“Even though most kids learn to stack up to three or four by the time they get to kindergarten, there are some kids who just don’t have the opportunity, and those kids are in really big trouble in kindergarten,” said Art Baroudy, professor emeritus at the University of Illinois in Urbana-Champaign, who has spent his career studying the best ways to teach counting, numbers and arithmetic concepts to young children. “The reason for this is that subitizing involves constructing an understanding of what is one, what is two, what is three, what is four, which is the conceptual basis of number sense. There are so many things that are based on that understanding.”
As children get older, subitizing can become more sophisticated and useful for understanding more complex mathematical concepts. Imagine a group of seven points like this:
Most people can’t plot that many points, especially when they’re not arranged in a familiar pattern, but they can detect subgroups. Some may see two groups of three plus an extra point. Others may see a group of four and a group of three. There are many possibilities.
Doug Clements and Julie Sarama, a research team at the University of Denver, call this rapid separation “conceptual subitizing.” It is said to help children add and factor numbers, which is useful when learning addition and subtraction facts. For example, 7 + 5 can be a difficult sum to learn. But a child who can subit can quickly see that 7 can be divided into 5 and 2. This makes it easy to add the two 5s to make 10 plus 2 more to get to 12. “That’s meaningful and don’t forget that,” Clements said.
Counting from 7 is, of course, another approach. But it can be mentally taxing to keep track of five more while a child counts from 7 to 12. Children who only know how to add by counting soon realize they don’t have time to count when presented with a long addition worksheet. These children often resort to memorizing a string of meaningless numbers (5, 7, 12) which they soon forget. “They missed the step of making sense of it,” Clements said. “In first grade, teachers will say, ‘He knew it last week.’ Now he forgot it. Well, he never knew it.
Subitizing seems to help with all sorts of mathematical concepts, according to its proponents. In 2014 another group of researchers described how third graders who could quickly group sets of small numbers, such as three sets of two, had a more sophisticated understanding of multiplication and could multiply more quickly. Some say that the ability to divide a number into smaller, manageable parts builds an understanding of the relationship between part and whole and helps with fractions.
Clements and Sarama recommend the use of subitizing exercises until the end of elementary school. For example, a 24-point array can support conceptual understanding of multiplication in grades three through five.
“Subitization is not a baby job to transition from,” Clements said. “You just keep developing it alongside all the other skills you’re developing.”
Well, sometimes it’s actually baby stuff. Some researchers believe that babies as young as five or six months master small numbers such as two and three. This belief is based on experiments in which infants hold their gaze longer in response to changes in quantities. As children grow older, they begin to match number words to collections of objects, such as “two socks,” and by age 2 many children can subitize to three elementsand 4- and 5-year-olds can put up with so much as five elements.
The researchers theorized that fostering this innate ability could help children build number sense. A case study of two students who struggled in math, published in 2009, found that subitizing was helpful. Baroody of the University of Illinois promotes the teaching of subitizing in a Practical Guide for 2013 on “Teaching Mathematics to Young Children” for the Institute of Education Sciences, the research arm of the US Department of Education.
Several studies have documented that children who have stronger subitizing abilities also tend to perform better in math. For example, a 2022 survey of more than 3,600 kindergartens in a southwestern school district found that children who were better subitizers also had much better arithmetic ability in first grade. A 2020 survey to 80 small childrenages 2 to 5, found that children who could subitize the number four and had been taught to count could usually answer the question “How many?” But children who could only subitize the number three demonstrated only partial understanding of quantity.
Yet the strong relationship between subitizing and math ability is not evidence that students will benefit from subitizing instruction. It is possible that stronger subitizers come from wealthier families who play more board games. These higher-income children also benefit from a number of other factors at home, from better nutrition to less stress, that could lead to their success in math.
Empirical evidence that it is worth investing classroom time in subitizing is scarce, at least so far. In a new but small study, 14 preschool children aged 3 to 5 years were randomly assigned to two different math interventions. Those given more practice in subitizing performed better on a quantity comprehension test. this studyavailable online, is scheduled to be published in the December 2024 issue. of Journal of Mathematical Behavior. More and larger studies are needed.
Even without this evidence, many elementary school teachers have been incorporating aspects of subitizing, sometimes called “quick imaging,” into their classrooms for decades. But teaching this practice more systematically seems to be gaining traction, based on interviews I’ve had with teachers, researchers, and the National Council of Teachers of Mathematics, a professional organization. One motivation is the desire to close the large achievement gap between rich and poor children by building stronger mathematical foundations in the earliest years. Subitizing is explicitly included in the The Illinois Mathematics Standards for kindergartens, while many other countries offer subitization as an example of one way to teach numbers. Curriculum publishers are increasingly printing dot images for arrangement in their materials.
Beth McDonald, an associate professor of early childhood mathematics education at Illinois State University, said subitization was a hot topic at the International Congress on Mathematics Education earlier this summer. “People come and say, ‘Oh, we want to study this,'” said McDonald, who wrote her dissertation on subitization more than a decade ago and is the author of several articles and studies on subitization that I read to write this article.
Despite the simplicity of subitizing, experts say they sometimes see it taught incorrectly. A common mistake is to have students count the points. Researchers say this undermines a child’s confidence and ability to develop mental images of sets.
Speed is of the essence. Often the dots are displayed for too long, long enough for children to count them. “That’s the biggest mistake,” said Clement Sarama’s partner, who explained to me that the brevity forces children to visualize the image on their own.
Sarama says that each point exercise can be performed in 10 seconds for a total of three minutes per day. Memory for quantities is built, researchers say, through brief but repeated exposures.
According to Milwaukee math coach Robinson, kids want to talk so much about all the different ways they see the dots that it’s hard for teachers to move on to the next activity. Kids really enjoy subitizing.
And many children learn to obey at home. The researchers say the practice doesn’t require fancy materials and parents don’t have to create dot maps. They advise parents to talk about quantities in daily activities. When folding laundry, talk about matching pairs of socks. Or at the end of lunch, ask who will eat the three fries left on the plate. Sometimes you don’t have to count them!
