Using eye movement measurement to tap into children’s implicit numerical magnitude representations (Heine, et al., 2010)

Heine, A., Thaler, V., Tamm, S., Hawelka, S., Schneider, M., Torbeyns, J., De Smedt, B., et al. (2010). What the eyes already “know”: using eye movement measurement to tap into children’s implicit numerical magnitude representations. Infant and Child Development, 19(2), 175-186. doi:10.1002/icd.640

The authors used eye tracking data to investigate the development of numerical magnitude understanding in primary school children (grades 1-3). Children were presented with two number line estimation tasks; one was restricted to behavioral measures, whereas the other included the recording of eye movement data. Gaze data indicates a quantitative increase as well as a qualitative change in children’s implicit knowledge about numerical magnitudes that precedes the overt (behavioural) demonstration of explicit numerical knowledge. Thus, eye movements may reveal more about the presence of implicit precursors of later explicit knowledge in the numerical domain than more traditional approaches, suggesting further exploration of eye tracking as a potential early assessment tool of individual achievement levels in numerical processing.

“The idea that cognitive growth can be characterized as a transition from implicit forms of knowledge to more explicit knowledge provides developmental researchers with a fascinating yet challenging starting point for translating the Vygotskyan ideas into the domain of knowledge acquisition.”

“Vygotsky’s concept of what he called a child’s zone of proximal development emphasizes the need to also consider children’s learning potential instead of merely focussing on what a child has explicitly mastered at any given point in his or her development. An important question is therefore how to assess children’s learning potential, that is, how to determine their level of knowledge that is not yet explicitly expressed (Brown & Ferrara, 1985).”

Dienes and Perner (1999): fully explicit knowledge – when the person is conscious of that knowledge and is in a position to entertain second-order thoughts about it. Eye movements can be used as sensitive indicators of implicit knowledge that precede more explicit forms of knowledge.

Susan Goldin-Meadow et al. conducted a number of influential developmental studies that aim at identifying children’s implicit or ‘budding knowledge’ (Vygotsky, 1978). “Focussing on the gestures that accompany children’s problem solving processes, Goldin-Meadow et al. were able to demonstrate subtle mismatches between children’s verbal and nonverbal behaviours (Alibali & Goldin- Meadow, 1993; Goldin-Meadow, Alibali, & Church, 1993). According to the authors, gesture–speech mismatches can be indicative of a child’s specific readiness-to-learn by revealing forms of knowledge that are there at an implicit level, but not yet explicitly available to the child (Goldin-Meadow, 2000; Goldin- Meadow & Sandhofer, 1999).”

Siegler et al.: primary schoolers’ representational patterns change from a logarithmic to a more adequate linear model of numerical magnitude between the first and the second grade. This transition is considered a crucial step that enables children to improve their achievement in a wide range of mathematical task domains.

“The question is whether the behaviourally established quantitative and qualitative changes can also be demonstrated using eye movement parameters, and, more importantly, whether eye movement measures that we assume to be able to tap into implicit knowledge add to what we already know about the development of numerical magnitude representation in children.”

Some results:

  • “While in first graders, the eyes seem to roam a wide range of numerical positions on the number line, in third graders, approximately two-thirds of all fixations fall within the immediate vicinity of the respective correct positions. The distributional pattern of second graders’ fixations lies somewhere between the two extremes.”
  • “In-depth analyses of the fits of number line fixations to the logarithmic and linear models determined in the behavioural setting revealed a somewhat different picture for the eye movement data compared with the behavioural data. Although first graders’ estimates were fitted substantially better by the logarithmic model than by the linear model in the behavioural task, this does not hold true for their eye movements. Comparable to the second and third graders, for the eye-tracking data the linear model provides a better fit in first graders as well.”
  • “However, tests of the residuals between the models and the gaze data revealed that in first graders the differences in the fit of the linear compared to the logarithmic functions are not significant. In second and third graders the linear functions provide the significantly better models.”
  • “Eye movement data demonstrate that even when, on first sight, children still appear to be stuck on a more immature level their implicit knowledge might already have undergone some subtle yet detectable changes. The eye movement data demonstrate that even in cases where no evidence of appropriate insight into a certain numerical magnitude can be found in children’s overt behavioural responses, their eye movements might still show manifestations of knowledge at work on a more implicit level. Thus, for instance, our analyses of children’s eye movements in error trials indicate that third graders seem to understand more about certain numerical magnitudes than first or second graders even when they do not actually appear to be more knowledgeable than their younger peers on the level of overt behaviour. The eye movement data reveal that compared with children from lower grade levels, older children shift their gaze significantly more often and also for a significantly longer time to the respective correct positions on the number line in trials where their explicit responses are consistently false.”

“While first graders’ overt behaviour suggests that their representation of numerical magnitude still follows an immature logarithmic model, a change towards more mature representational patterns might already be on the way, as revealed by measures that are capable to tap this emerging knowledge. There might, thus, be a transitional phase where both representational patterns exist in parallel. Such an idea is well in line with Siegler’s (1996) Overlapping the Wave model of change that proposes that during development ‘multiple ways of thinking coexist for prolonged periods’ (p. 89).”

References to read:

Clements, W., & Perner, J. (1994). Implicit understanding of belief. Cognitive Development, 9, 377–397.

Clements, W. A., Rustin, Ch. L., & McCallum, S. (2000). Promoting the transition from implicit to explicit understanding: A training study of false belief. Developmental Science, 3, 81–92.

Alibali, M., Flevares, L., & Goldin-Meadow, S. (1997). Assessing knowledge conveyed in gesture: Do teachers have the upper hand? Journal of Educational Psychology, 89, 183–193.

Alibali, M. W., Bassok, M., Solomon, K. O., Syc, S. E., & Goldin-Meadow, S. (1999). Illuminating mental representations through speech and gesture. Psychological Science, 10, 327–333. Alibali, M. W., & Goldin-Meadow, S. (1993). Gesture–speech mismatch and mechanisms of learning: What the hands reveal about a child’s state of mind. Cognitive Psychology, 25, 468–523.

Dienes, Z., & Perner, J. (1999). A theory of implicit and explicit knowledge. Behavioral and Brain Sciences, 22, 735–755. Garber, P., Alibali, M. W., & Goldin-Meadow, S. (1998). Knowledge conveyed in gesture is not tied to the hands. Child Development, 69, 75–84.

Goldin-Meadow, S. (2000). Beyond words: The importance of gesture to researchers and learners. Child Development, 71, 231–239. Goldin-Meadow, S., Alibali, M. W., & Church, R. B. (1993). Transitions in concept acquisition: Using the hand to read the mind. Psychological Review, 100, 279–297.

Goldin-Meadow, S., & Sandhofer, C. M. (1999). Gesture conveys substantive information about a child’s thoughts to ordinary listeners. Developmental Science, 2, 67–74.

Perner, J., & Dienes, Z. (1999). Deconstructing RTK: How to explicate a theory of implicit knowledge. Behavioural and Brain Sciences, 22, 790–808.

Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. Cambridge, MA: MIT Press.

[Karmiloff-Smith’s (1992) notion of representational re-description as a process by which ‘implicit information in the mind subsequently becomes explicit knowledge to the mind’ (p. 18).]

Garnham, W. A., & Perner, J. (2001). Actions really do speak louder than words—but only implicitly: Young children’s understanding of false belief in action. British Journal of Developmental Psychology, 19, 413–432.

Garnham, W. A., & Ruffman, T. (2001). Doesn’t see, doesn’t know: Is anticipatory looking really related to understanding of belief? Developmental Science, 4, 94–100.

Laski, E. V., & Siegler, R. S. (2007). Is 27 a big number? Correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison. Child Development, 76, 1723–1743.

Perry, M., Church, R. B., & Goldin-Meadow, S. (1988). Transitional knowledge in the acquisition of concepts. Cognitive Development, 3, 359–400.

Ruffman, T., Garnham, W., Import, A., & Connolly, D. (2001). Does eye gaze indicate knowledge of false belief: Charting transitions in knowledge. Journal of Experimental Child Psychology, 80, 201–224.


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