Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-19T10:11:13.731Z Has data issue: false hasContentIssue false

8 - The Role of Domain Knowledge in Creative Problem Solving

Published online by Cambridge University Press:  19 January 2010

Richard E. Mayer
Affiliation:
Department of Psychology, University of California at Santa Barbara, Santa Barbara, California
James C. Kaufman
Affiliation:
California State University, San Bernardino
John Baer
Affiliation:
Rider University, New Jersey
Get access

Summary

Consider the word problems presented in Table 8.1. Some people are able produce solutions to these problems, whereas others make errors, get frustrated, and fail to generate a correct answer. What do successful mathematical problem solvers know that less successful mathematical problem solvers do not know? This seemingly straightforward question motivates this chapter.

A review of research on mathematical problem solving supports the conclusion that proficiency in solving mathematical problems depends on the domain knowledge of the problem solver (Kilpatrick, Swafford, & Findell, 2001). In this chapter, I examine the research evidence concerning five kinds of knowledge required for mathematical problem solving: (1) factual knowledge, (2) conceptual knowledge, (3) procedural knowledge, (4) strategic knowledge, and (5) metacognitive knowledge.

Table 8.2 provides definitions and examples of each of the five kinds of knowledge relevant to mathematical problem solving. Factual knowledge refers to knowledge of facts such as knowing that there are 100 cents in a dollar. Conceptual knowledge refers to knowledge of concepts such as knowing that a dollar is a monetary unit and knowledge of categories such as knowing that a given problem is based on the structure (total cost) = (unit cost) = (number of units). Strategic knowledge refers to knowledge of strategies such as knowing how to break a problem into parts. Procedural knowledge refers to knowledge of procedures such as knowing how to add two decimal numbers.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, J. R., & Schunn, C. D. (2000). Implications of ACT-R learning theory: No magic bullets. In Glaser, R. (Ed.), Advances in instructional psychology (Vol. 5, pp. 1–33). Mahwah, NJ: Erlbaum.Google Scholar
Atkinson, R. K., & Derry, S. J. (2000). Computer-based examples designed to encourage optimal example processing: A study examining the impact of sequentially presented, subgoal-oriented worked examples. In Fishman, B. & O'Connor-Divelbiss, S. F. (Eds.), Proceedings of the Fourth International Conference of Learning Sciences (pp. 132–133). Mahwah, NJ: Erlbaum.Google Scholar
Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. W. (2000). Learning from examples: Instructional principles from the worked examples research. Review of Educational Research, 70, 181–214.CrossRefGoogle Scholar
Atkinson, R. K., Renkl, A., & Merrill, M. M. (2003). Transitioning from studying examples to solving problems: Combining fading with prompting fosters learning. Journal of Educational Psychology, 95, 774–783.CrossRefGoogle Scholar
Carpenter, T. P., Lindquist, M. M., Mathews, W., & Silver, E. A. (1983). Results of the third NAEP mathematics assessment: Secondary school. Mathematics Teacher, 76, 652–959.Google Scholar
Catrambone, R. (1995). Aiding subgoal learning: Effects on transfer. Journal of Educational Psychology, 87, 5–17.CrossRefGoogle Scholar
Catrambone, R. (1996). Generalizing solution procedures learned from examples. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, 1020–1031.Google Scholar
Catrambone, R. (1998). The subgoal learning model: Creating better examples so that students can solve novel problems. Journal of Experimental Psychology: General, 127, 355–376.CrossRefGoogle Scholar
Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13, 145–182.CrossRefGoogle Scholar
Davidson, J. E. (1995). The suddenness of insight. In Sternberg, R. J. & Davidson, J. E. (Eds.), The nature of insight (pp. 125–156). Cambridge, MA: MIT Press.Google Scholar
Dow, G. T., & Mayer, R. E. (2004). Teaching students to solve insight problems: Evidence for domain specificity in creativity training. Creativity Research Journal, 16, 389–402.CrossRefGoogle Scholar
Duncker, K. (1945). On problem solving. Psychological Monographs, 58(5), Whole No. 270.CrossRefGoogle Scholar
Ericsson, K. A. (2003). The acquisition of expert performance as problem solving: Construction and modification of mediating mechanisms through deliberate practice. In Davidson, J. E. & Sternberg, R. J. (Eds.), The psychology of problem-solving (pp. 31–86). New York: Cambridge University Press.CrossRefGoogle Scholar
Hegarty, M., Mayer, R. E., & Monk, C. A. (1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of Educational Psychology, 87, 18–32.CrossRefGoogle Scholar
Hinsley, D., Hayes, J. R., & Simon, H. A. (1977). From words to equations. In Carpenter, P. & Just, M. (Eds.), Cognitive processes in comprehension. Hillsdale, NJ: Erlbaum.Google Scholar
Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.Google Scholar
Lewis, A. B. (1989). Training students to represent arithmetic word problems. Journal of Educational Psychology, 81, 521–531.CrossRefGoogle Scholar
Low, R. (1989). Detection of missing and irrelevant information within algebraic story problems. British Journal of Educational Psychology, 59, 296–305.CrossRefGoogle Scholar
Low, R., & Over, R. (1990). Text editing of algebraic word problems. Australian Journal of Psychology, 42, 63–73.CrossRefGoogle Scholar
Luchins, A. S. (1942). Mechanization in problem-solving. Psychological Monographs, 54(6), Whole No. 248.CrossRefGoogle Scholar
Mayer, R. E. (1982). Memory for algebra story problems. Journal of Educational Psychology, 74, 199–216.CrossRefGoogle Scholar
Mayer, R. E. (1990). problem-solving. In Eysenck, M. W. (Ed.), The Blackwell dictionary of cognitive psychology. Oxford, UK: Basil Blackwell.Google Scholar
Mayer, R. E. (1992). Thinking, problem solving, cognition (2nd ed). New York: Freeman.Google Scholar
Mayer, R. E. (1995). The search for insight: Grappling with Gestalt psychology's unanswered questions. In Sternberg, R. J. & Davidson, J. E. (Eds.), The nature of insight (pp. 3–32). Cambridge, MA: MIT Press.Google Scholar
Mayer, R. E. (2003). Learning and instruction. Upper Saddle River, NJ: Merrill Prentice Hall.Google Scholar
Mayer, R. E. (in press). The role of knowledge in the development of mathematical reasoning. In R. Sternberg & Subotnik, R. (Eds.), The other 3 Rs: Reasoning, resilience, and responsibility. Greenwich, CT: Information Age.Google Scholar
Mayer, R. E. (2005). Cognitive theory of multimedia learning. In Mayer, R. E. (Ed.), Cambridge handbook of multimedia learning (pp. 31–48). New York: Cambridge University Press.CrossRefGoogle Scholar
Mayer, R. E., Sims, V., & Tajika, H. (1995). A comparison of how textbooks teach mathematical problem solving in Japan and the United States. American Educational Research Journal, 32, 443–460.Google Scholar
Paas, F., Renkl, A., & Sweller, J. (2003). Cognitive load theory and instructional design: Recent developments. Educational Psychologist, 38, 1–4.CrossRefGoogle Scholar
Quilici, J. H., & Mayer, R. E. (1996). Role of examples in how students learn to categorize statistics word problems. Journal of Educational Psychology, 88, 144–161.CrossRefGoogle Scholar
Quilici, J. H., & Mayer, R. E. (2002). Teaching students to recognize structural similarities between statistics word problems. Applied Cognitive Psychology, 16, 325–342.CrossRefGoogle Scholar
Reed, S. K. (1984). Estimating answers to algebra word problems. Journal of Experimental Psychology: Learning, Memory, and Cognition, 10, 778–790.Google Scholar
Reed, S. K. (1999). Word problems. Mahwah, NJ: Erlbaum.Google Scholar
Renkl, A. (1997). Learning from worked-out examples: Instructional explanations supplement self-explanations. Learning & Instruction, 12, 529–556.CrossRefGoogle Scholar
Renkl, A. (2005). The worked-out example principle in multimedia learning. In Mayer, R. E. (Ed.), Cambridge handbook of multimedia learning (pp. 229–246). New York: Cambridge University Press.CrossRefGoogle Scholar
Renkl, A., Stark, R., Gruber, H., & Mandl, H. (1998). Learning from worked-out examples: The effects of example variability and elicited self-explanations. Contemporary Educational Psychology, 23, 90–108.CrossRefGoogle ScholarPubMed
Riley, M., Greeno, J. G., & Heller J. I. (1982). The development of children's problem solving ability in arithmetic. In Ginsberg, H. (Ed.), The development of mathematical thinking. New York: Academic Press.Google Scholar
Schoenfeld, A. H. (1991). On mathematics and sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In Voss, J. F., Perkins, D. N., & Segal, J. W. (Eds.), Informal reasoning and education (pp. 311–343). Hillsdale, NJ: Erlbaum.Google Scholar
Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In Grouws, D. A. (Ed.), Handbook of research on mathematics and learning (pp. 334–370). New York: Macmillan.Google Scholar
Schunk, D. (1989). Self-efficacy and achievement behaviors. Educational Psychology Review, 1, 173–208.CrossRefGoogle Scholar
Schunk, D., & Hanson, A. R. (1985). Peer models: Influences on children's self-efficacy and achievement. Journal of Educational Psychology, 77, 313–322.CrossRefGoogle Scholar
Siegler, R. S., & Jenkins, E. (1989). How children develop new strategies. Hillsdale, NJ: Erlbaum.Google Scholar
Silver, E. A. (1981). Recall of mathematical problem information: Solving related problems. Journal of Research in Mathematics Education, 12, 54–64.CrossRefGoogle Scholar
Soloway, E., Lochhead, J., & Clement, J. (1982). Does computer programming enhance problem solving ability? Some positive evidence on algebra word problems. In Seidel, R. J., Anderson, R. E., & Hunter, B. (Eds.), Computer literacy. New York: Academic Press.Google Scholar
Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York: Free Press.Google Scholar
Sweller, J. (1999). Instructional design in technical areas. Camberwell, Australia: ACER Press.Google Scholar
Sweller, J. (2005). Implications of cognitive load theory for multimedia learning. In Mayer, R. E. (Ed.), Cambridge handbook of multimedia learning (pp. 19–30). New York: Cambridge University Press.CrossRefGoogle Scholar
Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2, 59–89.CrossRefGoogle Scholar
Verschaffel, L., Greer, B., & Corte, E. (2000). Making sense of word problems. Lisse, The Netherlands: Swets & Zeitlinger.Google Scholar
Wertheimer, M. (1959). Productive thinking. New York: Harper & Row.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×