By Lindsay Clare Matsumura and Kristin Klingensmith

Institute for Learning

Big Ideas from the Interview with Dr. Melissa Libertus

  • The seeds of mathematical reasoning begin in infancy, starting with the ability to discriminate between quantities.
  • Young children’s mathematical reasoning skills develop during everyday activities, such as grocery shopping, cooking, and even riding in a car. When young children engage in intentional talk about quantities that they see in everyday life, their reasoning deepens.
  • Adults can support children’s mathematical skills in activities such as arts and crafts by engaging in discussions about mathematical observations like the number of crayons in a box, which picture has the most flowers, patterns in beaded bracelets being made, how shapes fit together, etc.
What got you interested in studying young children’s math development?

I loved math as a child, but I saw that lots of other kids were struggling to learn it. I started tutoring math early in life and that got me to think about how best to explain concepts and solve problems in ways not regularly taught in class. One of my biggest joys was to see how concepts clicked for students. As an undergraduate, I majored in cognitive science and worked with professors studying math education and how to best teach kids algebra. I realized that by the time kids came to algebra, a lot has already happened that determines success, and I wanted to start younger. So, I worked with infants for my dissertation to understand the foundations of mathematical thinking—what happens before kids get to school and how that lays the foundations for math.

What does mathematical thinking and reasoning look like in infants? What are those early foundations? 

When it comes to understanding the basic foundations of mathematical thinking present in infancy, there are two main concepts that we believe are present. The first is that infants have an understanding of small quantities and their exact numbers. Essentially, they can tell that one ball is a different number of objects than two balls, and that’s different from three. But the exact understanding of being able to tell these different quantities apart is pretty much limited to those small sizes.

Secondly, while having a precise understanding of small quantities, they also have an appreciation for large quantities, but it’s imprecise. A lot of research shows that infants can discriminate between different quantities of objects, but it needs to be a certain kind of difference between the quantities for them to be able to do so. For example, they might be able to tell that a set of 24 balls is more than a set of 8, but they wouldn’t be able to tell the difference between a set of 24 and a set of 18 because those quantities are too similar.

How do thinking and reasoning change after infancy?

Over the first few years of life, approximate quantity representations become more precise. In addition, infants are able to use their understanding of quantities to learn about basic arithmetic operations. For example, if I show an infant about 10 objects and add another set of objects to this, the rough outcome might be something like 15 or so. Infants know that there are no longer 10 objects, and they know there are more than 10. And if I take something away, there are less than 10 objects. Thus, infants have a rough idea of where the results might end up, but it’s all just an approximate understanding of these quantities, not necessarily precise. Precision grows later on when kids learn to count and do arithmetic with quantities and symbols.

Could you tell us about your research looking at the development of children’s mathematical thinking and reasoning?

One of the most surprising findings of my early research was that we see individual differences in infants’ rudimentary understanding of quantities from a very young age. By the time they are six months old, we already see some infants who can better discriminate between quantities. Even though everybody has access to this knowledge, there are some who can tell that quantities that are close in size are actually different, and others who need quantities that vary significantly to be able to tell them apart.

One of my longitudinal studies showed that this difference in infancy predicts kids’ math abilities three years later, which was important. We tested infants and then retested them when they were in pre-school. We saw that those children who as infants, had an easier time discriminating between quantities were the ones who later scored higher on a standardized math assessment. So, we’re curious where these differences come from at this very young age. What happens in infancy that puts some kids on a trajectory to developing a more solid representation of these quantities and others to have more imprecision? And if that matters for math, what is it that we might be able to do from a very young age to help those kids who, for whatever reason, might be behind?

How might parents/families/caregivers influence the development of children’s mathematical thinking and reasoning? 

This is an interesting question, and we don’t know the full story yet. However, so far, we have explored a few different origins for variations in basic number concepts present from birth, and at least part of it seems to be running in families. So, we have some studies where we’ve found that parents who have more precise representations of approximate quantities tend to have kids with better and more precise approximate number representations. But then there seem to be other factors too. We know from a lot of other research that how parents talk about math with their kids, the activities they do, even in the context of everyday play and day-to-day activities, matters for the kids’ mathematical thinking. There may be a link between parents’ own math ability and what they do with their kids, how often they point out quantities in their environment, how often they talk about numbers and math, what kinds of activities they do with the kids that could foster an understanding of math, and parents’ attitudes about math.

There seems to be some parallelism between what you were saying about what possibly could be a connection with parents’ attitudes about math and the experiences that are afforded their children. We know that in reading, as well, there may be a nature component, but there is likely also a nurture component.    

Absolutely. I think what you’re saying is absolutely true. Yet, it’s important to know also that reading and math both need to be emphasized. We often see that children’s math abilities do not just rely on general cognitive abilities and appropriate cognitive stimulation in general, but that there’s also something specific to math. Talking to children or reading to them by itself does not foster math abilities. Instead, the interactions need to specifically targeted mathematical concepts. There are many ways to incorporate math concepts into daily activities with young kids that are age appropriate and fun, such as counting steps when walking up the stairs, comparing prices of produce when shopping, or finding as many numbers as possible when driving in the car.

Do you have to have a lot of formal math learning to provide these experiences, or can the average parent or family member engage young children in meaningful ways?

I think, especially when we’re talking about young children, it is not necessary to have a formal instruction in math through college. You don’t need to have taken calculus to be able to provide the kind of mathematical interactions that are appropriate for young kids. People who use math on a regular basis [i.e., carpenters, contractors, etc.] and see how math is really used in everyday life may have great ideas to offer beneficial experiences to young kids. Going back to what we were talking about earlier, there’s also something to be said about the importance of attitudes towards math that people bring to the table. If you approach the activities with which you engage your child with a sense of enjoyment, this could translate into really important lessons that children learn.

What are some things that adults can do to support children’s mathematical development?

There are many opportunities that naturally arise for parents, caregivers, and teachers to talk about math with young kids. Parents shopping with their kids, for example, can talk about the prices of things, compare what costs more, what costs less, what is heavier or what’s lighter, compare different sizes of produce, or count how many bananas are in a bunch. Cooking also provides great opportunities to talk about measurement and different units of measurement. Talking about concepts like time also helps kids develop their mathematical understanding. For example, you can ask kids ‘How long does it take us to get from A to B?’ or “How fast can we drive? “And then point out the speed limit on signs as you pass them. This helps kids learn about why numbers and measuring things are important and connect this understanding to the symbolic language that we use to express these kinds of concepts.

Activities that don’t look like math on the surface can also help kids develop their math skills. For example, making a necklace with a pattern in the beads is a great activity that kids enjoy that people would not necessarily identify as a math activity. Finding patterns and recognizing those and finding shapes and how you can make patterns out of different types of shapes are great activities for building mathematical skills that can be included in arts and crafts projects.

What are some things that adults can do to support very young children?

There are lots of math-related concepts you can talk about even with one- and two-year-olds. They might not be able to fully get a concept yet but pointing out things like “this is a big apple, and this is a small apple” is something that even a really young child can understand, and you can build on that from there.

I have to say though, that while I think that starting early and getting into the habit of seeing these opportunities probably would be helpful for very young kids, there hasn’t been much research about this. In my lab, we’re beginning some studies now looking at what parents of two-year-olds are doing, and then following these kids over time to understand what kinds of activities and what kind of input are appropriate and most beneficial for kids down the road. We really need solid evidence and more data to be able to give guidance and advice about what kinds of adult-child interactions in very early childhood best support kids’ math development.

For more information about Dr. Libertus’s research on numeracy and math sense in young children, check out her other publications.