Children's Understanding of the Natural Numbers' Structure.

When young children attempt to locate numbers along a number line, they show logarithmic (or other compressive) placement. For example, the distance between "5" and "10" is larger than the distance between "75" and "80." This has often been explained by assuming that children have a logarithmically scaled mental representation of number (e.g., Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, ; Siegler & Opfer, ). However, several investigators have questioned this argument (e.g., Barth & Paladino, ; Cantlon, Cordes, Libertus, & Brannon, ; Cohen & Blanc-Goldhammer, ). We show here that children prefer linear number lines over logarithmic lines when they do not have to deal with the meanings of individual numerals (i.e., number symbols, such as "5" or "80"). In Experiments 1 and 2, when 5- and 6-year-olds choose between number lines in a forced-choice task, they prefer linear to logarithmic and exponential displays. However, this preference does not persist when Experiment 3 presents the same lines without reference to numbers, and children simply choose which line they like best. In Experiments 4 and 5, children position beads on a number line to indicate how the integers 1-100 are arranged. The bead placement of 4- and 5-year-olds is better fit by a linear than by a logarithmic model. We argue that previous results from the number-line task may depend on strategies specific to the task.

Relational categories are more mutable than entity categories.

Across three experiments, we explore differences between relational categories-whose members share common relational patterns-and entity categories, whose members share common intrinsic properties. Specifically, we test the claim that relational concepts are more semantically mutable in context, and therefore less stable in memory, than entity concepts. We compared memory for entity nouns and relational nouns, tested either in the same context as at encoding or in a different context. We found that (a) participants show better recognition accuracy for entity nouns than for relational nouns, and (b) recognition of relational nouns is more impaired by a change in context than is recognition of entity nouns. We replicated these findings even when controlling for factors highly correlated with relationality, such as abstractness-concreteness. This suggests that the contextual mutability of relational concepts is due to the core semantic property of conveying relational structure and not simply to accompanying characteristics such as abstractness. We note parallels with the distinction between nouns and verbs and suggest implications for lexical and conceptual structure. Finally, we relate these patterns to proposals that a deep distinction exists between words with an essentially referential function and those with a predicate function.

Can statistical learning bootstrap the integers?

This paper examines Piantadosi, Tenenbaum, and Goodman's (2012) model for how children learn the relation between number words ("one" through "ten") and cardinalities (sizes of sets with one through ten elements). This model shows how statistical learning can induce this relation, reorganizing its procedures as it does so in roughly the way children do. We question, however, Piantadosi et al.'s claim that the model performs "Quinian bootstrapping," in the sense of Carey (2009). Unlike bootstrapping, the concept it learns is not discontinuous with the concepts it starts with. Instead, the model learns by recombining its primitives into hypotheses and confirming them statistically. As such, it accords better with earlier claims (Fodor, 1975, 1981) that learning does not increase expressive power. We also question the relevance of the simulation for children's learning. The model starts with a preselected set of15 primitives, and the procedure it learns differs from children's method. Finally, the partial knowledge of the positive integers that the model attains is consistent with an infinite number of nonstandard meanings-for example, that the integers stop after ten or loop from ten back to one.

From numerical concepts to concepts of number.

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas.

Do children learn the integers by induction?

According to one theory about how children learn the meaning of the words for the positive integers, they first learn that "one," "two," and "three" stand for appropriately sized sets. They then conclude by inductive inference that the next numeral in the count sequence denotes the size of sets containing one more object than the size denoted by the preceding numeral. We have previously argued, however, that the conclusion of this Induction does not distinguish the standard meaning of the integers from nonstandard meanings in which, for example, "ten" could mean set sizes of 10, 20, 30,... elements. Margolis and Laurence [Margolis, E., & Laurence, S. (2008). How to learn the natural numbers: Inductive inference and the acquisition of number concepts. Cognition, 106, 924-939] believe that our argument depends on attributing to children "radically indeterminate" concepts. We show, first, that our conclusion is compatible with perfectly determinate meanings for "one" through "three." Second, although the inductive inference is indeed indeterminate - which is why it is consistent with nonstandard meanings - making it determinate presupposes the constraints that the inference is supposed to produce.

Giving the boot to the bootstrap: how not to learn the natural numbers.

According to one theory about how children learn the concept of natural numbers, they first determine that "one", "two", and "three" denote the size of sets containing the relevant number of items. They then make the following inductive inference (the Bootstrap): The next number word in the counting series denotes the size of the sets you get by adding one more object to the sets denoted by the previous number word. For example, if "three" refers to the size of sets containing three items, then "four" (the next word after "three") must refer to the size of sets containing three plus one items. We argue, however, that the Bootstrap cannot pick out the natural number sequence from other nonequivalent sequences and thus cannot convey to children the concept of the natural numbers. This is not just a result of the usual difficulties with induction but is specific to the Bootstrap. In order to work properly, the Bootstrap must somehow restrict the concept of "next number" in a way that conforms to the structure of the natural numbers. But with these restrictions, the Bootstrap is unnecessary.