/systematic instruction and, 217–219, 220, 221–222 (figures)
  • implementation guidelines for, 215–216
  • levels of response and, 215
  • mathematical concepts, visual representations of, 220
  • motivational strategies and, 223–224
  • multiplication/division situations and, 222 (figure)
  • progress monitoring practice and, 223
  • quality core instruction and, 224
  • retrieval of arithmetic facts and, 220, 223
  • screening process and, 216–217
  • whole number operations, focus on, 217
  • word problems, common foundational structures and, 219–220, 221–222 (figures)
  • See also Differentiated instruction; Teaching practices
• Restating strategy, 147–148
• Retention, 226
• Revoicing strategy, 146
• Rhodes, J., 191
• Rigor, 3
• Rote learning, 29, 31, 82, 103

• Salen, K., 191
• Scaffolded learning, 42–43, 90, 93, 116
• Schor, J., 191
• Scott, P., 21
• Sefton-Green, J., 191
• Self-directed learning, 33
• Self-questioning, 109, 119, 148, 185–187, 186 (figure)
• Self-reflection, 187–188, 188 (figure)
• Self-verbalization, 20
• Sheffield, L. J., 225
• Show-and-tell instruction, 3, 116
• Simpson, A., 120
• Simulations, 22–23
• Small group collaboration. See Deep mathematics
• Smith, M. S., 23, 24, 26, 41, 82, 116
• Social learning intentions, 51–55, 55 (figure)
• Sociomathematical norms, 150–151
• SOLO (structure of observed learning outcomes) model, 27, 28 (figure)
  • surface mathematics and, 106, 106 (figure)
  • See also Guided learning; Learning
• Sousa, D., 174
• Spaced practice, 73, 128–130
• Spray-and-pray instruction, 24
• Standards for Mathematical Practice (MP), 7, 53–54, 136, 137, 240–241 (appendix)
• Stein, M. K., 23, 24, 26, 41, 82, 116
• Success criteria, 27, 39–40, 41, 56
  • buy-in for, 61–64
  • exit tickets, use of, 64
  • group collaboration assessment and, 62, 63 (figure)
  • “I can” statements, 59, 60 (figure), 61
  • preassessments, role of, 66–67
  • proximal goals and, 58–59
  • rich mathematical tasks, rubric for, 56–57, 62, 64, 65 (figure)
  • self-reported grades/student expectations and, 57
  • specificity/clarity of, 57–58
  • student motivation and, 56–61, 64
  • See also Assessment; Learning intentions; Surface mathematics; Teaching practices
• Summative evaluation, 208–211, 223
• Surface learning, 23, 27, 29–30, 34–35, 34 (figure)
  • See also Surface mathematics
• Surface mathematics, 100
  • computational/procedural skills, direct instruction and, 117–119
  • conceptual understanding, initial acquisition of, 100–103
  • direct instruction and, 116–119
  • evidence-based instructional practices and, 105
  • exit tickets, use of, 101, 102 (figure), 103
  • feedback just-in-time/just-for-me and, 129–130
  • graphic organizers and, 125, 126 (figure)
  • guided questions and, 109–113, 110–111 (figures)
  • In the Doghouse game and, 100–103, 101–102 (figures), 105
  • learning intentions/success criteria and, 100, 101, 103, 104, 106 (figure)
  • manipulatives, strategic use of, 125–128, 127 (figure)
  • mathematical talk and, 104, 106–120
  • mathematical tasks for, 105–106, 106 (figure)
  • metacognition, mathematical talk and, 119–120, 121 (figure)
  • mirror neuron systems, role of, 106–107
  • mistakes, opportunities of, 113, 115, 130
  • mnemonics and, 130–131
  • number talk, utility of, 107–109
  • self-questioning prompts and, 114 (figure), 115
  • self-questioning/self-verbalizing skills and, 119–120
  • SOLO framework and, 106, 106 (figure)
  • spaced practice with feedback and, 128–130
  • surface learning, nature of, 103–105
  • teaching methods for, 104–105
  • vocabulary instruction, strategic use of, 120, 122–123
  • word walls and, 123–125
  • worked examples and, 113–116, 114 (figure)
  • See also Deep mathematics; Mathematics instruction; Surface learning; Teaching practices; Transfer learning; Visible learning

• Talk. See Classroom discussion; Dialogic approach; Discourse; Guided learning
• Teacher clarity, 38, 40, 231
• Teaching practices, 38–39
  • ability grouping and, 154–155, 226–228
  • achievement variation and, 230–231
  • effective teaching strategies and, 244–245 (appendix)
  • expert blind spot and, 40–41
  • grade-level retention and, 226
  • grouping strategies and, 154–157, 156 (figure), 212, 226–228
  • homework and, 230
  • ineffective strategies and, 226–231
  • inviting/disinviting lessons and, 45–48, 47 (figure)
  • language learning intentions and, 48–51, 50 (figure)
  • learning, barriers to, 38
  • learning cycle, elements in, 41
  • learning intentions, student ownership of, 43
  • learning intentions/success criteria, clarity in, 27, 38, 39–40
  • learning styles, matched instruction and, 228–229
  • mathematics instruction, learning intentions for, 40–43
  • micro-teaching, improved teaching practices and, 113
  • online resources for, 246–247 (appendix)
  • precision teaching and, 26
  • prior knowledge, activation of, 34 (figure), 44
  • problem-solving teaching and, 192–193
  • quality core instruction and, 224, 230
  • reciprocal teaching and, 193–194
  • social learning intentions and, 51–55, 55 (figure)
  • teacher clarity and, 38, 40, 231
  • teacher mind frames and, 233, 233 (figure)
  • test prep instruction and, 6, 6 (figure), 229–230
  • Tier 2/Tier 3 vocabulary and, 49, 50 (figure)
  • visible teaching/visible learning and, 231–233, 232 (figure)
  • See also Assessment; Deep mathematics; Differentiated instruction; Feedback; Guided learning; Learning intentions; Mathematics instruction; Response to Intervention (RTI); Success criteria; Surface mathematics; Transfer learning; Visible learning; Vocabulary instruction
• Test prep instruction, 6, 6 (figure), 229–230
• Texas Essential Knowledge and Skills for Mathematics (TEKS), 54, 136, 137
• Think-Pair-Square activity, 162–163
• Timperley, H., 43
• Tomlinson, C. A., 211
• Transfer learning, 27, 32–35, 34 (figure), 174–175, 231
  • conditions necessary for, 183–184
  • connected learning and, 191–192
  • connections, encouragement of, 181, 189–192
  • far transfer and, 177, 178–179
  • generalizations and, 182
  • hugging/bridging methods for, 179–181, 180 (figure)
  • knowledge, application of, 175, 176–177, 183–184
  • mathematical talk and, 188–189
  • mathematical tasks for, 181–182
  • metacognitive awareness and, 176, 185–188, 186 (figure), 188 (figure)
  • model creation and, 175
  • nature of, 175–179
  • near transfer and, 177–178
  • paths for, 179–181, 180 (figure)
  • peer tutoring programs and, 190–191
  • pre-lesson/post-lesson questioning and, 185–187, 186 (figure)
  • prior knowledge and, 174, 177, 182
  • problem-solving teaching and, 192–193
  • reciprocal teaching and, 193–194
  • relevancy and, 183
  • self-directed learning, potential for, 176
  • self-questioning and, 185–187, 186 (figure)
  • self-reflection and, 187–188, 188 (figure)
  • student ownership of learning and, 191–192
  • transfer, definition of, 175
  • writing prompts and, 187
  • See also Guided learning; Learning; Visible learning

• Visible learning, 14
  • barometer of influence and, 6–7 (figures), 21–22, 22 (figure)
  • classroom discussion and, 15, 85–96, 92 (figure), 95–96 (figures)
  • collaborative learning opportunities and, 15
  • deep learning and, 23, 27, 30–31
  • definition of, 14
  • differentiated instructional strategies and, 27
  • direct instruction vs. dialogic approach and, 23–26, 25–26 (figure)
  • effect sizes and, 19–20, 21
  • evidence-based approach of, 18–20
  • hinge point and, 21
  • influential strategies and, 20–21
  • intentional instruction and, 14
  • learning intentions/success criteria, clarity in, 27, 38, 39–40
  • learning, recursive nature of, 16
  • meta-analyses and, 18–19, 21
  • mistakes, opportunity of, 16–17, 18
  • mobilization of, 4–7, 6–7 (figures)
  • precision teaching and, 26
  • quality instruction, practices of, 17–18
  • simulations and, 22–23
  • surface learning and, 23, 27, 29–30, 34–35, 34 (figure)
  • traditional pedagogical approaches and, 14–17
  • transfer learning and, 27, 32–35, 34 (figure)
  • visible teaching/visible learning and, 231–233, 232 (figure)
  • See also Deep mathematics; Guided learning; Mathematics instruction; SOLO (structure of observed learning outcomes); Surface mathematics; Teaching practices; Visible learning evidence
• Visible learning evidence, 4–5
  • effect size, aggregated information and, 5–6
  • effect size, examples of, 6–7, 6–7 (figures)
  • meta-analysis, role of, 5, 18–19
  • Visible Learning database and, 5
  • See also Mathematics instruction; Visible learning
• Vocabulary instruction, 120, 122–123, 124 (figure)
  • graphic organizers and, 125, 126 (figure)
  • language support decisions and, 123, 124 (figure)
  • reciprocal teaching and, 193–194
  • teaching options for, 122–123
  • tiers of words and, 122
  • vocabulary knowledge, dimensions of, 122
  • vocabulary support and, 123
  • word walls and, 123–125
  • See also Surface mathematics; Teaching practices
• Vygotsky, L. S., 51, 212

• Watkins, S. C., 191
• Whole class collaboration. See Deep mathematics
• Wiliam, D., 203
• Wood, L., 21
• Word walls, 123–125
• Worked examples, 113–116, 114 (figure)

• Yackel, E., 150
• Yates, G., 78

• Zone of proximal development, 212