Skip to main content

Advertisement

SpringerLink
Log in

Effect of worked examples and Cognitive Tutor training on constructing equations

  • Published:
Instructional Science Aims and scope Submit manuscript

Abstract

Algebra students studied either static-table, static-graphics, or interactive-graphics instructional worked examples that alternated with Algebra Cognitive Tutor practice problems. A control group did not study worked examples but solved both the instructional and practice problems on the Cognitive Tutor (CT). Students in the control group requested fewer hints and made fewer errors on the CT practice problems but required more learning time on the instructional examples. There was no difference among the four groups in constructing equations on a paper-and-pencil posttest or on a delayed test that included training and transfer problems. However, students who studied worked examples with a table were best at identifying the meaning of the equation components. The concept of transfer-appropriate processing (the overlap between instructional task and assessment task) aided our interpretation of the findings. Although the CT had a short-term effect on reducing errors and hint requests on CT practice problems, the worked examples were as effective on delayed paper-and-pencil tests. The subsequent construction of a new module for the Animation Tutor (Reed and Hoffman, Animation Tutor: Mixtures. Instructional software, 2011) used both the interactive-graphics and static-table worked examples to take advantage of the complementary strengths of different representations (Ainsworth, Learn Instr 16:183–198, 2006).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Adams, L. T., Kasserman, J. E., Yearwood, A. A., Perfetto, G. A., Bransford, J. D., & Franks, J. J. (1988). Memory access: The effects of fact-oriented versus problem-oriented acquisition. Memory & Cognition, 16, 167–175.

    Article  Google Scholar 

  • Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16, 183–198.

    Article  Google Scholar 

  • Aleven, V., McLaren, B. M., Sewall, J., & Koedinger, K. R. (2009). A new paradigm for intelligent tutoring systems: Example-tracing tutors. International Journal of Artificial Intelligence in Education (IJAIED), 19, 105–154.

    Google Scholar 

  • Anderson, J. R. (1990). Analysis of student performance with the LISP tutor. In N. Frederiksen, R. Glaser, A. Lesgold, & M. Shafto (Eds.), Diagnostic monitoring of skill and knowledge acquisition (pp. 27–50). Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Anderson, J. R., Corbett, A. T., Koedinger, K. R., & Pelletier, R. (1995). Cognitive tutors: Lessons learned. The Journal of the Learning Sciences, 4, 167–207.

    Article  Google Scholar 

  • Anderson, J. R., & Fincham, J. M. (1994). Acquisition of procedural skills from examples. Journal of Experimental Psychology. Learning, Memory, and Cognition, 20, 1322–1340.

    Article  Google Scholar 

  • Anthony, L. (2008). Developing handwriting-based intelligent tutors to enhance mathematics learning. Unpublished doctoral dissertation, Carnegie Mellon University.

  • Atkinson, R. K., & Renkl, A. (2007). Interactive example-based learning environments: Using interactive elements to encourage effective processing of worked examples. Educational Psychology Review, 19, 375–386.

    Article  Google Scholar 

  • Barsalou, L. (1999). Perceptual symbol systems. Behavioral & Brain Sciences, 22, 577–660.

    Google Scholar 

  • Berthold, K., Eysink, T. H. S., & Renkl, A. (2009). Assisting self-explanation prompts are more effective than open prompts when learning with multiple representations. Instructional Science, 37, 345–363.

    Article  Google Scholar 

  • Berthold, K., & Renkl, A. (2009). Instructional aids to support a conceptual understanding of multiple representations. Journal of Educational Psychology, 101, 70–87.

    Article  Google Scholar 

  • Chi, M. T. H. (2009). Active–constructive–interactive: A conceptual framework for differentiating learning activities. Topics in Cognitive Science, 1, 73–105.

    Article  Google Scholar 

  • Cooper, G., & Sweller, J. (1987). The effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology, 79, 347–362.

    Article  Google Scholar 

  • Corbett, A. T., & Anderson, J. R. (2001). Locus of feedback control in computer-based tutoring: Impact on learning rate, achievement and attitudes. In Proceedings of ACM CHI ’2001 conference on human factors in computing systems (pp. 245–252). New York, ACM Press.

  • Corbett, A., McLaughlin, M., Scarpinatto, K. C., & Hadley, W. (2000). Analyzing and generating mathematical models: An Algebra II Cognitive Tutor design study. In G. Gauthier, C. Frasson, & K. VanLehn (Eds.), Intelligent tutoring systems: 5th international conference (pp. 314–323). New York: Springer.

    Google Scholar 

  • Corbett, A. T., Wagner, A., Lesgold, S., Ulrich, H., & Stevens, S. (2007). Modeling students’ natural language explanations. In C. Conati, K. McCoy, & G. Paliouras (Eds.), User Modeling 2007: Proceedings of the eleventh international conference (pp. 117–126). Berlin: Springer.

    Google Scholar 

  • Corbett, A., Wagner, A., & Raspat, J. (2003). The impact of analysing example solutions on problelm solving in a pre-algebra tutor. Proceedings of AIED 2003: The 11th international conference on artificial intelligence and education.

  • Engelkamp, J. (1998). Memory for actions. Hove: Psychology Press.

    Google Scholar 

  • Fuson, K. C., & Willis, G. B. (1989). Second grader’s use of schematic drawings in solving addition and subtraction problems. Journal of Educational Psychology, 81, 514–520.

    Article  Google Scholar 

  • Gamo, S., Sander, E., & Richard, J. F. (2010). Transfer of strategy use by semantic recoding in arithmetic problem solving. Learning and Instruction, 20, 400–410.

    Article  Google Scholar 

  • Gerjets, P., Scheiter, K., & Catrambone, R. (2004). Designing instructional examples to reduce intrinsic cognitive load: Molar versus modular presentation of solution procedures. Instructional Science, 32, 33–58.

    Article  Google Scholar 

  • Goldstone, R. L., Landy, D. H., & Son, J. Y. (2009). The education of perception. Topics in Cognitive Science, 2, 265–284.

    Google Scholar 

  • Goldstone, R. L., & Son, J. Y. (2005). The transfer of scientific principles using concrete and idealized simulations. The Journal of the Learning Sciences, 14, 69–110.

    Google Scholar 

  • Heffernan, N. T., & Koedinger, K. R. (1998). A developmental model for algebra symbolization: the results of a difficulty factors assessment. In Proceedings of the Twentieth Annual Conference of the Cognitive Science Society (pp. 484–489). Hillsdale, NJ: Erlbaum.

  • Hmelo-Silver, C. E., Duncan, R. V., & Chinn, C. A. (2007). Scaffolding and achievement in problem-based and inquiry learning: A response to Kirschner, Sweller, and Clark (2006). Educational Psychologist, 42, 99–107.

    Article  Google Scholar 

  • Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). The expertise reversal effect. Educational Psychologist, 38, 23–31.

    Article  Google Scholar 

  • Kalyuga, S., Chandler, P., & Sweller, J. (2001a). Learner experience and efficiency of instructional guidance. Educational Psychology, 21, 5–23.

    Article  Google Scholar 

  • Kalyuga, S., Chandler, P., Tuovinen, J., & Sweller, J. (2001b). When problem solving is superior to studying worked examples. Journal of Educational Psychology, 93, 579–588.

    Article  Google Scholar 

  • Kirschner, P. A., Sweller, J., & Clark, R. C. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41, 75–86.

    Article  Google Scholar 

  • Koedinger, K. R., & Aleven, V. (2007). Exploring the assistance dilemma in experiments with cognitive tutors. Educational Psychology Review, 19, 239–264.

    Article  Google Scholar 

  • Koedinger, K. R., Anderson, J. R., Hadley, W. H., & Mark, M. A. (1997). Intelligent tutoring goes to school in the big city. International Journal of Artificial Intelligence in Education, 8, 30–43.

    Google Scholar 

  • MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation. Educational Studies in Mathematics, 33, 1–19.

    Article  Google Scholar 

  • Mayer, R. E. (2001). Multimedia learning. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • McLaren, B. M., Lim, S., & Koedinger, K. R. (2008). When and how often should worked examples be given to students? New results and a summary of the current state of research. In B. C. Love, K. McRae, & V. M. Sloutsky (Eds.), Proceedings of the 30th annual conference of the cognitive science society (pp. 2176–2181). Austin, TX: Cognitive Science Society.

    Google Scholar 

  • McNeil, N. M., Uttal, D. H., Jarvin, L., & Sternberg, F. J. (2009). Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. Learning and Instruction, 19, 171–184.

    Article  Google Scholar 

  • Moreno, R., & Mayer, R. E. (2007). Interactive multimodal learning environments. Educational Psychology Review, 19, 309–326.

    Article  Google Scholar 

  • Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40, 282–313.

    Google Scholar 

  • Paas, F. G. W. C. (1992). Training strategies for attaining transfer of problem-solving skill in statistics: A cognitive-load approach. Journal of Educational Psychology, 84, 429–434.

    Article  Google Scholar 

  • Paas, F., Renkl, A., & Sweller, J. (2004). Cognitive load theory: Instructional implications of the interaction between information structures and cognitive architecture. Instructional Science, 32, 1–8.

    Article  Google Scholar 

  • Paas, F. G. W. C., & Van Merrienboer, J. J. G. (1994). Variability of worked examples and transfer of geometrical problem-solving skills: A cognitive-load approach. Journal of Educational Psychology, 86, 122–133.

    Article  Google Scholar 

  • Paivio, A. (1986). Mental representations: A dual coding approach. New York: Oxford University Press.

    Google Scholar 

  • Pashler, H., Bain, P., Bottge, B., Graesser, A., Koedinger, K., McDaniel, M., & Metcalfe, J. (2007). Organizing instruction and study to improve student learning. Washington, DC: National Center for Education Research, Institute of Education Sciences, U.S. Department of Education.

  • Philipp, R. A. (1992). The many uses of algebraic variables. In B. Moses (Ed.), Algebraic thinking, Grades K-12: Readings from NCTM’s School-based journals and other publications (pp. 157–162). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Reed, S. K. (1987). A structure-mapping model for word problems. Journal of Experimental Psychology. Learning, Memory, and Cognition, 13, 124–139.

    Article  Google Scholar 

  • Reed, S. K. (1999). Word problems: Research and curriculum reform. New York: Erlbaum.

    Google Scholar 

  • Reed, S. K. (2005). From research to practice and back: The Animation Tutor project. Educational Psychology Review, 17, 55–82.

    Article  Google Scholar 

  • Reed, S. K. (2006). Cognitive architectures for multimedia learning. Educational Psychologist, 41, 87–98.

    Article  Google Scholar 

  • Reed, S. K. (2008). Manipulating multimedia materials. In R. Zheng (Ed.), Cognitive effects of multimedia learning (pp. 51–66). New York: IGI Global.

    Chapter  Google Scholar 

  • Reed, S. K. (2010). Thinking visually. New York: Taylor & Francis.

    Google Scholar 

  • Reed, S. K., & Ettinger, M. (1987). Usefulness of tables for solving word problems. Cognition and Instruction, 4, 43–59.

    Article  Google Scholar 

  • Reed, S. K., & Hoffman, B. (2011). Animation Tutor: Mixtures. Instructional software. San Diego: San Diego State University.

    Google Scholar 

  • Reed, S. K., & Saavedra, N. C. (1986). A comparison of computation, discovery, and graph procedures for improving students’ conception of average speed. Cognition and Instruction, 3, 31–62.

    Article  Google Scholar 

  • Renkl, A., & Atkinson, R. K. (2003). Structuring the transition from example study to problem solving in cognitive skills acquisition: A cognitive load perspective. Educational Psychologist, 38, 15–22.

    Article  Google Scholar 

  • Ritter, S., Anderson, J. R., Koedinger, K. R., & Corbett, A. (2007). Cognitive Tutor: Applied research in mathematics education. Psychonomic Bulletin & Review, 14, 249–255.

    Article  Google Scholar 

  • Salden, R. J. C. M., Aleven, V., Schwonke, R., & Renkl, A. (2010). The expertise reversal effect and worked examples in tutored problem solving. Instructional Science, 38, 289–307.

    Article  Google Scholar 

  • Schwonke, R., Renkl, A., Krieg, C., Wittwer, J., Aleven, V., & Salden, R. (2009). The worked-example effect: Not an artifact of lousy control conditions. Computers in Human Behavior, 25, 258–266.

    Article  Google Scholar 

  • Schwonke, R., Renkl, A., Salden, R., & Aleven, V. (2011). Effects of different ratios of worked solution steps and problem solving opportunities on cognitive load and learning outcomes. Computers in Human Behavior, 27, 58–62.

    Article  Google Scholar 

  • Siegel, S. (1956). Nonparametric statistics for the behavioral sciences. New York: McGraw-Hill.

    Google Scholar 

  • Sweller, J. (2003). Evolution of human cognitive architecture. In B. Ross (Ed.), The psychology of learning and motivation (Vol. 43, pp. 215–266). San Diego: Academic Press.

    Google 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.

    Article  Google Scholar 

  • Tarmizi, R. A., & Sweller, J. (1988). Guidance during mathematical problem solving. Journal of Educational Psychology, 80, 424–436.

    Article  Google Scholar 

  • van Gog, T., Paas, F., & Van Merrienboer, J. J. G. (2004). Process-oriented worked examples: Improving transfer performance through enhanced understanding. Instructional Science, 32, 83–98.

    Article  Google Scholar 

  • Ward, M., & Sweller, J. (1990). Structuring effective worked examples. Cognition and Instruction, 7, 1–39.

    Article  Google Scholar 

  • Weitz, R., Salden, R. J. C. M., Kim, R. S., & Heffernan, N. T. (2010). Comparing worked examples and tutored problem solving: Pure vs. mixed approaches. In S. Ohlsson & R. Catrambone (Eds.), Cognition in flux: Proceedings of the thirty-second annual meeting of the cognitive science society (pp. 2876–2881). Austin TX: Cognitive Science Society.

    Google Scholar 

  • Wilson, M. (2002). Six views of embodied cognition. Psychonomic Bulletin & Review, 9, 625–636.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CRMSE, San Diego State University, 6475 Alvarado Road, San Diego, CA, 92182, USA

    Stephen K. Reed & Bob Hoffman

  2. Carnegie Mellon University, Pittsburgh, PA, USA

    Albert Corbett, Angela Wagner & Ben MacLaren

Authors
  1. Stephen K. Reed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Albert Corbett
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bob Hoffman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Angela Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ben MacLaren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephen K. Reed.

Appendices

Appendix 1: Worked example problems

Session 1

Arithmetic type 1 interest

  1. 1.

    You have a MasterCard with a balance of $532 at a 21% interest rate. You also have a Visa credit card with a balance of $841 at a 16% interest rate. How much money are you paying in total annual interest?

$$.21\,\times\,\$ 532\, +\, .16\,\times\,\$841= {\text{Total Interest}} $$

Arithmetic type 1 ore

  1. 2.

    The Acme Mine yielded 67 tons of low-grade (35% iron) ore in the first year of operation. In the second year of operation, they yielded 62 tons of high-grade (75% iron) ore. How much pure iron did the company mine in two years?

$$ .35\,\times\,67\,{\text{ tons}}\, +\, .75\,\times\,62\,{\text{ tons}} = {\text{Total Iron}} $$

Arithmetic type 2 interest

  1. 3.

    At the end of her first year of using credit cards, Shelly owed $475 in total interest on her MasterCard and Visa accounts. Her MasterCard charges 19% interest and her Visa Card charges 22% interest. She paid the interest on her Visa Card debt of $1100. How much interest does she still owe on her MasterCard?

$$ \$ 475-. 22\,\times\,\$ 1100 = {\text{Remaining Interest}} $$

Arithmetic type 2 ore

  1. 4.

    A jeweler ordered 40 oz of pure gold. They supplier extracted the 40 oz from sylvanite gold ore that contains 28% pure gold and calaverite gold ore that contains 40% pure gold. If the supplier used 55 oz of calaverite ore, how much gold came from the sylvanite ore?

$$ 40\,{\text{ounces}}-. 40\,\times\,55\,{\text{ounces}} = {\text{Gold from Sylvanite Ore}} $$

Session 2

Algebra type 1 interest

  1. 1.

    You have an American Express credit card with a balance of $715 at an 11% interest rate and a Visa credit card with a 15% interest rate. If you pay a total of $165 in annual interest, what is the balance on your Visa card?

$$ .11\,\times\,\$ 715\, +\, .15\,\times\,V = \$165$$

Algebra type 1 ore

  1. 2.

    The Acme Mine yielded 67 tons of low-grade (35% iron) ore. How many tons of high-grade (75% iron) ore will they need in order to yield 70 total tons of pure iron?

$$ .35\,\times\,67\,{\text{ tons}}\, + \, .75\,\times\,{\text{H\;tons}} = 70\,{\text{tons}} $$

Algebra type 2 interest

  1. 3.

    You have a total balance of $1405 on two different credit cards— an American Express credit card with a 12% interest rate and a Discover credit card with a 24% interest rate. If you owe a total of $224 in annual interest, what is your balance on the Discover card?

$$.24\,\times\,D\, +\, .12\,\times\,\left({\$ 1405-D} \right) = \$224$$

Algebra type 2 ore

  1. 4.

    You have 51 oz of gold ore. The sylvanite gold ore contains 28% pure gold and the calaverite gold ore contains 40% pure gold. If you have 16 total ounces of pure gold, how many ounces of calaverite gold ore do you have?

$$ .40\,\times\,{\text{C\;ounces}}\, +\, .28\,\times\,\left({51 - C} \right)\;{\text{ounces}} =16\;{\text{ ounces}} $$

Appendix 2

CT Model Analysis problems

Each model analysis problem presents both a real-world situation and a symbolic model of the situation. Students are asked to describe what each hierarchical component of the symbolic model represents in the problem situation.

 

Answers

Equation part

English description

L

The number of tons of low-grade ore they have

.32

The percent of copper in the low-grade ore

0.32L

The number of tons of pure copper in their low-grade ore

91

The total tons of copper ore they have

(91 − L)

The number of tons of high-grade ore they have

.82

The percent of copper in the high-grade ore

.82(91 − L)

The number of tons of copper in their high-grade ore

0.16L + .82(91 − L)

The total number of tons of pure copper they have

42

The total number of tons of pure copper they have

Rights and permissions

Reprints and permissions

About this article

Cite this article

Reed, S.K., Corbett, A., Hoffman, B. et al. Effect of worked examples and Cognitive Tutor training on constructing equations. Instr Sci 41, 1–24 (2013). https://doi.org/10.1007/s11251-012-9205-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11251-012-9205-x

Keywords

Search

Navigation