SMINC

Examining the building blocks of mathematical ability While numerical ability has long been thought to be innate, recent findings suggest that other factors also play a role in the development of numerical sense. We spoke to Professor Avishai Henik about his research into the building blocks of numerical cognition, including the ability to perceive the size of objects A good grasp of numbers is essential to many aspects of everyday life, yet it has been estimated that between 3 and 6 percent of the global population suffers from dyscalculia, a deficiency in numerical cognition. This is an issue which lies at the core of the SMiNC project’s research agenda. “We aim to examine the building blocks of numerical cognition,” says Professor Avishai Henik, the project’s Principal Investigator. There are parallels here with previous research on reading. “Previously researchers looked at the building blocks of reading, and this helped build an understanding of the underlying processes involved. It also helped researchers understand what happens in dyslexia cases or other types of reading disability,” explains Professor Henik. “The hope with numerical cognition was similar – that if we invested time and effort in understanding the building blocks of numerical cognition, then we would be able to know what is going on and what’s the underlying basis of these deficits. From there, we may eventually be able to suggest ways to work with people who have difficulties with numerical cognition.”

The most prominent current line of thinking on the development of numerical cognition is that we are born with an innate sense of numbers that

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forms the basis of our arithmetical ability. However, Professor Henik and his colleagues take a different view. “We suggest that actually number sense is not innate but rather that it develops, and that we learn to recognise numbers by paying attention to amounts, sizes and so on, and to visual or auditory stimuli. There are examples of connections between recognising the sizes of objects and numerical cognition,” he explains.

Congruent

Incongruent

Size Matters Researchers now aim to investigate the underlying processes involved in number cognition. One method Professor Henik has used in research is to present numbers to subjects in an experiment, and ask them to identify which is larger. “When you look at 3 and 5, you of course know which number is larger, and if the two numbers are further apart – like 3 and 8 – then people are faster to say that 8 is larger than 3 than to say 5 is larger than 3.

Congruent condition A violin is larger than a banana both physically and conceptually.

Incongruent condition The conceptually larger violin is physically smaller than the banana.

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The fact that the distance is 5 in one case and 2 in another is unimportant – you are not asked to pay attention to it but you cannot ignore it,” he says. Besides their numerical value, numbers also have a physical appearance, which is a factor in numerical cognition. “The 3 can be small or large, in a physical sense,” points out Professor Henik. “When you’re asked to pay attention to the numerical value then you might think that the physical appearance shouldn’t do anything. But if you change the physical appearance in an experiment, you can see that it does in fact affect performance. The influence of physical sizes on judgment of numerical values suggests that there is an intimate relationship between physical appearance and numerical values, which is described by the numerical Stroop effect.” (https:// e n .w i k i p e d i a . o r g / w i k i / N u m e r i c a l _ Stroop_effect) The project has since built further on these findings, with researchers mixing congruent and incongruent trials to gain deeper insights into the relationship between size perception and numerical cognition. In a congruent trial, the numerically smaller number is also physically smaller, while in an incongruent trial, the numerically smaller number is physically larger. “Now, we’ve mixed these two types of trials, and changed the digits and the sizes. If people are asked to pay

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attention to the numerical value and ignore the physical size, then they can do it but you get a congruency effect. That is, the response to the incongruent trial is slower than the response to the congruent trial,” outlines Professor Henik. This on the surface may not seem very surprising, as we almost always notice physical appearance; however, Professor Henik says the reverse has also been observed.

whereas the numerical value is discrete, the physical dimension – the size – is continuous and not countable. While researchers have historically paid a great deal of attention to discrete variables like numbers and arrays of dots, or other mathematical symbols, Professor Henik says there has been less interest in the physical appearance of this stimuli and in non-countable dimensions. “What the

The hope is that if we invest time and effort in understanding the building blocks of numerical cognition, then we will be able to know what is going on and what the underlying basis of deficits in numerical cognition is “If I ask you to look at the two digits, and tell me which is physically larger – ignore the numerical values, they are not important – then we again see the congruency effect. So people cannot ignore numerical values, even when they are told they are not relevant,” he says.

Researchers have found that we are typically slower to respond to numerical values than to physical size. While it might be inferred from this that numerical cognition processes would not interfere with the perception of physical size, this is not in fact the case and there is a close relationship between the two. Another feature of the task described above is that

SMiNC project suggests is that in order to build a complete understanding of numerical cognition, we need to take a look at the non-countable dimensions as well,” he says.

Archer fish The behaviour of the archer fish, a fish which preys on insects and other small animals on trees above water level by spitting water at them, is of great interest in this regard. The archer fish is relatively easy to study, with researchers training it to shoot in response to a specific stimulus.

“We can measure the time from the appearance of the stimulus to when the

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archer fish starts spitting,” explains Professor Henik. One of the questions Professor Henik and his colleagues have been investigating is how archer fish respond to both discrete variables – such as an array of dots – and continuous variables, like the size of a square. “For example, we can put two differentlysized squares in front of the fish, and train the fish to shoot at the larger square,” he explains. Researchers have found that the larger the difference between the two squares, the faster the fish responds. “So, again you have a distance effect, but now with physical, non-countable dimensions,” continues Professor Henik. “Now why is that important? We usually think that cortical areas are the main areas of the brain involved in numerical cognition.

However, fish do not have a cortex, they have a mid-brain. This suggests that recognising the size of objects may appear very early in development.”

Professor Henik suggests that recognising the size of objects is an important factor in the development of numerical cognition. While areas of the brain that play a role in numerical cognition have been identified, Professor Henik believes that other factors also need to be considered to build a fuller picture.

“Other factors are involved in the development of numerical cognition, like these continuous variables. Size perception in general, assessing the size of objects and so on, is important to the development of numerical cognition,” he stresses.

“There is also an object congruity effect think of a small violin and large banana vs. a large violin and a small banana.

These stimuli create a size congruity

The figure depicts a small 8 climbing the ladder to the top of a large 3. It represents the fact that numbers have both physical features (e.g., physical size) and numerical value. These two aspects may affect one another.

effect. When participants are asked to describe which is larger conceptually and ignore the objects’ physical size, they cannot ignore the objects’ irrelevant physical sizes. When they are asked to pay attention to the physical sizes and ignore the conceptual sizes, a similar effect appears.”

If size perception is central to the development of numerical cognition, then it could enable clinicians to diagnose individuals susceptible to dyscalculia at an earlier stage. “This could even be before a child starts learning arithmetic,” says Professor Henik. “Currently, in order to find out if a person has dyscalculia, you have to wait until they start learning arithmetic. Only then, when they start falling behind their peers, can they be diagnosed. But if size really matters in numerical cognition, then we might be able to diagnose dyscalculia at an earlier stage and target interventions at an earlier stage.”

This could have a significant impact, given that arithmetic abilities at an early age are known to be a good predictor of later academic achievement. This is not an immediate prospect however, and for now researchers are looking more to publicise their findings, in both research papers and a book published earlier this year in the academic press called Continuous Issues in Numerical Cognition – How Many or How Much? “There are chapters in the book written by different experts,” outlines Professor Henik. “One of my students has also written a review paper on number sense versus our understanding of the magnitude sense, which will be published in Behaviour and Brain Sciences.”

At a glance Full Project Title Size Matters in Numerical Cognition (SMiNC) Project Objectives Numerical cognition is essential to many aspects of life and arithmetic abilities predict academic achievements better than reading.

Acquiring a solid sense of numbers and being able to mentally manipulate numbers are at the heart of this ability. In recent years research has been marked by looking for the underlying mental operations, an effort to unravel the neural tissue that supports these operations, and identifying low-level deficits that underlie deficiencies in numerical cognition such as developmental dyscalculia.

Project Funding Funded under: FP7-IDEAS-ERC. ERC Advanced Grant - The Human Mind and its complexity (Project ID: 295644).

Contact Details Professor Avishai Henik Department of Psychology Building 95, Room 003 Ben-Gurion University of the Negev Beer-Sheva, Israel T: + 972 8 6477209 E: henik@bgu.ac.il W: http://in.bgu.ac.il/en/Labs/CNL/Pages/ staff/AvishaiHenik.aspx

Professor Avishai Henik

Professor Avishai Henik was born in Tel Aviv in 1945. He received his undergraduate degree in psychology and education from Ben-Gurion University of the Negev in 1971. He then moved to the Hebrew University of Jerusalem to study for his MA and PhD degrees under the supervision of Nobel Laureate Daniel Kahneman.

He received his PhD in 1979. In 1980, Henik received a Rothschild post-doctoral fellowship and spent two years in Eugene, Oregon, in the laboratory of Michael I. Posner, considered a leading pioneer in building the field of cognitive neuroscience, and the neuropsychology laboratory led by Michael I. Posner and Oscar S.M. Marin in Portland, Oregon, U.S.A.

The Cognitive Neuropsychology Laboratory

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