| Figures | ||
| 2.1. | Ternary diagram for shifts in Stance (diagram by Adrian Simpson) | 71 |
| 2.2. | Kate’s and Lisa’s development of Stance in the video-intervention | 72 |
| 2.3. | Changes in topic in Kate’s and Lisa’s responses to tasks (ordered chronologically; 01 – pre-task, 03 – post-task, 04/06 – observation of the teaching of others at the LS/US teaching practice) | 73 |
| 2.4. | Kate’s and Lisa’s results in the tasks across the video-intervention (100% is the maximum number of points for each task) | 74 |
| 2.5. | Nature of Kate’s and Lisa’s reflection across the tasks | 76 |
| 4.1. | The five emphases of the school mathematics curriculum in Costa Rica | 118 |
| 4.2. | The curriculum syllabi in three columns, the third one with specific suggestions (translated from the Spanish) | 121 |
| 4.3. | Frontpage project’s website. Three of the main sections: Mini MOOCs, blended courses, documents | 122 |
| 4.4. | Mini MOOCs (thumbs of eight courses within the edX platform) | 129 |
| 5.1. | A schema of a documentational genesis | 143 |
| 5.2. | A screen of the French Digital Educational Resources Bank. For the keyword “proportionality,” 271 resources are available, and can be used to build lessons | 144 |
| 5.3. | The “Chinese abacus at school” training path | 147 |
| 5.4. | LaboMEP, choosing interactive exercises about functions | 151 |
| 5.5. | A screenshot from the PRIMAS platform | 156 |
| 6.1. | Drawing a midpoint of a given segment | 172 |
| 6.2. | Drawing a perpendicular line to a segment from a given point | 173 |
| 6.3. | Points C, D and E are on the circle with centre at A and FH, FG and GH are tangents lines to the circle | 173 |
| 6.4. | The loci of midpoints when tangent points move along the circle | 174 |
| 6.5. | Connecting the midpoint property to draw the equilateral triangle | 175 |
| 6.6. | Representing the problem geometrically | 177 |
| 6.7. | The locus of the slope of line EH provides key information to solve the task | 178 |
| 6.8. | Generating a family of triangles by moving vertex C on the circle | 179 |
| 6.9. | Point Q has the same x-coordinate as point C (the mobile point) and as y-coordinate the area of triangle ABC. The locus of point Q when point C is moved on the circumference shows the area variation of the family of generated triangles | 180 |
| 6.10. | The locus of point Q shows the area variation of the family of isosceles triangles and the largest area is reached when triangle ABC becomes equilateral | 181 |
| 6.11a. | The construction of an equilateral triangle via the use of congruent circles | 182 |
| 6.11b. | The use of any equilateral triangle to draw the required triangle | 182 |
| 6.12. | Two intersecting lines and a point Q on one line | 183 |
| 6.13. | Simpler problem: drawing a tangent circle to the given line that passes through P | 183 |
| 6.14. | Drawing a mobile point Q, a perpendicular to line CD through Q and the perpendicular bisector of PQ | 183 |
| 6.15. | The locus of point E when point Q moves along line CD is a parabola | 183 |
| 6.16. | The intersection of the parabolas are centres of the tangent circles | 184 |
| 6.17. | Exploring the case when point P lies on one line | 184 |
| 6.18. | A framework to design and implement interactive problem-solving activities | 186 |
| 6.19. | Sketch of a cat on a ladder (from Gutenmacher & Vasilyev, 2004, p. 18) | 187 |
| 6.20. | Photo that shows main elements of the problem statement | 187 |
| 6.21. | A dynamic model of the problem | 187 |
| 6.22. | What is the path left by point P (middle point of AB) when point A is moved along the x-axis? | 187 |
| 6.23. | Completing a rectangle from triangle AOB | 188 |
| 6.24. | Determining the segment associated with the arc left by the segment midpoint | 188 |
| 6.25. | Constructing an algebraic model | 189 |
| 6.26. | What is the locus of points P when point A is moved along the x-axis? | 190 |
| 6.27. | The use of a Cartesian system to represent the problem | 190 |
| 7.1. | (a) Musical note patterns; (b) colour mat patterns; (c) dabbed patterns; (d) patterns on grids | 202 |
| 7.2. | Repeating patterns computational modelling environment | 203 |
| 7.3. | More complex repeating pattern | 204 |
| 7.4. | Using a nested loop to represent the fraction 3/9 | 204 |
| 7.5. | Abstraction | 206 |
| 7.6. | Growing patterns with constants and variables | 210 |
| 7.7. | Measurement and circular functions | 210 |
| 7.8. | Coin tossing algorithms in Grade 1 classroom | 211 |
| 7.9. | Binomial distribution in Grade 1 classroom | 212 |
| 7.10. | Coding simulation | 213 |
| 7.11. | Sample scratch code and output | 214 |
| 7.12. | Drawing with Python | 214 |
| 7.13. | Listing terms of a numerical pattern | 215 |
| 7.14. | Plotting the seeds pattern with Python | 215 |
| 7.15. | Rotation and reflection coding environment | 217 |
| 7.16. | “Bumper” symmetries of the square | 217 |
| 7.17. | Result when eight symmetries of the square bump and transform one another | 218 |
| 8.1. | Instructional exchanges | 227 |
| 8.2. | A simple model of activity, according to Engeström | 229 |
| 8.3. | Blending activity and instructional exchanges in the case of students’ learning | 229 |
| 8.4. | Blending activity and instructional exchanges in the case of teacher learning | 230 |
| 8.5. | The teacher learning activity system including the student learning activity system | 231 |
| 8.6. | Diagram given along with the problem in the tangent circle | 241 |
| 8.7. | (a) The student’s solution. (b) Assuming the construction has been made | 242 |
| 9.1. | Three grain levels for examining the use of theory in mathematics teacher education | 256 |
| 9.2. | Broadening the interpretation of the theory-practice themes | 266 |
| 10.1. | Number of articles reported using each component of reducing complexity of implementation | 296 |
| 10.2. | Types of artifacts for professional discussions for the synthesized articles | 299 |
| 12.1. | Professional task used to notice students’ fractional thinking (from Ivars et al., 2016b, pp. 111–112) | 344 |
| 12.2. | Degrees of prospective teachers noticing in the domain of derivative concept (from Sánchez-Matamoros et al., 2015, p. 1325) | 348 |
| 12.3. | An example of a typical problem used by Alice | 353 |
| 12.4. | A sequence of four typical problems used by John | 353 |
| 13.1. | The major categories constituting the horizontal of our map of learning theories | 364 |
| 13.2. | The major regions constituting our map of learning theories | 365 |
| 13.3. | Our evolving map of learning theories | 366 |
| 13.4. | “Raveling”: Braiding the strands that make the rope (and strands that make the strands, and so on) | 369 |
| 13.5. | The Raveling-PID matrix. (PID refers to cycles of prompting awareness of critical discernments, interpreting learner awareness of those discernments, and deciding how the lesson might proceed) | 370 |
| 13.6. | Using contrast to highlight critical discernments | 375 |
| 13.7. | The train problem | 376 |
| Tables | ||
| 1.1. | Design considerations for video-based activity system | 26 |
| 1.2. | Design decisions for using video: Considerations by purpose and audience | 29 |
| 2.1. | Kate’s and Lisa’s teaching reports within the teaching practice periods | 65 |
| 2.2. | Categories present in Kate’s and Lisa’s self-reflective essays about their teaching practice | 67 |
| 2.3. | Stance for Kate and Lisa for chronologically ordered tasks | 75 |
| 4.1. | Design and implementation of the mathematics school curriculum in Costa Rica: A synthesis | 132 |
| 7.1. | Ten affordances of computational modelling | 201 |
| 7.2. | Performing/modelling activities with code | 209 |
| 9.1. | Mathematics teaching and learning frameworks as theoretical tools | 270 |
| 9.2. | Multiple theoretical perspectives on a teaching practice | 271 |
| 9.3. | Concepts in mathematics and mathematics teaching and learning | 272 |
| 9.4. | Artefacts used in teaching experiments | 277 |
| 10.1. | Categories for reducing complexity during controlled implementations | 295 |
| 12.1. | Examples of the three groups differing in the discourse generated – Answers to task presented in Figure 12.1 (from Ivars et al., 2017, pp. 29–30) | 345 |
| 15.1. | Tools and their uses | 424 |
Figures and Tables
in International Handbook of Mathematics Teacher Education: Volume 2
Herausgeber:innen:
und
Salvador Llinares
Salvador Llinares
Search for other papers by Salvador Llinares in
Current site
Google Scholar
PubMed
Search for other papers by Salvador Llinares in
Current site
Google Scholar
PubMed
Olive Chapman
Olive Chapman
Search for other papers by Olive Chapman in
Current site
Google Scholar
PubMed
Search for other papers by Olive Chapman in
Current site
Google Scholar
PubMed