Analysis of constructed mathematical responses by numeric tests for equivalence

ABSTRACT

A method is described to determine if a constructed mathematical response is equivalent to the correct answer specified by the course author. Based on a combination of random evaluation and operator analysis, the method is theoretically justified for a general class of functions. Possible breakdowns due to computer arithmetic are discussed as well as the difficulties encountered when a function fails to fall in the general class. For CAI applications where a response is considered correct if it is equivalent to the correct answer, the method offers two possible advantages: (1) Only one form of the correct answer is specified. (2) Extensive and time consuming manipulation of expressions is avoided.

REFERENCES

• 1: Bellman, R., "On Proving Theorems in Plane Geometry via Digital Computer", The American Mathematical Monthly, vol. 73, pp. 1107-1109, November, 1966.
• 2: Bochner, S. and Martin, W.T., "Several Complex Variables", Princeton University Press, 1948.
• 3: Caviness, B.F., "On Canonical Forms and Simplification", Doctoral Dissertation, Carnegie-Mellon University, May, 1968.
• 4: Engeli, M.E., "Achievements and Problems in Formula Manipulation", IFIP Congress 68, pp. 79-84.
• 5: Feingold, S.L. and Frye, C.H., "User's Guide to PLANIT", TM-3055/000/01, System Development Corporation, October, 1966.
• 6: Gunning, R.C. and Rossi, H. "Analytic Functions of Several Complex Variables", Prentice-Hall, Inc., 1965.
• 7: Korfhage, R., Hochgesang, G., Oldehoeft, A., Mitzell, M., "PICLS, Purdue Instructional and Computational Learning System", CSD TR 28, Purdue University, October, 1968.
• 8: Manacher, G.K., "A Content-Evaluating Mode of Computer-Aided Instruction", Interactive Systems for Experimental Applied Mathematics. pp. 286-293, Academic Press, 1968.
• 9: Martin, W.A., "Symbolic Mathematical Laboratory", Doctoral Dissertation, MAC-TR-36, Massachusetts Institute of Technology, January, 1967.
• 10: Sammet, J.E., "An Annotated Descriptor Based Bibliography on the Use of Computers for Non-Numerical Mathematics", Computing Reviews, vol. 7, no. 4, pp. B1-B31, July-August, 1966.
• 11: Jean E. Sammet, Survey of formula manipulation, Communications of the ACM, v.9 n.8, p.555-569, Aug. 1966 [doi>10.1145/365758.365762]

INDEX TERMS

Primary Classification:
  K. Computing Milieux
  K.3 COMPUTERS AND EDUCATION
      K.3.1 Computer Uses in Education
          Subjects: Computer-assisted instruction (CAI)

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Computer arithmetic

General Terms:
Theory