Arava Y. Kallai

Holistic representation of negative numbers is formed when needed for the task

Past research suggested that negative numbers are represented in terms of their components—the polarity marker and the number (e.g., Fischer & Rottmann, 2005; Ganor-Stern & Tzelgov, 2008). The present study shows that a holistic representation is formed when needed for the task requirement. Specifically, performing the numerical comparison task on positive and negative numbers presented sequentially required participants to hold both the polarity and the number magnitude in memory. Such a condition resulted in a holistic representation of negative numbers, as indicated by the distance and semantic congruity effects. This holistic representation was added to the initial components representation, thus producing a hybrid holistic-components representation.

When meaningful components interrupt the processing of the whole: the case of fractions.

Numerical fractions are composed of a numerator and a denominator that are natural numbers. These components influence processing of the fraction. This study was conducted to test whether eliminating the fractional components would result in the processing of fractions as unique numerical entities. Participants that learned to relate fractional values to arbitrary figures in a training task showed automatic processing of the numerical values of the new figures. The processing of fractions written in regular form improved following training, but did not show automatic processing. The results suggest that eliminating the influence of the fractional components allowed individual fractions to be represented in long-term memory.

The place-value of a digit in multi-digit numbers is processed automatically.

The automatic processing of the place-value of digits in a multi-digit number was investigated in 4 experiments. Experiment 1 and two control experiments employed a numerical comparison task in which the place-value of a non-zero digit was varied in a string composed of zeros. Experiment 2 employed a physical comparison task in which strings of digits varied in their physical sizes. In both types of tasks, the place-value of the non-zero digit in the string was irrelevant to the task performed. Interference of the place-value information was found in both tasks. When the non-zero digit occupied a lower place-value, it was recognized slower as a larger digit or as written in a larger font size. We concluded that place-value in a multi-digit number is processed automatically. These results support the notion of a decomposed representation of multi-digit numbers in memory.

Decimals are not processed automatically, not even as being smaller than one.

Common fractions have been found to be processed intentionally but not automatically, which led to the conclusion that they are not represented holistically in long-term memory. However, decimals are more similar to natural numbers in their form and thus might be better candidates to be holistically represented by educated adults. To test this hypothesis, we investigated the automatic processing of decimals by college students in 4 experiments. When decimals were presented in a familiar form (e.g., 0.3, 0.05) the length of the stimuli (i.e., the number of digits) dominated performance rather than the decimal value. When controlling for the number of digits and their location within the digit string, using the place-value task, decimals were not processed automatically in either a numerical comparison task or a physical comparison task. Under the same conditions, natural numbers were processed automatically. We conclude that decimals are not represented holistically. Results of mixed pairs of a decimal and a natural number suggest that, unlike common fractions, decimals are not automatically perceived as smaller than natural numbers. We conclude that decimal place-values (e.g., tenths, hundredths) are not represented well enough to be automatically activated, and we discuss possible explanations.

Size Perception and the Foundation of Numerical Processing

Research in numerical cognition has led to a widely accepted view of the existence of innate, domain-specific, core numerical knowledge that involves the intraparietal sulcus in the brain. Much of this research has revolved around the ability to perceive and manipulate discrete quantities (e.g., enumeration of dots). We question several aspects of this accepted view and suggest that continuous noncountable dimensions might play an important role in the development of numerical cognition. Accordingly, we propose that a relatively neglected aspect of performance—the ability to perceive and evaluate sizes or amounts—might be an important foundation of numerical processing. This ability might even constitute a more primitive system that has been used throughout evolutionary history as the basis for the development of the number sense and numerical abilities.

Mental arithmetic activates analogic representations of internally generated sums

The internal representation of numbers generated during calculation has received little attention. Much of the mathematics learning literature focuses on symbolic retrieval of math facts; in contrast, we critically test the hypothesis that internally generated numbers are represented analogically, using an approximate number system. In an fMRI study, the spontaneous processing of arithmetical expressions was tested. Participants passively viewed a sequence of double-digit addition expressions that summed to the same number. Adaptation was found in number-related regions in a fronto-parietal network. Following adaptation, arrays of dots were introduced, differing in their numerical distance from the sum of the addition expressions. Activation in voxels that showed adaptation to a repeated sum was also sensitive to the distance of the dot quantity from the sum. We conclude that participants exhibited adaptation to an internally generated number, that adapted representations were analogic in nature, and that these analogic representations may undergird arithmetic calculation.

What Do We Measure When We Measure Magnitudes

Using nonsymbolic representations of magnitudes (eg, arrays of items) is a common way to study the processing of the number of items in a set. This ability is considered to be very basic, innate, and automatic. However, arrays of items always include noncountable continuous magnitudes such as the density of the items, their total surface area, and so forth. These continuous magnitudes can affect performance when comparing numerosities (eg, where are more dots?) and when comparing proportions (eg, where are more blue dots compared with yellow dots?). Numerosities and continuous magnitudes are so inherently correlated that it is impossible to be certain that the required task indeed measured what we intended to measure (ie, numerosity or proportion processing). Researchers in the field are aware of this problem and employ different methods to reduce possible influences that continuous magnitudes might have on performance. In this chapter we will introduce such methods and discuss the different and often contradictory conclusions they lead to. We will also suggest—in light of recent studies—that great benefit can come from directly studying (1) how continuous magnitudes affect comparison of numerosity and proportions, and (2) the reciprocal interaction between numerosities and continuous magnitudes. Studying these issues during different stages of development and in typically and atypically developed adults can shed new light on the most elementary building blocks of our mathematical abilities.