Sequencing learning with multiple representations of rational numbers (Aleven, Rummel, & Rau)

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Learning with Multiple Representations in a Complex, Real-world Domain: Intelligent Tutoring Systems for Fractions

Vincent Aleven, Nikol Rummel, and Martina Rau

Summary Table

PIs Vincent Aleven & Nikol Rummel
Other Contributers Graduate Students: Martina Rau (CMU HCII)
2008 study N = 132 6th-grade students
2009 study N = 388 5th- and 6th-grade students
2010 study N = 690 4th- and 5th-grade students
Study Start Date September 1st, 2008
Study End Date August 31st, 2012
Total Number of Students to date N = 1210
Total Participant Hours ~6000
Data available in DataShop Dataset: Fraction Study Spring 2009 (log data only)
Dataset: Fraction Study Spring 2010 (fractions portion only)
  • Pre/Post Test Score Data: No
  • Paper or Online Tests: 2008 & 2009 experiments: online; 2010 experiment: paper
  • Scanned Paper Tests: No
  • Blank Tests: No
  • Answer Key: No


We investigate a key issue in coordinative learning, namely, how learning with multiple graphical representations should bue used to effectively support students’ conceptual understanding of fractions. In a previous experiment (Rau, Aleven, & Rummel, 2009), we demonstrated that students benefit from learning with multiple graphical representations when compared to a single graphical representation, provided that they were prompted to relate the graphical representations to the symbolic representation of fractions (e.g., 1/2). In two consecutive studies, we investigated how multiple representations should be sequenced. Prior research on contextual interference has demonstrated that interleaving different types of learning tasks can foster a deep understanding of the underlying concepts. Do the same advantages apply to interleaving representations? In future studies, we plan to investigate ways to explicitly support students in relating the different graphical representations to one another. We focus on fractions as a challenging topic area for students in which multiple representations are often used and likely to support robust learning. This research will contribute to the literature on early mathematics learning, learning with multiple representations, and learning with intelligent tutoring systems. It will also add to the portfolio of studies in the PSLC’s coordinative learning cluster.

Background & Significance

A quintessential form of coordinative learning occurs when learners work with multiple external representations (MERs) of subject matter. Accumulating evidence points towards the promise of learning with MERs (Ainsworth, Bibby, & Wood, 2002; Larkin & Simon, 1987; Seufert, 2003), and also to the need for students to make sense out of the different representations by connecting and abstracting from them (Ainsworth, 1999).
This research focuses on a difficult area of early mathematics learning: fractions. Both teachers’ experiences and research in educational psychology show that students have difficulties with fraction arithmetic and with the various representations for fractions (e.g. Brinker, 1997; Callingham & Watson, 2004; Caney & Watson, 2003; Person et al., 2004; Pitta-Pantazi, Gray & Christou, 2004). Coordinating between MERs is regarded as a key process for learning across areas of mathematics (Kilpatrick, Swafford, & Findell, 2001; NCTM, 2000), including fractions (e.g. Kieren, 1993; Moss & Case, 1999; Martinie & Bay-Williams, 2003; Thompson & Saldanha, 2003).
A number of authors have argued, based on observational studies, that MERs can lead to deeper conceptual understanding of fractions (Corwin et al., 1990; Cramer et al., 1997a, 1997b; Steiner & Stoeckling, 1997). However, we know of no experimental studies that have investigated the advantages of instruction with multiple (graphical) fraction representations over instruction that focuses on a single representation, with one exception: an in vivo experiment, in which 132 6th-grade students used four versions of CTAT-built tutors (Rau, Aleven, & Rummel, 2009). Students learning with MERs and prompted to self-explain performed best on a posttest and delayed posttest assessing procedural and conceptual knowledge of fractions.
At this point, however, we do not know enough about the circumstances that may influence the effectiveness of learning with multiple representations of fractions, a criticism that has been leveraged against the existing body of research on learning with MERs more generally (Ainsworth, 2006; Goldman, 2003). Learning with multiple representations is challenging. An important pre-requisite for benefiting from the multiplicity of multiple graphical representations is that students conceptually understand each one of them (Ainsworth, 2006).
When designing intelligent tutoring systems that use multiple graphical representations, designers must decide how to temporally sequence the GRs. How often should the curriculum alternate between multiple graphical representations? Practice schedules are likely to impact how students understand each GR. In particular, it may matter whether items with the same attributes (e.g., task types) are practiced in a “blocked” manner (e.g., A – A – B – B) or are interleaved with practice of other item types (e.g., A – B – A – B). Research on contextual interference shows that interleaving task types leads to better learning results than blocked practice [5, 6]. A common interpretation of this finding is that interleaved practice encourages deep processing [6]. Since students cannot hold all relevant knowledge components in working memory, they must reactivate task-specific knowledge components as they come up again in the task sequence.
The presented research investigates the effect of sequencing multiple graphical representations on students' learning of fractions.

Research questions

  1. Which task attribute(s) should designers of intelligent tutoring systems interleave? Should we interleaved task types or multiple graphical representations?
  1. Sequencing multiple graphical representations Do students benefit most from blocked or interleaved multiple graphical representations when task types are interleaved?


  • We hypothesize that a mix of these two designs (i.e., an intermediate position on the continuum between highly infrequent and highly frequent switching between the representations) would be best as it allows learners to gain some experience with one representation before moving on to the next, but also facilitates making connections across representations as the (temporal) distance between representations is smaller than in the highly infrequently switching design.
  • We hypothesize that gaining fluency with each of the representations is more important at the beginning of a tutoring session than towards the end. Therefore, we expect a sequence that transitions from infrequent to frequent switching between representations to be more effective than the extremes of the continuum between highly infrequent and highly infrequent switching between the representations.

2009 Study

Dependent variables

  • Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on:
    • Representational knowledge
    • Operational knowledge
  • Log data collected during tutor use, used to assess:
    • Learning curves
    • Time on task
    • Error rates
    • Hint usage

Independent Variables

  1. Blocked representations / interleaved topics – representations are blocked while topics are interleaved (students switch topics after every 18 problems)
  2. Fully interleaved representations / blocked topics – representations are highly interleaved (students switch representations after each problem) while topic types are blocked
  3. Moderately interleaved representations / blocked topics – representations are moderately interleaved (students switch representations after every three problems) while topic types are blocked
  4. Increasingly interleaved representations / blocked topics – the length of the blocks of representations is gradually reduced (at the beginning, students switch topics after every twelve problems, at the end they switch after each single problem) while topic types are blocked


We contrasted the effects of interleaving task types (while blocking multiple graphical representations) and several versions of interleaving multiple graphical representations (while blocking task types) in an ITS for fractions. Our results show an advantage for interleaving task types over interleaving multiple graphical representations.

2010 Study

Dependent variables

  • Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on:
    • Area model problems
    • Number line problems
    • Fraction comparison
    • Proportional reasoning
  • Log data collected during tutor use, used to assess:
    • Learning curves
    • Time on task
    • Error rates
    • Hint usage

Independent Variables

  1. Blocked representations – students switch representations after 36 problems
  2. Moderately interleaved representations – students switch representations after every six problems
  3. Fully interleaved representations –students switch representations after each problem
  4. Increasingly interleaved representations – the length of the blocks is gradually reduced from twelve problems at the beginning to a single problem at the end
  5. Circle-only control – students work with only the circle representation
  6. Rectangle-only control – students work with only the rectangle representation
  7. Number-line-only control – students work with only the number line representation


Students in the multiple representation conditions improved their test scores from pretest to posttest, and pertained their learning gains at the delayed posttest. This was not true of the single representation conditions. We found a slight advantage for interleaving multiple representations: On different dependent measures, it was either the fully interleaved, or the increasingly interleaved condition that outperformed the remaining conditions, but never the blocked condition. Finally, the multiple representation conditions significantly outperformed the single representation conditions (including the number line only condition) on number line test items.


  • Feenstra, Laurens; Aleven, Vincent; Rummel, Nikol; & Taatgen, Nils. Multiple interactive representations for fractions learning. I10th international conference on intelligent tutoring systems (ITS), 221-3. 2010.
  • Rau, Martina; Aleven, Vincent; Rummel, Nikol, Tunc-Pekkan, Zelha; Pacilio, Laura. How to schedule multiple graphical representations? A classroom experiment with an intelligent tutoring system for fractions. Under review.
  • Rau, Martina; Aleven, Vincent; Rummel, Nikol. Blocked versus Interleaved Practice With Multiple Representations in an Intelligent Tutoring System for Fractions. 10th International Conference of Intelligent Tutoring Systems (ITS), 413-422. 2010.
  • Rau, Martina; Aleven, Vincent; Rummel, Nikol. Intelligent Tutoring Systems with Multiple Representations and Self-Explanation Prompts Support Learning of Fractions. 14th International Conference on Artificial intelligence in Education (AIED), 441-448. 2009.
  • Tunç-Pekkan, Zelha; Zeylikman, Lyubov; Aleven, Vincent; Rummel, Nikol. Fifth Graders’ Conception of Fractions on Numberline Representations. The annual meeting of North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, Ohio. 2010.
  • Tunc-Pekkan, Zelha; Rau, Martina; Aleven, Vincent; Rummel, Nikol. External Representations and Fractional Knowledge. Third Annual inter-Science of Learning Center (iSLC) Conference For Students and Postdoctoral Fellows at the Science of Learning Centers, Boston, MA. 2010.


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