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

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Learning with Multiple Reprsentations 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: Mathtutor Fractions MERs Spring 2009 (revised)
  • 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.


  • Conceptual knowledge: knowledge about the rationale of a solution procedure
  • Procedural knowledge: knowledge of the components of a correct procedure involving knowledge about step-by-step actions for solving problems

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.

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


  • 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. Switch frequently – students will switch representations after every other tutor problem.
  2. Switch with moderate frequency – students will switch representations after every 5 tutor problems
  3. Switch infrequently – within each unit of the tutor curriculum, the representations are used in a “blocked” manner: students first work with one representation, then with the next, then with the third
  4. Switch with gradually-increasing frequency – students start out within each unit by switching representations with moderate frequency (as in the second condition), but the frequency increases gradually until at the end of each unit it reaches the same frequency as the first condition

Findings (2009)

We contrasted the effects of interleaving task types (while blocking multiple graphical representations) and 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.


Data collection is still in progress.

Further Information


Annotated Bibliography


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