Development of Preschoolers' Understanding of Zero

Is zero more than nothing? Relations between concepts of zero and integer understanding

The concept of zero is challenging for many children. This study investigated conceptions of zero in fifth- through seventh-grade children (N = 72) in the United States and examined how children's conceptions of zero related to other aspects of their integer knowledge. At the outset of the study, many participants held a null conception of zero (i.e., zero as "nothing"), and some participants held a conception of zero as the symmetry point between the positive and negative integers. Participants in higher grades were more likely to hold a symmetry conception. We hypothesized that participants' conceptions of zero would be related to other aspects of their integer knowledge. Relative to participants who held a null conception of zero, participants who held a symmetry conception demonstrated greater knowledge of the additive inverse principle (for every number x, there exists an inverse, -x, such that the two numbers sum to zero) and greater integer arithmetic skill. Conceptions of zero were not related to integer magnitude understanding. We also examined whether a brief lesson focusing on zero as the symmetry point would lead to shifts in participants' conceptions of zero and gains in understanding of the additive inverse principle. Participants were randomly assigned to receive a lesson about zero as null or zero as the symmetry point between positive and negative integers. Relative to the lesson about zero as null, the lesson about zero as the symmetry point did not lead to substantial changes in conceptions of zero or additive inverse knowledge.

Symbolic and non-symbolic representations of numerical zero in the human brain

Representing the quantity zero as a symbolic concept is considered a unique achievement of abstract human thought.1,2 To conceptualize zero, one must abstract away from the (absence of) sensory evidence to construct a representation of numerical absence: creating "something" out of "nothing."2,3,4 Previous investigations of the neural representation of natural numbers reveal distinct numerosity-selective neural populations that overlap in their tuning curves with adjacent numerosities.5,6 Importantly, a component of this neural code is thought to be invariant across non-symbolic and symbolic numerical formats.7,8,9,10,11 Although behavioral evidence indicates that zero occupies a place at the beginning of this mental number line,12,13,14 in humans zero is also associated with unique behavioral and developmental profiles compared to natural numbers,4,15,16,17 suggestive of a distinct neural basis for zero. We characterized the neural representation of zero in the human brain by employing two qualitatively different numerical tasks18,19 in concert with magnetoencephalography (MEG) recordings. We assay both neural representations of non-symbolic numerosities (dot patterns), including zero (empty sets), and symbolic numerals, including symbolic zero. Our results reveal that neural representations of zero are situated along a graded neural number line shared with other natural numbers. Notably, symbolic representations of zero generalized to predict non-symbolic empty sets. We go on to localize abstract representations of numerical zero to posterior association cortex, extending the purview of parietal cortex in human numerical cognition to encompass representations of zero.10,20.

Counting many as one: Young children can understand sets as units except when counting

Young children frequently make a peculiar counting mistake. When asked to count units that are sets of multiple items, such as the number of families at a party, they often count discrete items (i.e., individual people) rather than the number of sets (i.e., families). One explanation concerns children's incomplete understanding of what constitutes a unit, resulting in a preference for discrete items. Here we demonstrate that children's incomplete understanding of counting also plays a role. In an experiment with 4- and 5-year-old children (N = 43), we found that even if children are able to name sets, group items into sets, and create one-to-one correspondences with sets, many children are nevertheless unable to count sets as units. We conclude that a nascent understanding of the abstraction principle of counting is also a cause of some children's counting errors.