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Many recent studies in numerical cognition have moved beyond the use of purely chronometric techniques in favor of methods which track the continuous dynamics of numerical processing. Two examples of such techniques include eye tracking and hand tracking (or computer mouse tracking). To reflect this increased concentration on continuous methods, we have collected a group of 5 articles that utilize these techniques to answer some contemporary questions in numerical cognition. In this editorial, we discuss the two paradigms and provide a brief review of some of the work in numerical cognition that has profited from the use of these techniques. For both methods, we discuss the past research through the frameworks of single digit number processing, multidigit number processing, and mental arithmetic processing. We conclude with a discussion of the papers that have been contributed to this special section and point to some possible future directions for researchers interested in tracking the continuous dynamics of numerical processing.

The last 20 years have seen a great increase in the interest surrounding fundamental questions in numerical and mathematical cognition. Along with this increased interest, methods for approaching these questions have increased in complexity and diversity (c.f.,

Though chronometric methods still comprise a large portion of the current work in numerical cognition, there is a burgeoning literature that uses continuous, dynamic measures of cognitive processing to work on open problems in our field. Such methods include eye tracking (

The added value of these methods is crucial in order to advance the understanding of the mechanisms underlying simple number processing and mental arithmetic. For simple number processing (e.g., number comparison), continuous dynamical methods have allowed us to tackle a number of difficult problems, such as probing the temporal dynamics of mental number line activation (

Against this background, we have edited a special collection of articles for the

In this review and editorial, we have two immediate purposes. First, we wish to provide a brief review of the two dominant continuous dynamic methods in numerical cognition: eye tracking and hand movement tracking. Second, we will briefly describe the contributions of the authors in this special collection and conclude with an evaluation of the current state of affairs with respect to continuous methods in numerical cognition.

Eye tracking studies are based on the online recording of eye movements during an experimental task. These movements, consisting of rapid jumps called

Though eye tracking has a long history (

Early research in this area focused on basic aspects of symbolic number processing, such as processing speed and reading time. For example,

While most of the early research on the mental number line was based on chronometric analysis, several recent studies have shown that eye movements also show this spatial-numerical association. For example,

Whereas single-digit number tasks seem to evoke a tight correspondence between directional eye movement and spatial number representation, studies involving numbers with multiple components (e.g., multidigit Arabic numerals, such as 24, and fractions, such as 3/7) have used eye tracking to uncover other interesting results. One of the early studies in this regard was

Along a similar line of research,

Several researchers have also used eye tracking to study fractions, for which questions similar to those pursued for two-digit numbers can be asked (e.g.,

Other recent studies have used eye movements to infer participants’

In a similar vein, the analysis of eye movements also revealed different strategies used in a number line task (

Compared to single digit and multidigit number representation, the use of eye tracking to investigate mental arithmetic processes has received less attention (see

Other researchers have used eye tracking to provide a more sensitive measure of strategy use in mental arithmetic (e.g.,

Compared to eye tracking methods, the use of hand movement measures in cognitive research has been a much more recent phenomenon. Consequently, the collection of work in this area is much less developed. Much of this research paradigm owes its origin to the groundbreaking work of

Though the work of

In the field of numerical and mathematical cognition, the types of problems investigated with hand and/or mouse tracking has largely mirrored that of eye tracking. Indeed, the recent literature can be broadly classified as we did above with our review of the eye tracking literature. To this end, we will describe some of the recent papers that have used hand tracking to investigate numerical and mathematical processes.

Some of the first studies to use the hand tracking methodology in the context of numerical cognition were aimed at investigating phenomena with single digit number stimuli. The first of these was

This led to a follow-up study by

Another context in which computer mouse tracking has been used to arbitrate between competing theories of numerical representation is the size congruity effect (

In all, these studies demonstrate one of the primary advantages of the hand tracking paradigm. Specifically, recorded hand trajectories can provide rich data to arbitrate between competing theories of numerical representation. In addition to these studies, several other recent experiments have used computer mouse tracking to investigate the nature of basic number processing, including investigations of the dynamics of the SNARC effect (

Similar efforts with hand tracking have been employed to attack problems in multidigit number processing.

Similarly,

As with eye tracking, hand tracking has also been used to investigate the dynamics of fraction representations.

The literature employing hand tracking to understand mental arithmetic processes is considerably less developed compared to numerical representations. However, there are a few important first strides in this area. The first such study to examine manual dynamics of mental arithmetic was

This conclusion is also supported by the recent work of

The papers that we have collected for this special section of

Two of the papers used some version of hand tracking to investigate number processing. The article by

The second paper that used hand tracking to investigate numerical processing was offered by

Along with these papers on hand tracking, we also collected two papers that used eye tracking to study the dynamics of number processing. As described earlier, decisions with fractions pose special cognitive processing demands as a symbolic fraction represents a (rather abstract) relative magnitude. People often struggle with interference between holistic and decomposed strategies which treat the numerator and denominator either as a single whole number or as separate quantities.

The second paper of our collection that uses eye tracking reports on research by

Finally, the paper by

The papers collected in this special section add to an ever-growing literature focused on tracking the continuous dynamics of numerical representation. As this field of inquiry is still relatively young, there is plenty of room for researchers to apply these methods in a wide variety of empirical contexts, ranging from description to theory testing. The next decade will be an exciting time indeed, as the techniques and tools for measuring continuous dynamics in numerical representation will mature and become more widely available as time progresses. We think the present collection of papers will form a core set of work that will be widely cited in many of these future studies. We further hope that the ideas presented in these papers will generate a wide variety of new questions and techniques for future research.

We thank the Editor in Chief and the Editorial Board for their support in the development of this Special Section of the Journal of Numerical Cognition. We appreciate the efforts of the authors and reviewers and thank them for their positive contributions to this work.

M. H. was funded by the Swiss National Science Foundation [grant number PZ00P1_167995].

The authors have declared that no competing interests exist.