Paul Goldenberg has over 40 years of experience in K–12 mathematics curriculum development, research, and professional development. He leads a wide range of EDC projects that foster a love of and enthusiasm for mathematics in learners from early childhood through adulthood, and has taught self-contained elementary (Grades 2 and 4), middle and high school mathematics, and college and graduate school mathematics and psychology. In this post, the first in a series that will invite readers to take a look inside children's early mathematics explorations and epiphanies, Paul describes the connection between oral and written mathematics through the eyes of six-year-old Aaron.
I often write about how much clearer the structure and logic behind certain computations can be when those computations are spoken than when they are written. For one example, I recently asked a kindergartner what she thought five-eighths plus five-eighths might be and she immediately chirped “Ten ayfs! What’s an ayf?” By contrast, it’s not uncommon for older kids, even after they’ve got some idea about the meaning of 1/8 to respond to 5/8 + 5/8 by writing 10/16, as if the plus sign meant “add everything in sight.” The spoken language of “five-eighths plus five-eighths” evokes a logic we use for all additions of like objects—five goats plus five goats, five fingers plus five fingers—and kindergartners love the idea of adding three-hundred plus two-hundred or five-million plus five-million. The answer that makes logical sense to them is also the correct answer. Similarly, when the problem is presented orally, second graders want “twenty-eight” minus the “twenty” to be “eight” (just as “Tom Sawyer” minus “Tom” is “Sawyer”). That is the answer that makes sense when they hear the words, and the oral version gives them expectations that the written form 28 – 20 does not.
This linguistic “trick” is not mathematics. But it’s also not a trick. Numbers weren’t born with names. We named them precisely so that calculations of this kind would be easy, and we can use these “easy” ones as a foundation for other harder ones. If “twenty-eight minus eight” feels so obviously like “twenty,” then “twenty-eight minus nine” is less than twenty by just a little bit. With opportunities to play with nearby cases like that, second graders are pretty good at building logic that they can apply and generalize. Language doesn’t help with little numbers. There’s nothing in the language of “four plus eight” that makes “twelve” sound tempting; that requires mathematical strategies, like counting on, or partitioning the four plus eight into eight plus two plus two. But for children who get enough experience playing with the language and not just rules for on-paper addition, if there is a number called “twenty-eight,” twenty plus eight should make it! There are lots of other examples where verbal/mental calculations expose the logic more clearly than the same calculations shown on paper.
But this morning, six-year old Aaron gave me the first inkling that there are times when the written notation clarifies things that the spoken language does not!
Aaron is advanced in many ways. For kindergarten, he is an excellent reader and generally has the self-control and generosity to others of a well-adjusted 28-year-old. Though way ahead of his classmates in skills, he never lords it over them and, though astonishingly self-controlled, he still giggles (and wiggles) like a little kid.
He is also facile at addition and subtraction, with such a clear model in his head of what these operations mean that he can easily use facts he knows to perform new computations. This morning, for example, several children were playing a “How Many Are Hiding?” game. They made a collection of up to ten blocks (their choice). Another person covered some, and they were to figure out how many were hidden, using only how many they still saw. Aaron made a stick ten blocks long, I covered five, and he instantly (without counting) “saw” five and announced that five were hidden. Nor was it a challenge for him to know that seven were covered when he saw three left, so we started playing the game with “pretend cover-up.” Still pretending we were using his collection of ten, I’d tell him what he “saw” and he’d tell me what I had “covered up.” That, too, was easy. When I said “nine,” he said “one,” and we pushed the speed up with each of us just saying a number.
He enjoyed it, but enjoys seeking greater challenge even more. Actually, most young children who aren’t feeling insecure or judged do that, which may explain why they try to hop across the room on one foot instead of walking the easy way.
So Aaron added three more blocks to his stick and announced that he could still do it but would be a bit slower “because I’m not used to adding up to odd numbers.” Still playing the “pretend cover-up” style, I first gave him an easy throw-away “three” to which he instantly, and happily, responded “ten.” Then I said “five.” He thought for a moment, said “ten minus two” quietly to himself, and then said “eight.” Then I said “nine.” He paused, looked up for a second, and then said “four.”
This is all about verbal, not written, mathematics, but Aaron had come in that morning saying something he couldn’t do. “I can count up to nine-thousand ninety-nine, but I don’t know what comes next.” He also said that he had thought about a million but that was something like a thousand thousands, and then his thought was interrupted by other things going on. The immediate needs of the class called for morning time and three “center” activities (including the “how many are hidden” game I described earlier), so his teachers (Sharon Miller and Kate Bernstein) and I had space to think what might be responsive to Aaron's interest and implicit question.
Why was Aaron uncertain? He knew what number came after eight-thousand-ninety-nine (eight-thousand-one-hundred). Isn’t his new challenge just the same? It occurred to me that maybe language was getting in the way. After all, Aaron knows that after nine-hundred-ninety-nine comes a brand new word: one thousand. Nine-thousand-ninety-nine has the same verbal structure. Despite the counting, Aaron really had limited experience with hundreds and thousands; without a secure sense of the difference, perhaps he wasn’t sure whether, after nine-thousand-ninety-nine, he should again expect a brand new word (like the million he had mentioned in the morning, but then dismissed) just as there was a totally new word after nine-hundred-ninety-nine.
I couldn’t think how to check out my theory, but Sharon did. She invited Aaron back from another activity, figuring this was more worth his time, and asked how he thought nine-thousand-ninety-nine was written. He said “I thought it is” and then wrote 9,99. Both the emphasis on the past tense—“I thought” rather than “I think”—and his expression when he looked at 9,99 said that he wasn’t happy with that. He then wrote 999,999, (with the comma at the end as well as in the middle) without saying anything about it. When Sharon asked what number that was, he said he didn’t know, but it did make clear he had some sense that commas have something to do with three digits (though clearly he wasn’t quite getting the whole rule).
What to do? In kindergarten, he certainly doesn’t need to know more about large-integer notation or about how to count beyond 9,099. But he was interested, and implicitly asked. So, she showed, him, speaking “nine-thousand” as she wrote “9,” and “ninety-nine” as she wrote the remaining 099. He showed all the polite, casual interest of a six-year-old who has many other enticing things to do and thanked her and went back to his other activity. There was no “aha!” or other sign of new revelation. We’ll have to wait to see where, if anywhere, this goes. But he’s Aaron, and he’ll come back with something else for us all to think about!
Watch this playlist of four videos from EDC's Games for Young Mathematicians project, led by Jessica Young and Kristen Reed, in which Paul discusses the "How Many Are Hiding" game that he plays with Aaron in this post.
- View a recent book co-authored by Paul and Douglas Clements and published by the National Council of Teachers of Mathematics: Developing Essential Understanding of Geometry and Measurement for Teaching Mathematics in Pre-K-Grade 2.
Last Updated: May 19, 2016