Mental Number Representations are Spatially Mapped both by Their Magnitudes and Ordinal Positions

One of the most fundamental effects used to investigate number representations is the Spatial-Numerical Association of Response Codes (SNARC) effect showing that responses to small/large numbers are faster with the left/right hand, respectively. However, in recent years, it is hotly debated whether the SNARC effect is based upon cardinal representation of number magnitude or ordinal representation of number sequence in working memory. However, one problem is that evidence comes from different paradigms, e.g., evidence for ordinal sequences comes usually from experiments, where ordinal sequences have to be learnt and it has been ar-gued that this secondary task triggers the effect. Therefore, in this preregistered study we em-ployed a SNARC task, without secondary ordinal sequence learning, in which we can dissociate ordinal and magnitude accounts by careful manipulation of experimental stimulus sets and com-pare magnitude and ordinal models. The results indicate that even though the observed data is better accounted for by the magnitude model, the ordinal position seems to matter as well. Thus, it appears that the mechanisms described by both accounts play a significant role when mental numbers are temporarily mapped onto space even when no ordinal learning is involved.

A Field Guide to Whole Number Representations in Children’s Books

Trade books are a common resource used to teach children mathematical ideas. Yet, detailed analyses of the mathematics content of such books to determine potential impacts on learning are needed. This study investigated how trade books represent whole numbers. A two-pronged approach was used a) one team documented every way 197 books represented numerical ideas and b) another team used standards to identify ideal representations. A third team validated the traits on 67 books. Greater variation than expected was documented (103 traits identified) and organized into a field guide for researchers to consult to design studies about how particular traits influence number learning. Studies could investigate how a particular trait supports learning or experimentally compare a selected combination of the 45 pictorial, 45 written symbol, 10 tactile, 2 kinesthetic, and 1 auditory trait. Implications for practice include recognizing what representations are present or missing from books used in classrooms. The study also serves as an example of how the field of mathematics education would benefit from adopting structures from disciplinary science, such as field guides, to inform how we organize phenomena of mathematics learning.

Mechanisms Underlying Transfer from Domain-Specific and Domain-General Cognitive Training to Children’s Math Skills

Math achievement is one of the strongest predictors of later life outcomes, and much of what comprises later math is decided by the time children enter kindergarten. Individual differences in precision of approximate representations of number and mapping between non-symbolic and symbolic number representations predict math achievement and honing these representations improves math skills. The goal of this registered report is to disentangle potential mechanisms of transfer. Approximately 324 preschool-aged children will be assigned to one of three, 5-week computerized, teacher-facilitated training conditions to target their approximate number system, symbolic number skills, and executive function to better understand whether changes in approximate number system acuity, mapping between number representations, or attention to number underlie successful transfer of skill training.

Exact number concepts are limited to the verbal count list

Previous studies suggest that mentally representing exact numbers larger than four depends on a verbal count routine (e.g. “one, two, three...”). However, these findings are controversial, as they rely on comparisons across radically different languages and cultures. We tested the role of language in number concepts within a single population – the Tsimane’ of Bolivia – where knowledge of number words varies across individual adults. We used a novel data analysis model to quantify the point at which participants switched from exact to approximate number representations during a simple numerical matching task. The results show that these behavioral switchpoints were bounded by participants’ verbal count ranges; their representations of exact cardinalities were limited to the number words they could recite. Beyond that range, they resorted to numerical approximation. These findings resolve competing accounts of previous findings and provide unambiguous evidence that large exact number concepts are enabled by language.

What is to be learnt? Critical aspects of elementary arithmetic skills

In this paper, we present a way of describing variation in young children’s learning of elementary arithmetic within the number range 1–10. Our aim is to reveal what is to be learnt and how it might be learnt by means of discerning particular aspects of numbers. The Variation theory of learning informs the analysis of 2184 observations of 4- to 7-year-olds solving arithmetic tasks, placing the focus on what constitutes the ways of experiencing numbers that were observed among these children. The aspects found to be necessary to discern in order to develop powerful arithmetic skills were as follows: modes of number representations, ordinality, cardinality, and part-whole relation (the latter has four subcategories: differentiating parts and whole, decomposing numbers, commutativity, and inverse relationship between addition and subtraction). In the paper, we discuss particularly how the discernment of the aspects opens up for more powerful ways of perceiving numbers. Our way of describing arithmetic skills, in terms of discerned aspects of numbers, makes it possible to explain why children cannot use certain strategies and how they learn to solve tasks they could not previously solve, which has significant implications for the teaching of elementary arithmetic.

Neuroethology of number sense across the animal kingdom

ABSTRACT Many species from diverse and often distantly related animal groups (e.g. monkeys, crows, fish and bees) have a sense of number. This means that they can assess the number of items in a set – its ‘numerosity’. The brains of these phylogenetically distant species are markedly diverse. This Review examines the fundamentally different types of brains and neural mechanisms that give rise to numerical competence across the animal tree of life. Neural correlates of the number sense so far exist only for specific vertebrate species: the richest data concerning explicit and abstract number representations have been collected from the cerebral cortex of mammals, most notably human and nonhuman primates, but also from the pallium of corvid songbirds, which evolved independently of the mammalian cortex. In contrast, the neural data relating to implicit and reflexive numerical representations in amphibians and fish is limited. The neural basis of a number sense has not been explored in any protostome so far. However, promising candidate regions in the brains of insects, spiders and cephalopods – all of which are known to have number skills – are identified in this Review. A comparative neuroscientific approach will be indispensable for identifying evolutionarily stable neuronal circuits and deciphering codes that give rise to a sense of number across phylogeny.

Standard Number Representations in the Teaching of Arithmetic

The paper describes the specific approach towards the choice and use of number representations as developed by the project Mathe 2000.