Early Mathematics Learning in Family and Community Contexts was the 2019 topic of the Promising Math biennial conference. Framing the event were three plenary presentations that brought cognitive, contextual, and collaborative lenses to the topic. In her presentation “Cognition and Early Childhood Numeracy: How Number Concepts are Built and Why Input Matters” (presentation PDF), Kelly Mix bridged research and practice in her discussion of math language and learning.
We talked with Mix in order to understand better the connections between her research and how children learn mathematics.
Kelly, in your talk you identified four key cognitive processes that help young children learn about math: statistical learning, structure mapping, language, and spatial cognition. I have a question about statistical learning.
The way I understand it, statistical learning is about an ability we all have to detect similarities. That is, in the case of number learning, we eventually come to associate the word “three” with “three-ness” because the word and the cardinal amount are frequently paired. When they are paired again and again, we notice the pairing, and distinguish it from random noise. What do you think this suggests about the kinds of family interactions that will help develop numeracy?
Good question! Yes, you have it right – frequency is a big part of this learning mechanism. Children need lots of experiences that help them pair quantities and number words. In statistical learning terminology, we like to talk about dense co-occurrences and regularity. Dense co-occurrences means how often and how close together in time experiences are paired. So, if a parent labels a set of 3 items such as three tea cups repeatedly in the same play session, that would be an example of dense co-occurrences. If they label a set of 3 on Tuesday and then do it again on Friday, that would provide co-occurrences, but they would not be dense. Regularity means that the words and experiences are structured in a clear, predictable way. For example, if parents count and label sets every time they comment on number (e.g. “1, 2, 3…3 tea cups”), that language structure becomes regular and predictable, and children can readily see how it applies to lots of different situations.
There are many ways parents and caregivers can help by talking about quantities when there is some way for kids to simultaneously see or experience those words and quantities. For example, adults can mention that we need four forks to set the table and involve kids in getting four forks out of the drawer, or saying “just three more pushes on the swing” and then counting out “one, two, three” – these are both the kinds of everyday interactions that activate statistical learning and help children discover the meaning of the number words. Because of how the statistical learning process works, finding these opportunities in your daily interactions with children and presenting them with frequency and regularity can make a big difference.
Another of the processes you discussed, structure mapping, suggests that when we compare things, we tend to see the similarities and differences between them in greater detail, and the similarities become generalized. For example, when we compare a car and a motorcycle, we might think about four wheels versus two wheels, but we might also think about the idea of transportation. Can you expand a bit on how this might be relevant to the way a child learns number outside of school?
The thing about number is it’s fairly difficult to “see.” Think about trying to explain to a visitor from space what we mean by “two.” You might point to two mittens, two cookies, and two trees, saying “these are all two.” This is a good approach, but there is so much detail and information in each of these kinds of objects, that it’s hard to focus on the quantity. Partly that’s because the “two-ness” is held by the sets of things, rather than by the things themselves; each mitten by itself is not “two.”
Because of this difficulty, comparison processes may be particularly important for number learning. For example, children can get a sense of “two-ness” by comparing object sets. A good way to “see” number by comparing is to demonstrate one-to-one correspondence. For example, when children see three dogs and three dog dishes, they can match the objects one-to-one and see that the sets are equivalent. Another way is to hear the sets labeled with a number word, and then look to see what attribute the label refers to. If multiple sets are called, “two,” they must have something in common—what is it? One of the things we know about structure mapping is that it works best when the two structures are alignable. A set of dogs and dog dishes is easier to align than a set of airplanes and a set of whales in the ocean, or a set of claps and a set of glasses on the table. So, adults can support structure mapping by pointing out numerical equivalence in sets that are easily aligned, positioning the items in space to show the correspondence (like, lining up the dogs and dog dishes), and using gestures to point back and forth between the items in each set.
Just like statistical learning, this process of structure mapping highlights the importance of adult input for children’s learning. In this case, it’s because adults can use words or actions to prompt comparisons, and these comparisons help children abstract the relevant numerical structures: the more transparent and alignable the structures we present, the more likely children are to make these comparisons and figure out what the number words mean!
Number names, counting, and written numerals are all different symbols for quantities that children must link to each other and to the underlying meaning. Both statistical learning and structure mapping support this grounding process by helping children notice that words and written symbols go together, or that the same quantities are called “three,” are counted, “1-2-3,” and can be labeled by the same written numeral.
In your talk, you suggested it is helpful to think of developing early numeracy as a “symbol grounding” task. Can you clarify what this means and how it connects to children’s interactions with adults?
The central idea of symbol grounding as it applies to early numeracy is that each child must make connections between the symbols we use to talk about quantity and the underlying meaning of these symbols. That is, the word “five” is itself a symbol, and children must learn to map it to examples of five in the world or to the abstract idea of “fiveness”. An easy way to think about symbol grounding is to think about how children learn other words, like words for objects. The word, “tree,” for example, is an arbitrary symbol that has no meaning until it’s linked to examples of trees in the world or to the abstract concept of trees. Symbol grounding is what happens when people make these connections between symbol and meaning. To learn to think mathematically, then, the child is exposed to lots of math-related symbols, and must connect each one to its meaning. When a symbol is “grounded” to its meaning in the thinking of a child, rather than floating freely in an ocean of symbols children have yet to understand, it can be used meaningfully by the child.
It’s for this reason that adult input is so important. Number names, counting, and written numerals are all different symbols for quantities that children must link to each other and to the underlying meaning. Both statistical learning and structure mapping support this grounding process by helping children notice that words and written symbols go together, or that the same quantities are called “three,” are counted, “1-2-3,” and can be labeled by the same written numeral. In the very early years, we can almost think of “symbol grounding” as “language grounding” instead, since language is the child’s most powerful symbol system, both for learning and for being understood. Children are busily grounding words they hear in their environment all day long.
In terms of advice for parents and caregivers, we can boil this down to: talk with your child about small numbers! Whether it’s conversation about the number of fingers on each hand, the number of school days before the weekend, or the number that is one more than four, give your child abundant opportunities to connect the symbols for quantity to their meanings.
Two Ways That Language Scaffolds Understanding of Cardinality
At Promising Math 2019, Kelly Mix gave a talk about how language promotes the development of numerical cognition in young children. Dr. Mix is the Chair of the Human Development and Quantitative Methodologies (HDQM) department at the University of Maryland.