The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra
Abstract
The knowledge required to solve algebra manipulation problems and procedures designed to hasten knowledge acquisition were studied in a series of five experiments. It was hypothesized that, as occurs in other domains, algebra problem-solving skill requires a large number of schemas and that schema acquisition is retarded by conventional problem-solving search techniques. Experiment 1, using Year 9, Year 11, and university mathematics students, found that the more experienced students had a better cognitive representation of algebraic equations than less experienced students as measured by their ability to (a) recall equations, and (b) distinguish between perceptually similar equations on the basis of solution mode. Experiments 2 through 5 studied the use of worked examples as a means of facilitating the acquisition of knowledge needed for effective problem solving. It was found that not only did worked examples, as expected, require considerably less time to process than conventional problems, but that subsequent problems similar to the initial ones also were solved more rapidly. Furthermore, decreased solution time was accompanied by a decrease in the number of mathematical errors. Both of these findings were specific to problems identical in structure to the initial ones. It was concluded that for novice problem solvers, general algebra rules are reflected in only a limited number of schemas. Abstraction of general rules from schemas may occur only with considerable practic…
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Students learn more efficiently and more robustly when more frequent study of worked examples is interleaved with problem solving practice as opposed to practice that is all problem solving.
“Students learn more efficiently and more robustly when more frequent study of worked examples is interleaved with problem solving practice as opposed to practice that is all problem solving.”
From The Knowledge‐Learning‐Instruction Framework: Bridging the Science‐Practice Chasm to Enhance Robust Student Learning (2012) · cites this paper as Sweller & Cooper (1985)
More complex variable-variable KCs imply instruction that includes comparison/blocking and more worked example study.
“More complex variable-variable KCs (e.g., designing a controlled experiment, see Table 2) imply instruction that includes comparison/blocking (Gick & Holyoak, 1983;Gentner, et al., 2009) and more worked example study (Sweller & Cooper, 1985).”
From The Knowledge‐Learning‐Instruction Framework: Bridging the Science‐Practice Chasm to Enhance Robust Student Learning (2012) · cites this paper as Sweller & Cooper, 1985
Students work through interleaved worked examples and problems more quickly and perform better on corresponding problems in a posttest.
“Students work through the interleaved worked examples and problems more quickly and perform better on corresponding problems in a posttest.”
From The Knowledge‐Learning‐Instruction Framework: Bridging the Science‐Practice Chasm to Enhance Robust Student Learning (2012) · cites this paper as Sweller and Cooper (1985)
Alternating worked examples with standard problems leads to more efficient and more effective learning than having students solve all the problems in the set.
“Sweller and Cooper (1985) demonstrated that alternating worked examples with standard problems leads to more efficient and more effective learning than having students to solve all the problems in the set”
From The Knowledge‐Learning‐Instruction Framework: Bridging the Science‐Practice Chasm to Enhance Robust Student Learning (2012) · cites this paper as Sweller and Cooper (1985)
Sweller and Cooper (1985) found that worked examples aided in reducing acquisition time and improved achievement on the test phase of the experiments.
“Throughout the experiments, Sweller and Cooper (1985) also found that worked examples aided in reducing the acquisition time and improved achievement on the test phase of the experiments.”
From Worked Example Mid-Semester Intervention in College Algebra (2022) · cites this paper as Sweller and Cooper (1985)
Worked examples reduce cognitive load on working memory and build better schema acquisition.
“One way to reduce the cognitive load on the working memory of a student and build better schema acquisition is through worked examples (Sweller & Owen, 1989;Sweller & Cooper, 1985).”
From Worked Example Mid-Semester Intervention in College Algebra (2022) · cites this paper as Sweller & Cooper, 1985
Sweller and Cooper (1985) concluded that worked examples assist students when faced with similar problems, but the advantage does not extend to dissimilar problems.
“Sweller and Cooper (1985) concluded, while”
From Worked Example Mid-Semester Intervention in College Algebra (2022) · cites this paper as Sweller and Cooper (1985)
A few students mentioned that it took them less time to learn the material, which corresponds with the findings of Sweller and Cooper (1985).
“A few also mentioned that it took them less time to learn the material, which corresponds with the findings of Sweller and Cooper (1985).”
From Worked Example Mid-Semester Intervention in College Algebra (2022) · cites this paper as Sweller and Cooper (1985)
Algebra students learned more studying algebra worked examples than solving the equivalent problems.
“The worked-example effect was first demonstrated by Sweller and Cooper (1985) and Cooper and Sweller (1987), who found that algebra students learned more studying algebra worked examples than solving the equivalent problems.”
From Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching (2006) · cites this paper as Sweller & Cooper, 1985
Cited by
Sorted by citations
Cognitive Load During Problem Solving: Effects on Learning
Sweller, J. (1988) · Cognitive Science · doi:10.1016/0364-0213(88)90023-7 · Cited by 11,048 · see citing works
Self-citationKirschner, P. A., Sweller, J., & Clark, R. E. (2006) · Educational Psychologist · doi:10.1207/s15326985ep4102_1 · Cited by 6,887 · see citing works
Self-citationCognitive Architecture and Instructional Design
Sweller, J., van Merrienboer, J. J. G., & Paas, F. G. W. C. (1998) · Educational Psychology Review · doi:10.1023/a:1022193728205 · Cited by 6,023 · see citing works
Self-citationCognitive load theory, learning difficulty, and instructional design
Sweller, J. (1994) · Learning and Instruction · doi:10.1016/0959-4752(94)90003-5 · Cited by 3,801 · see citing works
Self-citationCognitive Load Theory and the Format of Instruction
Chandler, P., & Sweller, J. (1991) · Cognition and Instruction · doi:10.1207/s1532690xci0804_2 · Cited by 2,894 · see citing works
Self-citationSelf-explanations: How students study and use examples in learning to solve problems
Chi, M. (1989) · Cognitive Science · doi:10.1016/0364-0213(89)90002-5 · Cited by 2,287 · see citing works
Cognitive Architecture and Instructional Design: 20 Years Later
Sweller, J., van Merriënboer, J. J. G., & Paas, F. (2019) · Educational Psychology Review · doi:10.1007/s10648-019-09465-5 · Cited by 2,337 · see citing works
Self-citationKalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003) · Educational Psychologist · doi:10.1207/s15326985ep3801_4 · Cited by 1,927 · see citing works
Self-citationSelf‐Explanations: How Students Study and Use Examples in Learning to Solve Problems
Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989) · Cognitive Science · doi:10.1207/s15516709cog1302_1 · Cited by 1,714 · see citing works
Why Some Material Is Difficult to Learn
Sweller, J., & Chandler, P. (1994) · Cognition and Instruction · doi:10.1207/s1532690xci1203_1 · Cited by 1,455 · see citing works
Self-citationDevelopment of an instrument for measuring different types of cognitive load
Leppink, J., Paas, F., Van der Vleuten, C. P. M., Van Gog, T., & Van Merriënboer, J. J. G. (2013) · Behavior Research Methods · doi:10.3758/s13428-013-0334-1 · Cited by 1,066 · see citing works
Instructional control of cognitive load in the training of complex cognitive tasks
Paas, F. G. W. C., & Van Merriënboer, J. J. G. (1994) · Educational Psychology Review · doi:10.1007/bf02213420 · Cited by 968 · see citing works
Cognitive load theory and educational technology
Sweller, J. (2020) · Educational Technology Research and Development · doi:10.1007/s11423-019-09701-3 · Cited by 911 · see citing works
Self-citationKoedinger, K. R., Corbett, A. T., & Perfetti, C. (2012) · Cognitive Science · doi:10.1111/j.1551-6709.2012.01245.x · Cited by 772 · see citing works
“• More complex variable-variable KCs (e.g., designing a controlled experiment, see Table 2) imply instruction that includes comparison ⁄ blocking (Gentner et al., 2009; Gick & Holyoak, 1983) and more worked example study (Sweller & Cooper, 1985).”
BackgroundEffects of schema acquisition and rule automation on mathematical problem-solving transfer.
Cooper, G., & Sweller, J. (1987) · Journal of Educational Psychology · doi:10.1037/0022-0663.79.4.347 · Cited by 688 · see citing works
Self-citationSkulmowski, A., & Xu, K. M. (2022) · Educational Psychology Review · doi:10.1007/s10648-021-09624-7 · Cited by 678 · see citing works
Cognitive load as a factor in the structuring of technical material.
Sweller, J., Chandler, P., Tierney, P., & Cooper, M. (1990) · Journal of Experimental Psychology: General · doi:10.1037/0096-3445.119.2.176 · Cited by 629 · see citing works
Self-citationEvidence for Cognitive Load Theory
Sweller, J., & Chandler, P. (1991) · Cognition and Instruction · doi:10.1207/s1532690xci0804_5 · Cited by 543 · see citing works
Self-citationRenkl, A., & Atkinson, R. K. (2003) · Educational Psychologist · doi:10.1207/s15326985ep3801_3 · Cited by 540 · see citing works
When problem solving is superior to studying worked examples.
Kalyuga, S., Chandler, P., Tuovinen, J., & Sweller, J. (2001) · Journal of Educational Psychology · doi:10.1037/0022-0663.93.3.579 · Cited by 548 · see citing works
Self-citationEffects of pairs of problems and examples on task performance and different types of cognitive load
Leppink, J., Paas, F., van Gog, T., van der Vleuten, C. P. M., & van Merriënboer, J. J. G. (2014) · Learning and Instruction · doi:10.1016/j.learninstruc.2013.12.001 · Cited by 520 · see citing works
Evolution of human cognitive architecture
Sweller, J. (2003) · Psychology of Learning and Motivation · doi:10.1016/s0079-7421(03)01015-6 · Cited by 511 · see citing works
Self-citationToward an Instructionally Oriented Theory of Example‐Based Learning
Renkl, A. (2014) · Cognitive Science · doi:10.1111/cogs.12086 · Cited by 509 · see citing works
Paas, F., & Sweller, J. (2012) · Educational Psychology Review · doi:10.1007/s10648-011-9179-2 · Cited by 502 · see citing works
Self-citationSweller, J. (2004) · Instructional Science · doi:10.1023/b:truc.0000021808.72598.4d · Cited by 470 · see citing works
Self-citationWorked-out examples: Instructional explanations support learning by self-explanations
Renkl, A. (2002) · Learning and Instruction · doi:10.1016/s0959-4752(01)00030-5 · Cited by 452 · see citing works
Guidance during mathematical problem solving.
Tarmizi, R. A., & Sweller, J. (1988) · Journal of Educational Psychology · doi:10.1037/0022-0663.80.4.424 · Cited by 464 · see citing works
Self-citationIntegrating information: A meta-analysis of the spatial contiguity and temporal contiguity effects
Ginns, P. (2006) · Learning and Instruction · doi:10.1016/j.learninstruc.2006.10.001 · Cited by 403 · see citing works
Learning from Worked-Out Examples: The Effects of Example Variability and Elicited Self-Explanations
Renkl, A., Stark, R., Gruber, H., & Mandl, H. (1998) · Contemporary Educational Psychology · doi:10.1006/ceps.1997.0959 · Cited by 374 · see citing works
A model of the self-explanation effect.
VanLehn, K., Jones, R. M., & Chi, M. T. H. (1992) · Journal of the Learning Sciences · doi:10.1207/s15327809jls0201_1 · Cited by 305 · see citing works
The evolution of cognitive load theory and its application to medical education
Leppink, J., & Van den Heuvel, A. (2015) · Perspectives on Medical Education · doi:10.1007/s40037-015-0192-x · Cited by 299 · see citing works
Peer reviewers learn from giving comments
Cho, Y. H., & Cho, K. (2011) · Instructional Science · doi:10.1007/s11251-010-9146-1 · Cited by 294 · see citing works
Domain-Specific Knowledge and Why Teaching Generic Skills Does Not Work
Tricot, A., & Sweller, J. (2014) · Educational Psychology Review · doi:10.1007/s10648-013-9243-1 · Cited by 273 · see citing works
Self-citationGerjets, P., Scheiter, K., & Catrambone, R. (2004) · Instructional Science · doi:10.1023/b:truc.0000021809.10236.71 · Cited by 251 · see citing works
Marcus, N., Cooper, M., & Sweller, J. (1996) · Journal of Educational Psychology · doi:10.1037/0022-0663.88.1.49 · Cited by 244 · see citing works
Self-citationClarke, T., Ayres, P., & Sweller, J. (2005) · Educational Technology Research and Development · doi:10.1007/bf02504794 · Cited by 227 · see citing works
“To avoid such problem-solving approaches and enhance learning, instructional designers have successfully employed worked examples in a number of domains, including mathematics (see Cooper & Sweller, 1987; Paas & van Merrienboer, 1994; Sweller & Cooper, 1985 ).”
MethodSelf-citationAttention guidance during example study via the model’s eye movements
van Gog, T., Jarodzka, H., Scheiter, K., Gerjets, P., & Paas, F. (2009) · Computers in Human Behavior · doi:10.1016/j.chb.2009.02.007 · Cited by 224 · see citing works
Sweller, J. (2023) · Educational Psychology Review · doi:10.1007/s10648-023-09817-2 · Cited by 220 · see citing works
Self-citationChen, O., Paas, F., & Sweller, J. (2023) · Educational Psychology Review · doi:10.1007/s10648-023-09782-w · Cited by 224 · see citing works
“The effect suggests that asking novices to study examples improves learning more than having them solve the equivalent problems (Sweller & Cooper, 1985).”
BackgroundSelf-citationNückles, M., Roelle, J., Glogger-Frey, I., Waldeyer, J., & Renkl, A. (2020) · Educational Psychology Review · doi:10.1007/s10648-020-09541-1 · Cited by 213 · see citing works
Cognitive Load Theory: Historical Development and Relation to Other Theories
Moreno, R., & Park, B. (2010) · Cognitive Load Theory · doi:10.1017/cbo9780511844744.003 · Cited by 204 · see citing works
Cognitive load theory and individual differences
Sweller, J. (2024) · Learning and Individual Differences · doi:10.1016/j.lindif.2024.102423 · Cited by 214 · see citing works
Self-citationThe Expertise Reversal Effect is a Variant of the More General Element Interactivity Effect
Chen, O., Kalyuga, S., & Sweller, J. (2017) · Educational Psychology Review · doi:10.1007/s10648-016-9359-1 · Cited by 200 · see citing works
“…learners provided with high levels of guidance in the form of worked examples perform better on subsequent test problems than learners presented with the same material as problems to be solved (Cooper and Sweller 1987; Paas 1992; Paas and Van Merriënboer 1994; Renkl 2014; Sweller and Cooper 1985).”
BackgroundSelf-citationWorking Memory, Long-term Memory, and Instructional Design
Sweller, J. (2016) · Journal of Applied Research in Memory and Cognition · doi:10.1016/j.jarmac.2015.12.002 · Cited by 192 · see citing works
Self-citationThe worked example effect, the generation effect, and element interactivity.
Chen, O., Kalyuga, S., & Sweller, J. (2015) · Journal of Educational Psychology · doi:10.1037/edu0000018 · Cited by 167 · see citing works
Self-citationStark, R., Kopp, V., & Fischer, M. R. (2011) · Learning and Instruction · doi:10.1016/j.learninstruc.2009.10.001 · Cited by 154 · see citing works
Reisslein, J., Atkinson, R. K., Seeling, P., & Reisslein, M. (2006) · Learning and Instruction · doi:10.1016/j.learninstruc.2006.02.008 · Cited by 138 · see citing works
Working memory, fluid intelligence, and science learning
Yuan, K., Steedle, J., Shavelson, R., Alonzo, A., & Oppezzo, M. (2006) · Educational Research Review · doi:10.1016/j.edurev.2006.08.005 · Cited by 124 · see citing works
Attali, Y., & van der Kleij, F. (2017) · Computers & Education · doi:10.1016/j.compedu.2017.03.012 · Cited by 119 · see citing works
Using cognitive theory to facilitate medical education
Qiao, Y. Q., Shen, J., Liang, X., Ding, S., Chen, F. Y., Shao, L., Zheng, Q., & Ran, Z. H. (2014) · BMC Medical Education · doi:10.1186/1472-6920-14-79 · Cited by 111 · see citing works
A randomized controlled trial of interleaved mathematics practice
Rohrer, D., Dedrick, R. F., Hartwig, M. K., & Cheung, C.-N. (2020) · Journal of Educational Psychology · doi:10.1037/edu0000367 · Cited by 90 · see citing works
In Academe, What Is Learned, and How Is It Learned?
Sweller, J. (2015) · Current Directions in Psychological Science · doi:10.1177/0963721415569570 · Cited by 85 · see citing works
Self-citationRelations between the worked example and generation effects on immediate and delayed tests
Chen, O., Kalyuga, S., & Sweller, J. (2016) · Learning and Instruction · doi:10.1016/j.learninstruc.2016.06.007 · Cited by 82 · see citing works
Self-citationW. M. Van Gerven, F. G. W. C. P. (2000) · Educational Gerontology · doi:10.1080/03601270050133874 · Cited by 81 · see citing works
Machine learning based feedback on textual student answers in large courses
Bernius, J. P., Krusche, S., & Bruegge, B. (2022) · Computers and Education: Artificial Intelligence · doi:10.1016/j.caeai.2022.100081 · Cited by 75 · see citing works
Driscoll, D. M., Craig, S. D., Gholson, B., Ventura, M., Hu, X., & Graesser, A. C. (2003) · Journal of Educational Computing Research · doi:10.2190/q8cm-fh7l-6hju-dt9w · Cited by 73 · see citing works
State of the art research into Cognitive Load Theory
Ayres, P., & Gog, T. V. (2009) · Computers in Human Behavior · doi:10.1016/j.chb.2008.12.007 · Cited by 72 · see citing works
Evaluating the Efficiency and Effectiveness of Adaptive Parsons Problems
Ericson, B. J., Foley, J. D., & Rick, J. (2018) · Proceedings of the 2018 ACM Conference on International Computing Education Research · doi:10.1145/3230977.3231000 · Cited by 70 · see citing works
Collaborative learning effects when students have complete or incomplete knowledge
Retnowati, E., Ayres, P., & Sweller, J. (2018) · Applied Cognitive Psychology · doi:10.1002/acp.3444 · Cited by 66 · see citing works
Self-citationEndres, T., Lovell, O., Morkunas, D., Rieß, W., & Renkl, A. (2023) · British Journal of Educational Psychology · doi:10.1111/bjep.12563 · Cited by 57 · see citing works
Exploring the Impact of Worked Examples in a Novice Programming Environment
Zhi, R., Price, T. W., Marwan, S., Milliken, A., Barnes, T., & Chi, M. (2019) · Proceedings of the 50th ACM Technical Symposium on Computer Science Education · doi:10.1145/3287324.3287385 · Cited by 55 · see citing works
Pillay, H. K. (1994) · Instructional Science · doi:10.1007/bf00892159 · Cited by 52 · see citing works
When should guidance be presented in physics instruction?
Hsu, C.-Y., Kalyuga, S., & Sweller, J. (2015) · Archives of Scientific Psychology · doi:10.1037/arc0000012 · Cited by 44 · see citing works
Self-citationWouters, P., Paas, F., & van Merriënboer, J. J. G. (2010) · Instructional Science · doi:10.1007/s11251-008-9079-0 · Cited by 37 · see citing works
Darejeh, A., Marcus, N., & Sweller, J. (2021) · Educational Technology Research and Development · doi:10.1007/s11423-021-10024-5 · Cited by 29 · see citing works
Self-citationThe Split-Attention Principle in Multimedia Learning
Ayres, P., & Sweller, J. (2021) · The Cambridge Handbook of Multimedia Learning · doi:10.1017/9781108894333.020 · Cited by 18 · see citing works
Self-citationManaging Element Interactivity in Equation Solving
Ngu, B. H., Phan, H. P., Yeung, A. S., & Chung, S. F. (2018) · Educational Psychology Review · doi:10.1007/s10648-016-9397-8 · Cited by 16 · see citing works
Pyke, W., Lunau, J., & Javadi, A.-H. (2025) · Quarterly Journal of Experimental Psychology · doi:10.1177/17470218241308143 · Cited by 15 · see citing works
“As with the goal-free effect , a wealth of studies lend support to the effectiveness of worked examples in terms of improving memorisation (Carroll, 1994; Cooper & Sweller, 1987; Paas & Van Merriënboer, 1994; Sweller & Cooper, 1985; Trafton & Reiser, 1993; Zhu & Simon, 1987).”
Williams, A. E., Denaro, K., Dennin, M. B., & Sato, B. K. (2021) · Journal of Applied Research in Higher Education · doi:10.1108/jarhe-10-2019-0272 · Cited by 9 · see citing works
Non-restricted Access to Model Solutions: A Good Idea?
Nygren, H., Leinonen, J., & Hellas, A. (2019) · Proceedings of the 2019 ACM Conference on Innovation and Technology in Computer Science Education · doi:10.1145/3304221.3319768 · Cited by 7 · see citing works
Experimenting with Model Solutions as a Support Mechanism
Nygren, H., Leinonen, J., Pirttinen, N., Leinonen, A., & Hellas, A. (2019) · Proceedings of the 1st UK & Ireland Computing Education Research Conference on - UKICER · doi:10.1145/3351287.3351288 · Cited by 4 · see citing works
Effects of concept maps and worked examples in learning skills in mathematics
Van Orman, D. S. J., Wong, R. M., Carbonneau, K. J., & Adesope, O. O. (2022) · School Science and Mathematics · doi:10.1111/ssm.12524 · Cited by 3 · see citing works
Worked Example Mid-Semester Intervention in College Algebra
Miller, D., & Schraeder, M. (2022) · Journal of Research in Mathematics Education · doi:10.17583/redimat.9645
- Authors
- Sweller, J., & Cooper, G. A.
- Affiliations
- UNSW Sydney (AU)
- Venue
- Cognition and Instruction
- Publisher
- Informa UK Limited
- Published
- 1985-03
- Volume
- 2
- Issue
- 1
- Pages
- 59-89
- ISSN
- 0737-0008 (ISSN-L)
- 1532-690X
- OpenAlex
- W2046675442
- Cited by
- 1,214
- References
- 0
- Citations by year
- 2026: 53 · 2025: 43 · 2024: 35 · 2023: 26 · 2022: 41 · 2021: 49 · 2020: 60 · 2019: 72 · 2018: 44 · 2017: 43 · 2016: 52 · 2015: 70 · 2014: 53 · 2013: 59 · 2012: 53
- FWCI
- 6.12
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Cognition and Instruction (publishedVersion)
- Funders
- NSW Department of Education
- Topics
- Social Sciences › Psychology › Experimental and Cognitive Psychology › Visual and Cognitive Learning Processes
- Physical Sciences › Mathematics › Applied Mathematics › Mathematics Education and Pedagogy
- Social Sciences › Education › Mathematics Education and Teaching Techniques
- Keywords
- worked examples, algebraic problem solving, schema acquisition, solution time, mathematical errors, mathematics learning
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
