Головна » Статті » АКТУАЛЬНІ ПРОБЛЕМИ МАТЕМАТИКИ ТА МЕТОДИКИ НАВЧАННЯ МАТЕМАТИКИ

Voskoglou Michael Gr. MODES OF THINKING IN PROBLEM SOLVING
Voskoglou Michael Gr. [mvoskoglou@gmail.com]
Graduate Technological Educational Institute of Western Greece, Patras, Greece
Download in PDF: http://fmo-journal.fizmatsspu.sumy.ua/journals/2020-v3-25-1/2020_3-25-Voskoglou_FMO.pdf

MODES OF THINKING IN PROBLEM SOLVING

Formulation of the problem: Problem Solving affects our daily lives in a direct or indirect way for centuries. Volumes of research have been written about it and attempts have been made by scientists to make it accessible to all in various degrees. The modes of thinking used in Problem Solving is, therefore, a very interesting and timely subject to study. The present paper discusses the main modes of thinking involved in PS, which are Critical Thinking (CrT), Computational Thinking (CT) and Statistical Thinking (ST). Emphasis is given to ST and in particular to Bayesian Reasoning due to its great importance for everyday life and science that only recently has been fully recognized. We start with a brief description of the use of CrT in PS. Next ST and the necessity of combining it with CrT in PS is discussed, while Bayesian reasoning is studied separately afterwards. Next the role of CT for solving complex technological problems is examined and the paper closes with the final conclusion.
Materials and methods. The methods of analysis used are based on a synthesis of already reported researches and on suitable examples illustrating our results.
Results. The article studies the role of Statistical Thinking in Problem Solving, where the problem is considered with its wide meaning (not mathematical problems only). Particular emphasis is given to Bayesian Reasoning, whose importance in everyday life and science applications has been only recently fully recognized. Critical and Computational Thinking, the other two main modes of thinking used in Problem Solving, are also discussed.
Conclusions. Problem Solving is a complex cognitive process that needs the combination of several modes of thinking in order to be successful. Those modes, apart from the simple spontaneous thinking, include Critical, Statistical and Computational Thinking.
KEY WORDS: Problem Solving (PS), Critical Thinking (CrT), Statistical Thinking (ST), Computational Thinking (CT), Bayesian Reasoning, Fuzzy Logic (FL).

СПОСОБИ МИСЛЕННЯ ПРИ ВИРІШЕННІ ПРОБЛЕМ
Майкл Г. Воскоглу
Вищий технологічний освітній інститут Західної Греції, Греція

Постановка проблеми. Вирішення проблем впливає на наше повсякденне життя прямим чи опосередкованим чином протягом століть. Способи мислення, що використовуються при вирішенні проблем, є актуальним предметом дослідження.
Матеріали та методи. Використані методи аналізу засновані на синтезі вже відомих досліджень та відповідних прикладів, що ілюструють наші результати.
Результати. У статті вивчається роль статистичного мислення при вирішенні проблем, де проблема сприймається у широкому розумінні (не лише математичні проблеми). Особливий наголос приділяється Байєсовому мисленню, значення якого у повсякденному житті та науці застосовується повністю лише нещодавно. Також обговорюються інші два основні способи мислення, що використовуються при вирішенні проблем, – критичне та обчислювальне мислення.
Висновки. Вирішення проблем – це складний пізнавальний процес, який для досягнення успіху обумовлює інтегроване використання кількох способів мислення. Ці способи, крім простого спонтанного мислення, включають критичне, статистичне та обчислювальне мислення.
Ключові слова: вирішення проблеми, критичне мислення, статистичне мислення, обчислювальне мислення, байєсівське міркування, нечітка логіка.

References

  1. Schoenfeld, A. (1983), The wild, wild, wild, wild world of problem solving: A review of sorts, For the Learning of Mathematics, 3, 40-47.
  2. Polya, G. (1973), How I solve it: A new aspect of mathematical method, New Jersey: Princeton University Press.
  3. Green, A. J. K. & Gillhooly, K. (2005), Problem solving, in Braisby, N.  & Gelatly, A. (Eds.), Cognitive Psychology, Oxford University Press, Oxford.
  4. Martinez, M. (2007), What is meta cognition?  Teachers intuitively recognize the importance of metacognition, but may not be aware of its many dimensions, Phi Delta Kappan, 87(9), 696-714.
  5. Halpern, D. (2003), Thought and knowledge: An introduction to critical thinking, 4th edition, Mahwah, Earlbaum, NJ, USA.
  6. Mc Guinness, C. (1993), Teaching thinking: New signs for theories of cognition, Educational Psychology, 13(3-4), 305-316.
  7. Williams, R. L. (2005), Targeting critical thinking within teacher education: The potential impact on Society, The Teacher Educator, 40(3), 163-187.
  8. Mc Peck, J. E. (1981), Critical thinking and education, Martin Robinson, Oxford.
  9. Brookfield, S.D. (1987), Developing critical thinkers: Challenging adults to explore alternative ways of thinking and acting, Open University Press, England.
  10. Pascarella, E. T. & Terenzini,  P. (1991), How college affects students, Jossey-Bass, San Francisco.
  11. Martinez, M. (2007), What is metacognition?  Teachers intuitively recognize the importance of metacognition, but may not be aware of its many dimensions”, Phi Delta Kappan, 87(9), 696-714.
  12. Tiwari, A., Lai, P., So, M. & Yuen, K. (2006), A comparison of the effects of problem-based learning and lecturing on the development of students’ critical thinking, Med. Educ.,41(2), 156-174.
  13. Jaynes, E.T. (2011), Probability Theory: The Logic of Science, 8th Printing, Cambridge University Press, UK.
  14. Mumford, D. (2000), The Dawning of the Age of Stochasticity, in V. Amoid, M. Atiyah, P. Laxand & B. Mazur (Eds.), Mathematics: Frontiers and Perspectives, AMS, 197-218.
  15.  Kosko, B. (1993), Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion, NY, USA.
  16. Zadeh, L.A. (1965), Fuzzy Sets, Information and Control, 8, 338–353.
  17. Voskoglou, M.Gr. (2019), Methods for Assessing Human-Machine Performance under Fuzzy Conditions, Mathematics, 7, article 230.
  18. Changingminds.org, Full alphabetic list of Fallacies, available online: http://changingminds.org /disciplines/argument/fallacies/fallacies_alpha.htm (accessed on 23 June 2020).
  19. Athanassopoulos, E. & Voskoglou, M.Gr. (2020), Quantifying the Aristotle’s Fallacies, Mathematics, 8, article 1399.
  20. Gardener, J. & Resnik, D. (2002), The misuse of statistics, concepts, tools and a research agenda, Accountability in Research: Policies and Quality Assurance, 9(2), 65-74.
  21. Wason, P. C. (1977), Self-contradictions, in Johnson-Laird, P. N.; Wason, P. C. (Eds.), Thinking: Readings in cognitive science, Cambridge, Cambridge University Press.
  22. Evans, J.St.B.T., Newstead, S. E., Byrne, R.M.J. (1993), Human Reasoning: The Psychology of Deduction, Psychology Press, East Sussex, UK.
  23. Tversky, A., & Kahneman, D. (1982), Judgment under uncertainty: Heuristics and biases, Cambridge University Press, Cambridge, UK.
  24. Gould, S. J. (1992), Bully for brontosaurus: Reflections in natural history ,  Norton, New York.
  25. Aristidou, M. (2013), Irrationality Re-Examined: A Few Comments on the Conjunction Fallacy, Open Journal of Philosophy, 3(2), 329-336.
  26. Schuler, J. & Lipschutz, S. ((2010), Schaum’s Outline of Probability, 2nd Edition, McGraw Hill, NY.
  27. Horgan, J. (2015), Bayes’ Theorem: What is the Big Deal?”, available in http//:blogs. scientificamerican.com /cross-check/ bayes-s-theorem-what-s-the-big-deal.
  28.  Bertsch McGrayne, S. (2012), The Theory that would not die, Yale University Press, New Haven and London.
  29. Athanassopoulos, E. & Voskoglou, M.Gr. (2020), A Philosophical Treatise on the Connection of Scientific Reasoning with Fuzzy Logic, Mathematics, 8, article 875.
  30. What do you think about machines that think? (2015), available in http://edge.org/response-detail/26871.
  31. Jeffreys, H. (1973), Scientific Inference, 3d Edition, Cambridge University Press, UK, 1973.
  32. Stone, L.D., Keller, C.M., Kratzke, T.M. & Strumpfer, J.F. (2014), Search for the Wreckage of the Air France Flight AF 447, Statistical Science, 29(1), 69-80.
  33. Thrope, W.H (1979), The origins and rise of ethology: The science of the natural behavior of animals, Praeger, London-NY.
  34. Gingerich, O. (1993), The Eye of the Heaven – Ptolemy, Copernicus, Kepler, American Institute of Physics, NY, USA.
  35. Singh, S. (2005), Bing Bang - The Origin of the Universe, Harper Perennian Publishers, NY, USA.
  36. Papert, S. (1996), An exploration in the space of Mathematics Education, International Journal of Computers for Mathematics, 1(1), 95-123.
  37. Wing, J. M. (2006), Computational thinking, Communications of the ACM, Vol.49, 33-35.
  38. Liu, J.  & Wang, L. (2010), Computational Thinking in Discrete Mathematics, IEEE 2nd International Workshop on Education Technology and Computer Science, 413-416.
  39. Voskoglou, M. Gr. & Buckley, S. (2012), Problem Solving and Computers in a Learning Environment, Egyptian Computer Science Journal, 36(4), 28-46.
  40. Kazimoglu, C., Kiernan, M., Bacon, L. & MacKinnon, L. (2011), Understanding Computational Thinking Before Programming:  Developing Guidelines for the Design of Games to Learn Introductory Programming Through Game-Play, International Journal of Game-Based Learning, 1(3), 30-52.
Розділ: АКТУАЛЬНІ ПРОБЛЕМИ МАТЕМАТИКИ ТА МЕТОДИКИ НАВЧАННЯ МАТЕМАТИКИ
Додано: 26.11.2020 | Переглядів: 53 | Рейтинг: 0.0/0
Статті з теми:
Всього коментарів: 0
avatar