| Figures | ||
| 1.1 | Growth tendency in journal articles on mathematics and embodiment. | 2 |
| 1.2 | Situated embodiment. | 4 |
| 1.3 | Gestures in situated embodiment. | 7 |
| 2.1 | Diagram generated by Gabriella and Marcos during the episodes selected for this paper. | 17 |
| 2.2 | Point A projects onto point A’ and vice versa. | 23 |
| 2.3 | Alberti’s Window to draw on and the Eyepiece to look through. | 24 |
| 2.4 | Projection of points located in the horizontal plane (e.g., A, B, and C) onto the vertical plane (i.e., A’, B’, and C’). | 25 |
| 2.5 | Students laying out a rope on a football field along a parabolic function by joining points marked with orange cones. | 26 |
| 2.6 | Marcos and Gabriella projected first from behind the vertex. Later, in Episodes 2 and 3, they projected from a position ahead of the vertex. | 26 |
| 2.7 | Tracing the rope in a standing position. | 27 |
| 2.8 | Tracing the rope lying on the ground. | 27 |
| 2.9 | Marcos’ projection of the parabolic shape. | 27 |
| 2.10 | Gabriella looks at the rope-parabola through the eyepiece. | 28 |
| 2.11 | Note written at the bottom of the page. | 30 |
| 2.12 | Projection above Gabriella’s eyes. | 30 |
| 2.13 | Marcos traces an arc upwards; they are both looking backwards towards the vertex. | 31 |
| 2.14 | Marcos obtains the projection of the section of the parabola behind them by “flipping” it vertically. | 32 |
| 2.15 | “This would be at an angle, right?”. | 33 |
| 2.16 | “Your eye would be at a bigger angle here, right”. | 33 |
| 2.17 | Gabriella traces an arc upwards. | 33 |
| 2.18 | The side of the rope on her left side. | 33 |
| 2.19 | Upper point on the opposite side of the Alberti’s Window. | 34 |
| 2.20 | Marcos added the upper branch corresponding to the part of the parabola that includes the vertex and is behind them. | 35 |
| 2.21 | “They go off”. | 36 |
| 2.22 | “Definitely not converging”. | 37 |
| 2.23 | If the parabola stayed below the divergent lines. | 37 |
| 2.24 | Diagonals on a tiled floor created earlier in the classroom. | 38 |
| 2.25 | Gabriella looking through the eyepiece. | 39 |
| 2.26 | “If this parabola goes on forever”. | 39 |
| “then it will converge”. | 39 | |
| 2.28 | “We have this angle and it goes through your eye”. | 39 |
| 2.29 | “This is the angle that goes from there to your eye”. | 40 |
| 2.30 | “It is not the same as this angle”. | 40 |
| 2.31 | “There’s going to be a space of about this much”. | 40 |
| 2.32 | “This goes on to infinity”. | 42 |
| 2.33 | “and that side doesn’t [go to infinity]”. | 42 |
| 2.34 | “If we had a parabola that went to infinity way over there”. | 42 |
| 2.35 | “then the angle from there [way back there]”. | 42 |
| 2.36 | “Would be the same as the horizon line over there”. | 42 |
| 2.37 | Gabriella traces the horizon line. | 42 |
| 3.1 | Students participate in performatively evoking y = 2x. (Reproduced from Brady et al., 2013, Figure 6, page 110, with permission from Springer Nature). | 50 |
| 3.2 | Marcos and Gabriella operating with the Alberti’s Window apparatus. Nemirovsky, this volume, figure 2.7 (Ch 2, Section 4). | 56 |
| 3.3 | Gabriella and Marcos begin working on projecting the region of the parabola shaded in gray. A projection ray for one point from the right “branch” of the graph is shown. | 62 |
| 3.4 | Schematic rendering of Gabriella’s and Marcos’s work in the “second phase” of the task. | 63 |
| 3.5 | In the final phase of the activity, Gabriella and Marcos project the portion of the parabola behind their eye-line. | 64 |
| 4.1 | The first terms of a sequence explored in a Grade 1 class. | 77 |
| 4.2 | Nicolas building Term 2. | 79 |
| 4.3 | Nicolas building Term 3. | 80 |
| 4.4 | The construction of Term 4. Pic 1: Nicolas starts by building two inverted delta (Δ) triangles connected by a common vertex. Pic 2: Krista starts building Term 4. Pics 3 and 4: Nicolas tries to continue the construction of Term 4. Pic 5: Krista starts adding a triangle. Pic 6: Krista’s final Term 4. | 91 |
| 4.5 | The teacher and Nicolas working together on the construction of Term 4. | 94 |
| 4.6 | The next part of the construction of Term 4 and the final construction of Term 5 and Term 6. | 95 |
| 5.1 | Diagram provided to the students. Radford this volume, figure 4.1 (Ch 4, Section 2). | 110 |
| 5.2 | Term 4 of the pattern, with individual triangles numbered and with labels to allow reference to specific lines and points. | 111 |
| 6.1 | A reflex circle: the coupling between sensory afferentation and anticipatory motor efference as a unit of the complex functional system. (BASED ON Bernstein, 1947, p. 384). | 129 |
| 6.2 | The Mathematics Imagery Trainer for proportion. Here the screen is green when one cursor is two times higher above the bottom of the screen than another cursor, for a 1:2 ratio. Art acknowledgment: Virginia J. Flood. | 133 |
| Dual eye-tracking experimental setting where two participants discuss an image on a shared screen. | 144 | |
| 6.4 | An action-based embodied design for parabolas. The triangle is green when it is isosceles with BC = AC, where B runs along the horizontal dashed line, A is the parabola’s focus, and the student manipulates Vertex C. By keeping the triangle green while moving Vertex C, the student would effectively be inscribing a parabola. 6.4a presents a non-target state and 6.4b presents a target state of the screen. Note that the labels (A, B, C) as well as the dashed lines in this figure are only used here to illustrate the design for readers of this text—these lines did not appear for the students as they engaged in the activity. | 145 |
| 6.5 | The second stage of the action-based embodied design for parabolas. 6.5a presents a non-target state and 6.5b presents a target state of the screen. Axes and markers for coordinates are introduced. (The dashed line was not exposed to the participants.). | 146 |
| 6.6 | An intermediate stage of the solution. (The blue inscriptions were not exposed to the participants, but discussed with the tutor). | 147 |
| 6.7 | Synchronous eye movements of the student (yellow, lighter) and the tutor (red, darker), as the student manipulates Vertex C (the blue triangle’s top vertex). Each node-like point on a gaze-path line represents one sample, and the duration between two points is about 17 ms. The circles represent current eye-positions at the moment of this video frame (two circles mean that the video camera is slower than the eye tracker), thus Figures 6.7a, 6.7b, and 6.7c show the sequential development of gaze path. | 148 |
| 6.8 | The tutor’s (red) AAs are manifested in rapid, iterated, back-and-forth saccadic eye movements, while the student’s eye movements (yellow) are initially restricted to continuously tracing only the focal object they are manipulating (Vertex C of the moving triangle). | 150 |
| 6.9 | A student’s AAs are reflected in rapid iterative saccadic eye movements either along the median of the triangle (a, b) or along one of its sides (c). (In the interest of this figure’s clarity, the tutor’s overlapping gazes—the red lines—have been removed from this image). | 152 |
| 6.10 | Intercorporeal coupling between student and tutor: forming a distributed perception–action system. The dashed line indicates imaginary simulation of action. (a) A student’s model—a perception–action loop in regulation of enactment; (b) a tutor’s model—a perception–simulated-action loop; and (c) coupling of two perception–action systems upon the joint operational point. | 152 |
| 6.11 | The student (yellow) and tutor (red) perform iterative eye movements, but these movements are different, signifying different AAs: Whereas the student performs saccades along the median of the triangle, the tutor performs a three-step repetitive eye-movements: she traces one side of the triangle (on the left) and then goes to the other side and back to the manipulated Vertex С. (Note that the tutor’s red gaze path should be interpreted as offset to the left of its actual position, due to constraints of instrument sensitivity. The “C” notation of the vertex was added to this figure for clarity). | 153 |
| The student focuses attention on the triangle: (a) She explores the right side of the triangle; (b) The student momentarily pauses her actions while attempting to recall the geometrical term for the class of triangles that her actions have (unwittingly) been generating (viz. isosceles triangles); (c) The student answers the tutor’s question about the meaning of the term “isosceles triangle.” (In the interest of this figure’s clarity, the tutor’s overlapping gazes—the red lines—have been removed from this image). | 154 | |
| 6.13 | (a, Turn 01) The student (yellow) and the tutor (red) run their eyes in synchrony along the rectangle’s vertices. (b, Turn 03) The student misses a tutor’s gesture along the x-axis. (c, Turn 03) The student wrongly conjoins the tutor’s gesture along the vertical side of the triangle with the tutor’s verbal notation X in Turn 03. The white arrow overlaid on these eye-tracking images indicates the location of the tutor’s pointer, which appears as a gray line. All gray lines shown here actually appeared on the screen. | 156 |
| 6.14 | Visual joint attention to the sides of the triangle, while the tutor moves the pointer along them and verbally points at them in Turns 11 and 13. | 157 |
| 6.15 | In the circle of mutual co-regulation, the student simulates (dashed blue line) the tutor’s statements (red line) and confirms whether the utterance matches the simulation or asks for re-explanation (blue line). | 158 |
| 7.1 | An example of flocking. The panels show the time evolution of the development of a flock with 15 irregular BQ particles. (REPRINTED WITH PERMISSION). | 181 |
| 7.2 | The two columns of panels on the left show the response of a flock to the introduction of a cold probe (panels A, C, and E) and corresponding thermal images. The two columns of panels on the right show the response of a flock to the introduction of a warm probe and corresponding thermal images. (REPRINTED WITH PERMISSION). | 181 |
| 7.3 | The response of the flock to the introduction of a low-energy magnetic field. The single sensor particle is highlighted in red. (REPRINTED WITH PERMISSION). | 182 |
| 7.4 | Mathematicians collaborating. Screen shots separated by 10 seconds (still from video). | 189 |
| 7.5 | A network representation of a precalculus textbook showing the flow of ideas. The nodes refer to specific topics covered in the book and edges reveal a deliberate meaningful connection made by the author. | 190 |
| A sample precalculus knowledge network by two students at the end of the course. | 192 | |
| 7.7 | Mean Clustering Coefficient (a) and Mean Average Path Length (b) as a function of students’ performance in the course. | 192 |
| 8.1 | Slide from presentation of Graphs and Gestures project findings. | 210 |
| 8.2 | Weaving at small scale. | 213 |
| 8.3 | Weaving at medium scale. | 213 |
| 8.4 | Weaving at large scale. | 213 |
| 8.5 | PH4 in licorice strings and modelling clay. | 215 |
| 8.6 | PH4 as a collaboratively-made visual collage. | 216 |
| 8.7 & 8.8 | PH4 as a performative four-person poem and diagrammatic poem. | 216 |
| 8.9 | Small-scale paper cutout Wurzelschnecke. | 217 |
| 8.10 | Wurzelschnecke wearables?. | 218 |
| 8.11 | The Fedora of Theodorus. | 218 |
| 8.12 | Foamhenge and sunburst from two Wurzelschnecke. | 218 |
| 8.13 | Mathematics educators experiment with Wurzelschnecke. | 219 |
| 8.14 | Grade 8 students experiment with recycled cardboard Wurzelschnecke. | 220 |
| 8.15 | Biscuit tin Wurzelschnecke jewelry. | 220 |
| 8.16 | Furniture-scale foam Wurzelschnecke inspires architectural inventiveness and mathematical noticing (see also the “Large Wurzelschnecke Dominos” video at https://vimeo.com/458426910). | 221 |
| 8.17 | Still frame from video “Dancing Euclidean Proofs”. | 221 |
| 9.1 | Two taskscapes. (Left) Ingold uses Pieter Bruegel’s The Harvesters from 1565 to illustrate the concept of a taskscape, a living landscape of interconnected human activity. (Right) A scene from Gerofsky’s mathematical taskscape in “Dancing a Braid into Being,” (still from video) found at https://vimeo.com/269321443. | 230 |
| 10.1 | Sequence of inscriptions on blackboard. | 259 |
| 10.2 | “mapping into” (still from video). | 261 |
| 10.3 | “opposite signs” (still from video). | 263 |
| 10.4 | “right there” (still from video). | 266 |
| 10.5 | “cross this line” (said while drawing diagonal in 2D) (still from video). | 266 |
| 10.6 | “at zero or one” (still from video). | 267 |
| 10.7 | “bigger than zero” (still from video). | 267 |
| 10.8 | “crosses the line”. | 270 |
| 10.9 | “the picture’s completely clear” (still from video). | 270 |
| 10.10 | “where it has different signs” (still from video). | 272 |
| “So, here, g(0)’s negative” (still from video). | 273 | |
| 10.12 | “it’s negative here” (still from video). | 273 |
| 10.13 | “it’s positive there” (still from video). | 274 |
| 10.14 | “not fixed at either of the end points” (still from video). | 275 |
| 10.15 | “Then it has to be like that” (still from video). | 275 |
| 10.16 | “from negative” (still from video). | 276 |
| 10.17 | “to positive” (still from video). | 276 |
| 10.18 | Gestures accompanying “you’d go from [negative] to positive” (still from video). | 284 |
| 10.19 | Gestures accompanying “opposite signs” (left) and “different signs” (right) (still from video). | 285 |
| 11.1 | Representation of speech intensity, pitch, and resulting words of the later part of turn 16. | 303 |
| 11.2 | Representation of the sound during the writing of two lines of inscription (“f(0) ≠ 0,” “f(1) ≠ 1”) and the beginning of another line (“f(“) which, after an instance of staring at it, was erased. | 304 |
| 11.3 | Snapshots of the event following the production of two mathematical statements, the production of the beginning of a new statement, which is then erased prior to a movement that foreshadows a new and different statement. (The chalk lines have been enhanced for visibility) (still from video). | 305 |
| 11.4 | Seemingly searching where to go, the hand movements provide an inkling of the movement of thinking, thinking in movement. The verbal transcription including pauses appear above the photographs. (Chalk lines have been enhanced for visibility) (still from video). | 307 |
| 11.5 | Another type of a more-complete transcription, which includes here includes both interlocutors. (Grey background marks attending to and hearing, white background areas constitute what is traditionally seen in transcriptions). | 310 |
| Tables | ||
| 5.1 | Nature of teacher’s scaffolding moves. | 116 |
| 5.2 | Level of participants’ contributions to construction of each triangle in term 4. | 116 |
| 10.1 | Segment A: Sharing existing knowledge. | 260 |
| 10.2 | Segment B: Making plausible assertions. | 261 |
| 10.3 | Segment C: Adding visual anchors. | 265 |
| 10.4 | Segment D: Clarifying the argument. | 267 |
| 10.5 | Segment E: Adding written symbols. | 269 |
| 10.6 | Segment F: Attempts at formalizing. | 272 |
| 10.7 | Segment G: Restating and concluding. | 274 |
In: The Body in Mathematics
Editors:
and
Laurie D. Edwards
Laurie D. Edwards
Search for other papers by Laurie D. Edwards in
Current site
Google Scholar
PubMed
Search for other papers by Laurie D. Edwards in
Current site
Google Scholar
PubMed
Christina M. Krause
Christina M. Krause
Search for other papers by Christina M. Krause in
Current site
Google Scholar
PubMed
Search for other papers by Christina M. Krause in
Current site
Google Scholar
PubMed