Children fall for a surprisingly simple numerical illusion — and it grows stronger with age

A new study published in Developmental Science suggests that both children and adults are susceptible to a visual illusion that makes connected dots appear less numerous than unconnected ones. The illusion, known as the “connectedness illusion,” appears as early as age 5 and tends to grow stronger into adulthood. Interestingly, the individuals who show the most accurate visual number sense are also the most vulnerable to the illusion.

The findings provide insight into how our brains approximate number and offer evidence that the human visual system tends to operate on bounded objects when estimating quantity. This feature may reflect how the system is optimized for interpreting the environment but also introduces distortions under specific conditions.

“We often think of mathematics as the height of the human intellect – something that’s only done by smart, educated adults. However, lots of evidence has emerged, suggesting that even newborn infants have a basic sense of number,” explained study author Sam Clarke, an assistant professor of philosophy at the University of Southern California.

“For instance, young infants will reliably discriminate two collections of dots when they differ in number by a suitably large ratio, and notice the surprising coincidence when two collections happen to match in number. This suggests that our basic grasp of number might be innately hardwired, rather than learnt!”

“Of course, this isn’t uncontroversial,” Clarke explained. “One concern has been that when children or infants identify that one collection of dots is larger than another, they might not be tracking or responding to their number but rather some confound in the displays.”

“For instance, the total surface area of the dots on the screen. In fact, it’s really hard to rule these kinds of confounds out, because when you control for one – e.g. by matching the total surface area of the dots in both collections while continuing to vary their number – this requires that other differences in the displays are exacerbated – e.g. that the spatial density of the displays will now be more different. So, if kids continue to discriminate the displays, it’s possible that this still has nothing to do with number – they are now simply responding to the newly exacerbated differences.”

“This raises the question: How could we know that it’s number that children are tracking and representing in these kinds of experiment?” Clarke continued. “There are many ways of approaching this question. Our approach takes inspiration from the philosopher, Gottlob Frege. More than a hundred years ago, Frege noted that numbers are special kinds of quantity in that they have a ‘second-order character’; i.e. that in order to identify the number of items in a collection, you need to first decide how they are being carved up and individuated.”

“For instance, a single pile of cards might be thought of as one item if it’s the number of decks that we’re interested in counting, four items if it’s the number of suits that we’re interested in, or 52 items if it’s number of individual cards. What Frege noticed is that no other quantities seem to be like this. If you want to know how much the pile of cards weighs, or its volume, it doesn’t matter whether you think of it as a single deck, four suits, or 52 cards – when you measure the pile or chuck it on the scales the answer will be the same, regardless.

“In an article from 2021, co-authored with Jacob Beck, we noted that when people discriminate collections or estimate their quantity in the ways that young children and infants appear to, they seem to be specifically in the business of tracking and representing quantities with this second-order character that Frege noted is unique to number.”

“We gave a few reasons for this, but some of the clearest evidence for this came from a visual illusion, known as ‘the connectedness illusion,’ which was discovered concurrently by Lixia He and colleagues and Steven Franconerri and colleagues back in 2009. In the connectedness illusion, collections of dots that are connected with thin lines – effectively turning pairs of dots into single dumbbell shaped items – have their number or quantity systematically underestimated, even when participants explicitly try to ignore the connections and focus only on the dots. Indeed, you can see this for yourself.”

“To cut a long story short, this occurs because your visual system seems to take a stand on how items in the displays are to be carved up and individuated,” Clarke said. “In other words, your visual system seems to automatically try to keep track of the quantity of bounded objects in the display. Thus, by adding connecting lines (and thus more surface area to the collection) we reduce the apparent quantity of items in the display, because these connecting lines turn pairs of dots into single bounded objects.”

“This much isn’t inevitable – it could have been the case that your visual system would individuate the items differently, or not at all. Regardless, the fact that your visual system does take a stand on how the items should be carved up and individuated, coupled with the fact that this does seem to affect the perceived quantity of the collection in precisely the ways we would expect if it were tracking and representing a quantity with Frege’s second order character, seems to demonstrate that it really is some kind of number or numerical quantity that is being tracked and represented in these studies. After all, we know from Frege that non-numerical quantities just don’t work like this!

“With all of this in view, our study asked whether we might find evidence of the connectedness illusion, even in young children who have not learnt or been taught to count. Before our study, this is something that had not been tested.”

The researchers tested 43 children (aged 5 to 12) and 57 adults. Participants were recruited through a university child development lab and a university community, respectively. Children received gift cards for participating, while adults received course credit. All participants had normal or corrected-to-normal vision.

Participants completed an online task using visual arrays of blue dots. In some arrays, dots were connected into dumbbell-like pairs with straight lines. In others, the dots were either unconnected or arranged with lines that didn’t form connections. Across 450 trials, participants viewed two dot arrays side-by-side and were asked to judge which side had more dots, while being told to ignore the lines.

Before beginning, participants received simple, child-friendly instructions and practice trials. The researchers reminded them to ignore the connecting lines and to focus only on the number of blue dots. Each trial presented the arrays for a brief moment, followed by a response screen where participants selected the side they thought had more dots.

Clarke and his colleagues found that children as young as 5 years old showed signs of the connectedness illusion. They were more likely to judge an array of connected dots as having fewer items compared to an unconnected array, even when the total number of dots was the same. This suggests that the visual system at that age is already organizing visual input into discrete units—treating connected pairs as single entities.

The strength of the illusion increased with age. On average, children perceived unconnected arrays as about 3.4% more numerous than connected ones. In adults, this difference rose to about 7%. Statistical modeling confirmed that the illusion was significantly stronger in adults, and the trend held across the full sample—older participants tended to show stronger illusion effects.

“When we first designed the study, I hypothesized that the strength of the illusion might be stronger during early development,” Clarke told PsyPost. “In other words, I expected that young children might more susceptible to the illusion.”

“I’d hypothesized as much, because I’d been struck by the fact that while the connectedness illusion is really robust in adults, and can be readily appreciated just by looking at a figure, connecting pairs of dots doesn’t come anywhere close to halving their perceived number, which is what we might expect if the visual system was simply carving the display up into its bounded items and then enumerating these. I wondered if this was because adult participants were being asked to ignore the lines entirely, and were thereby able to suppress the effect to some extent by trying to attend to the dots independently of the lines.

“Alas, my prediction was not borne out – we found precisely the opposite of what I’d suspected. The illusion gets stronger with age! Indeed, we found that – across all age groups – people with better numerical acuity were more susceptible to the illusion. This suggests that, rather than thinking of the connectedness illusion as an unfortunate quirk of our visual makeup, it actually reflects the system’s optimal functioning.”

The researchers also found that individuals with sharper number discrimination skills—those better at identifying which array had more dots—also tended to be more susceptible to the connectedness illusion. This association remained even after controlling for age. This pattern suggests that susceptibility to the illusion is not a sign of faulty processing, but instead reflects an aspect of optimal functioning in the visual number system.

The findings offer support for theories that the visual system processes number by first organizing visual input into discrete, bounded objects. The illusion occurs because connecting two dots makes the brain treat them as one object, reducing the perceived count. This supports a “direct” model of number perception, in which number is extracted from clearly defined objects, rather than being estimated from continuous properties like area or density.

The illusion also challenges alternative models that suggest number is inferred indirectly from visual features such as total surface area or spatial density. Arrays with connected dots have the same total area and density as arrays without connections, yet they are perceived differently. This discrepancy indicates that number perception may not simply rely on these features but instead depends on how the visual system segments the scene into countable units.

“Our main takeaway was that a connectedness illusion is found in all of the age groups we tested, right down to children as young as five years old, but that the strength of the illusion varied across development,” Clarke said. “Perhaps surprisingly, younger children were less susceptible to the illusion than adults.”

But the study, like all research, includes some caveats to consider. The experiment was conducted online, which limited control over participants’ screen sizes and viewing conditions. Although steps were taken to ensure stimulus consistency across devices, variations in display could have introduced noise into the data.

“Our experimental paradigm didn’t allow us to test really young babies – thus, the youngest children tested in our study were five years old,” Clarke noted. “This is because it had to framed as a fun game, in which it could be explained to the participating children that they needed to try and ignore the connecting lines. So, while our results suggest that the connectedness illusion is a ubiquitous feature of our visual number sense, our results don’t enable us to confirm this.”

Future studies may investigate whether attention, working memory, or other cognitive factors contribute to individual differences in illusion strength. Researchers might also explore whether training can alter susceptibility to the illusion, or whether similar patterns appear in infants or in populations with neurodevelopmental differences.

“My broader interest is in understanding how children learn,” Clarke explained. “That might sound surprising given the emphasis on innate (unlearnt) numerical abilities in this study. But children need to have some innate mechanisms and abilities if, unlike rocks and many other things, they are to be capable of learning anything at all.”

“The hypothesis that I’m exploring is that infants are born with surprisingly rich innate numerical abilities, and that these enable them to learn in surprisingly sophisticated and targeted ways – something which researchers in AI might do well to take heed of. For instance, I’ve argued that they enable children to focus their attention of statistically surprising events and to keep track of how often grammatical rules are and aren’t violated when learning a language.”

The study, “Children’s Number Judgments Are Influenced by Connectedness,” was authored by Sam Clarke, Chuyan Qu, Francesca Luzzi, and Elizabeth Brannon.