About the link between numbers and the eyes

How do we process elements of an arithmetic problem? How do innumerate adults or dyscalculic children differ in their strategies when solving numerical tasks? Such questions remain unclear when looking at reaction times or accuracy alone. Analysis of eye movements during numerical cognition can provide fundamental insight in the underlying cognitive processes. Eye tracking enables an immediate access to participant’s mental processes by capturing the focus of attention with high spatial and temporal resolution,Footnote 1 in a noninvasive and unobtrusive way (e.g., Van Gompel 2007).

Beside these obvious advantages of eye tracking as a research method, studying the possible link between numbers and eyes is particularly interesting for the following reasons: Numbers are cognitively represented in a spatial format, the so-called mental number line, with small numbers being located to the left of large numbers (in Western cultures). Spatial–numerical associations have been found to influence performance in various tasks, from single-digit categorization to mental arithmetic (see Fischer and Shaki 2014, for a recent review) and can be observed across effectors, including hands, feet (Schwarz and Müller 2006), and also the eyes (e.g., Schwarz and Keus 2004). Based on the hypothesis of a spatially oriented mental number line, it has been suggested that perceiving numbers causes shift in spatial attention (Fischer et al. 2003), and such shifts can be observed in eye movements (Hartmann et al. 2015; Knops et al. 2009; Loetscher et al. 2008, 2010). Moreover, numbers are represented within a generalized magnitude system located in the parietal cortex, allowing for interactions with other action-related magnitude information, such as object size and time (e.g., Bueti and Walsh 2009). The parietal cortex is also critically involved in the control of eye movements (e.g., Cohen and Andersen 2002; Nyffeler et al. 2008). Interestingly, the perception of numerosity can be modulated by eye movements (Binda et al. 2011, 2012; Irwin and Thomas 2007). Thus, beyond a conceptual link between numbers and space, eye movement planning and number processing interact at the neuronal level (Burr et al. 2010).

Despite these striking links between eye movements and number processing, there are still relatively few studies using eye tracking to advance the understanding of numerical cognition. Some of the most important findings and promising future directions are discussed in the next sections.Footnote 2

Eye movements provide insights into basic aspects of numerical cognition

The development of the number sense

A classical task for assessing a child’s number representation is the “number-to-position” task where a given number needs to be mapped onto an appropriate spatial position on a horizontal line, usually by pointing or marking (Siegler and Opfer 2003). Eye tracking revealed different age-related strategies used to solve this task (Schneider et al. 2008): Young children start from the endpoint of the line and count upward (or downward) in whole units until they reached the target position. This task strategy is reflected by a linear distribution of fixations along the line (from one end to the final position). Older children start to count from the midpoint when the target position is closer to the midpoint than to the endpoints of the line. An increased use of such a midpoint strategy is associated with greater arithmetic competence (Schneider et al. 2008). Finally, adults use a fully proportion-based strategy, and the translation of numerical information to space is a rapid process: Fixations are distributed along reference points that are associated with the proportion-based strategy (e.g., endpoint, midpoint, points between the endpoint and midpoint), and early eye position on the line predicts final estimation performance (Sullivan et al. 2011). Thus, eye movements contribute to the understanding of the number sense by identifying specific strategies that are associated with different levels of its development.

Numerical cognition in adults

How do humans extract and comprehend information from digit strings? Eye tracking significantly contributed to the understanding of some basic aspects of numerical cognition in adults, for example how multi-digit numbers are perceived (e.g., Moeller et al. 2009), and how elements of an arithmetic problem are processed (Green et al. 2007; Moeller et al. 2011; Schneider et al. 2012; Susac et al. 2014). For example, multi-digit numbers are not perceived as an entity but are decomposed and processed in a parallel fashion (Moeller et al. 2009), as evident by more and earlier fixations on the largest entity (decade) and fewer fixations on the smaller entity (unit digit). When two two-digit numbers are added, fixation duration increases for the unit digits of the summands (Moeller et al. 2011), because the unit digits determine whether or not a carry is needed (i.e., when the sum of the unit digits is larger than 10). Interestingly, longer fixations for carry problems are evident already for the first fixation (i.e., during the initial encoding of the problem), suggesting that computation of the unit sum and the recognition whether or not a carry operation is required occur very early in the arithmetic process (Moeller et al. 2011). Moreover, when possible solutions of an arithmetic problem are provided along with the problem, fixation paths revealed that most participants scan these solutions before or during solving the problem (Susac et al. 2014).

Eye movements also contribute to the understanding of the spatial nature of number representation. Initially conceptualized as horizontal line, new evidence is accumulating that the mental number line has also a vertical orientation (e.g., Hartmann et al. 2012), but results from bilateral responses are not always clear and may depend on the exact response mapping rules (cf. Hartmann et al. 2014; Müller and Schwarz 2007). Clear evidence for a concurrent involvement of both horizontal and vertical spatial dimensions in number representation comes from eye movements. Loetscher et al. (2010) analyzed spontaneous eye movements in the dark when participants called out numbers in a random sequence. They found that the direction and magnitude of saccades were correlated with the relative numerical distance from one number to the next, in a small-left/down and large-right/up manner. Using a similar approach, we found that eye gaze on a blank screen shifts rightward and upward during upward counting (Hartmann et al. in revision; see also Hartmann et al. 2015). These results suggest that spatial concepts of magnitudes such as “more is right” (as derived from the horizontal mental number line) and “more is up” as derived from sensorimotor experiences (cf. Lakoff and Johnson 1980) are more than metaphors; they are actually reflected in eye movements during number processing.

Outlook

As described above, eye movement patterns capture different strategies in the number-to-position task (Schneider et al. 2008) and may provide a more fine-grained picture of the number sense than the final estimation alone. Further establishing the exact properties of age-specific strategies in the number-to-position task (and other tasks) will enable the use of eye tracking as valid tool to diagnose children with mathematical disabilities. Preliminary evidence in this direction has been reported by van Viersen et al. (2013), showing that a child with developmental dyscalculia made less fixations on the reference points associated with a proportion-based strategy (e.g., endpoints, midpoint). Particularly, the child had a high proportion of saccades that could not be attributed to a specific strategy, as if the child did not know where to look at. Thus, a specific training of eye movements (e.g., guiding eye movements toward critical reference points) could help children to develop a more efficient task strategy and eventually improve their number sense.

It should also be highlighted that previous research investigated normal adults and children with and without impairments in number processing. Future studies could investigate the other end of the spectrum: high-performing individuals and mathematicians. Mathematicians differ in their task strategies (e.g., Dowker 1992) and eventually in their spatial representation of numbers (Hoffmann et al. 2014). Eye movement data can provide a more reliable description of the applied task strategy than participants’ metacognitive reports (Susac et al. 2014). Eye movement patterns could therefore help to further clarify differences in task strategy between high and average performers, and this knowledge can then be used to improve performance of average performers.

Another interesting question for future research is whether the eye movements accompanying number processing in the absence of a visual stimulus (Hartmann et al. 2015; Hartmann et al. in preparation; Loetscher et al. 2010) are only epiphenomenal, or whether they play a functional role for the understanding of number magnitudes and arithmetic processes. For example, do rightward eye movements facilitate mental addition because the direction of gaze is congruent with the assumed mental rightward movement along the number line? This question can be addressed by explicitly manipulating eye movements during number processing in a congruent or incongruent way (cf. Wiemers et al. 2014) or by comparing numerical performance under free-viewing conditions versus inhibited viewing conditions. Inhibition of free viewing can, for example, be achieved by instructing participants to fixate a static object, by temporarily paralyzing eye muscle, or by inducing involuntary eye movements through optokinetic (e.g., Ranzini et al. 2014) or caloric (e.g., Ferrè et al. 2013) vestibular stimulation. Impaired numerical performance in incongruent or inhibited eye movement conditions would suggest a role of the eyes as “slave” system for the mind that “spatializes” abstract thoughts to support understanding and manipulation of its contents.

This paper showed how eye movements can reveal new insights into number processing, and will hopefully motivate researchers to make use of the added value of eye tracking in the fascinating and fast-growing field of numerical cognition.