Absolute or relative size: What do we perceive when we look at a glass that is half full?

Given that both children and adults struggle with fractions in mathematics education, we investigated the processing of nonsymbolic fractions in a continuous form of part-of-the-whole. Continuous features of nonsymbolic numbers (e.g., the size of dots in an array) were found to influence numerosity judgment, but it should be noted that the (continuous) size of a part can be processed relative to a whole or as an absolute size. This study tested which of these size types (i.e., absolute and relative) influences comparison of parts. In two Stroop-like comparison tasks, we measured the interference of each size type on the processing of the other. In Experiment 1, stimuli were three-dimensional-like partially filled glasses of water. In both tasks, congruent trials (in which the larger absolute size was also the larger part-of-the-whole) were processed more efficiently than incongruent trials (in which the larger absolute size was the smaller part-of-the-whole). In Experiment 2, where stimuli were two-dimensional rectangles, this result was replicated under improved experimental control. We conclude that both absolute size and relative size of a part are automatically processed. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

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.

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.

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.

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.

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.

A generalized fraction: an entity smaller than one on the mental number line.

The representation of fractions in long-term memory (LTM) was investigated by examining the automatic processing of such numbers in a physical comparison task, and their intentional processing in a numerical comparison task. The size congruity effect (SiCE) served as a marker of automatic processing and consequently as an indicator of the access to the primitives of numerical representation in LTM. Mixed pairs composed of a natural number and a fraction showed both a SiCE and a distance effect. The SiCE for mixed pairs was stable across relative sizes of natural numbers compared to the fraction digits (Experiment 4). However, comparing pairs of fractions revealed a strong influence of fractional components: An inverse SiCE was found for pairs of unit fractions (Experiment 1), while no SiCE was found for pairs of non-unit fractions (Experiments 2-3). This leads to the conclusions that: (1) there are no unique representations of distinct fraction values in LTM, and (2) there is a representation of a "generalized fraction" as an "entity smaller than one" that emerges from the notational structure common to all fractions.