Numbers on the Brain

Francesca Luzzi, C’24, spent much of her junior and senior years showing pictures of dots and lines to children and adults. In one image, the lines connected the dots, in the other they did not, and Luzzi wanted to understand which one the viewers thought had more dots. The goal: to use a numerical quirk called the connectedness illusion to learn more about how our brains rapidly compute quantities without counting.

Researchers think we are born able to grasp the approximate number of items in a collection—people in a crowd, for instance, or grapes in a bowl—quickly and without need of language. This is called the approximate number system, or ANS, and has been observed even in infants. What’s unclear, however, is whether the ANS is the basis of our mathematical ability, a question Luzzi spent most of her undergraduate career trying to answer, at least in part.

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Working in the lab of Elizabeth Brannon, Edmund J. and Louise W. Kahn Term Professor in the Natural Sciences in the psychology department, Luzzi studied the number system and how it develops. By the time she graduated this past May as a member of Phi Beta Kappa with a psychology major and chemistry minor, she had completed two experiments of her own and coauthored a paper just published in the journal Cognition. The work confirmed that how precisely we understand concepts of quantity improves with age. It also showed that susceptibility to the illusion—thinking that one of the images shows more dots when, in fact, they have equal amounts—might not mean a weaker understanding of the number system.

1 Interest + 1 Interest = 1 Lab

Luzzi came to Penn with two main interests. “I’ve always been very interested in the mind in general, how different parts of the mind work, and how it creates our subjective experiences,” she says. “And also working with kids. I love learning about development, how children grow and change as they get older.”

When she looked for research opportunities at the University, Brannon’s Developing Minds lab seemed ideal, covering both of those areas. That team focuses on understanding how people represent number before and even after they learn to count and use their culture’s numeric system. (Researchers use the singular “number” or the word “numerosity” to refer to the concept of quantity.)

“In school, we are educated to do all sorts of complex, symbolic mathematics,” says Brannon. “However, even before we learn any of that, we have a primitive number sense that we can observe in newborn infants.” Specifically, Brannon is trying to discern how we build our “uniquely human, symbolic, mathematical system. What does it harness onto? What are the initial primitive representations that we build upon when we start learning math?”

Luzzi thought such questions would be a “really good place to start research,” she says. “I ended up just loving it and sticking with it the whole time” she was at Penn.

16 Single Dots = 16 Dots in Pairs

Luzzi started her Penn career remotely due to COVID-19, but she reached out to Developing Minds her first semester. When she got to campus in the spring she joined the lab. She started doing basic tasks, but by her junior year was ready to create and conduct experiments to explore her own research questions. Then-postdoc Sam Clark (now an assistant professor at the University of Southern California) had been working on an experiment involving the connectedness illusion, in which people systematically underestimate the quantity of items in a set when they are connected by lines, as compared to the same number without such lines. 

The illusion intrigued Luzzi.

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“The connectedness illusion has been demonstrated many times before, but nobody had studied whether it is present in young children, whether it’s something that requires experience for it to emerge, or whether it’s something there from the start,” Brannon says. “Francesca wanted to explore that.”

Researchers studying how the brain or body is physically changing can use structural imaging, Luzzi explains, but “when it comes to a system in the mind, the way to see its change is by seeing how performance on a task changes” over time. In regard to the connectedness illusion, she adds, “I wanted to know what kids were capable of at a young age.” She worked with Clark, Brannon, and graduate student Chuyan Qu to design an experiment to see whether and how susceptibility to the illusion might change with age.

5 Year Olds > 18 Year Olds 

To test the connectedness illusion, researchers show a pair of images to the subject for an instant, and the subject chooses which array appears to have more items. In Luzzi’s experiment, the child and adult groups received the same instructions and images. “The children and the adults are both told to ignore the lines and only attend to the number of dots,” says Brannon. “That requires inhibition and executive control, and children have a hard time with that. So, we initially predicted they would have trouble ignoring the lines and therefore would show less of a connectedness effect.”

Yet the results were a surprise: Children did show the connectedness effect, but adults were more susceptible to the illusion than children, and older children more susceptible than younger, indicating a linear trend as we age.

These new data points add to those the lab is using to build computational models that try to explain number processing non-verbally and predict all the different kinds of illusions. “When we discover any kind of systematic bias on number, that tells us something about how the system is working,” says Brannon. “If we create a model that doesn’t make the right predictions, then we know we’re doing something wrong. These empirical findings outline important constraints that any model we build must conform to.”

X x 2 Minutes = X x 5 Minutes

During her senior year, Luzzi conducted a follow-up experiment focusing on adults, to see if the amount of time they looked at the arrays could account for a disparity in correctly identifying the quantities. She found that the longer she displayed the images, the more the illusion influenced number judgments. However, further analyses revealed that how long the adults spent viewing the images didn’t account for differences in their susceptibility to the connectedness illusion compared to children.

Even though we normally think of illusions as something that tricks our visual system or tricks our minds, we don’t think that the connectedness illusion is necessarily some sort of mistake of our number system. Now we actually think that this sensitivity to those connecting lines is probably a functional property of the approximate number system—a sign that it’s working well.

Beyond that, regardless of age, participants with a higher numerical acuity—those better at working with numbers—tended to be more prone to the illusion, a counterintuitive finding because visual illusions are often thought of as moments during which we perceive the world incorrectly.  Rather, the findings suggest that the connectedness illusion may not be a failure of the approximate number system but instead a feature of it.

“Even though we normally think of illusions as something that tricks our visual system or tricks our minds, we don’t think that the connectedness illusion is necessarily some sort of mistake of our number system,” says Luzzi. “Now we actually think that this sensitivity to those connecting lines is probably a functional property of the approximate number system—a sign that it’s working well.”

“Our visual systems evolved to pay attention to whole objects, and even though we are told to ignore the lines that connect dots our visual system enumerates over the parts of the image that are perceived as discrete whole objects,” adds Brannon.

While in Brannon’s lab, Luzzi was also involved in the discovery of a new illusion that showed when items in an array have a coherent orientation (all pointing in the same direction), they appear more numerous than randomly oriented items. She is also a middle author on a recently published Cognition paper, “Rational Number Representation by the Approximate Number System,” which provides evidence that the ANS helps us estimate quantities without explicitly counting and offers information about how we perceive ratios.

Luzzi is currently taking a gap year to apply to medical school and working as a patient care technician in a hospital near her home in Connecticut. She plans to get involved in research at med school “and really look at this area of developmental psychology, cognition, vision science.”