| Figures | ||
| 1.1. | Diagrams with open array for equation | 32 |
| 2.1. | Adapted mathematical tasks framework | 71 |
| 3.1. | Three possible models for ½ ÷ ⅙ | 89 |
| 3.2. | A model that differs from (A) in Figure 3.1 and connects with the complex fraction division algorithm | 90 |
| 5.1. | The complexity of relationships in mathematics education (adapted from Xie & Carspecken, 2008, p. 17) | 137 |
| 7.1. | The division of fractions item | 196 |
| 8.1. | Ball’s model of Mathematical Knowledge for Teaching (Ball et al., 2008, p. 403) | 214 |
| 10.1. | Inquiry in three layers | 278 |
| 10.2. | Inquiry in three layers at university level | 279 |
| 11.1. | The activity system (adapted from Engeström, 2001a) | 313 |
| 12.1. | Constitutive elements of Mathematical Discourse in Instruction (adapted from Adler & Ronda, 2015) | 333 |
| 12.2. | Mathematics tasks for quadratic equations and trigonometric notation | 340 |
| 12.3. | Learner example set involving application of the distributive law | 342 |
| 12.4. | Teacher example set for application of the distributive law | 342 |
| 12.5. | Teachers’ extensions of the given example set | 343 |
| 12.6. | Example set for learner task to match equations and graphs | 344 |
| 12.7. | Teacher task for function matching task | 345 |
| 12.8. | Learner task for operations on integers | 346 |
| 12.9. | Successive adaptations of the object of learning and example sets by teachers | 347 |
| 13.1. | Flowchart of the literature search and annotation | 357 |
| 13.2. | How techniques are nested within practices of different grain sizes which are nested within the domain of “Assessing Student Thinking” (based on Sleep & Boerst, 2012) | 363 |
| 14.1. | Prospective elementary school teachers looking at school mathematical issues | 392 |
| 14.2. | Prospective elementary school teachers looking at school mathematical issues through the lens of children’s mathematical thinking see richer and different mathematics | 393 |
| 14.3. | An approach to the topic of fraction division as an integration of concepts, procedures, reasoning, and problem solving | 395 |
| 14.4. | A solely procedural approach to the topic of fraction division | 395 |
| 14.5. | Andrew’s work in solving 63 – 25 | 403 |
| 14.6. | Recomposing the minuend or subtrahend and compensating for the change to show Andrew’s reasoning | 404 |
| 14.7. | Circles of Caring. A model of growth, by way of children’s mathematical thinking, from prospective elementary school teachers’ caring about children to their caring about mathematics (Philipp et al., 2007, p. 441) | 410 |
| 14.8. | Subtraction of 321 – 80 using an equal-addition algorithm | 411 |
| 14.9. | Many paths exist for developing mathematical knowledge for teaching, represented in this figure by the paths ending with arrows. Some paths provide higher views offering a more comprehensive view of the landscape, and others do not. The path that combines mathematics with children’s mathematical thinking, represented by the solid path, is a particularly productive path yielding a comprehensive view of the landscape | 413 |
| Tables | ||
| 2.1. | An example of mathematics authentic assessment task (adapted from Lim, 2011) | 47 |
| 2.2. | Definitions of the six mathematical competencies in the item analysis scheme (Turner et al., 2015) [as cited in Pettersen & Braeken, 2019, p. 408] | 49 |
| 2.3. | Cognitive demands or complexity of mathematical tasks | 50 |
| 2.4. | Authentic intellectual quality criteria for mathematics authentic assessment tasks | 52 |
| 2.5. | An example of patchwork text assessment approach (Koh et al., 2015) | 63 |
| 2.6. | PT-A’s grade 7 authentic mathematics task | 65 |
| 3.1. | Summary results of sampled elementary school teachers’ mathematics conceptual knowledge for teaching in fraction division | 99 |
| 4.1. | The Knowledge Quartet (adapted from Rowland et al., 2005) | 116 |
| 4.2. | The Knowledge Quartet – Dimensions and contributory codes | 120 |
| 5.1. | Comparison between sections of the Chinese lesson plan and the Italian lesson plan | 143 |
| 6.1. | Types of relationships between creativity and excellence | 167 |
| 6.2. | Inferred beliefs related to each category | 168 |
| 7.1. | Framework for Pedagogical Content Knowledge (PCK) for teaching school mathematics | 191 |
| 8.1. | Elements of the numeracy model developed by Goos and colleagues | 227 |
| 9.1. | Characterization of exclusionary/inclusive mathematics teaching practices (Louie, 2017, p. 496) | 247 |
| 10.1. | Teaching Triad analysis of teaching-learning in the ESUM project | 289 |
| 10.2. | Teaching Triad analysis of activity in the SYMBoL Project | 291 |
| 10.3. | Teaching Triad analysis of learning in the Catalyst Project | 295 |
Figures and Tables
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Despina Potari
Despina Potari
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