Elaborating heuristic reasoning and rigor with mathematical games

ABSTRACT

Mathematical games arouse enthusiasm and challenge. They usually involve clear and simple rules, with physical, visual, or numerical entities, which raise motivation and intuition. The development of their playing strategies requires both heuristic reasoning and rigor. In order to win, one should recognize and capitalize on number patterns, such as parity and symmetry, as well as invariant patterns of repeated algorithmic actions. The search for patterns involves essential problem solving heuristics, and the validation of devised algorithmic actions requires a rigorous, scientific point of view. While games are known to be stimulating, their utilization in textbooks is very limited, if at all. In this paper, we offer an instructional approach, of using mathematical games, for elaborating fundamental notions that are apparent and relevant already at the very basic levels of computer science (CS) studies. We display our approach, illustrate it, and describe our experience in applying it in class.

REFERENCES

• 1: Ben-David Kolikant, Y. & Pollack, S., Establishing computer science professional norms among high-school students, Computer Science Education, 14(1), (2004), 21--35.
• 2: Berlekamp, E. R., Conway, J. H., & Guy, R. K., Winning ways for you Mathematical Plays, Academic Press, (1982).
• 3: Kim B. Bruce , Robert L. Scot Drysdale , Charles Kelemen , Allen Tucker, Why math?, Communications of the ACM, v.46 n.9, September 2003 [doi>10.1145/903893.903918]
• 4: Edsger Wybe Dijkstra, A Discipline of Programming, Prentice Hall PTR, Upper Saddle River, NJ, 1997
• 5: Peter J. Denning, A debate on teaching computing science, Communications of the ACM, v.32 n.12, p.1397-1414, Dec. 1989 [doi>10.1145/76380.76381]
• 6: David Ginat, The greedy trap and learning from mistakes, Proceedings of the 34th SIGCSE technical symposium on Computer science education, February 19-23, 2003, Reno, Navada, USA
• 7: Ginat, D., On novices' local views of algorithmic characteristics, Informatics Education Conference -- ISSEP, Springer Verlag, (2006), 127--137.
• 8: Stuart Hansen, The game of set®: an ideal example for introducing polymorphism and design patterns, Proceedings of the 35th SIGCSE technical symposium on Computer science education, March 03-07, 2004, Norfolk, Virginia, USA
• 9: John M. D. Hill , Clark K. Ray , Jean R. S. Blair , Curtis A. Carver, Jr., Puzzles and games: addressing different learning styles in teaching operating systems concepts, Proceedings of the 34th SIGCSE technical symposium on Computer science education, February 19-23, 2003, Reno, Navada, USA
• 10: Anany Levitin , Mary-Angela Papalaskari, Using puzzles in teaching algorithms, Proceedings of the 33rd SIGCSE technical symposium on Computer science education, February 27-March 03, 2002, Cincinnati, Kentucky
• 11: Polya, G., How to Solve it, Princeton University Press, (1945).
• 12: Richard Rasala , Jeff Raab , Viera K. Proulx, The SIGCSE 2001 Maze Demonstration program, Proceedings of the 33rd SIGCSE technical symposium on Computer science education, February 27-March 03, 2002, Cincinnati, Kentucky
• 13: Schoenfeld, A. H., Learning to think mathematically: problem solving, metacognition and sense-making in mathematics. In Grouws, D. (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, (1992), 334--370.
• 14: Spohrer, J. G., Soloway, E., & Pope, E., A goal/plan analysis of buggy Pascal programs, in Soloway, E. & Spohrer, J. G. (eds.), Studying the Novice Programmer, Lawrence Erlbaum, (1989), 355--400.

INDEX TERMS

Primary Classification:
  K. Computing Milieux
  K.3 COMPUTERS AND EDUCATION
      K.3.2 Computer and Information Science Education
          Subjects: Computer science education

General Terms:
Algorithms

Keywords:
games, heuristics, problem-solving, rigor