Geometrical Diagrams as Representation and Communication: A Functional Analytic Framework Jehad Alshwaikh PhD Institute of Education, University of London Abstract: Although diagrams are considered part and parcel of mathematics, mainstream mathematicians exhibit prejudice against the use of diagrams in public. Adopting a multimodality social semiotics approach, I consider diagrams as a semiotic mode of representation and communication which enable us to construct mathematical meaning. Mathematics is a multimodal discourse, where different modes of representation and communication are used, such as (spoken and written) language, algebraic notations, visual forms and gestures. These different modes have different meaning potentials. I suggest an analytic framework that can be used as a tool to analyse the kinds of meanings afforded by diagrams in mathematical discourse, focusing on geometry. Starting from characteristics of diagrams identified in the literature, I construct the framework using an iterative methodology tested with data from classrooms in the UK and the Occupied Palestinian territories and from textbooks. The classroom data consist of approximately 350 written mathematical texts in English and Arabic produced by 13- and 14-year-old students as a response to two geometrical problems, accompanied by audio and video records of their verbal and gestural interactions with each other while solving the problems. I then present the critical aspects of the development journey of the framework followed by a discussion of each of the three (meta)functions: ideational, interpersonal and textual. Each of these functions is illustrated by examples of diagrams from mathematical texts collected from the empirical data, textbooks and the Internet. Because I consider mathematics to be a social and cultural practice, I discuss the issue of culture and language in relation to the meanings of diagrams. Lastly, I discuss the implications of the study on representation and communication in mathematical discourse, with possible applications for the framework in learning and teaching mathematics. 2 Declaration and Word Count I hereby declare that, except where explicit attribution is made, the work presented in this thesis is entirely my own. Word count (exclusive of list of figures and tables, references of figures and bibliography): 75,221 words. Jehad Alshwaikh February 2011 3 Here, where the hills slope before the sunset and the chasm of time near gardens whose shades have been cast aside we do what prisoners do we do what the jobless do we sow hope Mahmoud Darwish (A State of Siege) For Ward, my son With love and hope Table of Contents Acknowledgments 12 1 Introduction 14 2 Communication and language in mathematics 21 3 Extending the semiotic landscape of mathematics: Diagrams and gestures 35 4 Methodology - Aim and design of the study: An iterative approach 58 5 From directionality to temporality: Development of the diagrammatic framework 107 6 Narrative diagrams: Designing mathematical activity 117 7 Conceptual diagrams: Designing mathematical objects 138 8 Diagrams as interaction: Designing the position of the viewer 172 9 Visual cohesion: The textual meaning 194 10 Multimodal communication and representation: An analysis 219 11 Conclusions and implications of the study 240 References of figures 261 Bibliography 266 5 List of Figures Figure 1-1 : A summary of the thesis 20 Figure 2-1 : Multimodal mathematical text 30 Figure 3-1 : YBC 7289 (The Yale tablet) - an approximation to the square root of 2 39 Figure 3-2: BM 15285: An Old Babylonian 'textbook' (Robson, 2008a) 40 Figure 3-3: Problem 14 in the Moscow Papyrus 41 Figure 4-1: A sketch of the iterative design of the study 62 Figure 4-2: Cycles and iterations in the methodology 65 Figure 4-3: Different types of arrows in geometry 69 Figure 4-4: The Trapezoid Field task (TF) 86 Figure 4-5: The Proof task (Pf) 88 Figure 4-6: A mathematical text in English 90 Figure 4-7: A mathematical text in Arabic 92 Figure 4-8: Carly's TF diagram (Year 9, unique) 104 Figure 5-1: Narrative and conceptual diagrams based on the directionality characteristic 111 Figure 5-2: Different possible uses of arrows in geometry 111 Figure 5-3: Different uses for dotted lines in 2D geometry 113 Figure 5-4: Interpersonal and textual functions 116 Figure 6-1: Israeli soldiers threatening Palestinian schoolchildren 118 Figure 6-2: Schematic figure for 'The British used guns' picture (Kress & Van Leeuwen, 2006, p. 49) 119 Figure 6-3: Schematic figure for the 'Israeli soldiers threatening Palestinian schoolchildren' picture 119 Figure 6-4: Carly's TF diagram (Year 9, unique) 120 6 Figure 6-5: Translation process 122 Figure 6-6: The vertex angles of a pentagram sum to 180° (Nelson, 1993, p. 14) 123 Figure 6-7: Arrowed diagrams 124 Figure 6-8: Measurement diagrams 124 Figure 6-9: Measuring the length of a guitar 125 Figure 6-10: Dotted diagrams 125 Figure 6-11: Shaded diagrams 126 Figure 6-12: Sequence of diagrams 127 Figure 6-13: Construction diagrams 128 Figure 6-14: Claire's TF solution text (Year 9, typical) 129 Figure 6-15: Sarah's solution to task 2 (Year 9, typical) 130 Figure 6-16: Mandy's diagram for task 2 (Year 8, unique) 131 Figure 6-17: Elimination of the human figure in mathematical discourse (O'Halloran, 2005, 43-45) 133 Figure 6-18: Applications (of congruent triangles, similar triangles and Pythagoras theorem) show human figures, physical objects and context in a Palestinian textbook (Grade 7, part 2) 135 Figure 6-19: Human figure, physical objects and context represented in diagrams 135 Figure 6-20: The draft and the final diagram for two students 136 Figure 7-1: Two different diagrams represent the 'same' theorem 138 Figure 7-2: The relationships between different polygons 149 Figure 7-3: The relationships between different quadrilaterals 150 Figure 7-4: Lettered diagrams with capital letters 152 Figure 7-5: Lettered diagrams with small letters 152 Figure 7-6: Indexical attributive arrows 153 Figure 7-7: Indexical identifying arrows 153 Figure 7-8: Narrative and indexical arrows 153 7 Figure 7-9: Symbolic attributive 154 Figure 7-10: Symbolic Identifying 154 Figure 7-11: Positional relations: An illustrative example 158 Figure 7-12: Linear measurement-based size relation 161 Figure 7-13: Angular measurement-based size relation 162 Figure 7-14: Area measurement-based size relation 162 Figure 7-15: Measurement-based size relations: An illustrative example 163 Figure 7-16: Labels as size relational processes 164 Figure 7-17: Colour as size relational processes 165 Figure 7-18: Angle bisector and incentre as new created objects 166 Figure 7-19: Proof without words of Pythagoras Theorem as relational processes. 167 Figure 7-20: Indexical relations in students' diagrams 168 Figure 7-21: Students' symbolic diagrams 168 Figure 7-22: Labels in students' diagrams 169 Figure 7-23: A summary of relational processes in geometric diagrams 170 Figure 7-24: Mixed diagrams 170 Figure 8-1: Laundry detergent advertisment 173 Figure 8-2: Question marks in diagrams 175 Figure 8-3: Indicative exchange information 177 Figure 8-4: Question mark as demand 178 Figure 8-5: Indirect demand labels 179 Figure 8-6: Diagram and the accompanying verbal text pose the problem 179 Figure 8-7: Labels as interpersonal aspects and relational processes 180 Figure 8-8: Labels as geometric relationships 181 Figure 8-9: Labels (numbers) as specific quantities 181 Figure 8-10: Colour as offer 182 8 Figure 8-11: Arrows and words as offer 182 Figure 8-12: Neat diagrams 184 Figure 8-13: Rough diagrams 184 Figure 8-14: Labels express specific quantities 185 Figure 8-15: Modality markers and values in geometric diagrams 189 Figure 8-16: Two diagrams of a pile of rods 191 Figure 8-17: Labels: Specific quantities and variables 191 Figure 9-1: Given and New in geometric problems 198 Figure 9-2: Given and New in an Arabic geometric text (Grade 8, part 1, p. 54) 198 Figure 9-3: The first problem in Figure 9-2 (translated from Arabic) 199 Figure 9-4: Cumulative Given-New structure (Kress & Van Leeuwen, 2006, p. 185) 199 Figure 9-5: Cumulative Given-New structure in diagrams (http://vvww.rnathsisfim.comitriangle.html) 200 Figure 9-6: 'Interesecting chords' lesson in a Palestinian textbook (Grade 9, part 1, pp. 70-71) 201 Figure 9-7: Verbal ideal in Ideal-Real structures in a participant's text (Year 8, Wendy, typical) 202 Figure 9-8: Visual ideal in Ideal-Real structures in a participant's text 203 Figure 9-9: The Gougu theorem 204 Figure 9-10: Centre-Margin composition in Euclid's Element 205 Figure 9-11: The dimensions of the visual space 205 Figure 9-12: An English investigation text of Task 2 (Year 8, unknown name) 212 Figure 9-13: An Arabic investigation text of Task 2 (Grade 8, Sarni) 215 Figure 10-1: Drawing/sliding as a narrative action 222 Figure 10-2: Two fingers drawing a radius 223 Figure 10-3: Symbolic gestures showing parallelism 224 9 Figure 10-4: Modelling a diagram by two fingers or two hands 224 Figure 10-5: Pointing at the whole diagram or parts of it 225 Figure 10-6: A fixed distance (length) between two fingers/hands 226 Figure 10-7: Length of a segment as shown visually in mathematics texts 226 Figure 10-8: A 'gestured' square 227 Figure 10-9: The verbal part of Richard's final text 233 Figure 10-10: A square 233 Figure 10-11: A triangle 234 Figure 10-12: A trapezium and a triangle 235 Figure 10-13: Another triangle