Tight, bidirectional links between mathematical cognition and spatial processing are documented (Cipora et al., 2015). For instance, there is a relationship between mathematical and spatial development (Young et al.). Individuals who perform well on spatial tasks perform well on math tasks as well (Mix et al., 2016, for a review). Individuals with impairments in mathematical processing are more prone to interference on spatial tasks, even when such tasks do not contain numerical components (Eidlin-Levy and Rubinsten). Apart from correlations between spatial and numerical skills and deficits, there is a broad range of phenomena linking numerical and spatial processing referred to as “Spatial-Numerical Associations” (SNAs). SNAs are not just correlates of numerical processing and math skills, it is also supposed that they may be the key to hidden, deep properties of numerical representations and processes operating on them.

In this context, the question not only of how SNAs correlate with numerical development but also what their role in this development is arises. Lifetime development of SNAs and their functional role is therefore a focus of this Research Topic. As one can see from this collection of 27 papers, there is a considerable variety of SNAs. Our guidance through this variety is based on a taxonomy of SNAs we have proposed and extended (Patro et al., 2014; Cipora et al., 2015, 2018b).

The major distinction in this taxonomy of SNAs is between Extension and Direction SNAs. In extension SNAs, numbers are associated with (one- or multi-dimensional) extensions in space: larger numbers are associated with larger extensions. There are two subcategories within extension SNAs. In Approximate Extension SNAs “more” in one domain corresponds to “more” in the other one (e.g., in the numerical Stroop task larger font size is associated with larger numerical magnitude, but there is no relationship between specific font size and specific numerical magnitude). In Exact Extension SNAs, the accuracy of a relationship between number magnitude and spatial extension is examined. The variable of interest is usually the deviation from the exact isomorphy of numerical and spatial magnitude. For instance, in the number line estimation task the deviation of a marked spatial position on the number line and the position corresponding to the exact numerical magnitude given is analyzed.

In case of Direction SNAs, relationships between numbers and specific directions in space (left-right, up-down, near-far) are investigated. The association can either be explicit (as revealed in overt controllable actions such as ordering objects in a certain dimension), or implicit (e.g., a reaction time pattern; Dehaene et al., 1993; Nuerk et al., 2005b). Importantly, several aspects of a number can be associated with space: cardinality, ordinality, functions, and place-value structures.

The rationale for such a taxonomy and distinction is conceptual and is not only aimed at emphasizing peculiarities of different tasks used to measure SNAs. SNA types differ in several fundamental aspects: (1) their relationship with math skill, (2) their potential for being used in interventions, and (3) the extent to which they are prone to situated influences (Cipora et al., 2018c). We are glad to see that papers published in the Research Topic cover all SNA types from the taxonomy (except two types, which we theoretically postulated, but could not find any existing study supporting their existence).

This SNA category was also thoroughly investigated in the past and is very well-covered in the Research Topic. Studies on this SNA type fall into three groups. Firstly, some studies have aimed to investigate the nature of this SNA type by proposing new experimental paradigms and developing new theories. Thompson et al. show that finding a specific page in a book correlates with number line estimation scores (even after controlling other variables), and this new task is postulated to tap into the same processes as the traditional number line estimation task (Siegler, 2009). Van't Noordende et al. show that children dynamically change strategies employed when solving the number line task. Secondly, Dowker and Li also utilized the number line estimation task, and show that although Chinese children outperform their English peers in more language-loaded task, the performance on the number line estimation task does not differ between the groups, suggesting less reliance of this SNA on language (see Helmreich et al., 2011 for opposite effects). Thirdly, Sella et al. as well as Opfer et al. focus on the functional role of this SNA type. Sella et al. show that preschoolers' ability to compare Arabic numbers (but not number words) relates to both ordinal and cardinal knowledge as well as the ability to order numbers in space. In the case of number words, magnitude comparison depended only on ordinal knowledge. Opfer et al. show that accuracy of linear mappings of numbers in the number line estimation task relates to multiple measures of memory for numbers and thus may provide a cornerstone or helpful tool for number memory.

All of these dissociations and impacts observed within this SNA type call for more fine-grained analyses and careful generalizations of conclusions.

Papers referring to this SNA category in the present Research Topic were mostly investigating the nature and structure of this SNA type, but also employed novel paradigms. Kucian et al. argue for the presence of a general magnitude system responsible for both spatial and numerical processing and attempt to observe its developmental trajectory. Their empirical evidence suggests that the structure of such a magnitude system is rather complex and consists of dissociated but related representations of continuous and discrete magnitudes. Rugani et al. provide evidence that numerical primes not only influence reaction time patterns but also affect motor execution, especially the reaching component of the movement. In sum, evidence coming from new task types supports SNA construct and criterion validity.

Many, but not all the papers in this volume fit into the taxonomy. Nevertheless, they provide important insights into our understanding of human number processing. Pixner et al. investigate the development of the understanding of the cardinality of small numbers and zero. Zero is a special number and its role is hard to overestimate (see Nuerk et al., 2004; Nieder, 2016). Thus, it is crucial to understand the developmental trajectories of the understanding of the zero concept. Eventually, in the opinion piece, Fischer et al. provides multiple arguments for why numbers are embodied concepts. While it plays a crucial role in some extension-exact SNAs (especially embodied number line trainings,Dackermann et al., 2017, for a review), its role for some directional SNAs, and especially place value-processing might be more limited.

Where are we with the SNA research? What are SNAs and what is their function in math skills and number processing? Summarizing existing studies, especially considering papers published within this Research Topic, we propose the following agenda for future SNA research.

Firstly, at least for now, we need to suspend our hopes about making conclusions as to what SNAs are in general since they seem to be a too broad range of phenomena and underlying representations. One can characterize some common features of some but not all SNA types. These common features may partially overlap. However, for the moment, the conclusions about SNAs, and their role need to be limited to specific SNA instances.

Secondly, both previous research (see Cipora et al., 2018a) and papers published in this Research Topic (e.g., Lohmann et al.; Bender et al.) show that SNAs are not stable or fixed, but they can be influenced by factors such as language, culture or even simple experimental manipulation. These inter- and intra-individual differences need to be taken into account when the functional role of SNAs is investigated.

We wish to summarize with the claim that space is an extremely important tool for many aspects of number processing and representation. However, not all SNAs (e.g., Directional Implicit Cardinality SNAs) might be equally important for math skills. We still need to learn when space is important for math, but also when math is important for space: after all, in most of the studies we are looking at correlations. The example in which math skills influence strategies and also performance in spatial tasks has already been demonstrated (see Van ‘t Noordende et al.). For future investigation of the reciprocal interaction between SNA and math skills, it is crucial to specify both the SNAs and math skill under scrutiny, and carefully generalize the conclusions to other SNA types.

KC and H-CN wrote the first draft, which was then read, commented, and approved by FD and MH. All authors approved the final version of the manuscript.