Problem solving, working backwards, and graphic proof representation

Teaching Philosophy

Volumn 23, Issue 2, 2000, Pages 169-187

  Croy, Marvin J ^a  

^a UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 57749114034     PISSN: 01455788     EISSN: 21536619     Source Type: Journal    
DOI: 10.5840/teachphil200023226     Document Type: Article

References (31)

1. Allan Newell and Herbert Simon, Human Problem Solving (Englewood Cliffs: Prentice Hall, 1972).
2. Richard Burton, "Diagnosing Bugs in a Simple Procedural Skill," in Intelligent Tutoring Systems, ed. Derek Sleeman and John Seely Brown (New York: Academic Press, 1982), 157-83.
3. Derek Sleeman, "Assessing Aspects of Competence in Basic Algebra," in Intelligent Tutoring Systems, ed. Derek Sleeman and John Seely Brown (New York: Academic Press, 1982), 185-99.
4. This finding is based on undergraduates working problems from Richard Butrick's Deduction and Analysis, revised edition (Washington, DC: University Press of America, 1981)
5. Irving Copi's Introduction to Logic, 7th ed. (New York: Macmillan, 1986)
6. Howard Kahane's Logic and Philosophy, 5th ed. (Belmont, CA: Wadsworth, 1986)
7. Virginia Klenk's Understanding Symbolic Logic, 2nd ed. Englewood Cliffs: Prentice Hall, 1989).
8. See the reports contained in Marvin Croy, "CAI and Rule Applications for Deductive Proof Construction," Collegiate Microcomputer, VI, l988, 5l-6
9. Marvin Croy, "The Use of CAI to Enhance Human Interaction in the Learning of Deductive Proof Construction," Computers and the Humanities, 22(4), 1988, 277-84
10. Marvin Croy, "Integrating CAI Development and Empirical Research: Opportunities and Responsibilities," Computerized Logic Teaching Bulletin, IV, 1991, 2-12.
11. Marvin Croy, "Empirical Explorations of Deductive Proof Construction," Computers and Philosophy Newsletter, 4, 1989, 111-27. This and other relevant articles are presently available at: http//:www.uncc.edu/ colleges/arts-and-sciences/ philosophy/faculty/croy.html.
12. John Anderson, Cognitive Psychology and its Implications, 3rd ed. (New York: Freeman and Company, 1990).
13. See Merrie Bergmann, James Moor, and Jack Nelson, The Logic Book, 3rd ed. (New York: McGraw-Hill, 1998) for a text which presents the intelim rule set and which emphasizes the working backward method.
14. As for texts using the inference/ replacement rule set, the working backwards method is not mentioned in Robert Johnson's A Logic Book, Fundamentals of Reasoning (Belmont, CA: Wadsworth Publishing Company, 1999).
15. Steven DeHaven's The Logic Course, 2nd ed. (Peterborough, Ontario: Broadview Press, 1998).
16. The method is mentioned with little or no development in Irving Copi and Carl Cohen's An Introduction to Logic, 10th ed. (Englewood Cliffs: Prentice Hall, 1998)
17. C. Stephen Laymon's The Power of Logic (Mountain View, CA: Mayfield Publishing Company, 1999).
18. Working backwards is described in somewhat more detail in David Kelly's The Art of Reasoning 3rd expanded ed. (New York: Norton and Company, 1998)
19. Patrick Hurley's A Concise Introduction to Logic, 7th ed. (Belmont, CA: Wadsworth Publishing Company, 1999)
20. Paul Tidman and Howard Kahane's Logic and Philosophy, A Modern Introduction, 8th ed. (Belmont, CA: Wadsworth, 1999).
21. Virginia Klenk's Understanding Symbolic Logic, 3rd ed. (Englewood Cliffs: Prentice Hall, 1994) provides the most detailed discussion of both working backwards and working forwards during proof construction.
22. As put forward in several texts, the rules included here are Modus Ponens, Modus Tollens, Disjunctive Syllogism, Simplification, Addition, Conjunction, Hypothetical Syllogism, and Constructive Dilemma. Absorption, sometimes included as an inference rule, is actually a rule of replacement, given that it expresses a truth functional equivalence, and will not be discussed here.
23. The three-step procedure would initially suggest extracting the conclusion from the first premise via Modus Ponens, and instantiating this rule leads to a fruitless search for 'A'. Once this dead end is discovered, one should backtrack to the conclusion and consider constructing it via Hypothetical Syllogism.
24. The results referred to here are part of a four-semester study in which a computer program was used to collect and analyze working backwards errors. A report on this analysis is forthcoming in the Journal of Computers in Mathematics and Science Teaching.
25. Working backwards runs into difficulties when confronting proof problems containing inconsistent premises. For example, given 'A' and '~A' as premises and 'B' as the conclusion, it is quickly seen that the conclusion can neither be extracted nor constructed. Nevertheless, even here the standard strategy of using Addition and Disjunctive Syllogism to generate a solution can provide a basis for postulating the last two steps in the proof, thereby characterizing the types of expressions to be aimed for when working forwards.
26. Howard Kahane, Logic and Philosophy: A Modern Introduction, sixth edition (Belmont, CA: Wadsworth Publishing Company, 1990), 78.
27. Hurley, 1999, 399. This warning appears in all editions of A Concise Introduction to Logic.
28. Tidman and Kahane, 1999, 99-100.
29. In earlier editions of Kahane's Logic and Philosophy, Arabic numerals were used for working backwards in place of the Roman numerals advocated in later editions. Unfortunately, using Arabic numerals for working both forwards and backwards allows two different lines to have the same line number. This probably enhanced confusion over working backwards and may have contributed to Kahane's restriction of its use.
30. See Kahane, Logic and Philosophy, 2nd edition, 1973, 55.
31. Richard Scheines and Wilfried Sieg, "Computer Environments for Proof Construction," Interactive Learning Environments 4(2), 1994, 159-69.