This commentary reviews each of the three content chapters in the integers section and offers questions to promote further discussion. In addition to the themes raised in the three chapters, I introduce the role of formal mathematical structure in generalizing systems of number, from natural numbers to integers, and analogously, from real numbers…

In this commentary, I discuss why, despite the existence of gradience in phonetics and phonology, there is still a need for abstract representations. Most proponents of exemplar models assume multiple levels of abstraction, allowing for an integration of the gradient and the categorical. Ben Ambridge's dismissal of generative models such as…

During language acquisition children generalise at multiple layers of granularity. Ambridge argues that abstraction-based accounts suffer from lumping (over-general abstractions) or splitting (over-precise abstractions). Ambridge argues that the only way to overcome this conundrum is in a purely exemplar/analogy-based system in which…

Individuals draw conclusions about possibilities from assertions that make no explicit reference to them. The model theory postulates that assertions such as disjunctions refer to possibilities. Hence, a disjunction of the sort, "A or B or both," where "A" and "B" are sensible clauses, yields mental models of an…

This article reviews two aspects of human learning: (1) people draw inferences that appear to rely on hierarchical conceptual representations; (2) some categories are much easier to learn than others given the same number of exemplars, and some categories remain difficult despite extensive training. Both of these results are difficult to reconcile…

Humans are sensitive to the properties of individual items, and exemplar models are useful for capturing this sensitivity. I am a proponent of an extension of exemplar-based architectures that I briefly describe. However, exemplar models are very shallow architectures in which it is necessary to stipulate a set of primitive elements that make up…

The syntactic behaviour of human beings cannot be explained by analogical generalization on the basis of concrete exemplars: analogies in surface form are insufficient to account for human grammatical knowledge, because they fail to hold in situations where they should, and fail to extend in situations where they need to. [For Ben Ambridge's…

Ambridge argues that the existence of exemplar models for individual phenomena (words, inflection rules, etc.) suggests the feasibility of a unified, exemplars-everywhere model that eschews abstraction. The argument would be strengthened by a description of such a model. However, none is provided. I show that any attempt to do so would immediately…

In this response to commentators, I agree with those who suggested that the distinction between exemplar- and abstraction-based accounts is something of a false dichotomy and therefore move to an abstractions-made-of-exemplars account under which (a) we store all the exemplars that we hear (subject to attention, decay, interference, etc.) but (b)…

The authors argue that Ambridge's radical exemplar account of language cannot clearly explain all syntactic priming evidence, such as inverse preference effects ("greater" priming for less frequent structures), and the contrast between short-lived lexical boost and long-lived abstract priming. Moreover, without recourse to a level of…

This commentary critiques Ambridge's radical exemplar model of language acquisition using research from the Longitudinal Study of Early Language, which has tracked the language development of 30+ children with Autism Spectrum Disorders (ASD) since 2002. This research has demonstrated that the children's capacity for abstraction at the grammatical…

The visual-manual modality of sign languages renders them a unique test case for language acquisition and processing theories. In this commentary the authors describe evidence from signed languages, and ask whether it is consistent with Ambridge's proposal. The evidence includes recent research on collocations in American Sign Language that reveal…

The three target articles presented in this special issue converged on an emerging theme: the importance of spatial proportional reasoning. They suggest that the ability to map between symbolic fractions (like 1/5) and nonsymbolic, spatial representations of their sizes or "magnitudes" may be especially important for building robust…

Ambridge argues that there is widespread agreement among child language researchers that learners store linguistic abstractions. In this commentary the authors first argue that this assumption is incorrect; anti-representationalist/exemplar views are pervasive in theories of child language. Next, the authors outline what has been learned from this…

The papers in this issue describe recent collaborative research into the role of inhibition of intuitive thinking in mathematics education. This commentary reflects on this research from a mathematics education perspective and draws attention to some of the challenges that arise in collaboration between research fields with different cultures,…