The influence of an educational intervention involving mathematical modelling on the visualisation of engineering students

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(2) THE INFLUENCE OF AN EDUCATIONAL INTERVENTION INVOLVING MATHEMATICAL MODELLING ON THE VISUALISATION OF ENGINEERING STUDENTS by. JOHANNA HENDRINA KOTZE. A thesis. Submitted in fulfillment of the requirements for the degree. DOCTOR OF PHILOSOPHY. in. MATHEMATICS EDUCATION. in the. FACULTY OF EDUCATION. at the UNIVERSITY OF JOHANNESBURG. SUPERVISOR: Dr E D Spangenberg CO-SUPERVISOR: Prof C Long. JANUARY 2018.

(3) DECLARATION. I hereby declare that the work submitted here is the result of my own independent investigation. Where help was sought, it has been acknowledged. I further declare that this work is submitted for the first time at this university, the University of Johannesburg, towards the PHILOSOPHIAE DOCTOR (EDUCATIONIS) degree in Mathematics Education. Furthermore, I declare that this work has never been submitted to any other university for any purpose.. 31 January 2018 ______________ J.H. Kotze. _____________________ Date. iii.

(4) ACKNOWLEDGEMENTS My appreciation goes to the educational trust SANTRUST which enabled me to develop my doctoral proposal through their preparation programme. I am truly grateful for the additional financial support from SANTRUST which facilitated welcome teaching relief. I want to thank Juliana van Staden from STATKON who patiently guided me in analysing quantitative data with IBM SPSS Statistics. To my co-supervisor, Prof Caroline Long, for taking my hand and giving me confidence. To my supervisor, Dr Erica Spangenberg, for being a super advisor and mentor, who walked the journey with me and never stopped believing in me. To my family, my beautiful daughters Lucille and Nicole, for your selfless love. To my mother, who always encouraged and prayed for me, I know you would have been proud. And to Dave…. iv.

(5) ABSTRACT Many engineering subjects rely on the interpretation of symbolic, numeric and graphic representations. Engineering students have difficulties with the interpretation of representations generated with a computer algebra system (CAS). For professional engineers, the interpretation of multiple representations is a daily activity and inherent to problem solving. The ability to reason visually and to fluently interpret multiple representations is a cognitive process referred to as visualisation. The use of CAS such as Mathematica has stimulated new mathematical tools desirable for work-place engineers; these include programming, mathematical modelling and visualisation. Step-by-step processes that are automated by CAS create a pragmatic-epistemic gap, an underlying principle of visualisation not yet sufficiently researched in mathematics education. This study assesses the influence of mathematical modelling on the visualisation of engineering diploma students at the University of Johannesburg (UJ), South Africa (SA). The aim is to contribute to research on visualisation within the vocational field, both nationally and internationally. The research is driven by the question: What is the influence of an educational intervention involving mathematical modelling on the visualisation of engineering students? An explanatory sequential mixed methods design was used in three phases. In the first phase, a preliminary pilot study involved two mathematical modelling tasks and an open-ended questionnaire. For the second phase, a pilot study involved a quasiexperiment with a pre-test, mathematical modelling intervention – using the same modelling tasks as in phase one – and a post-test. The quasi-experiment was augmented by two semi-structured reflective group interviews. Phase two was repeated in its entirety for the main study, which was phase three. In each phase, the sample was a different cohort of engineering diploma students in their second year of study at UJ. Quantitative data were analysed with descriptive and inferential statistics, using the Statistical Package for the Social Sciences (SPSS). All qualitative data were analysed with content analysis. Four visualisation dimensions were identified namely translation, visual reasoning, new insights and intuition. While both the experimental and control group could smoothly translate from one representation to another, the experimental group benefited mostly. v.

(6) from the mathematical modelling intervention on the dimension labelled intuition. Concurrently, intuition was the weakest visualisation dimension of the control group. Tests revealed that the experimental group performed significantly differently from their control counterparts in the post-test, suggesting that the mathematical modelling intervention had a moderate influence on engineering students’ visualisation. Qualitative findings offered evidence that students came to appreciate 1) the vital role of paper-and-pen techniques in a CAS environment; 2) visualisation as an indispensable engineering tool and 3) the added benefits of team work including improved mathematical judgement and robust engineering discourse. It is argued that a suitable mathematical modelling intervention has potential to impact on engineering students’ visualisation, in particular the dimensions of intuition and visual reasoning. The implication for engineering education holds firm: classroom reforms which integrate CAS technology can benefit from a mathematical modelling approach to advance visualisation as a cognitive but vital skill for engineers of the 21st century. A limitation of the study is that it only explored visualisation in a CAS environment; this being a construct that finds gestalt in various digital learning environments. The competencies embedded in mathematical modelling and visualisation are in high demand in the engineering practice and arguably also in other mathematical domains; further investigations will broaden our understanding.. vi.

(7) TABLE OF CONTENTS AFFIDAVIT ………………………………………………………………………………….ii DECLARATION ……………………………………………………………………………iii ACKNOWLEDGEMENTS ………………………………………………………………..iv ABSTRACT …………………………………………………………………………………v LIST OF TABLES …………………………………………...……………………………xxi LIST OF FIGURES ………………………………………………………………………xxiv LIST OF ABBREVIATION AND ACRONYMS ………………………………………xxvii CHAPTER 1:. INTRODUCTION AND ORIENTATION ........................................... 1. 1.1 BACKGROUND ................................................................................................ 1 1.1.1. South Africa’s Higher Education sector: Challenges and expectations 1. 1.1.2. Engineering mathematics and visualisation with technology ............... 3. 1.1.3. The authentic exposure of engineers in training .................................. 5. 1.1.4. Mathematical modelling and its potential for engineering students ..... 7. 1.1.5. Gaps in the literature ......................................................................... 10. 1.2 ORIGIN OF THE RESEARCH IDEA: PERSONAL PERSPECTIVES ............. 11 1.3 SIGNIFICANCE OF THE STUDY .................................................................... 14 1.4 CHARTING THE RESEARCH PROBLEM ...................................................... 15 1.4.1. Background to the research problem and underlying rationale ......... 15. 1.4.2. The main research question .............................................................. 20. 1.4.3. Research sub-questions .................................................................... 20. 1.5 PURPOSE OF THE STUDY ............................................................................ 20 1.5.1. Research aim .................................................................................... 20. 1.5.2. Research objectives .......................................................................... 21. 1.5.3. Linking the research questions and objectives .................................. 21. vii.

(8) 1.6 THEORETICAL ASSUMPTIONS AND CONCEPTUAL FRAMEWORK ......... 22 1.6.1. Demarcation of the field of study ....................................................... 22. 1.6.2. Main theoretical underpinnings ......................................................... 23. 1.6.2.1. APOS theory .................................................................................. 24. 1.6.2.2. The Pirie-Kieren model .................................................................. 25. 1.6.2.3. Realistic Mathematics Education ................................................... 26. 1.6.2.4. Source-target framework ................................................................ 27. 1.6.3. Clarification of key concepts .............................................................. 28. 1.6.3.1. Visualisation ................................................................................... 28. 1.6.3.2. Mathematical modelling ................................................................. 29. 1.6.3.3. Technology-rich teaching and learning environment ...................... 29. 1.6.3.4. Engineering students ..................................................................... 30. 1.6.4. Conceptual framework for the study .................................................. 30. 1.7 RESEARCH DESIGN AND METHODS .......................................................... 32 1.7.1. Framing the researcher’s approach to the study ............................... 32. 1.7.2. The research strategy and paradigm ................................................. 33. 1.7.3. Philosophical assumptions ................................................................ 34. 1.7.3.1. Ontological assumptions ................................................................ 35. 1.7.3.2. Epistemological assumptions ......................................................... 35. 1.7.3.3. Axiological assumptions ................................................................. 35. 1.7.3.4. Rhetorical assumptions .................................................................. 35. 1.7.3.5. Methodological assumptions .......................................................... 36. 1.7.4. Research method: Mixed methods .................................................... 36. 1.7.5. Unit of analysis .................................................................................. 37. 1.7.6. Sampling, participants and research design ...................................... 38. 1.7.7. Methods of data analysis ................................................................... 39. 1.7.7.1. Quantitative data analysis .............................................................. 39. 1.7.7.2. Qualitative data analysis ................................................................ 40. 1.7.8 1.7.8.1. Quality issues in mixed methods ....................................................... 41 Validity ........................................................................................... 41. viii.

(9) 1.7.8.2. Reliability ........................................................................................ 42. 1.7.8.3. Trustworthiness .............................................................................. 42. 1.7.9. Ethical considerations ....................................................................... 43. 1.8 ORIENTATION TO CHAPTERS ..................................................................... 43 CHAPTER 2:. THEORETICAL PERSPECTIVES ON VISUALISATION .............. 45. 2.1 INTRODUCTION ............................................................................................. 45 2.2 PROFILING ENGINEERING GRADUATES OF THE 21ST CENTURY ........... 45 2.2.1. Demands and expectations from industry ......................................... 45. 2.2.2. Desired classroom culture ................................................................. 47. 2.2.3. Competencies ................................................................................... 52. 2.2.4. Learning with a computer algebra system (CAS) .............................. 54. 2.3 THEORIES UNDERPINNING VISUALISATION ............................................. 56 2.3.1. Introduction and overview ................................................................. 56. 2.3.2. The contribution of Sfard ................................................................... 57. 2.3.3. The contribution of Vinner ................................................................. 59. 2.3.4. Source-target framework ................................................................... 62. 2.3.5. APOS theory ..................................................................................... 64. 2.3.6. Pirie-Kieren model of mathematical understanding ........................... 68. 2.4 VISUALISATION IN TEACHING AND LEARNING WITH CAS ...................... 72 2.4.1. Overview and challenges .................................................................. 72. 2.4.2. Pedagogical map for learning with CAS ............................................ 77. 2.4.2.1. Task level opportunities ................................................................. 78. 2.4.2.2. Classroom level opportunities ........................................................ 81. 2.4.2.3. Goals.............................................................................................. 83. 2.4.3. Visualisation with technology ............................................................ 85. 2.4.3.1. Outsourcing algorithms .................................................................. 85. 2.4.3.2. Paper-and-pen techniques vs technology ...................................... 87. 2.4.3.3. Cognitive technologies ................................................................... 90. ix.

(10) 2.5 MEASURING VISUALISATION ...................................................................... 90 2.6 VISUALISATION DIMENSIONS ..................................................................... 92 2.6.1. Multiple representations .................................................................... 92. 2.6.2. Visual reasoning ................................................................................ 93. 2.6.3. The role of visual justification in the construction of knowledge ........ 94. 2.6.4. New insights ...................................................................................... 95. 2.6.5. Visual narration ................................................................................. 95. 2.7 SYNTHESIS .................................................................................................... 96 CHAPTER 3:. THEORETICAL. MODELLING. 98. PERSPECTIVES. ON. MATHEMATICAL. 3.1 INTRODUCTION ............................................................................................. 98 3.2 TRADITIONAL VERSUS PROBLEM-BASED APPROACH ........................... 99 3.3 WHAT IS MATHEMATICAL MODELLING? ................................................. 102 3.3.1. Models and modelling ..................................................................... 102. 3.3.2. Use of technology in modelling........................................................ 103. 3.3.3. Classification of mathematical modelling tasks ............................... 106. 3.4 THEORIES UNDERPINNING MATHEMATICAL MODELLING .................... 109 3.4.1. Realistic Mathematics Education (RME) ......................................... 109. 3.4.1.1. Historical background and philosophy.......................................... 109. 3.4.1.2. Reasoning and cognition in Realistic Mathematics Education ..... 111. 3.4.1.3. Didactics of Realistic Mathematics Education .............................. 112. 3.4.2. Framework of Galbraith and Stillman (2006) ................................... 116. 3.5 MATHEMATICAL MODELLING COMPETENCIES FOR ENGINEERS ....... 119 3.5.1. Desired modelling competencies for engineers ............................... 119. 3.5.2. Modelling competencies elicited ...................................................... 119. 3.5.2.1. Cognitive and metacognitive skills ............................................... 120. 3.5.2.2. Soft skills ...................................................................................... 126. 3.5.2.3. Technology skills .......................................................................... 126. x.

(11) 3.6 POTENTIAL AND CHALLENGES OF MATHEMATICAL MODELLING ...... 128 3.6.1. The objectives of mathematical modelling ....................................... 128. 3.6.2. A potential link between mathematical modelling and visualisation. 129. 3.6.3. Challenges ...................................................................................... 130. 3.6.3.1. Curricular ..................................................................................... 131. 3.6.3.2. Didactical ..................................................................................... 132. 3.7 MODELLING WITH DIFFERENTIAL EQUATIONS ...................................... 133 3.8 SYNTHESIS .................................................................................................. 134 CHAPTER 4:. RESEARCH DESIGN AND METHODOLOGY ............................ 135. 4.1 INTRODUCTION ........................................................................................... 135 4.2 JUSTIFYING THE RESEARCH METHODOLOGY ....................................... 135 4.2.1. Introduction and link to the research question ................................. 135. 4.2.2. The connection between research problem and research design ... 136. 4.2.3. Closing the gap between engineering education and practice ........ 138. 4.2.3.1. Mathematical modelling connecting three worlds ......................... 138. 4.2.3.2. Institutional constraints................................................................. 140. 4.2.3.3. Philosophical considerations in closing the gap ........................... 141. 4.2.4. A rationale for mixed methods ......................................................... 143. 4.3 EXPERIMENTAL AND QUASI-EXPERIMENTAL DESIGNS ....................... 144 4.3.1. Origins and historical overview ........................................................ 144. 4.3.2. Definitions and aim of experiments ................................................. 145. 4.3.2.1. Variables ...................................................................................... 145. 4.3.2.2. Experiments ................................................................................. 146. 4.3.2.3. Aim and strengths of experiments ................................................ 147. 4.3.3. The randomised or true experimental design .................................. 148. 4.3.4. Quasi-experimental designs ............................................................ 150. 4.3.5. Difference-in-difference design........................................................ 152. 4.3.6. Case control matching ..................................................................... 155. xi.

(12) 4.4 QUALITATIVE DESIGN ................................................................................ 155 4.4.1. Introduction ..................................................................................... 155. 4.4.2. Strengths and weaknesses ............................................................. 156. 4.4.3. Qualitative methods......................................................................... 157. 4.4.3.1. Worksheet documents ................................................................. 158. 4.4.3.2. Open-ended questionnaire ........................................................... 158. 4.4.3.3. Semi-structured reflective group interviews ................................. 159. 4.5 THE EMPIRICAL ENVIRONMENT................................................................ 160 4.5.1. Purpose ........................................................................................... 160. 4.5.2. The curriculum................................................................................. 161. 4.5.3. The curricular approach to numerical DEs ...................................... 162. 4.5.4. Setting the scene for experimentation ............................................. 163. 4.6 OVERVIEW OF DATA COLLECTION AND TIME FRAME .......................... 165 4.7 THE PILOT STUDIES.................................................................................... 167 4.7.1. Definition and aim............................................................................ 167. 4.7.2. Pilot study 1 ..................................................................................... 167. 4.7.3. Pilot study 2 ..................................................................................... 169. 4.7.4. The modelling tasks: design, aim and appropriateness ................... 170. 4.7.4.1. Task 1 .......................................................................................... 170. 4.7.4.2. Task 2 .......................................................................................... 174. 4.7.5. Open-ended questionnaire .............................................................. 177. 4.7.6. Semi-structured reflective group interviews ..................................... 179. 4.7.7. Conclusion ...................................................................................... 180. 4.8 THE MAIN STUDY ........................................................................................ 181 4.8.1. Participants ..................................................................................... 181. 4.8.2. Phase 1a: The pre-test .................................................................... 182. 4.8.3. Phase 1b: Demographic survey ...................................................... 182. 4.8.4. Phase 1c: Case control matching .................................................... 184. xii.

(13) 4.8.5. Phase 2a: Worksheet documents.................................................... 186. 4.8.6. Phase 3: The post-test .................................................................... 186. 4.9 DATA ANALYSIS: PROCEDURES AND TESTS ......................................... 187 4.9.1. Demographic survey ....................................................................... 187. 4.9.2. The quasi-experiment...................................................................... 188. 4.9.2.1. Descriptive statistics..................................................................... 189. 4.9.2.2. Principle component analysis ....................................................... 190. 4.9.2.3. Custom tables .............................................................................. 191. 4.9.2.4. Inferential statistics....................................................................... 191. 4.9.2.5. Case control matching ................................................................. 192. 4.9.3. Content analysis .............................................................................. 192. 4.9.3.1. Quantitative content analysis ....................................................... 193. 4.9.3.2. Quantification of qualitative data .................................................. 194. 4.9.3.3. Qualitative content analysis ......................................................... 195. 4.9.3.4. Codes and types of coding ........................................................... 195. 4.9.3.5. Documents collected from modelling tasks .................................. 197. 4.9.3.6. Open-ended Questionnaires ........................................................ 201. 4.9.3.7. Semi-structured reflective group interviews ................................. 201. 4.10 MIXING OF QUANTITATIVE AND QUALITATIVE FINDINGS ..................... 202 4.11 RELIABILITY, VALIDITY AND TRUSTWORTHINESS................................. 205 4.11.1. Introduction ..................................................................................... 205. 4.11.2. Reliability ......................................................................................... 205. 4.11.3. Validity ............................................................................................. 207. 4.11.3.1 Content validity ............................................................................ 207 4.11.3.2 Construct validity .......................................................................... 208 4.11.3.3 Internal validity ............................................................................. 211 4.11.3.4 External validity ............................................................................ 213 4.11.3.5 Face validity ................................................................................. 213 4.11.4. Trustworthiness ............................................................................... 214. 4.11.4.1 Credibility ..................................................................................... 214 4.11.4.2 Transferability .............................................................................. 215 xiii.

(14) 4.11.4.3 Dependability ............................................................................... 215 4.11.4.4 Confirmability ............................................................................... 216 4.12 ETHICAL CONSIDERATIONS ...................................................................... 216 4.13 SYNTHESIS .................................................................................................. 218 CHAPTER 5:. ANALYSIS AND INTERPRETATION OF QUANTITATIVE DATA 219. 5.1 INTRODUCTION ........................................................................................... 219 5.2 DESCRIPTIVE STATISTICS ......................................................................... 219 5.2.1. Both workshops group ..................................................................... 221. 5.2.2. First workshop only group ............................................................... 222. 5.2.3. Control group................................................................................... 222. 5.2.4. Differences between pre-test and post-test mean scores ............... 222. 5.3 FACTOR ANALYSIS – PRINCIPAL COMPONENT ANALYSIS .................. 224 5.3.1. Suitability of data ............................................................................. 224. 5.3.2. Data extraction and criteria ............................................................. 225. 5.3.3. Extraction and labelling ................................................................... 227. 5.3.3.1. Component 1 - translation ............................................................ 227. 5.3.3.2. Component 2 – visual reasoning .................................................. 228. 5.3.3.3. Component 3 – new insights ........................................................ 228. 5.3.3.4. Component 4 – intuition ............................................................... 228. 5.3.4. Component score coefficients ......................................................... 230. 5.3.4.1. Inter-dimensionality of pre-test items ........................................... 230. 5.3.4.2. Inter-dimensionality of post-test items .......................................... 232. 5.3.4.3. Bi-plots ......................................................................................... 233. 5.4 CUSTOM TABLES ........................................................................................ 238 5.4.1. Pre-test and post-test analysis of visualisation dimension: translation 239. 5.4.1.1. Comparing pre-test item A1.1 with post-test item B1.1 ................ 239. 5.4.1.2. Comparing pre-test item A1.4 with post-test item B1.4 ................ 240. xiv.

(15) 5.4.1.3. Comparing pre-test item A1.3 with post-test item B1.3 ................ 241. 5.4.1.4. Comparing pre-test item A1.2 with post-test item B1.2 ................ 242. 5.4.1.5. Interpretation: visualisation dimension – translation ..................... 243. 5.4.2. Pre-test and post-test analysis of visualisation dimension: visual. reasoning ....................................................................................................... 245 5.4.2.1. Comparing pre-test item A4.1 with post-test item B4.2 ................ 245. 5.4.2.2. Comparing pre-test item A4.2 with post-test item B4.4 ................ 247. 5.4.2.1. Comparing pre-test item A4.3 with post-test item B4.3 ................ 248. 5.4.2.2. Unpaired post-test item B4.1........................................................ 249. 5.4.2.3. Interpretation: visualisation dimension – visual reasoning ........... 249. 5.4.3. Pre-test and post-test analysis of visualisation dimension: new insights 252. 5.4.3.1. Comparing pre-test item A2.1 with post-test item B2.1 ................ 252. 5.4.3.2. Comparing pre-test item A2.2 with post-test item B2.2 ................ 253. 5.4.3.3. Comparing pre-test item A2.3 with post-test item B2.3 ................ 254. 5.4.3.4. Comparing pre-test item A2.4 with post-test item B2.4 ................ 255. 5.4.3.5. Interpretation: visualisation dimension – new insights.................. 258. 5.4.4. Pre-test and post-test analysis of visualisation dimension: intuition 260. 5.4.4.1. Comparing pre-test item A3.1 with post-test item B3.1 ................ 260. 5.4.4.2. Comparing pre-test item A3.2 with post-test item B3.2 ................ 261. 5.4.4.3. Unpaired pre-test item A4.4 ......................................................... 262. 5.4.4.4. Interpretation: visualisation dimension – intuition ......................... 262. 5.4.5. Summary: all visualisation dimensions ............................................ 265. 5.5 INFERENTIAL STATISTICS ......................................................................... 266 5.5.1. Meeting the assumptions ................................................................ 266. 5.5.2. Kruskal-Wallis H test ....................................................................... 268. 5.5.3. Post-hoc test ................................................................................... 269. 5.6 CASE CONTROL MATCHING ...................................................................... 270 5.6.1. Wilcoxon signed ranks test analyses............................................... 273. 5.6.2. Comparing matched and unmatched analysis ................................ 278. xv.

(16) 5.7 SYNTHESIS .................................................................................................. 279 CHAPTER 6:. ANALYSIS AND INTERPRETATION OF QUALITATIVE DATA 280. 6.1 INTRODUCTION ........................................................................................... 280 6.2 VISUALISATION. PROCESSES INITIATED BY THE. MATHEMATICAL. MODELLING INTERVENTION .............................................................................. 280 6.2.1. Introduction ..................................................................................... 280. 6.2.2. APOS theory ................................................................................... 281. 6.2.3. Pirie-Kieren model ........................................................................... 282. 6.2.4. Mathematical modelling Task 1 ....................................................... 283. 6.2.4.1. Rubric for assessment of task ...................................................... 283. 6.2.4.2. A prototype solution ..................................................................... 285. 6.2.4.3. APOS theory analysis .................................................................. 290. 6.2.4.4. Pirie-Kieren analysis .................................................................... 293. 6.2.4.5. Internalisation ............................................................................... 296. 6.2.4.6. Representation ............................................................................. 297. 6.2.4.7. Justification .................................................................................. 300. 6.2.5. Mathematical modelling Task 2 ....................................................... 303. 6.2.5.1. A prototype solution ..................................................................... 303. 6.2.5.2. APOS theory analysis .................................................................. 307. 6.2.5.3. Pirie-Kieren analysis .................................................................... 308. 6.2.5.4. Internalisation ............................................................................... 312. 6.2.5.5. Representation ............................................................................. 316. 6.2.5.6. Justification .................................................................................. 319. 6.3 STUDENTS’ EXPERIENCES ........................................................................ 322 6.3.1. General experiences: quantified ...................................................... 323. 6.3.2. General experiences ....................................................................... 325. 6.3.2.1. Nature of the modelling task ........................................................ 325. 6.3.2.2. Value of the modelling task .......................................................... 330. 6.3.3 6.3.3.1. Collaboration experiences ............................................................... 333 A team approach in engineering .................................................. 333. xvi.

(17) 6.3.4. Visualisation experiences ................................................................ 337. 6.3.4.1. Cognitive technologies ................................................................. 337. 6.3.4.2. Different representations .............................................................. 339. 6.4 THE NATURE OF THE MODELLING TASKS .............................................. 341 6.4.1. Category 1: Nature of the modelling tasks ...................................... 342. 6.4.1.1. No single correct answer.............................................................. 342. 6.4.1.2. A story behind it ........................................................................... 344. 6.4.1.3. Collaboration ................................................................................ 345. 6.4.2. Category 2: Value of the modelling tasks ........................................ 347. 6.4.2.1. Mathematics is everywhere .......................................................... 347. 6.4.2.2. Done practically............................................................................ 348. 6.4.2.3. Metacognition ............................................................................... 349. 6.4.3. Category 3: Critique towards the status quo ................................... 352. 6.4.3.1. Told what to do ............................................................................ 352. 6.4.3.2. Abilities of an engineer ................................................................. 353. 6.4.3.3. Black or white ............................................................................... 355. 6.4.3.4. More modelling ............................................................................ 356. 6.4.3.5. Apply with purpose ....................................................................... 356. 6.5 SYNTHESIS .................................................................................................. 358 CHAPTER 7:. RESULTS AND FINDINGS ......................................................... 360. 7.1 INTRODUCTION ........................................................................................... 360 7.2 INTERPRETATION OF FINDINGS ............................................................... 361 7.2.1. Sub-question one ............................................................................ 361. 7.2.1.1. The pre-intervention nature of visualisation ................................. 361. 7.2.1.2. The post-intervention nature of visualisation ................................ 365. 7.2.1.3. Confirmation of findings as revealed by inferential statistics ........ 371. 7.2.2. Sub-question two............................................................................. 372. 7.2.2.1. Mathematising initiatives .............................................................. 372. 7.2.2.2. Intuition advantages ..................................................................... 376. 7.2.2.3. Visual reasoning initiatives ........................................................... 379. 7.2.2.4. Partially developed visualisation processes ................................. 381. xvii.

(18) 7.2.3. Sub-question three .......................................................................... 382. 7.2.3.1. A natural interest .......................................................................... 383. 7.2.3.2. Consolidation of knowledge ......................................................... 383. 7.2.3.3. Individual contributions to group work .......................................... 384. 7.2.3.4. Emergent engineering skills ......................................................... 384. 7.2.4. Sub-question four ............................................................................ 385. 7.2.4.1. Contexts as navigational tool ....................................................... 385. 7.2.4.2. The role of group work ................................................................. 386. 7.2.4.3. Future creators and innovators .................................................... 386. 7.2.4.4. Mathematics is everywhere .......................................................... 387. 7.2.5. Merged findings ............................................................................... 387. 7.3 IMPLICATIONS OF THE RESEARCH .......................................................... 392 7.3.1. Influence of the modelling intervention ............................................ 392. 7.3.2. Emergent visualisation processes ................................................... 393. 7.3.3. Mathematical modelling experiences .............................................. 395. 7.3.4. The nature of the mathematical modelling intervention ................... 395. 7.3.5. Collective implications ..................................................................... 395. 7.4 GUIDELINES FOR A MATHEMATICAL MODELLING INTERVENTION ..... 397 7.5 SYNTHESIS .................................................................................................. 400 CHAPTER 8:. CONCLUSION ............................................................................. 402. 8.1 INTRODUCTION ........................................................................................... 402 8.2 OVERVIEW OF CHAPTERS ......................................................................... 402 8.3 SUMMARY OF IMPLICATIONS .................................................................... 405 8.4 CONTRIBUTIONS OF THE STUDY .............................................................. 406 8.4.1. Contribution from quasi-experimental design .................................. 406. 8.4.2. Contributions to theory .................................................................... 407. 8.4.2.1. Mathematical modelling framework of Galbraith and Stillman (2006) 408. 8.4.2.2. Pirie-Kieren model........................................................................ 410. xviii.

(19) 8.4.2.3 8.5. APOS theory ................................................................................ 412. LIMITATIONS OF THE STUDY ..................................................................... 413. 8.6 RECOMMENDATIONS FOR FURTHER RESEARCH .................................. 413 8.7. CONCLUDING REMARKS ........................................................................... 414. 8.8 PERSONAL REFLECTION ON THE STUDY ............................................... 415 REFERENCES …………………………………………………………...………………416 APPENDIX A: ETHICAL CLEARANCE CERTIFICATE ……………………………445 APPENDIX B: NATIONAL GEOGRAPHIC ARTICLE ON WORLD POPULATION ……………………………………………………………………………………………...44 APPENDIX C: MODELLING WORKSHOP TASK 1 ………………………………..447 APPENDIX D: MODELLING WORKSHOP TASK 2 ………………………………..448 APPENDIX E: OPEN-ENDED QUESTIONNAIRE FOR PILOT STUDY …………..449 APPENDIX F: PRE-TEST ………………………………………………………………450 APPENDIX G: DEMORAPHIC SURVEY ……………………………………………..454 APPENDIX H: POST-TEST …………………………………………………………….455 APPENDIX I: INTERVIEW QUESTIONS ……………………………………………..459 APPENDIX J: CORRELATION MATRICES FOR PRE-TEST AND POST-TEST ……………………………………………………………………………………………...460 APPENDIX K: VIGNETTE 1 ……………………………………………………………461 APPENDIX L: VIGNETTE 2 ……………………………………………………………464 APPENDIX M: VIGNETTE 3 …………………………………………………………...467 APPENDIX N: VIGNETTE 4 ……………………………………………………………472 APPENDIX O: VIGNETTE 5 ……………………………………………………………474. xix.

(20) APPENDIX P: VIGNETTE 6 ……………………………………………………………476 APPENDIX Q: VIGNETTE 7 ……………………………………………………………479 APPENDIX R: VIGNETTE 8 ……………………………………………………………481 APPENDIX S: VIGNETTE 9 ……………………………………………………………483 APPENDIX T: VIGNETTE 10 …………………………………………………………..484 APPENDIX U: VIGNETTE 11 ………………………………………………………….485 APPENDIX V: THEMES, CATEGORIES, CODES AND VERBATIM RESPONSES ……………………………………………………………………………………………..487 APPENDIX W: PROOFREADING REPORT …………………………………………499. xx.

(21) LIST OF TABLES Table 1.1: Research questions and objectives. ........................................................ 22 Table 2.1: Operational and structural concepts (Sfard, 1991:33). ............................ 58 Table 2.2: Translation process within four representations (Janvier, 1987:28). ....... 63 Table 2.3: Changes in didactical contract due to CAS (Heugl, 1997:147). ............... 82 Table 2.4: Processes required to sketch a parabola with paper-and-pen and with CAS. ................................................................................................................................. 88 Table 2.5: DIPT framework of Wattanawaha (1977). ............................................... 91 Table 2.6: Contextual connectivity rating according to Samson (2011:90). .............. 92 Table 3.1: Classification of modelling tasks (Maaß, 2010a:296). ........................... 108 Table 3.2: Didactical approaches to mathematising (Freudenthal, 1991:133)........ 115 Table 4.1: Aligning the research questions with the research design. .................... 137 Table 4.2: Strengths and weaknesses of qualitative research. .............................. 156 Table 4.3: Grouping of participants in pilot study 2. ............................................... 169 Table 4.4: Task 1, delineated into subtasks with rationale and expected response 172 Table 4.5: Task 2, delineated into subtasks with rationale and expected response 177 Table 4.6: Summary of open-ended questionnaire. ............................................... 178 Table 4.7: Participation in two semi-structured reflective group interviews. ........... 180 Table 4.8: Grouping of participants in main study. ................................................. 181 Table 4.9: Translation between different representations in pre-test items. ........... 182 Table 4.10: Descriptive statistics for demographic data. ........................................ 187 Table 4.11: Codes for analysing the pre-test and post-test. ................................... 193 Table 4.12: Rating scores and performance criteria for three visualisation categories. ............................................................................................................................... 194 Table 4.13: Processes elicited in promoting visualisation in modelling Task 1. ...... 198 Table 4.14: Processes elicited in promoting visualisation in modelling Task 2. ...... 200 Table 4.15: Components in the pre-test and post-test with KR-20 statistics. ......... 206 Table 4.16: Visualisation components implied in pre-test and post-test items. ...... 209 Table 5.1: Descriptive statistics of pre-test and post-test. ...................................... 219 Table 5.2: Calculation of difference in differences between the two experimental group means and the control group means. ..................................................................... 223. xxi.

(22) Table 5.3: Rotated component matrices with principal components extracted for pretest and post-test. ................................................................................................... 227 Table 5.4: Descriptive statistics showing missing data for pre-test and post-test. .. 229 Table 5.5: Comparing pre-test item A1.1 and post-test item B1.1. ......................... 240 Table 5.6: Comparing pre-test item A1.4 and post-test item B1.4. ......................... 241 Table 5.7: Comparing pre-test item A1.3 and post-test item B1.3. ......................... 242 Table 5.8: Comparing pre-test item A1.2 and post-test item B1.2. ......................... 243 Table 5.9: Comparing pre-test item A4.1 and post-test item B4.2. ......................... 246 Table 5.10: Comparing pre-test item A4.2 and post-test item B4.4. ....................... 247 Table 5.11: Comparing pre-test item A4.3 and post-test item B4.3. ....................... 248 Table 5.12: Statistics for post-test item B4.1. ......................................................... 249 Table 5.13: Comparing items pre-test item A2.1 and post-test item B2.1. ............. 252 Table 5.14: Comparing pre-test item A2.2 and post-test item B2.2. ....................... 253 Table 5.15: Comparing pre-test item A2.3 and post-test item B2.3. ....................... 254 Table 5.16: Comparing pre-test item A2.4 and post-test item B2.4. ....................... 256 Table 5.17: Exemplars of right and wrong answers to item A2.4 from different groups. ............................................................................................................................... 257 Table 5.18: Comparing pre-test item A3.1 and post-test item B3.1. ....................... 260 Table 5.19: Comparing pre-test item A3.2 and post-test item B3.2. ....................... 261 Table 5.20: Statistics for standalone item A4.4. ..................................................... 262 Table 5.21: Kolmogorov-Smirnov and Shapiro-Wilk tests for normality.................. 267 Table 5.22: Results from the Kruskal-Wallis related median test. .......................... 269 Table 5.23: Mann-Whitney U statistics comparing the Both workshops and Control group on the left and First workshop only and control group on the right. .............. 270 Table 5.24: Matching criteria used in separate case control matching analyses. ... 272 Table 5.25: Wilcoxon signed rank test for the data matched on pre-test scores only. ............................................................................................................................... 273 Table 5.26: Wilcoxon signed ranks tests performed on data sets according to indicated characteristics. ....................................................................................................... 275 Table 5.27: Comparing a matched and unmatched analysis with Wilcoxon signed ranks tests. ............................................................................................................. 279 Table 6.1: Group ratings of workshop 1 in terms of three visualisation categories: internalisation, representation and justification. ...................................................... 284. xxii.

(23) Table 6.2: Visualisation strategies used by main study Group 1 during Task 1. .... 285 Table 6.3: Framework of actions taken by main study Group 1 during Task 1. ...... 294 Table 6.4: Visualisation strategies used by main study Group 1 during Task 2. .... 303 Table 6.5: Actions taken by main study Group 1 during Task 2. ............................ 309 Table 6.6: Quantification of open-ended questionnaire items. ............................... 323 Table 6.7: Themes, categories and codes relating to open-ended questionnaire. . 324 Table 6.8: Group scores rating the internalisation, representation and justification of the cohort in pilot study 1. ...................................................................................... 332 Table 6.9: Quantification of individual involvement in team work. .......................... 335 Table 6.10: Codes and categories of the theme: mathematical modelling experiences. ............................................................................................................................... 342 Table 7.1: Summary for four items in new insights dimension. .............................. 363 Table 7.2: Merged findings of study. ...................................................................... 389. xxiii.

(24) LIST OF FIGURES Figure 1.1: Mathematical modelling cycle according to Blum and Leiß (2007:225).... 8 Figure 1.2: Three students’ examination solution curves and responses to the question: Is there a tollgate at the 50 km mark, why? .............................................. 13 Figure 1.3: Suggested overarching theoretical framework. ...................................... 24 Figure 1.4: The Pirie-Kieren model for the development of mathematical understanding (Pirie & Kieren, 1994:167). ............................................................... 25 Figure 2.1: Number of engineers per 10 000 citizens in SA servicing seven categories of municipalities (van Baalen et al., 2015:6). ............................................................ 46 Figure 2.2: Interplay between concept definition and concept image (Vinner, 1991:71). ................................................................................................................................. 60 Figure 2.3: Three alternative routes to progress to concept formation (Vinner, 1991:7273). ........................................................................................................................... 61 Figure 2.4: Structure of a schema according to Dubinsky (1991:107). ..................... 65 Figure 2.5: Visualisation as a language; the visual alphabet of a pattern (Hershkowitz, 2014:200). ................................................................................................................ 75 Figure 2.6: Pedagogical map for learning with CAS adapted from Pierce and Stacey (2010:6). ................................................................................................................... 77 Figure 2.7: A family of functions obtained for the parabola y  x 2  3x  c through variations in the parameter c. ................................................................................... 79 Figure 2.8: Multiple representations of a differential equation and its solution. ........ 80 Figure 2.9: Didactical functions of technology in mathematics education (Drijvers, 2015:137). ................................................................................................................ 83 Figure 2.10: Reflective learning cycle (Vicéns, Zamora & Ojados, 2016:2).............. 85 Figure 2.11: The human-computer interface, according to Tall (1993:394). ............. 86 Figure 3.1: Distributed meanings of a concept across different representations. ... 104 Figure 3.2: Distributed meanings of rabbit population growth across different representations. ..................................................................................................... 106 Figure 3.3: Stages and transitions in the modelling process (Galbraith & Stillman, 2006:144). .............................................................................................................. 117 Figure 4.1: Research questions – isolated or integrated (Yin, 2006:43). ................ 136 Figure 4.2: Connecting the three worlds in engineering education. ........................ 140. xxiv.

(25) Figure 4.3: A typical quasi-experiment design (Shadish et al., 2002:137). ............. 151 Figure 4.4: A quasi-experiment with a pre-test post-test control-group design....... 152 Figure 4.5: The difference-in-difference design, adopted from Somers et al. (2013:xv). ............................................................................................................................... 154 Figure 4.6: Design principles in the learning environment. ..................................... 163 Figure 4.7: Instrument-timeline of two pilot studies and main study. ...................... 166 Figure 4.8: Task 1 linking the three worlds: real world, mathematical world and technology world. ................................................................................................... 174 Figure 4.9: Tollgate problem depicting four tollgates along the N1 National highway. ............................................................................................................................... 175 Figure 4.10: Linking the three worlds with visualisation processes embedded in Task 2. ............................................................................................................................ 176 Figure 4.11: Composition of groups for case control matching. ............................. 185 Figure 4.12: Possible outcomes of pre-test – post-test non-equivalent comparison group design. ......................................................................................................... 190 Figure 4.13: Quantitative and qualitative content analysis with corresponding instruments............................................................................................................. 195 Figure 4.14: Three tiered cyclic process of coding used in qualitative content analysis. ............................................................................................................................... 196 Figure 4.15: Mixing of qualitative and quantitative research design components. .. 203 Figure 4.16: Triangulation of research questions, data sources and data analysis. 214 Figure 5.1: Boxplots comparing pre-test and post-test means of three groups. ..... 221 Figure 5.2: Pre-test and post-test group means. .................................................... 223 Figure 5.3: Scree plots for the pre-test (left) and post-test (right). .......................... 225 Figure 5.4: Eigenvalues and the respective total variance explained for the pre-test (left) and post-test (right). ....................................................................................... 226 Figure 5.5: Visual evidence of the inter-dimensionality of pre-test items. ............... 231 Figure 5.6: Visual evidence of the inter-dimensionality of post-test items. ............. 232 Figure 5.7: Concentration ellipses identifying correlations between translation and visual reasoning in the 2-D component space. ....................................................... 235 Figure 5.8: Concentration ellipses identifying correlations between new insights and intuition in 2-D component space. .......................................................................... 238 Figure 5.9: Mean scores for items in the dimension: translation. ........................... 244. xxv.

(26) Figure 5.10: Mean scores for items in the dimension: visual reasoning. ................ 250 Figure 5.11: Mean scores for items in the dimension: new insights. ...................... 259 Figure 5.12: Mean scores for items in the dimension: intuition............................... 264 Figure 5.13: Four visualisation dimensions in pre-test and post-test. ..................... 265 Figure 5.14: Composition of groups for Kruskal-Wallis and Case control matching analyses. ................................................................................................................ 271 Figure 5.15: Effect sizes contrasting the differences between the pre-test and posttest scores of matched pairs. ................................................................................. 276 Figure 6.1: Pirie-Kieren model of processes used by Group 1 in task 1. ................ 295 Figure 6.2: Pirie-Kieren model of processes used by Group 1 in task 2. ................ 310 Figure 6.3: Graphical evidence of erroneous solution curves. ................................ 314 Figure 6.4: Incorrect initial condition causes a negative number of vehicles. ......... 318 Figure 6.5: Analytical solution offers proof of intuition. ........................................... 320 Figure 7.1: Three tiered translation process, informed by textual and pictorial backdrop. ............................................................................................................... 366 Figure 7.2: Contrast between bottom-up construction and top-down solution of DE. ............................................................................................................................... 380 Figure 7.3: Interplay between mathematical modelling (MM), computer algebra system (CAS), differential equations (DEs) and visualisation (VIS). ................................... 397 Figure 7.4: The W-H-I framework to advance visualisation when supported by mathematical modelling. ......................................................................................... 399 Figure 8.1: The use of theories in this study. .......................................................... 408 Figure 8.2:. The modelling cycle showing transitions, modelling activities and. corresponding visualisation processes. .................................................................. 409 Figure 8.3: The effect of subtasks on understandings. ........................................... 411. xxvi.

(27) LIST OF ABBREVIATIONS AND ACRONYMS CAS. Computer algebra system. DE(s). Differential equation(s). UJ. University of Johannesburg. DFC. Doornfontein campus. xxvii.

(28) CHAPTER 1: INTRODUCTION AND ORIENTATION 1.1. BACKGROUND. 1.1.1 South Africa’s Higher Education sector: Challenges and expectations In the next decade, Higher Education Institutions (HEIs) in South Africa (SA) will have to accelerate the pace of producing quality engineers with sound technological and problem solving skills. The local engineering industry is desperate for skills as revealed in the National Scarce Skills List (Goga & Van der Westhuizen, 2012) where engineering occupations dominate the list of short supply in the South African economy. It is envisaged that student enrolments in South African HEIs will grow by 77% towards 2030 while the corresponding figure for technical and vocational education and training (TVET1) colleges is predicted to reach 400% (Department of Higher Education and Training [DHET], 2014). However, such growth will aggravate the existing shortage of qualified lecturers and further strain the already burdened student-to-academic staff ratio (John, 2014). The role of engineering through technological innovations has a substantial impact on society (The Royal Academy of Engineering, 2007), arguably far exceeding that of the agricultural or industrial revolutions. New technological innovations are being introduced into people’s daily lives at an unwavering pace. Inevitably, mathematics education has also been affected by new technologies that resulted in the implementation and research on innovative pedagogical approaches (Meagher, 2005). In their critical analysis on “being educated in the 21st century”, Prinsloo and Louw (2006:288) highlight the obligation of universities to create new contexts with new technologies that can lead to a new type of professionalism in global citizenship. HEIs therefore need to continuously revise course content and align teaching and learning strategies with the practices endorsed by the engineering industry (The Royal Academy of Engineering, 2007). Houston, Mather, Wood, Petocz, Reid, Harding, Engelbrecht and Smith, (2010:77) challenge mathematics educators “to create a dynamic curriculum and teaching and. 1. TVET colleges were formerly known as further education and training [FET] colleges.. 1.

(29) learning environment that inspire students to engage deeply with mathematical ideas”. While fostering such ideals, the reality facing South African learners in Basic Education is a world apart. Many South African schools experience a lack of educational and human resources and the effects thereof reverberates to mathematics departments in HEIs (Howie, Scherman & Venter, 2008). The inadequate access or exposure to computers at school level can potentially impede engineering students’ learning progress in a technology-rich teaching and learning environment. Berger (2010) shows that although first year mathematics major students who have no computer background could manage to successfully programme a Mathematica2 code, they were considerably less successful than their computer-literate classmates to interpret the computer generated outputs. Berger (2010) argues that different skills are needed to interpret computer graphs and tables as compared to the interpretation of graphs in a traditional paper-and-pen environment. The interpretation of computer graphs and tables demands conceptual skills which co-evolve with theoretical knowledge and technological skills (Kieran & Saldanha, 2008). This can be an uncharted terrain for many students who receive their schooling mostly on the procedural level (Engelbrecht, Bergsten & Kågesten, 2009). In fact, programming software usually relieves the user from the procedural computations as this is outsourced to the computer. Cognisant of this intricate interplay between technological and conceptual skills, Engelbrecht et al. (2009) call for suitable tasks and pedagogies to advance the procedural-conceptual gap. The interpretation of computer outputs is a vital skill when learning with technology and is regarded to be a prerequisite in the engineering world of work (Berger, 2010). The teaching of mathematics at South African HEIs is also burdened by the quality of students attracted to this discipline (Berger, 2010; Hockman, 2005). Researchers (Howie & Plomp, 2002; John, 2014a) ascribe this to a shortage of properly trained teachers. The quality of students relates to the quality of teaching which is inseparable to the content knowledge and pedagogic content knowledge of teachers (Hockman, 2005). In the absence of proper content knowledge, teachers teach mostly on lower cognitive levels which signifies a failure to a “deep approach” to learning (Laurillard,. 2. Mathematica is a state-of-the-art technical computing system used by educators, scientists, economists and engineers worldwide.. 2.

(30) 2005:135). Paradoxically, future engineers will be expected in their world of work to act independently, to make decisions and face novel problems where higher levels of cognitive thinking are expected. In sum, success in higher education is for school leavers the gateway to the engineering world of work. At the entrance gate are the school leavers in SA who are mostly short-changed due to poor teaching environments. This limitation can lead to the inadequate development of higher cognitive skills of school leavers and result in under-developed interpretational capabilities. The Engineering Council of South Africa (ECSA) raises concerns about the ability of the school system to produce quality candidates for engineering courses (Lourens, 2015). The exit gate releases students into an engineering industry that hungers for quality engineers who will face challenges that cannot even be anticipated today. The gravity of this dilemma appeals to academia in higher education to put innovative strategies in place and revise pedagogical approaches which can mediate between Basic Education and the engineering industry. 1.1.2 Engineering mathematics and visualisation with technology Some of the main challenges in engineering education culminate in the proper knowledge and application of mathematics (Ramkrishna & Amundson, 2004; Steyn & du Plessis, 2007). Mathematics is a core component of the engineering discipline and is anecdotally called the “mental tool” and “language” of scientists and engineers (Hamming, 1980:81-82; Hockman, 2005). The importance of mathematics in engineering can be traced back to ancient times and is still acknowledged today: “Engineering speaks through an international language of mathematics, science, and technology” (National Academy of Engineering, 2004:3). In contrast with pure mathematics, engineering diploma students study mathematics not for its academic scholarship value, but rather for its embeddedness in engineering applications. Engineering mathematics is taught as a service subject in vocational engineering qualifications and is consequently directed by developments in the engineering industry (The Royal Academy of Engineering, 2012). It can be argued that engineering graduates of the 21st century face a technologicallydriven world of work that is different from that of a few decades ago. This prompted mathematics educators to explore computer algebra systems (CAS) – such as. 3.

(31) Mathematica – which offer the user a symbolic, numeric and graphic interface (Dubinsky & Tall, 1991). Gradually, CAS has stimulated the use of other desirable mathematical tools for work-place engineers namely programming, mathematical modelling and visualisation (Aslaksen & Santosa, 2013). For students who are predominantly schooled in analytical procedures, such a technology-rich environment implicates a shift from traditional paper-and-pen-based techniques. With these different foci, the school-to-higher mathematics transition is problematic. Scholars predict “epistemological obstacles” as students require “changing lenses” to bridge the “cognitive discontinuity” caused by different tools, a new computer language and multiple representations (Yerushalmy, 2005:37). Many engineering subjects make use of. symbolic,. numeric and. graphic. representations; to master the interpretation of these representations is essential (Laurillard, 2005a). In practice, the interpretation of multiple representations is a daily activity for professional engineers and inherent to problem solving (Stavridou & Kakana, 2008). The ability to reason visually and to fluently interpret multiple representations is a cognitive process referred to as visualisation (Duval, 2014). In a CAS environment, visualisation is cognitively challenging as several registers – for example natural language, symbolic, numeric and visual registers - must be mobilised (Camacho & Depool, 2003). Typically, CAS responds to syntax-specific inputs, thereupon suppresses procedural processes and instantly displays outputs in the form of symbols, tables and graphs. Unlike in a paper-and-pen environment, the systematic step-by-step cognitive processes that would normally be followed, are outsourced to CAS. As such, students are confronted by a cognitive discontinuity (Yerushalmy, 2005). Those students who come from under-resourced teaching and learning backgrounds are often underprepared for the cognitive complexity imposed by visualisation (Duval, 2006). As Duval (2006:116) states “when we focus on visualization we are facing a strong discrepancy between the common way to see the figures, generally in an iconic way, and the mathematical way they are expected to be looked at”. As a result, students may be reluctant to connect different representations of the same concept or may even form connections that are incoherent (Van Dooren, de Bock & Verschaffel, 2013). As such, visual reasoning evokes cumulative visual processes (Fischbein, 1999) to forge inter-representational connections. Therefore,. 4.

(32) visualisation in a CAS environment imposes a “cognitive reorganisation” (Samson, 2011:19) of processes and mental activities. This reorganisation must also consider the specific ways in which actions and processes can be re-united. The centrality of visualisation must therefore go beyond the perceptual and also involve a deep engagement with the underlying concepts (Arcavi, 2003). It stands to reason that learning with technology can challenge engineering students on a metacognitive level. Within a single CAS task, students have to contextualise preknowledge from various mathematical domains into new situations. This recontextualisation requires a past-to-present continuum that is emphatically part of mathematical problem solving. Hjalmarson, Wage and Buck (2008) highlight that knowledge cannot remain detached in separate mathematical domains. Engineering students must be able to generate and interpret computer generated graphs by using various fields of mathematics in the final interpretation of graphs. Thus future engineers require metacognitive skills to integrate their mathematical knowledge with their technological skills in order to logically interpret and analyse computer graphs. These skills must also be transferred beyond the borders of mathematics classrooms to other engineering subjects. Schraw and Moshman (1995) are of the opinion that peer interaction can improve the metacognitive process. This notion is also supported by Mevarech, Tabuk and Sinai (2006) who pose that metacognitive instruction can enhance students’ abilities to solve problems within a cooperative setting. Metacognition can be a long-term advantageous skill of engineering students who will be entrenched in a community that expects them to plan, monitor and critically evaluate processes as a team. Dixon and Johnson (2012) are in accord when they call for curricula and teaching strategies to target metacognitive skills in future engineers. Such skills may also help students to relate different representations with each other and to form links between representations and associated real world situations. 1.1.3 The authentic exposure of engineers in training According to the national oversight body of professional engineers – the Engineering Council of South Africa (ECSA) – the core business of an engineer is to plan how to approach a given task using theoretical and practical knowledge in a proficient and professional manner (ECSA, 2013). In the quest to be globally competitive, ECSA. 5.

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