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Revision as of 01:07, 22 January 2007

Learning with Diagrams in Geometry: Strategic Support for Robust Learning

Vincent Aleven & Kirsten Butcher


The overall goal of this project is to gain a better understanding of 1) learning with visual and verbal knowledge components in a problem-solving environment and, 2) how such learning can best be supported in an intelligent tutoring system. Ultimately, we are interested coordinative and integrative aspects of robust learning processes in learning with visual and verbal knowledge components.

We are using the Geometry Cognitive Tutor as a research vehicle for our project. In geometry, visual information is represented in a problem diagram and verbal/symbolic information is represented in text that contains given and goal information as well as some information about the problem situation that is not (yet) expressed in the diagram. The goals of this research are to investigate how coordination between and integration of visual and verbal information influence robust learning processes, as measured by knowledge retention and transfer. By coordination, we mean the processes that support mapping between relevant visual and verbal information as well as the processes that keep relevant knowledge components active. For example, in geometry a student needs to map between text references to angles and their location in a diagram and will need to maintain the numerical (given or solved) value of that angle to use in problem solving. By integration, we mean knowledge construction events that involve both visual and verbal knowledge components. For example, in geometry a student may need to construct an understanding of linear angles that includes both a verbal (e.g., “two adjacent angles that form a line”) and a visual situation description (e.g., a visual representation of the two angles formed by intersection of a line). We are investigating scaffolds in the Geometry Cognitive Tutor that may influence the ways in which students coordinate and integrate visual and verbal knowledge during learning.


  • Contiguous Representation: Representations that include visual and verbal inforamtion in close temporal and physical proximity. Also see Contiguity.
  • Elaborated Explanations: Self-explanations that include visual references in addition to verbal explanations.
  • Mapping: The process by which learners find the visual refernce of a referring expression (e.g., in geometry, finding the angle in a diagram that is referred to in the given information for a problem).
  • Visual-verbal integration: The process by which learners combine visual and verbal knowledge into a coherent, flexible knowledge representation.

Research questions

  1. Do students learn more with contiguous representations that support direct interaction and reasoning with diagrams? Is a contiguous representation sufficient to support integration of visual and verbal knowledge components?
  2. Do students learn more when they produce elaborated explanations that connect visual and verbal knowledge components in geometry?
  3. What robust learning processes are affected by contiguous representations during problem-solving?

Background & Significance

In this research, we draw upon previous work in multimedia learning, self-explanations, and Cognitive Tutors. We hypothesize that three factors—drawn from theoretical issues in learning with multimedia and multiple representations—are relevant to supporting successful learning from visual and verbal representations. These factors are 1) contiguity, 2) elaborated explanations, and 3) integrated hints.

By contiguity we mean placing related representations such as a table and a diagram in close proximity or even eliminating one of them altogether. Previous research in multimedia learning (e.g., Mayer, 2001) has identified learning benefits when visual and verbal information is placed is close temporal and spatial proximity; these benefits have been hypothesized to result from the easing of cognitive load required to map between visual and verbal information. Although we predict that contiguous representations may be useful during geometry learning, we believe that contiguous presentations alone may not be sufficient to promote the kinds of cognitive processes necessary for robust learning with visual and verbal information. Ultimately, we believe that scaffolds that support active integration of visual and verbal knowledge components will show the strongest effects in supporting robust learning.

Thus, the other two factors (elaborated explanations and integrated hints) are hypothesized to help students in integrating visual and verbal knowledge components, as needed for robust learning in this kind of domain.

By elaborated explanations we mean justifications given by the student of their problem-solving steps in which they elaborate not only which problem-solving principle is being applied (as investigated in much previous research on self-explanation with interactive learning environments) but also how it applies. A rich body of prior research has demonstrated that students develop deeper understanding of instructional materials when they self-explain to themselves during learning (e.g., Bielaczyc, Pirolli, & Brown, 1995; Chi, Bassok, Lewis, Reimann, & Glaser, 1989; Chi, de Leeuw, Chiu, & LaVancher, 1994). An existing version of the Geometry Cognitive Tutor implements student self-explanations of their problem solving using a very simple process: after correctly answering a geometry problem step, students must select the geometry rule or theorem that justifies their answer from a glossary menu of terms. Despite the limitations of the menu-based explanations, they have been shown to promote student learning in the Geometry Cognitive Tutor (Aleven & Koedinger, 2002). Thus, our question is whether more explicit forms of communication that link verbal and visual knowledge components can be more successful at promoting student understanding. We argue that scaffolds that promote sense making using both visual and verbal knowledge may be of greater benefit to students than scaffolds that require student explanations using only verbal declarative knowledge.

By integrated hints we mean instructional hints that use visual means to highlight relations between visual and verbal representations. Previous research has shown that integrated representations can be useful for learning (e.g., Sweller & Chandler, 1994). Integrated hints may support robust learning better than verbal hints alone because they reduce the translation required for integration during problem-solving and potentially support active integration of knowledge.

Dependent variables

  • Pretest and immediate posttest, measuring student performance on:
    • Problem-solving items isomorphic to the practiced problems (near-term knowledge retention)
    • Problem-solving items unlike those seen during problem practice (near-term knowledge transfer)
  • Delayed posttest, measuring student performance on:
    • Problem-solving items isomorphic to the practiced problems (far-term knowledge retention)
    • Problem-solving items unlike those seen during problem practice (far-term knowledge transfer)
  • Log data collected during tutor use, used to assess:
    • Learning curves
    • Time on task
    • Error rates
    • Latency of responses

Independent Variables

1. Contiguity of Representation
Contiguous representation (students work in diagram) vs. Non-contiguous representation (students work in separate table)
2. Type of Explanation
Verbal explanations (students state geometry principles only) vs. Elaborated Explanations (students state geometry principles and their application to the diagram)
3. Type of Hints
Verbal hints (hints provided as text only) vs. Integrated Hints (hints provided as text embedded in and applied to a diagram)


  • Contiguous representations in geometry decrease cognitive load and increase strategic inferences during problem-solving, thus supporting immediate and long-term knowledge retention as well as transfer.
  • Elaborated explanations promote integration of visual and verbal knowledge components during problem-solving, thus supporting knowledge transfer.
  • Integrated hints promote sense making by promoting coordination through self-explanation and contiguity, thus supporting knowledge transfer.


Study 1

  • Summary
    • In Vivo Study: 10th grade geometry class in rural Pennsylvannia school
    • Domain: Angles curriculum in the Geometry Cognitive Tutor
    • Grade-matched pairs of students were randomly assigned to one of two conditions:
      • Diagram (Contiguous) Condition: Students interacted directly with geometry diagrams and accepted answers are displayed directly in the diagram
      • Table (Noncontiguous) Condition: Students work separate from the diagrams, in a distally located table
  • Findings
    • No overall effect of experimental condition on students' performance on geometry answers or reasons at posttest
    • Aptitude X Treatment interaction for students' performance on geometry rules at posttest (XX add stats). Although lower-knowledge participants perform better on rule-based explanations after interacting with diagrams during practice, higher-knowledge students did not.

Study 2

Study 3


Contiguous representations eliminate additional cognitive load involved in finding the referent of symbolic angle names, continually mapping between diagram and table, and maintaining relevant feature quantities. Thus, these representations should reduce student errors (where visual information is incorrectly mapped to verbal references) and will reduce the estimated effort of deep learning paths (by supporting strategic inferences and reasoning directly with the diagram). Further, we anticipate a path effect: students who are able to reason directly with diagram representations will attend more closely to the geometric features and relations to which geometry principles apply. This should impact meaningful learning by increasing feature validity of the visual and verbal knowledge components.

Elaborated explanations and integrated hints support coordination through self-explanation of visual and verbal information. Providing these multiple representations during learning likely aftects path choice. For example, when students are required to explain the application of geometry principles using diagrams, there will be only small differences in estimated effort of shallow and deep strategies since shallow strategies are unlikely to achieve the correct answer. Further, we anticipate that elaborated explanations and integrated hints also produce path effects: the processes that students employ via path choice are more effective when the materials support use of visual and verbal information during sense making. Specifically, scaffolds or materials that support sense-making with visual and verbal information should promote integration.

Annotated Bibliography


  • Bielaczyc, K., Pirolli, P. L., & Brown, A. L. (1995). Training in self-explanation and self-regulation strategies: Investigating the effects of knowledge acquisition activities on problem solving. Cognition & Instruction, 13, 221-252.
  • Chi, M. T., 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.
  • Chi, M. T. H., de Leeuw, N., Chiu, M.-H., & LaVancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18, 439-477.
  • Koedinger, K. R., & Anderson, J. R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14, 511-550.
  • Lovett, M. C., & Anderson, J. R. (1994). Effects of solving related proofs on memory and transfer in geometry problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition, 20, 366-378.
  • Mayer, R. E. (2001). Multimedia Learning. Cambridge, Cambridge University Press.
  • Sweller, J., & Chandler, P. (1994). Why some material is difficult to learn. Cognition and Instruction, 12, 185-233.