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

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Sequencing learning with multiple representations of rational numbers

Vincent Aleven, Nikol Rummel, and Martina Rau

Summary Table

Study 1

PIs Vincent Aleven & Nikol Rummel
Other Contributers Graduate Students: Martina Rau (CMU HCII)
Study Start Date September 1st, 2008
Study End Date August 31st, 2009
Number of Students ~350
Total Participant Hours ~2100
DataShop Log data is uploaded and available in the DataShop


We investigate a key issue in coordinative learning, namely, how learning with multiple external representations (MERs) should be sequenced to effectively support students’ conceptual understanding. In order to benefit from MERs, learners must attain some level of fluency in interpreting and manipulating the individual representations, and must also engage in sense making across the representations to relate them and abstract underlying concepts. The question arises how tasks involving different representations should be sequenced so that both these aspects of robust learning are realized. In particular, how frequently should students switch between representations? 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). The proposed research looks at how the development of fluency with any given representation interacts with sense making across representations. First, as Ainsworth (2006) points out, being able to interpret a particular type of representation is a prerequisite for learning from it. However, such ‘representational fluency’ does not just emerge by itself, but requires practice. Second, it is important that students engage in sense making across the different representations to relate them and integrate the information they provide (Ainsworth, 2006; Brinker, 1997; Paik, 2005; Uttal et al., 1997). According to cognitive flexibility theory (Spiro & Jehng, 1990), being presented with MERs challenges the learner to switch between different perspectives on the same concepts. Under this perspective, learning with MERs supports the development of robust – flexible and transferable – knowledge (Kaput, 1989), to the extent that learners coordinate between the representations, that is, cognitively link the information the MERs provide and abstract underlying conceptual knowledge. A key question is therefore whether learners should build up fluency with each representation first, before they engage in sense-making activities aimed at coordinating representations, or whether they develop more flexible knowledge when they become familiar with the different representations in parallel and continuously engage in sense making across representations. This potential conflict is inherent in designing instruction with MERs.


  • 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

When learning with multiple external representations, do students learn more robustly when new representations are introduced gradually, with (highly) infrequent switching between representations, or when the representations are introduced right from the start, with students switching often between representations? Or, is a transition from infrequent to frequent switching between representations most beneficial?


  • 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

Independent Variables

  • Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on:
    • Reproduction of conceptual knowledge
    • Reproduction of procedural knowledge
    • Transfer of conceptual knowledge
    • Transfer of procedural knowledge
  • Log data collected during tutor use, used to assess:
    • Learning curves
    • Time on task
    • Error rates
    • Hint usage
    • Latency of responses


Data collection is still in progress.


Data collection is still in progress.

Further Information


Annotated Bibliography


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