A large body of research has shown that human adults are fast and accurate at enumerating arrays of ~1-4 items. This phenomenon has been called subitizing. Above this range, enumeration is slower and less accurate. The subitizing range has been related to individual differences in variables such as mathematical abilities, working memory, etc. The two most common methods for calculating subitizing range today – bilinear fit and sigmoid fit – have their strengths and weaknesses. By combining these two methods, we overcome their biggest limitations and come up with a novel way for calculating Individual Subitizing Range (ISR). This paper introduces this new method as well as empirical studies designed to test the new method. We replicated classic effects from the literature and obtain a high correlation with the sigmoid fit method. This paper includes a Matlab code for easy calculation of ISR as well as a ready-to-use experimental file for testing ISR. We hope that these tools would be of use to researchers studying individual differences in the subitizing range.

A large body of evidence shows that when comparing non-symbolic numerosities, performance is influenced by irrelevant continuous magnitudes, such as total surface area, density, etc. In the current work, we ask whether the weights given to numerosity and continuous magnitudes are modulated by top-down and bottom-up factors. With that aim in mind, we asked adult participants to compare two groups of dots. To manipulate task demands, participants reported after every trial either (1) how accurate their response was (emphasizing accuracy) or (2) how fast their response was (emphasizing speed). To manipulate bottom-up factors, the stimuli were presented for 50 ms, 100 ms or 200 ms. Our results revealed (a) that the weights given to numerosity and continuous magnitude ratios were affected by the interaction of top-down and bottom-up manipulations and (b) that under some conditions, using numerosity ratio can reduce efficiency. Accordingly, we suggest that processing magnitudes is not rigid and static but a flexible and adaptive process that allows us to deal with the ever-changing demands of the environment. We also argue that there is not just one answer to the question ‘what do we process when we process magnitudes?’, and future studies should take this flexibility under consideration.

We make magnitude-related decisions every day, for example, to choose the shortest queue at the grocery store. When making such decisions, which magnitudes do we consider? The dominant theory suggests that our focus is on numerical quantity, i.e., the number of items in a set. This theory leads to quantity-focused research suggesting that discriminating quantities is automatic, innate, and is the basis for mathematical abilities in humans. Another theory suggests, instead, that non-numerical magnitudes, such as the total area of the compared items, are usually what humans rely on, and numerical quantity is used only when required. Since wild animals must make quick magnitude-related decisions to eat, seek shelter, survive, and procreate, studying which magnitudes animals spontaneously use in magnitude-related decisions is a good way to study the relative primacy of numerical quantity versus non-numerical magnitudes. We asked whether, in an animal model, the influence of non-numerical magnitudes on performance in a spontaneous magnitude comparison task is modulated by the number of non-numerical magnitudes that positively correlate with numerical quantity. Our animal model was the Archerfish, a fish that, in the wild, hunts insects by shooting a jet of water at them. These fish were trained to shoot water at artificial targets presented on a computer screen above the water tank. We tested the Archerfish's performance in spontaneous, untrained two-choice magnitude decisions. We found that the fish tended to select the group containing larger non-numerical magnitudes and smaller quantities of dots. The fish selected the group containing more dots mostly when the quantity of the dots was positively correlated with all five different non-numerical magnitudes. The current study adds to the body of studies providing direct evidence that in some cases animals’ magnitude-related decisions are more affected by non-numerical magnitudes than by numerical quantity, putting doubt on the claims that numerical quantity perception is the most basic building block of mathematical abilities.

How children learn basic skills such as discriminating between different quantities and counting? According to the dominant Approximate Number System (ANS) theory,humans are born with the ability to discriminate between discrete quantities (i.e.,numbers). Accordingly, early math curriculum should focus on discrete quantities. This theory guides many early-math curricula worldwide. We provide a review of empirical evidence challenging the ANS theory and introduced a more recent theoretical framework, the Approximate Magnitude System (AMS). This theory suggests that continuous magnitudes (such as area, density, volume, etc.) are more intuitive and acquired earlier then the ability to understand numbers. Using examples from early math education practices, we emphasize and demonstrate the potential benefits of taking the AMS approach and using magnitudes as a scaffolding for understanding numbers and more complex math concepts. Insights gained from studies that combines both cognitive psychology and educational research, with the active participation and contribution of early-math teachers, may be of assistance to both cognitive psychologists, interested in how math abilities develop, and to educators, looking to improve math curriculum. We hope that this article will inspire others to consider research in this direction and start a productive discussion on the role of continuous magnitudes in teaching math.