Note: An earlier version of this post was part of the Talking Math with Your Kids newsletter. The provider I used for that newsletter ceased operating a while back, but I saved all of the content and will republish some of that writing here on the blog from time to time. The newsletter itself lives on at tmwyk.beehiiv.com; click on through to subscribe.
Years ago, I had an opportunity to meet Michael Doyle (and Rafranz Davis and Frank Noschese; it was quite a weekend!) at a short consulting gig. Over beverages one evening, Michael—a deeply thoughtful high school science teacher—was intrigued by how much there is to learn about how children learn about fractions. He asked me if my own thinking about children’s thinking was influenced by Margaret Donaldson, and I had to report to him that I had no idea who that was.
So he recommended her book, Children’s Minds, published in 1978. I ordered it the next day. (Used; it’s long out of print), and shortly after that, I was blown away.
In case you are also unfamiliar, Donaldson was a student of Piaget’s, and she took issue with a number of the conclusions in his research. Many of the findings that Piaget attributed to developmental stages, Donaldson later attributed to social development. A number of the things that Piaget said young children can’t do, she and others demonstrated they can so long as the social context makes sense to them.
For example, there is a classic experiment in which children are shown four red flowers and two white flowers and asked, “Are there more red flowers or more flowers?” Five-year-olds will usually say that there are more red flowers. The Piagetian conclusion is that children of this age cannot consider both the part and the whole simultaneously.
I assure you that this experiment—and its results—are highly reproducible. Young children will in fact tell you that there are more Snickers than candy bars, and more dump trucks than vehicles.
But if you tweak that experiment just a bit, children are much more likely to be successful. Get yourself some toy cows—three black, one white, all lying down. Ask the child “Are there more black cows or more sleeping cows?” The black cows are sleeping just like the white one, remember. Children are much more likely to correctly state that there are more sleeping cows.
Additional tweaks, which include asking about steps along a path instead of cows or candy bars, lead to even greater success. Donaldson concludes that in the original version of the task, children are answering a question different from the one the experimenter asks. She argues that if we want to know what children can do, we need to formulate our tasks in ways that make sense to them.
This is what I find compelling about Donaldson’s writing: her clear interest in the richness of children’s minds. From time to time I revisit the book, as I did recently.
I want to share a passage with you, in a chapter about children’s language acquisition.
To Western adults, and especially to Western adult linguists, languages are formal systems. A formal system can be manipulated in a formal way. It is an easy but dangerous move from this to the conclusion that it is [only or primarily] also learned in a formal way.
So too with early mathematics.
Time and again, we learn that cultural exposure is a powerful force in early math learning. Children understand the rudiments of place value before they understand that it is a base-10 system. They consistently use the term diamond even though no formal definition exists for this mathematical object. They will tell you that a triangle is a shape with three sides, and also that a tall-skinny triangle is not a triangle because it doesn’t look like a triangle.
Because children learn a tremendous amount of mathematics informally, exposure to a wide range of experiences is important. Just as being surrounded by lots of language is key to language learning, having many opportunities for math talk and math play is key to early math learning.
By contrast, if you think that math is primarily learned in a formal way, then you might worry that you’re doing things wrong. You might worry that you don’t know the correct rules for things, or that you might misremember something, and that may lead you to avoid the subject entirely.
My hopes for early math learning are for liberation rather than obligation. I dream of daily playful math experiences for children and their caregivers. Those experiences can take many forms, such as helping Dad to remodel a bathroom.