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In their recent paper on “Challenges in mathematical cognition”, Alcock and colleagues (Alcock et al. [2016]. Challenges in mathematical cognition: A collaboratively-derived research agenda. Journal of Numerical Cognition, 2, 20-41) defined a research agenda through 26 specific research questions. An important dimension of mathematical cognition almost completely absent from their discussion is the cultural constitution of mathematical cognition. Spanning work from a broad range of disciplines – including anthropology, archaeology, cognitive science, history of science, linguistics, philosophy, and psychology – we argue that for any research agenda on mathematical cognition the cultural dimension is indispensable, and we propose a set of exemplary research questions related to it.

In a recent paper on “Challenges in mathematical cognition” (

The question of whether to include cultural aspects is not simply a matter of diverging research interests or a trade-off between comprehensiveness on the one hand and focus on the other. Rather than conceiving of cognition as taking place in the head of individuals and as being separate from the wider human context – and so rendering context seemingly optional for consideration – the two need to be understood as being intrinsically intertwined (

This is not to say that every single aspect of mathematical cognition is culturally mediated. It is generally accepted that humans share with other species two phylogenetically ancestral cognitive systems that may serve as preconditions for the development of numbers and hence for mathematical cognition in general: one for parallel individuation of small quantities in the subitizing range (i.e., for numbers smaller than 4) and one for magnitude approximation (e.g.,

Higher cognitive functions involving number presuppose cognitive tools; one fundamental instance is a conventionalized counting sequence for the exact assessment of quantities. Such tools emerged in human communities to serve specific purposes, they are socially transmitted, and have been developed over cultural evolutionary time in a continuous and iterative process of reproduction and alteration, involving microgenetic, ontogenetic, and sociogenetic changes (for a case study and a theoretical framework, see

In this paper, we review lines of research that, while taking widely different starting points, all converge on the essential role of culture for mathematical thinking. The review includes perspectives on conventional systems of number representations, on the contribution of individual cognitive processes for the reproduction and alteration of these systems, and on their interaction with mathematical cognition. We then identify challenges to the field arising from a neglect of the cultural dimension before formulating a set of questions for future research. As we largely focus on tools for representing and dealing with natural numbers, we use the term

The most fundamental cultural tool for numerical cognition are numeration systems, which come in different modalities: in a verbal modality as number words, in an embodied modality as finger counting or other body-based representations, in a written modality as notational systems, and in other external modalities such as tally sticks, quipus, and abaci. Each system has structural properties that affect its learning and use. For instance, the system of number words in the verbal modality does not provide a durable and manipulable external representation due to the ephemeral nature of vocal utterances, whereas numeration systems in other modalities are more or less durable and open up possibilities for external interactions. Of the numerous ways in which numerical cognition is saturated with culture, we focus on three aspects. We begin by reviewing the origin and variability of numeration systems. Then, we address properties of such systems and their interplay with numerical thinking. Finally, we consider processes of enculturation and the cultural context of numerical cognition.

The early archaeological evidence for numerical cognition is limited and therefore interpretations are hotly debated. The first unambiguous numbers – numerical notations in Mesopotamia and Egypt – do not appear until about 5,000 years ago. Earlier devices possibly used to accumulate, represent, and store numerical information include marine shell beads (Blombos Cave, South Africa, about 75,000 years old), notched bones (Border Cave, South Africa, about 42,000 years old), and hand stencils (Cosquer Cave, France, about 27,000 years old). Determining whether such devices represented numerical information has been challenging, because they might serve social purposes other than dealing with numbers (e.g., beads can be personal ornaments, notches decorations, and handprints part of rituals), and many of the more specific claims about their possible functions (e.g., as lunar calendars) have been discounted as unproven (

Two open questions are related to the role of language and communication: First, what was the interactional context for early symbol formation for number? Did notches or other semiotic forms emerge primarily in non-social settings in which individuals were trying to solve a problem for themselves, or in social settings in which individuals were trying to communicate an intended meaning, or both? A focus on primacy of communication would be consistent with Vygotskian and neo-Vygotskian perspectives (

Processes whereby new systems of numeration emerge in contemporary groups can provide some insights on origins. For instance, when studying an urban community of unschooled candy sellers,

Given that numerical abilities and practices may date back ten-thousands of years, it will be hardly surprising that present-day numeration systems differ tremendously across languages on a range of dimensions (detailed in the next subsection). While this diversity is a hallmark of cultural forces and hence significant in its own right, it also inspires more profound questions on the nature of numerical cognition. Importantly, it puts into perspective the observation that there are also striking commonalities among numeration systems, as attested to by a well-described set of regularities in the world’s languages (

Taking into account the extent of cross-cultural diversity in numeration systems will not only help us to delineate those aspects that are universal from those that are variable; mustering this information to reconstruct the changes in numeration systems over time, both on a small scale (e.g.,

As semiotic tools, numeration systems have properties that vary across languages and cultures (

One tremendously varying property is the extent of the system. Several languages contain number words that do not reach beyond the limit of subitizing, and few languages lack number words and numerical expressions altogether (

An even more important property concerns the representation of numerical information, which may give rise to

However, while such differences in representational format may affect cognitive processing (

A third property particularly relevant for children’s acquisition of the numeration system is the regularity in composition. For instance, in various East Asian languages, number word construction mirrors the regular structure of written Indo-Arabic number notation, whereas word construction in Indo-European languages such as English, German, or French comes with several irregularities (e.g.,

Taking seriously the specific properties of numeration systems is therefore indispensable when investigating the processes involved in numerical cognition, and even more so when seeking ways to support and improve mathematical performance in education. But the system-specific properties are by no means the only way in which numerical cognition is intrinsically linked to culture; arguably even more relevant is the cultural context in which numerical cognition takes place.

Acquisition of the numeration systems themselves (such as the sequence of number words or a conventionalized finger counting pattern) as well as an understanding of their precise numerical meaning requires participation in enculturating practices of collective life, whether in counting games or other activities with older siblings or adults (and later on also in schooling). Such collective practices vary markedly within and across cultural groups. For example, in a study of working and middle class families and their 2.5- and 4-year-old children,

The range of number word functions is also attested to by other studies, some of which point to the progressive arithmetization of number words to serve cardinality functions in early development (e.g.,

Learning to count and what to do with this ability also depends on how numbers are valued in a given community (

In sum, we suggest that a careful study of the processes of enculturation and the cultural context in which numerical cognition takes place are important. This work will provide us with crucial insights on the active role that the child plays in the learning process itself, and in shaping the practices of collective life that guide them, rather than being a simple recipient of math education.

Even though

Taking such WEIRD populations as reference may seem justified if one’s agenda is aimed at the education systems in North American and Western European countries, with “success in mathematics” being defined as mastery of the respective school curriculum. To the extent, however, that it is also intended as a research agenda on mathematical cognition more generally, neglecting the cultural nature of the phenomena investigated is problematic. If one’s goal is “a broad approach to understanding human mathematical cognition” (

Taking culture more seriously entails important implications. Most generally, it helps us elucidate the nature of mathematical thinking (the first of the six broad topics identified by

What are the regulative processes that constrain and enable the emergence, reproduction, and alteration of numeration systems in human communities, and are these processes similar across different communities? For example, in his studies of economic exchange in Oksapmin communities,

If these regulations are fundamental to the reproduction and alteration of numerical forms, how might they be manifest across situations when authority and power are similar or different across interlocutors, whether in adult-child interactions, peer interactions, or interactions between adult members of similar social positions? There is such a diversity of community-specific collective practices that the possibilities for detailed analysis are extraordinary. Examples include deaf children of hearing parents communicating with one another about number and new forms of representation that emerge as communication (e.g.,

Language appears to have a profound effect on the thinking of individuals (e.g.,

A particularly illuminative example is the bootstrapping process in which individuals deploy problems that involve multiple conceptual resources supporting their transcendence of particular cultural forms of representation. For instance, more than half of the world’s population speaks more than one language, with numeration systems in these languages most likely differing from each other in at least subtle, if not fundamental ways. Moreover, even if monolingual, most people nowadays still use several numeration systems in parallel, such as the sequence of number words in one’s mother tongue and one or more notational systems (e.g., Indo-Arabic digits and Roman numerals). While serving partly different functions, these systems differ significantly in how they represent the same type of information (

A related question concerns the treatment of external tools. When solving numerical problems, the work is distributed over the material tools with which one engages and cognitive (including cortical) processes recruited for numerical activities. When solving an arithmetic problem, in which ways then do these processes differ depending on whether one deploys a representational artefact like a digital calculator, paper and pencil, or mental arithmetic? Such variation in the way in which number is used in everyday life are commonplace, but understanding how cognition becomes distributed over the artefacts that we use and the cognitive processes involved with number (

What is the nature of and the variation in enculturating practices across communities?

Answers to questions like these will eventually allow us to modify and improve our models of numerical cognition such as the triple-code model described by

And finally, conceiving of cultural diversity in numeration systems and other ethno-mathematical patterns (for examples see

Culture is not only “out there”, in some exotic corners of the world, but everywhere around us and inherent in the material and conceptual systems we use and the practices in which they are embedded. We

Authors are listed in alphabetical order. This does by no means imply any order with regard to relevance. In fact, the manuscript is joint work, and all authors have contributed equally to its formation and current form.

This work was supported by a SPIRE grant of the University of Bergen for the workshop

We thank all participants of the SPIRE-funded workshop

The authors have declared that no competing interests exist.