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Making connections among multiple graphical representations of fractions: sense-making competencies enhance perceptual fluency, but not vice versa

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Abstract

Prior research shows that representational competencies that enable students to use graphical representations to reason and solve tasks is key to learning in many science, technology, engineering, and mathematics domains. We focus on two types of representational competencies: (1) sense making of connections by verbally explaining how different representations map to one another, and (2) perceptual fluency that allows students to fast and effortlessly use perceptual features to make connections among representations. Because these different competencies are acquired via different types of learning processes, they require different types of instructional support: sense-making activities and fluency-building activities. In a prior experiment, we showed benefits for combining sense-making activities and fluency-building activities. In the current work, we test how to combine these two forms of instructional support, specifically, whether students should first work on sense-making activities or on fluency-building activities. This comparison allows us to investigate whether sense-making competencies enhance students’ acquisition of perceptual fluency (sense-making-first hypothesis) or whether perceptual fluency enhances students’ acquisition of sense-making competencies (fluency-first hypothesis). We conducted a lab experiment with 74 students from grades 3–5 working with an intelligent tutoring system for fractions. We assessed learning processes and learning outcomes related to representational competencies and domain knowledge. Overall, our results support the sense-making-first hypothesis, but not the fluency-first hypothesis.

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Notes

  1. In addition to the assessments detailed below, we assessed eye-tracking data. Because the eye-tracking data did not yield results relevant to the research questions we investigate in this article, we do not report eye-tracking data. Results from the analysis of eye-tracking data are reported in Rau et al. (2014b).

  2. The procedure for the cued retrospective think-alouds changed midway during the experiment. The procedure change only affected the cued retrospective think-alouds (no other aspects of the experimental procedure, because the cued retrospective think-alouds came last), and it equally affected both experimental conditions. The change was necessary due to delayed arrival of eye-tracking equipment. Specifically, the first 38 (of 74) students were asked to do a retrospective think-aloud using video recordings without eye-gaze recordings. For the remaining 36 students, eye-tracking data were recorded with an unobtrusive remote eye-tracker. (Specifically, we used an SMI RED 250—which uses an infrared camera attached under a computer monitor to record eye-gaze behaviors. The interactions with the computer were no different than without the eye-tracker.) For the cued retrospective think-alouds for these 36 students, we used eye-gaze recordings as cues, following the method proposed by Van Gog et al. (2005). For each activity, the experimenter played back the recorded eye-gaze behaviors. The eye-gaze recordings depict the student’s eye-gaze as a circle, overlaid with a background-screen recording showing the student’s interactions with the problem-solving activity. In replaying the eye-gaze recording, the experimenter first explained what the eye-gaze circle shows, and then paused after each step for a think-aloud prompt. The remainder of the cued retrospective think-alouds proceeded as for the first 38 students.

  3. Tetrad, freely available at www.phil.cmu.edu/projects/tetrad, contains a causal model simulator, estimator, and over 20 model search algorithms, many of which are described and proved asymptotically reliable in Spirtes et al. (2000).

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Acknowledgements

This work was supported by the National Science Foundation, REESE-21851-1-1121307 and by IES, NCER-CASL Award No. R305A120734. We thank Richard Scheines, Ken Koedinger, Brian Junker, Jay Raspat, Michael Ringenberg, Angela McCarthy, Siyan Zhao, Lavender Yi, Jessica Han, Lisa Kwon, the Datashop and CTAT teams.

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  1. Department of Educational Psychology, University of Wisconsin-Madison, 1025W Johnson St, Madison, WI, 53706, USA

    Martina A. Rau

  2. Human-Computer Interaction Institute, Carnegie Mellon University, Pittsburgh, USA

    Vincent Aleven

  3. Institute of Educational Research, Ruhr-Universität, Bochum, Germany

    Nikol Rummel

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  1. Martina A. Rau
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  2. Vincent Aleven
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Correspondence to Martina A. Rau.

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Rau, M.A., Aleven, V. & Rummel, N. Making connections among multiple graphical representations of fractions: sense-making competencies enhance perceptual fluency, but not vice versa. Instr Sci 45, 331–357 (2017). https://doi.org/10.1007/s11251-017-9403-7

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