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The Begle Memorial Series on Research in Mathematics Education

  • Richard E. Snow
  • Herbert J. Walberg
  • Christine Keitel-Kreidt
  • Donald J. Dessart
  • L. Ray Carry
  • Jens Holger Lorenz
  • Nicholas A. Branca
  • Richard E. Mayer
  • Edward A. Silver
  • Robert B. Davis
  • Gunnar Gjone
  • John P. Keeves
  • Thomas J. Cooney

Abstract

This paper is one of a series of presentations at this Congress commemorating the work of Ed Begle. Begle’s (1979) last work, “Critical Variables in Mathematics Education”, served as a basis for these presentations. In particular, I will address “The use of critical variables to organize research, problems of synthesizing research, and the kind of empirical research that would be most useful to mathematics education.”

Keywords

Critical Variable Mathematics Classroom Mathematics Achievement Mathematical Ability Task Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Anderson, J.R. Language, memory, and thought. Hillsdale, N.J.: Erlbaum, 1976.Google Scholar
  2. Anderson, J.R., Kline, P.J., Beasley, C.M., Jr. Complex learning processes. In Snow, R.E., Federico, P-A, Montague, W.E. (Eds.) Aptitude, learning, and instruction Vol. 2: Cognitive process analyses of learning and problem solving. Hillsdale, N.J.: Erlbaum, 1980.Google Scholar
  3. Begle, E.G. Critical variables in mathematics education. Washington, D.C.: Mathematical Association of America and the National Council of Teachers of Mathematics, 1979.Google Scholar
  4. Crist-Whitzel, J.L., Hawley-Winne, B.J. Individual differences and mathematics achievement: An investigation of aptitude-treatment interactions in an evaluation of three instructional approaches. Paper presented at the meeting of the American Educational Research Association, San Francisco, 1976.Google Scholar
  5. Cronbach, L.J., Snow, R.E. Aptitudes and instructional methods. N.Y.: Irvington, 1977.Google Scholar
  6. Greeno, J.G. A study in problem solving. In Glaser, R. (Ed.) Advances in instructional psychology. Vol I: Hillsdale, N.J.: Erlbaum, 1978.Google Scholar
  7. Greeno, J.G. Some examples of cognitive task analysis with instructional implications. In Snow, R.E., Federico, P-A., Montague, W.E. (Eds.) Aptitudes, learning, and instruction Vol 2: Cognitive process analyses of learning and problem solving. Hillsdale, N.J.: Erlbaum, 1980.Google Scholar
  8. Kennedy, M.M. Findings from the Follow Through Planned Variation Study. Educational Researcher 1978, 7 (6), 3–11.Google Scholar
  9. Newell, A. Simon, H.A. Human problem solving. Englewood Cliffs, N.J.: Prentice-Hall, 1972.Google Scholar
  10. Norman, D.A. Rumelhart, D.E. Explorations in cognition. San Francisco: Freeman, 1975.Google Scholar
  11. Pellegrino, J.W. Glaser, R. Cognitive correlates and components in the analysis of individual differences. Intelligence, 1979, 1, 187–214.CrossRefGoogle Scholar
  12. Pelligrino, J.W. Glaser, R. Components of inductive reasoning. In Snow, R.E., Federico, P-A., Montague, W.E. (Eds.) Aptitude, learning, and instruction Vol I: Cognitive process analyses of aptitude. Hillsdale, N.J.: Erlbaum, 1980.Google Scholar
  13. Sharps, R. A study of interactions between fluid and crystallized abilities of two methods of teaching reading and arithmetic. Unpublished doctoral dissertation. Pennsylvania State University, 1973.Google Scholar
  14. Snow, R.E. Research on aptitude for learning: A progress report. In Shulman, L.S. (Ed.) Review of research in education. Vol 4. Itasca, Ill.: Peacock, 1977.Google Scholar
  15. Snow, R.E. Theory and method for research on aptitude processes: A prospectus. Intelligence, 1978, 2, 225–278.Google Scholar
  16. Snow, R.E., Yalow, E. Intelligence and education. In Sternberg, R.J. (Ed.) Handbook of Human Intelligence. N.Y.: Cambridge University Press, in press.Google Scholar
  17. Spearman, C. The nature of “intelligence” and the principles of cognition. London: MacMillan, 1923.Google Scholar
  18. Sternberg, R.J. Intelligence, information processing and analogical reasoning: The componential analysis of human abilities. Hillsdale, N.J.: Erlbaum, 1977.Google Scholar
  19. Sternberg, R.J. The nature of mental abilities. American Psychologist. 1979, 34, 214–230.CrossRefGoogle Scholar
  20. Sternberg, R.J., Guyote, M.J., Turner, M.E. Deductive reasoning. In Snow, R.E., Federico, PA., Mongague, W.E. (Eds.) Aptitude, learning, and instruction Vol. I: Cognitive process analyses of aptitude.Google Scholar
  21. Webb, N.M. Learning in individual and small group settings. Technical Report No. 7. Aptitude Research Project, School of Education, Stanford University, 1977.Google Scholar
  22. Begle, E.G. Critical Variables in Mathematics Education: Findings from a Survey of the Empirical Literature. Washington: Mathematical Association of America and National Council of Teachers of Mathematics, 1979.Google Scholar
  23. Begle, E.G. The Effects of Varyinq_the Number of Practice Problems, Number or Examples, and Location of the Practice Problems in Elementary School Geometry. Palo Alto, CA: Stanford University, 1976. ( ERIC Document Reproduction Service No. ED 142 408 )Google Scholar
  24. Bird, B.A. Effect of Systematic Drill System on Computational Ability of Primary Children (Doctoral Dissertation, Brigham Young University, 1977.) Dissertation Abstracts International, 1978, 34, 1317A. (University Microfilms No. 78–16188)Google Scholar
  25. Bright, G.W.; Harvey, J.G.; and Wheeler, M.M. Using Games to Retrain Skills with Basic Multiplicatin Facts. Journal for Research in Mathematics Education, 9 (2), 103–110.Google Scholar
  26. Butcher, J.E. Comparison of the Effects of Distributed and Massed Problem Assignments on the Homework of Ninth-Grade Algebra Students. (Doctoral Dissertation, Rutgers University, I 975.) Dissertation Abstracts International, 1976, 36 6586A–6588A. (University Microfilms No. 76–868Google Scholar
  27. Cranford, H.R. A Study of the Effects of Computer-Assisted Instruction in Mathematics on the Achievement and Attitude of Pupils in Grades Five and Six in a Rural Setting. (Doctoral Dissertation, University of Southern Mississippi, 1976.) Dissertation Abstracts International, 1977, 37A, 5660A. (University Microfilms No. 77–5932)Google Scholar
  28. Cummins, J.K. The Arithmetic Achievement of Sixth Grade Pupils and the Effect of Short Term, Well Designed Practice on Their Computational Abilities. (Doctoral Dissertation, University of California, Los Angeles, 1974.) Dissertation Abstracts International, 1975, 35, 403IB-40328. (University Microfilms No. 75–222Google Scholar
  29. Davidson, T.E. The Effects of Drill on Addition, Subtraction Learning with Implication of Piagetian Reversibility. (Doctoral Dissertation, Utah State University, 1975.) Dissertation Abstracts International, 1975, 36, 102A, (University Microfilms No. 75–14427)Google Scholar
  30. Dienes, A.B. The Time Factor in Computer-Assisted Instruction. (Doctoral Dissertation, University of Toronto, 1972.) Dissertation Abstracts International, 1974, 34 498IA. (National Library of Canada at Ottawa)Google Scholar
  31. Hohlfeld, J.F. Effectiveness of an Immediate Feedback Device for Learning Basic Multiplication Facts (Doctoral Dissertation, Indiana University, 1973.) Dissertation Abstracts International, 1974, 34, 4563A. (University Microfilms No. 74–2670)Google Scholar
  32. Horwitz, S. Effects of Amount of Immediate and of Delayed Practice on Retention of Mathematical Rules. Tallahassee, Florida: Florida State University, 1975. (ERIC Document Reproduction Service No. ED 120010)Google Scholar
  33. Howell, K.W. Using Peers in Drill-Type Instruction. Journal of Experimental Education, 46 (3), 52–56.Google Scholar
  34. Katz, W.H. Effects of Item Placement in Exercise Sets on Achievement in Elementary Algebra. (Doctoral Dissertation, The University of Connecticut, 1974.) Dissertation Abstracts International, 1974, 34, 369IA–3692A. (University Microfilms No. 749205)Google Scholar
  35. Malone, T.W. and Others. Projecting Student Trajectories in a Computer-Assisted Instruction Curriculum. Journal of Educational Psychology, 979, 71 (1), 75–84.Google Scholar
  36. McClung, C.J. The Effects of Cognitive Style on Type of Practice. (Doctoral Dissertation, University of Southern California, 1976.) Dissertation Abstracts International, 1977, 37, 5706A.Google Scholar
  37. Palmer, H. Three Evaluation Reports of Computer-Assisted Instruction in Drill-and-Practice Mathematics. Los Angeles, California: Los Angeles County Schools, 1973. ( ERIC Document Reproduction Service No. ED 087 422 )Google Scholar
  38. Parrish, D.C. An Investigation of the Effects of Required Drill Homework Versus No Homework on Attitudes Toward and Achievement in Mathematics. (Doctoral Dissertation, University of Houston, 1976.) Dissertation Abstracts International, 1976,.37 2040A. (University Microfilms No. 76–23369)Google Scholar
  39. Pence, B. and Begle, E.G. Effects of Varying the Number of Examples and Practice Problems. SMESG Working Paper No. 7. Palo Alto, California: Stanford University, 1974. ( ERIC Document Reproduction Service No. ED 142 405 )Google Scholar
  40. Starr, R.J. Modern Math Plus Computational Drills: Affective and Cognitive Results. School Science and Mathematics, 1977, 77, 601–604.CrossRefGoogle Scholar
  41. Suppes, P. and Others. Evaluation of Computer-Assisted Instruction in Elementary Mathematics for Hearing-Imparied Students. Palo Alto, California: Stanford University, 1973. ( ERIC Document Reproduction Service No. ED 084 722 )Google Scholar
  42. Suydam, M. and Dessart, D.J. Skill Learning. In R.J. Shumway (Ed.) Research in Mathematics Education. Reston, VA: National Council of Teachers of Mathematics, 1980.Google Scholar
  43. Taylor, S.C. The Effects of Mastery, Adaptive Mastery, and Non-Mastery Models on the Learning of a Mathematical Task. Tallahassee, Florida: Control Data Corporation, 1975. (ERIC Document Reproduction Service No. ED106 145)Google Scholar
  44. Weaver, J.R. The Relative Effects of Massed Versus Distributed Practice Upon the Learning and Retention of Eighth Grade Mathematics. (Doctoral Dissertation, The University of Oklahoma, 1976.) Dissertation Abstracts International, 1976, 37, 2698A. (University Microfilms No. 76–24394)Google Scholar
  45. Begle, E.G. Critical Variables in Mathematics Education. Washington: Mathematics Association of American and National Council of Teachers of Mathematics, 1979.Google Scholar
  46. Carry, L.R. Interaction of Visualization and General Reasoning Abilities with Instructional Treatment in Algebra. Doctoral Dissertaion, Stanford University, 1967.Google Scholar
  47. Cronbach, L.J. and Snow, R.E. Individual Differences in Learning Ability as a Function of Instructional Variables. Contract No. OCE 4–6–061269–1217 USOE, Stanford Unviersity, 1969.Google Scholar
  48. Cronbach, L.J. and Snow, R.E. Aptitudes and Instructional Methods. New York: Irvington, 1977.Google Scholar
  49. DuRapau, V.J. Interaction of General Reasoning Ability and Gestalt and Analytic Strategies of Processing Spatial Tasks with Transformational and Non-Transformational Treatments in Secondary School Geometry. Doctoral Dissertation, The University of Texas at Austin, 1979.Google Scholar
  50. Eastman, P.M. and Carry, L.R. Interaction of Spatial Visualization and General Reasoning Abilities with Instructional Treatment in Quadratic Inequalities: A Further Investigation. Journal for Reserach in Mathematics Education, 1975, 6 142–149.CrossRefGoogle Scholar
  51. Hickey, P.A. A Long Range Test of the Aptitude Treatment Interaction Hypothesis in College Level Mathematics. Doctoral Dissertation, The University of Texas at Austin, 1980.Google Scholar
  52. Larkin J. et al. Expert and Novice Performance in Solving Physics Problems. Science, 1980, 208, 1335–1342.CrossRefGoogle Scholar
  53. McLeod, D.B. and Briggs, J.T. Interactions of Field Independence and General Reasoning with Inductive Instruction in Mathemaatics. Journal for Research in Mathematics Education, 1980, 11, 94–103.CrossRefGoogle Scholar
  54. Romberg, T.A. and Wilson, J.W. The Development of Tests. NLSMA Report No. 7 (Wilson, Cahen, Begle, Ec—riT Stanford: School Mathematics Study Group, 1969.Google Scholar
  55. Salhab, M.T. The Interaction Between Selected Cognitive Abilities and Instructional Treatments on Absolute Value Equations. Doctoral Dissertation, The University of Texas at Austin, 1973.Google Scholar
  56. Skemp, R.R. Relational Understanding and Instrumnental Understanding. Arithmetic Teacher, 1980, 26, 9–15.Google Scholar
  57. Webb, L.F. and Carry, L.R. Interaction of Spatial Visualization and General Reasoning Abilities with Instructinal Treatment: A Follow-Up Study. Journal for Research in Mathematics Education, 1975, 6 132–140.CrossRefGoogle Scholar
  58. Begle, E.G. Critical Variables in Mathematics Education. Washington, D.C.: Mathematical Association of America and National Council of Teachers of Mathematics, 1979.Google Scholar
  59. Branca, N.A. Kilpatrick, J. The consistency of strategies in the learning of mathematical structures. Journal of Research in Mathematics Education, 1972, 3, 132–40.CrossRefGoogle Scholar
  60. Brown, J.S., Collins, A., Harris, G. Artificial intelligence and learning strategies. In H.F. O’Neil (Ed.), Learning strategies. New York: Academic Press, 1978.Google Scholar
  61. Days, H.C. Classifying algebra problems according to the complexity of their mathematical representation. Inn G.A. Goldin C.E. McClintock (Eds.), Task variables in mathematical problem solving. Columbus, Ohio: ERIC, 1980.Google Scholar
  62. Dienes, Z.P., Jeeves, J.A. Thinking in structures. London: Hutchinson Educational, 1965.Google Scholar
  63. Dienes, Z.P., Jeeves, M.A. The effects of structural relations on transfer. London: Hutchinson Educational, I970.Google Scholar
  64. Goldin, G.A. Structure variables in problem-solving. In G.A. Goldin C.E. McClintock (Eds.), Task variables in mathematical problem solving. Columbus, Ohio: ERIC, 1980.Google Scholar
  65. Kantowski, M.G. The use of heuristics in problem-solving: An exploratory study. (NSF Technical Report SED 77 18543 ). Gainesville, Florida: University of Florida, 1979.Google Scholar
  66. Klahr, D. Goal formation, planning, and learning by pre-school problem solvers or “My socks are in the dryer”. In R.S. Siegler (Ed.), Children’s thinking: What develops? Hillsdale, N.J.: Lawrence hrlbaum, 1978.Google Scholar
  67. Kruteskii, V.A. The psychology of mathematical abilities in school children, J. Kilpatrick I. Wirzup (Eds.). Chicago: University of Chicago Press, 1976.Google Scholar
  68. Kuhm, G. The classification of problem-solving research variables. In G.A. Goldin C.E. McClintock (Eds.), Task variables in mathematical problem solving. Columbus, Ohio: ERIC 1980.Google Scholar
  69. Landa, L.N. The ability to think how cm it be taught? Soviet Education, 1976, 5 4–66.Google Scholar
  70. Lester, F.K. Issues in mathematical problem-solving research. A paper presented at a Research Pression of the Annual Meeting of the National Council of Teachers of Mathematics, Seattle, WA, April 1980.Google Scholar
  71. Luger, G.F. Applications of problem structure. In G.A. Goldin C.E. McClintock (Eds.), Task variables in mathematical problem solving. Columbus, Ohio: ERIC, 1980.Google Scholar
  72. Malin, J.T. Strategies in mathematical problem solving. Journal of Educational Research, 1979, 73, 10I - 08.Google Scholar
  73. McClintock, C.E. Heuristic processes as task variables. In G.A. Goldin C.E. McClintock (Eds.), Task variables in mathematical problem solving. Columbus, Ohio: ERIC, 1980.Google Scholar
  74. Pereira-Mendoza, L. Heuristic strategies utilized by high school students. The Alberta Journal of Educational Research, 1979, 25, 213–20.Google Scholar
  75. Polya, G. How to solve it. ( 2nd ed. ). Garden City, N.Y.: Doubleday, 1957.Google Scholar
  76. Rigney, J.W. Learning strategies: A theoretical perspective. In H.F. O’Neil (Ed.), Learning strategies. New York: Academic Press, 1978.Google Scholar
  77. Schoenfeld, A.H. Heuristic behavior variables in instruction. In G.A. Goldin C.E. McClintock (Eds.), Task variables in mathematical problem solving. Columbus, Ohio: ERIC, 1980.Google Scholar
  78. Snow, R.E. Aptitude processes. In Snow, Federico, Montague (Eds.), Aptitude, learning instruction: V I Cognitive process analysis of aptitude. Hillsdale, N.J.: Lawrence Erlbaum, 1980.Google Scholar
  79. Waters, W. Concept acquisition tasks. In G.A. Goldin C.E. McClintock (Eds.), Task variables in mathematical problem solving. Columbus, Ohio: ERIC, 1980.Google Scholar
  80. Webb, N. A review of the literature related to problem-solving tasks and problem-solving strategies used by students in grades four, five and six. (Technical Report). Bloomington, Indiana: Mathematics Education Development Center, 1977.Google Scholar
  81. Wittman, E. Matrix strategies in heuristics. International Journal of Mathematical Education in Science and Technology. 1975, 6, 187–98.CrossRefGoogle Scholar
  82. Bobrow, D.G. Natural Language Input for a Computer Problem-Solving System. In M. Minsky (Ed.), Semantic information processing. Cambridge, Mass.: MIT Press, 1968.Google Scholar
  83. Brown, J.S. Burton, R.R. Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 1978, 2, 155–192.CrossRefGoogle Scholar
  84. Bundy, A. Analyzing mathematical proofs. Edinburgh: University of Edinburgh, Department of Artificial Intelligence, Research Report No. 2, 1975.Google Scholar
  85. Case, R. Implications of developmental psychology for the design of effective instruction. In A.M. Lesgold, J.W. Pellegrino, S.D. Fokkema R. Glaser (Eds.) Cognitive Psychology and Instruction. New York: Plenum, 1978.Google Scholar
  86. Carry, L.R., Lewis, C. Bernard, J.E. Psychology and equation solving: An information processing study. Austin: University of Texas, Department of Curriculum and Instruction, NSF Final Report 7822293, 1980.Google Scholar
  87. Davis, R.B. McKnight, C.C. Modeling and processes of mathematical thinking. The Journal of Children’s Mathematical Behavior, 1979, 2 91-I 13.Google Scholar
  88. Greeno, J.G. Trends in the theory of knowledge for problem solving. In D.T. Tuma F. Reif (Eds.), Problem solving and education: Issues in teaching and research. Hillsdale, N.J.: Erlbaum, 1980.Google Scholar
  89. Groen, G.J. Parkman, J.M. A chronometric analysis of simple addition. Psychological Review, 1972, 79, 329–343.CrossRefGoogle Scholar
  90. Hayes, J.R. Simon, H.A. Understanding written instructions. In L.W. Gregg (Ed.), Knowledge and cognition. Hillsdale, N.J.: Erlbaum, 1974.Google Scholar
  91. Hayes, J.R., Waterman, D.A. Robinson, C.S. Identifying relevant aspects of a problem text. Cognitive Science, 1977, 297–313.Google Scholar
  92. Heiler, J. Greeno, J.G. Semantic processing in arithmetic word problem solving. Paper presented at the Midwestern Psychological Association, 1978.Google Scholar
  93. Hinsley, D., Hayes, J.R. Simon, H.A. From words in equations. In P. Carpenter M. Just (Eds.), Cognitive processes in comprehension. Hillsdale, N.J.: Erlbaum, 1977.Google Scholar
  94. Hunt, E. Mechanics of verbal ability. Psychological Review, 1978, 85, 109–130.CrossRefGoogle Scholar
  95. Hunt, E., Lunneborg, C. Lewis, J. What does it mean to be high verbal? Cognitive Psychology, 1975, 2, 194–277.CrossRefGoogle Scholar
  96. Johnson, J., Ryan, K., Cook, L. Mayer, R.E. From words to equations: An instructional study. Santa Barbara: Department of Psychology, Series on Learning and Cognition, Report No. 80–4, 1980.Google Scholar
  97. Klatzky, R. Human memory: Second Edition. San Francisco: Freeman, 1980.Google Scholar
  98. Larkin, J.H. Information processing models and science instruction. In J. Lochhead J. Clement (Eds.), Cognitive process instruction. Philadelphia: Franklin Institute Press, 1979.Google Scholar
  99. Larkin, J.H. Teaching problem solving in physics: The psychological laboratory and the practical classroom. In D.T. Tuma and F. Reif (Eds.), Problem solving and education: Issues in teaching and research. Hillsdale, N.J.: Erlbaum, 1980.Google Scholar
  100. Lochhead, J. Clement, J. Cognitive process instruction. Philadelphia: Franklin Institute Press, 1979.Google Scholar
  101. Loftus, G.R. Loftus, E.F. Human memory: The processing of information. Hillsdale, N.J.: Erlbaum, 1976.Google Scholar
  102. Loftus, E.F. Suppes, P. Structural viables that determine problem-solving difficulty in computer-assisted instruction. Journal of Educational Psychology, 1972, 63, 53I - 542.CrossRefGoogle Scholar
  103. Luchins, A.S. Mechanization in problem solving. Psychological Monographs, 1942, 54:6, Whole No. 248.Google Scholar
  104. Matz, M. Towards a process model for high school algebra errors. Paper presented at Conference on Cognitive Processes in Algebra, University of Pittsburgh, 1979.Google Scholar
  105. Mayer, R.E. Information processing variables in learning to solve problems. Review of Educational Research, 1975, 45, 525–541.Google Scholar
  106. Mayer, R.E. Effects of meaningfulness on the representation of knowledge and the process of inference for mathematical problem solving. In R. Revlin R.E. Mayer (Eds.), Human Reasoning, Washington: Winston-Wiley, 1978.Google Scholar
  107. Mayer, R.E. Schemas for algebra story problems. Santa Barbara: Department of Psychology, Series in Learning Cognition, Report No. 80–3, 1980.Google Scholar
  108. Mayer, R.E. Bayman, P. Analysis of students’ intuitions about the operation of electronic calculators. Santa Barbara: Department of Psychology, Series in Learning Cogntion, Report No. 80–4, 1980.Google Scholar
  109. Mayer, R.E. Bromage, B. Recall of algebra story problems. Santa Barbara: Department of Psychology, Series in Learning and Cognition. Report No. 80–5, 1980.Google Scholar
  110. Mayer, R.E. Greeno, J.G. Structural differences between learning outcomes produced by different instructional methods. Journal of Educational Psychology, 1972, 63, 165–173.CrossRefGoogle Scholar
  111. Mayer, R.E. Greeno, J.G. Effects of meaningfulness and organization on problem solving and computability judgments. Memory Cognition, 1975, 3 356–362.CrossRefGoogle Scholar
  112. Mayer, R.E. Larkin, J.H. Kadane, J. Analysis of the skill of solving equations. Santa Barbara: Department of Psychology, Series in Learning Cognition, Report No. 80–2, 1980.Google Scholar
  113. Newell, A. Simon, H.A. Human problem solving. Englewood Cliffs, N.J.: Prentice-Hall, 1972.Google Scholar
  114. Paige, J.M. Simon, H.A. Cognitive process in solving algebra word problems. In B. Kleinmuntz (ed.), Problem solving: Research, method and theory. New York: Wiley, 1966.Google Scholar
  115. Polya, G. Mathematical discovery. New York: Wiley, 1968.Google Scholar
  116. Resnick, L.B. Task analysis in instructional design: Some cases from mathematics. In D. Klahr (Ed.), Cognition and instruction. Hillsdale, N.J.: Erlbaum, 1976.Google Scholar
  117. Riley, M.S. Greeno, J.G. Importance of semantic structure in the difficulty of arithmetic word problems. Paper presented at the Midwestern Psychological Association, 1978.Google Scholar
  118. Robinson, C.S. Hayes, J.R. Making inferences about relevance in understanding problems. In R. Revlin R.E. Mayer (Eds.), Human reasoning. Washington: Winston/Wiley, 1978.Google Scholar
  119. Simon, H.A. Problem solving and education. In D.T. Tuma F. Reif (Eds.),Problem solving and education: Issues in teaching and research. Hillsdale, N.J.: Erlbaum, 1980.Google Scholar
  120. Simon, D.P. Simon, H.A. Individual differences in solving physics problems. In R. Siegler (Ed.), Cihldren’s thinking: What develops? Hillsdale, N.J.: Erlbaum, 1978.Google Scholar
  121. Wickelgren, W. How to solve problems. San Francisco: Freeman, 1974.MATHGoogle Scholar
  122. Bobrow, D.G. Norman, D.A. Some principles of memory schemata. In D.G. Bobrow A Collins (Eds.), Representation and understanding. New York: Academic Press, 1975.Google Scholar
  123. Brown, A.L. Knowing when, where, and how to remember: A problem of metacognition. In R. Glaser (Ed.), Advances in instructional psychology (Vol. I ). Hillsdale, N.J.: Erlbaum, 1978.Google Scholar
  124. Brown, J.S. Burton, R.R. Diagnostic models for procedural bugs in mathematical skills. Cognitive Science, 1978, 2 155–192.CrossRefGoogle Scholar
  125. Carroll, J.B. Psychometric tests as cognitive tasks: “A new structure of intellect.” In L.B. Resnick (Ed.), The nature of intelligence. Hillsdale, N.J.: Erlbaum, 1976.Google Scholar
  126. Case, R. Gearing the demands of instructions to the developmental capacities of the learner. Review of Educational Research, 1975, 45, 59–87.Google Scholar
  127. Case, R. Intellectual development from birth to adulthood: A new-Piagetian interpretation. In R.S. Siegler (Ed.), Children’s thinking: What develops? Hillsdale, N.J.: Erlbaum, 1978.Google Scholar
  128. Davis, R.B., Jockusch, E., McNight, C. Cognitive processes in learning algebra. The Journal of Children’s Mathematical Behavior, 1978, 2 (1), 10320.Google Scholar
  129. Ericsson, K.A., Simon, H.A. Verbal reports as data. Psychological Review, 1980, 87 (3), 215–251.CrossRefGoogle Scholar
  130. Flavell, J.H. Metacognitive aspects of problem solving. In L.B. Resnick (Ed.), The nature of intelligence. Hillsdale, N.J.: Erlbaum, 1976.Google Scholar
  131. Flavell, J.H. Wellman, H.M. Metamemory. In R.V. Kail J.W. Hagen (Eds.), Perspectives on the development of memory and cognition. Hillsdale, N.J.: Erlbaum, 1977.Google Scholar
  132. Gagne, R.M. White, R.T. Memory structures and learning outcomes. Review of Educational Research, 1978, 48 (2), 187–222.Google Scholar
  133. Goldin, G.A. McClintock, C.E. Task variables in mathematical problem solving. Columbus, OH: ERIC/SMEAC, 1979.Google Scholar
  134. Greeno, J.G. Trends in the theory of knowledge for problem solving. In D.T. TUma F. Reif (Eds.), Problem solving and education. Hillsdale, N.J.: Erlbaum, 1980.Google Scholar
  135. Gurova, L.L. Schoolchildren’s awareness of their own mental operations in solving arithmetic problems (Originally published in 1959). In J. Kilpatrick I. Wirszup (Eds.) Soviet studies in the psychology of learning and teaching mathematics (Vol. 3 ). Stanford: SMSG, 1969.Google Scholar
  136. Hadamard, J. The psychology of invention in the mathematical field. Princeton, N.J.: Princeton University Press, 1945.Google Scholar
  137. Hinsley, D.A., Hayes, J.R., Simon, H.A. From words to equations - meaning and representations in algebra word problems. In M. Just P. Carpenter (Eds.), Cognitive processes in comprehension. Hillsdale, N.J.: Erlbaum, 1977.Google Scholar
  138. Hunt, E., Lunneborg, C., Lewis, J. What does it mean to be high verbal? Cognitive Psychology, 1975, 7 197–227.CrossRefGoogle Scholar
  139. Kalmykova, Z.I. Psychological analysis of the formation of a concept of a problem type (Originally published in 1947.). In J. Kilpatrick I Wirszup (Eds.). Soviet studies in the psychology of learning and teaching mathematics (Vol. 6 ). Stanford: SMSG, 1969.Google Scholar
  140. Krutetskii, V.A. The psychology of mathematical abilities in schoolchildren, J. Kilpatrick and I. Wirszup (Eds.). Chicago: University of Chicago Press, 1976.Google Scholar
  141. Larkin, J.I. Skilled problem solving in physics: A hierarchical planning model. Unpublished manuscript, University of California at Berkeley, September 1977.Google Scholar
  142. Paivio, A. Imagery and verbal processes. New York: Holt, Rinehart and Winston, 1971.Google Scholar
  143. Piaget, J. Inhelder, B. Memory and Intelligence. New York: Basic Books, 1973.Google Scholar
  144. Poincare, H. Mathematical creation. In The Foundations of Science. (Translated by G.H. Halstead.) New York: Science Press, 1913.Google Scholar
  145. Polya, G. How to solve it ( 2nd ed. ). Garden City, N.Y.: Doubleday, 1957.Google Scholar
  146. Potts, G.R. Information processing strategies used in the encoding of linear orderings. Journal of Verbal Learning and Verbal Behavior, 1972 I I, 727–740.Google Scholar
  147. Rumelhart, D.E. Ortony, A. The representation of knowledge in memory. In R.C. Anderson, R.J. Spiro, W.E. Montague (Eds.), Schooling and the acquisition of knowledge. Hillsdale, N.J.: Erlbaum, 1977.Google Scholar
  148. Shavelson, R.J. Porton, V.M. An information processing approach to research on mathematics learning and problem solving. Paper presented at the Conference on Modeling Mathematical Cognitive Development, Athens, GA., May, 1979.Google Scholar
  149. Silver, E.A. Recall of mathematical problem information: Solving related problems. Journal for Research in Mathematics Education, in press.Google Scholar
  150. Silver, E.A. Student perceptions of relatedness among mathematical verbal problems. Journal for Research in Mathematics Education, 1979, 10, 195–210.CrossRefGoogle Scholar
  151. Simon, H.A. Problem solving and education. In D.T. Tuma F. Reif (Eds.), Problem solving and education. Hillsdale, N.J.: Erlbaum, 1980.Google Scholar
  152. Simon, H.A. Gilmartin, K. A simulation of memory for chess positions. Cognitive Psychology, 1973, 5 29–46.CrossRefGoogle Scholar
  153. Simon, D.P. Simon, H.A. Individual differences in solving physics problems. In R. Siegler (Ed.), Children’s thinking: What develops? Hillsdale, N.J.: Erlbaum, 1978.Google Scholar
  154. Sternberg, R.J. Intelligence, information processing, and analogical reasoning: The componential analysis of human abilities. Hillsdale, N.J.: hrlbaum, 1976.Google Scholar
  155. Beth E., Piaget, J. Mathematical Epistemology and Psychology. Dordrecht, Holland: D. Riedel, 1966.Google Scholar
  156. Bruner, J.A. The Process of Education. Cambridge, Massachusetts: Harvard University Press, 1960.Google Scholar
  157. Lesh, R.A. An interpretation of advanced organizers. Journal for Research in Mathematics Education, 1976, 7 69–74.CrossRefGoogle Scholar
  158. Lesh, R. Mathematical learning disabilities: Considerations for identification, diagnosis, and remediation. In R. Lesh, D. Mierkiewicz, M.C. Kantowski (Eds.), Applied Mathematical Problem Solving. Columbus, Ohio: ERIC/Clearinghouse for Science, Mathematics, and Environmental Education, 1979, Lesh, R. Applied mathematical problem solving. To appear in Educational Studies in Mathematics.Google Scholar
  159. Lesh, R., Landau, M., Hamilton, E. Rational number ideas and the role of representational systems. To appear in Proceedings of the Fourth International Conference for the Psychology of Mathematics Education, August, 1980.Google Scholar
  160. I. Bauersfeld, H.: Hidden dimensions in the so-called reality of a mathematics classroom. Educational Studies in Mathematics 11, (1980), pp. 23–41.Google Scholar
  161. 2.
    Begle, E.G.: Critical Variables in Mathematics Education. Mathematical Association of American and National Council of Teachers of Mathematics, Washington, 1979.Google Scholar
  162. 3.
    Bussmann, H., Heymann, H.-W., Lorenz, J.-H., Reiss, V., Scholz, R.W. and Seeger, F.: “Begle, E.G.: Critical Variables in Mathematics Education”. Zentralblatt fur Didaktik der Mathematik, Jhrg, 12, Heft I, 1980.Google Scholar
  163. 4.
    Dienes, Z.P.: Learning mathematics, In Wain, G.T. (Ed.): Mathematical Education, Van Nostrand Reinhold Col, Wokingham, 1978.Google Scholar
  164. 5.
    Freudenthal, H.: Address to the first conference of I.G.P.M.E., at Utrecht 29 August 1977. Educational Studies in Mathematics 9 (1978), pp. 1–5.CrossRefGoogle Scholar
  165. 6.
    Freudenthal, H. Weeding and sowing. Preface to A Science of Mathematical Education, D. Reidel, Dordrecht, 1978.Google Scholar
  166. 7.
    Freudenthal, H.: New math or new education. Prospects, Vol. IX, No. 3, (1979), pp. 321–331.CrossRefGoogle Scholar
  167. 8.
    Jackson, P.W.: Life in Classrooms. Holt, Rinehart and Winston, New York, 1968.Google Scholar
  168. 9.
    NCTM: An Agenda for Action. National Council of Teachers of Mathematics, Reston, 1980.Google Scholar
  169. Australian Council for Educational Research. Background in Mathematics: A Guidebook to Elementary Mathematics for Teachers ih Primary Schools. Melbourne: Department of Education, Victoria, 1966.Google Scholar
  170. Begle, E.G. Critical Variables in Mathematics Education. Washington, D.C.: Mathematical Association of America, 1978.Google Scholar
  171. Blakers, A.L. Change in Mathematics Education since the late 1950’s - Ideas and Realisation. Australia. Educational Studies in Mathematics, 1978, 9 147158.Google Scholar
  172. Bloom, B.S. Alterable variables: the new direction in educational research. Edinburgh: SCRE., 1979.Google Scholar
  173. Carroll, J.B. A Model of School Learning. Teachers College Record 1963, 64, 723–733.Google Scholar
  174. Glass, G.V. and Smith, M.L. Meta-analysis of Research on the Relationship of Class size and Achievement. San Francisco: Far West Laboratory for Educational Research and Development, 1978.Google Scholar
  175. Husen, T. (ed.) International Study of Achievement in Mathematics (2 vols.) New York: Wiley, and Stockholm: Almqvist and Wiksell, 1967.Google Scholar
  176. Keeves, J.P. Educational Environment and Student Achievement. Stockholm: Almqvist and Wiksell, and Melbourne: ACER, 1972.Google Scholar
  177. Keeves, J.P. The Performance Cycle: Motivation and Attention as Mediating Variables in School Performance. Melbourne: ACER, 1974.Google Scholar
  178. Keeves, J.P. Curricular Factors Influencing School Learning: Time and Opportunity to Learn. Melbourne: SCER, 1976.Google Scholar
  179. McGaw, B., Keeves, J., Sorbom, D. and Cumming, Joy. The Mediated Influence of Prior Performance on Subsequent Performace: An Analysis of Linear Structural Relationships. (Paper presented at Australian Association for Research in Education Conference, Melbourne 1979 ).Google Scholar
  180. Rosier, M.J. Changes in Secondary School Mathematics in Australia: 1964 to 1978. Hawthorn, Victoria: ACER (in press).Google Scholar
  181. Ausubel, D.P. Educational psychology: A cognitive view. New York: Holt, Rinehart, and Winston, Inc., 1968.Google Scholar
  182. Berliner, D.C. Allocated time, engaged time, and academic learning time in elementary school mathematics instruction. Paper presented at the 56th Annual Meeting of the National Council of Teacher of Mathematics, San Diego, April 1978.Google Scholar
  183. Brophy, J.E. Evertson, C.M. Learning from teaching: A developmental perspective. Boston: Allyn Bacon, 1976.Google Scholar
  184. Bush, A.J., Kennedy, J.J., Cruickshank, D.R. An empirical investigation.of teacher clarity. Journal of Teacher Education. March-April 1977, 28 (2), 5358.Google Scholar
  185. Cooney, T.J., Davis, E.J., Henderson, K.B. Dynamics of teachin. secondary school mathemtics. Boston: Houghton-Mifflin, 1975.Google Scholar
  186. Cruickshank, D., Kennedy, J., Myers, B., Bush, A. Teacher clarity What is it? Paper presented at the Conference on Innovative Practices in Teacher Education, Atlanta, January 1976.Google Scholar
  187. Evertson, C.M., Emmer, E.T. Brophy, J.E. Predictors of effective teaching in junior high mathematics classrooms. Journal of Research in Mathematics Education, May 1980, 11 (3), 167–178.CrossRefGoogle Scholar
  188. Good, T. Grouws, D. Teaching effects: A process-product study in fourth-grade mathematics classrooms. Journal of Teacher Education, 1977, 28, 49–54.CrossRefGoogle Scholar
  189. Heath, R.W., Nielson, M.A. The research basis for performance-based teacher education. Review of Educational Research. Fall 1974, 44 (4), 463–484.Google Scholar
  190. Kolb, J.R. A predictive model for teaching strate.ies research. Part I: Derivation of the model. Athens: The Georgia Center for the Study of Learning and Teaching Mathematics, 1977.Google Scholar
  191. Kounin, J.S. Discipline and group management in classrooms. New York: Holt, Rinehart Winston, 1970.Google Scholar
  192. Rosenshine, B., Furst, N. Research in teacher performance criteria in B.O. Smith (Ed.), Symposium on research in teacher education. Englewood Cliffs, N.J.: Prentice-Hall, 1971.Google Scholar
  193. Stiff, L.V. C. (Doctoral dissertation, North Carolina State University, 1978 ). Dissertation Abstracts International. 1978, 39, 2803-A.Google Scholar
  194. Thornton, C.D. An evaluation of the mathematics-methods program involving the study of teaching characteristics and pupil achievement in mathematics. Journal for Research in Mathematics Education. January 1977, 8 (I), 17–25.CrossRefGoogle Scholar
  195. Tikunoff, W.J., Berliner, D.C. Rist, R.C. An ethnographic study of the forty classrooms of the beginning teacher evaluation study known sample (Tech. Rep. 75–5). San Francisco: Far West Laboratory, 1975.Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1983

Authors and Affiliations

  • Richard E. Snow
    • 1
  • Herbert J. Walberg
    • 2
  • Christine Keitel-Kreidt
    • 3
  • Donald J. Dessart
    • 4
  • L. Ray Carry
    • 5
  • Jens Holger Lorenz
    • 6
  • Nicholas A. Branca
    • 7
  • Richard E. Mayer
    • 8
  • Edward A. Silver
    • 7
  • Robert B. Davis
    • 9
  • Gunnar Gjone
    • 10
  • John P. Keeves
    • 11
  • Thomas J. Cooney
    • 12
  1. 1.Stanford UniversityStanfordUSA
  2. 2.University of IllinoisChicagoUSA
  3. 3.Institut fur Didaktik der MathematikBielefeldFederal Republic of Germany
  4. 4.The University of TennessesKnoxvilleUSA
  5. 5.The University of Texas at AustinUSA
  6. 6.University of BielefeldBielefeldFederal Republic of Germany
  7. 7.San Diego State UniversitySan DiegoUSA
  8. 8.University of CaliforniaSanta BarbaraUSA
  9. 9.Laboratoire de Recherche PedagogiqueL’Universite d’Illinois a Champaign-UrbanaUSA
  10. 10.Pedagogisk SeminarOsloNorway
  11. 11.Australian Council for Educational ResearchHawthornAustralia
  12. 12.University of GeorgiaAthensUSA

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