Ordinality and the Nature of Symbolic Numbers

Abstract

The view that representations of symbolic and nonsymbolic numbers are closely tied to one another is widespread. However, the link between symbolic and nonsymbolic numbers is almost always inferred from cardinal processing tasks. In the current work, we show that considering ordinality instead points to striking differences between symbolic and nonsymbolic numbers. Human behavioral and neural data show that ordinal processing of symbolic numbers (Are three Indo-Arabic numerals in numerical order?) is distinct from symbolic cardinal processing (Which of two numerals represents the greater quantity?) and nonsymbolic number processing (ordinal and cardinal judgments of dot-arrays). Behaviorally, distance-effects were reversed when assessing ordinality in symbolic numbers, but canonical distance-effects were observed for cardinal judgments of symbolic numbers and all nonsymbolic judgments. At the neural level, symbolic number-ordering was the only numerical task that did not show number-specific activity (greater than control) in the intraparietal sulcus. Only activity in left premotor cortex was specifically associated with symbolic number-ordering. For nonsymbolic numbers, activation in cognitive-control areas during ordinal processing and a high degree of overlap between ordinal and cardinal processing networks indicate that nonsymbolic ordinality is assessed via iterative cardinality judgments. This contrasts with a striking lack of neural overlap between ordinal and cardinal judgments anywhere in the brain for symbolic numbers, suggesting that symbolic number processing varies substantially with computational context. Ordinal processing sheds light on key differences between symbolic and nonsymbolic number processing both behaviorally and in the brain. Ordinality may prove important for understanding the power of representing numbers symbolically.

Beyond magnitude: Judging ordinality of symbolic number is unrelated to magnitude comparison and independently relates to individual differences in arithmetic

“…Complementary to Turconi and colleagues' (2004) results, Lyons and Beilock (2013), in a functional MRI (fMRI) study using a symbolic magnitude comparison task and an ordering task, no overlapping regions of activation were found.. However, in contrast to Turconi et al (2004) and Lyons and Beilock (2013), Franklin and Jonides (2009) found common activation of the intraparietal sulcus (IPS) for both the magnitude comparison and ordering tasks.…”

“…It is called the reverse distance effect because it exhibits a relationship that is opposite to that of the NDE revealed during number comparison: decreased accuracy and increased reaction time as the numerical distance between the target numbers increases (Franklin & Jonides, 2009). The RDE has been replicated with adult data using the three digit task (Lyons & Beilock, 2013) and has also been demonstrated in the two digit task by Turconi and colleagues (2006).…”

“…For example, much research has investigated how children learn that the Arabic symbol "5" refers to five items (for a review see Ansari, 2008). This referent of symbolic numbers is called the symbol's cardinality, or the number of items in a set that a symbol represents (e.g., Lyons & Beilock, 2013). …”

“…Ordinality refers to the sequencing of number symbols, for example five is the fifth number -it comes after four and before six (Lyons & Beilock, 2013). In order to fully characterize the cognitive nature of symbolic number processing it is critical to learn more about the differences and similarities between ordinal and cardinal processing of symbolic number.…”

A Relational View of Mathematical Concepts

“…Lyons and Beilock (2013) tested subjects performing basic number tasks such as comparison of size and judgment of whether numerals were in order (4, 5, 6 are in order; 4, 6, 5 are not). What they discovered is that similar patterns of brain activity are present when comparing the sizes of numerals, or collections of dots and even when judging if three sets of dots are in order of size (perhaps all these are examples of Goutard's 'empirical' thinking) and that a different pattern of brain activity is aroused by being asked to make (ordinal) comparisons of whether three numerals are in sequence (perhaps requiring more structural or relational awareness).…”