Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T22:22:00.537Z Has data issue: false hasContentIssue false

8 - The Role of Domain Knowledge in Creative Problem Solving

from SECTION TWO - CREATIVITY AND REASON IN COGNITION AND NEUROSCIENCE

Published online by Cambridge University Press:  05 February 2016

By
Richard E. Mayer
Edited by
James C. Kaufman  and
John Baer
Richard E. Mayer
Affiliation:
University of California Santa Barbara
James C. Kaufman
Affiliation:
University of Connecticut
John Baer
Affiliation:
Rider University, New Jersey
Get access

Summary

Consider the word problems presented in Table 8.1. Some people are able to 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: factual knowledge, conceptual knowledge, procedural knowledge, strategic knowledge, and metacognitive knowledge (Mayer, 2008, 2011, 2013).

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 knowldege of procedures such as knowing how to add two decimal numbers. Metacognitive knowledge refers to knowledge of about how to assess and adjust one's thinking, including attitudes and beliefs concerning one's competence in solving mathematics word problems. In this chapter, after reviewing definitions of key terms, I examine research evidence concerning the role of each of these kinds of knowledge in supporting mathematical problem solving.

Definitions

An important first step is to define key terms such as problem, problem solving, and mathematical problem solving.

What Is a Problem?

In his classic monograph entitled “On-Problem Solving,” Duncker (1945, p. 1) eloquently wrote that a problem arises when a problem solver “has a goal but does not know how this goal is to be reached.” More recently, we have expressed this idea by saying “a problem occurs when a problem solver wants to transform a problem situation from the given state to the goal state, but lacks an obvious method for accomplishing the transformation” (Mayer & Wittrock, 2006, p. 288).

Type
Chapter
Information
Creativity and Reason in Cognitive Development , pp. 147 - 163
Publisher: Cambridge University Press
Print publication year: 2016

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: Lawrence 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.Google 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
Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems. Cognitive Science, 5, 121–152.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.Google Scholar
Ericsson, K. A. (2009). Enhancing the development of professional performance: Implications from the study of deliberate practice. In Ericsson, K. A. (Ed.), Development of professional expertise (pp. 405–431). New York: Cambridge University Press.CrossRefGoogle Scholar
Grobe, C. S., & Renkl, A. (2007). Finding and fixing errors in worked examples: Can this foster learning outcomes?Learning and Instruction, 17, 612–634.Google 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: Lawrence Erlbaum.Google Scholar
Kilpatrick, J., Swafford, J., & Findell, B., Eds. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies 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. (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. (2006). The role of knowledge in the development of mathematical reasoning. In Sternberg, R. & Subotnik, R. (Eds.), The other 3 Rs: Reasoning, resilience, and responsibility (pp. 145–156). Greenwich, CT: Information Age Publishing.Google Scholar
Mayer, R. E. (2008). Learning and instruction, 2nd ed. Upper Saddle River, NJ: Merrill Prentice Hall.Google ScholarPubMed
Mayer, R. E. (2011). Applying the science of learning. New York: Pearson.Google Scholar
Mayer, R. E. (2013). Problem solving. In Reisberg, D. (Ed.), Oxford handbook of cognitive psychology (pp. 769–778). New York: Oxford University Press.Google Scholar
Mayer, R. E. (2014). Cognitive theory of multimedia learning. In Mayer, R. E. (Ed.), Cambridge handbook of multimedia learning (2nd ed., pp. 43–71). 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
Mayer, R. E., & Wittrock, M. C. (2006). Problem solving. In Alexander, P., Winne, P., & Phye, G. (Eds.), Handbook of educational psychology (pp. 287–303). Mahwah. NJ: Lawrence Erlbaum.Google Scholar
National Research Council (2012). Education for life and work: Developing transferable knowledge and skills in the 21st century. Washington, DC: National Academies Press.
Paas, F., & Sweller, J. (2014). Implications of cognitive load theory for multimedia learning. In Mayer, R. E. (Ed.), Cambridge handbook of multimedia learning (2nd ed., pp. 27–42). New York: Cambridge University Press.Google 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., (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: Lawrence Erlbaum.Google Scholar
Renkl, A. (2011). Instruction based on examples. In Mayer, R. E. & Alexander, P. A. (Eds.), Handbook of research on learning and instruction (pp. 272–295). New York: Routledge.Google Scholar
Renkl, A. (2014). The worked-out example principle in multimedia learning. In Mayer, R. E. (Ed.), Cambridge handbook of multimedia learning (2nd ed., pp. 391–412). New York: Cambridge University Press.Google 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, (1982). The development of children's problem solving ability in arithmetic. In Ginsberg, H. (Ed.), The development of mathematical thinking (pp. 153–200). 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: Lawrence 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: Lawrence Erlbaum.Google Scholar
Silver, E. A. (1981). Recall of mathematical problem information: Solving related problems. Journal of Research in Mathematics Education, 12, 54–64.Google 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., Ayres, P., & Kalyuga, S. (2011). Cognitive load theory. New York: Springer.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., & De 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
×