Reprinted with permission from https://blogs.ams.org/matheducation/2018/09/02/ideas-under-construction-children-saying-what-they-know/.
Alli entered kindergarten quite skilled at mental addition and proud of her skill. Subtraction followed quickly. Near the end of her kindergarten year, Alli bounced into class and said that her father had taught her about negative numbers. To assure that I knew about them, she explained, “If you subtract 20 from 10, you get negative 10.” I asked, “And what if you subtract ten from seven?” She thought a second and chirped “Negative three.” Then she explained how to write a negative number—“Just put a minus in front”—and added “There are negative numbers and positive numbers.” And that was it. As with many conversations with 5-year-olds, this one ended as abruptly as it began.
Later, the teacher showed the kids a mathematical tug-of-war game. Each pair of children would have a single die, a small plastic bear, and a number line laid out like this.
The bear starts on the 10 and children take turns rolling the die, one child moving the bear that many steps toward 20 and the other child moving the bear toward 0. Each child also each had a sheet to record the bear’s moves, one sheet with addition templates the other with subtraction . Using this format, the children were to record where the bear had started when their turn began, the size of their move, and where the bear landed.
They all understood the mechanics—roll the die and move the bear that many spaces toward their side. I was surprised that several didn’t seem to understand that they were playing one game, together, rather than taking turns re-starting the bear at 10 and rolling their die to see how far it went this time. It was no surprise, though, that only a few recorded their jumps. Frankly, that made sense. The recording step may (or may not!) serve learning but, to the children, it was simply an arbitrary rule with no logical role in the game. Nothing about the game was enhanced by recording it.
We played, cleaned up, and then it was snack time.
During snack time, Alli asked me “how do I write positive three?” I thought, of course, of her early morning announcement about negative numbers. Her question was so clear and specific that I didn’t think (as I always should) to say (as I often do) “I’m not sure I understand. Tell me more.” I too quickly assumed that I knew what she meant.
“Well, we usually just write three, just the way you always write it.”
“But I mean positive three.”
I should have realized right then that I’d mistaken what she had in mind, but I plowed on.
“Just 3—we could put a plus sign in front, but we don’t usually.”
“No but I was on 17 and I rolled 6. How do I write positive 3?”
“Well, Alli, what is seventeen plus six?”
“Twenty-three. But how do I write positive three?”
Now I understood.
Communication with kindergarteners can feel like a string of non-sequiturs when we don’t see the connective tissue, the theory in their mind that they assume we know and that they therefore don’t bother communicating.
It turns out that what Alli meant tells us a lot about the theory she had constructed when her father told her about negative numbers. Prior to hearing about them, Alli had never heard of positive numbers, either. There were just numbers. Now she knew there are kinds of numbers. I don’t know what her father did or didn’t say, but it’s easy to believe that he, like I, would have assumed that nothing further needed to be said about positive numbers; after all, Alli was already quite adept with them. But for Alli, it wasn’t yet clear that the familiar numbers were just getting a new name, positive. For all she knew, the designation positive might well be reserved only for some special use.
And that does explain her question. She learned that going below zero called for negative numbers, and that they contrasted with positive numbers somehow. Perhaps she first thought that positive numbers were all the numbers she had already known (or, less likely, that 0 was yet a third category), but in the context of the number line tug of war game, she built a competing theory. The line contained the numbers from 0 to 20—just plain numbers. She knows that there are other numbers, not shown. Now she knows that below 0 were negative numbers. Perhaps the designation positive also refers to numbers not shown, but above 20. In other words, the categories she created were not “above and below zero,” but “above and below the range we’re attending to.” With astonishing ease for a kindergarten child, she mentally computed 17 + 6 = 23, but now she assumed that “positive three” was the way to express that excess above 20 and she wanted to know how to write it.
The point of relating this story is not to show how impressively smart kindergarteners can be. And it’s certainly not to note a “misconception.” It’s to illustrate what I think is a subtle aspect of teaching mathematics. As teachers, we can’t fully control what ideas our students build, even if we believe we are being are quite clear and precise. What people (children and adults) put in their minds is what they construct, not what someone else says or even shows, and it combines what they already know with their interpretation of what they are currently seeing and hearing. Because that construction combines current experience with past, our “clear and precise” communication will reach different people differently: each makes something of it, but not necessarily what someone else would make, and not necessarily what we expected would be made. We say/write what’s in our mind; what gets in the mind of the listener/reader isn’t conveyed there but built there. Communication is not high-fidelity.
Alli was working out a piece of mathematics. That’s where her dad was no doubt focused when he mentioned negative numbers and that’s where I focused as I tried (and at first failed) to answer Alli’s question. But Alli was also working out a piece of English, a definition. In many contexts, we do report how far some value is above or below a range. Although she’s unlikely to have examples like blood-pressure or cholesterol levels, any kindergartener does already know that some categories name whole ranges of numbers above and below another range of numbers. For example, with no particular precision about which numbers demarcate the categories, they know that babies are below a certain age and adults are above a certain other age and in between are children. Alli has no information yet from which to conclude that this isn’t how the words negative and positive are used when referring to numbers. But it could be, whence Alli’s interest in knowing how to (or whether we should) treat 23 as “positive three.”
In this story, the uncertainty about the meaning of a word is of no real consequence. Though someone might wonder why knowing about “negative” was insufficient to clarify for her what “positive” meant, there’s no risk that Alli’s confusion would lead anyone to conclude that she’s “bad at math.” And, aside from her own interest, there’s no rush for her to know: she is, after all, still in kindergarten and will surely sort this all out in time.
But there are times when the vagaries of communication cause mischief. In US elementary schools, it’s common (probably close to universal) practice for teachers to instruct children to pronounce numbers like 3.12 as “three and twelve hundredths,” not as “three point one two,” what I call a spelling pronunciation. (In my opinion, the insistence on a fraction pronunciation in school is not helpful—for one thing, just think how you’d be expected to pronounce 3.14159—but I’ll save my many reasons for a later blog post.) In one fourth grade classroom that I was supporting, the teacher asked the students to read 3.12, and then wanted to check their understanding of the place value names, so she asked “how many ones?”
The class chorused “Three!”
“How many tenths?”
“And how many hundredths?”
Then a timid “two?” and a more timid “twelve?”
The context “how many ones, how many tenths” seemed to call for the answer two, which is what we know the teacher wanted to hear, despite the loose wording of her question. But children don’t yet have a way to be sure. They’d just read the number as “three and twelve hundredths,” so twelve was a sensible answer. Nobody, of course, answered “three hundred twelve,” which would have been a delightful response showing deep understanding, just as nobody answered the earlier questions with “3.12 ones” and “31.2 tenths.” All of these answers are mathematically correct but they’re “wise guy” answers because they violate norms for communication. They are correct, but clearly not what the teacher meant by the question. In the case of “how many hundredths,” however, students might well be unsure which the teacher meant.
Because the teacher didn’t recognize the source of the confusion—just as I had not at first understood the source of Alli’s confusion—she heard the hesitation and mixed answers as evidence that the class didn’t really understand the mathematics. I had the luxury of being the observer, hearing and following up individual children’s queries rather than having the full responsibility of the teacher addressing and trying to manage the entire class. What I heard and saw made it clear that virtually all of the children did understand the mathematics; the confusion was only about which of two very reasonable interpretations of the teacher’s question was the one she intended.
Unlike the story of Alli, this miscommunication did have consequences. One consequence was a review that was unnecessary, and therefore a turn-off, and that still didn’t clarify the question (the English) and so left several children feeling like they “don’t get it,” despite being able to respond correctly to unambiguous questions on the same content. The worst consequence, in my opinion, is that the lesson some children are getting is not about decimals but that they “just don’t get math.”
So what can we do to reduce negative consequences of missed communications?
At times, I read laments about teachers’ imprecision in language; these are decent examples and I’ll say a bit more about the issue, but later.
In my view (and in all kinds of circumstances), we give students a valuable message when we try to figure out what is sensible about their responses and explicitly state it: “Ah, you were thinking about the twelve hundredths we had just read, and [to the other student] you were thinking about just the number shown in that hundredths place.” In a case like this, it’s valuable even to acknowledge that can now see why they hesitated to answer and that we didn’t at first understand: “Oops, I wasn’t clear about which of those I meant.” Such responses from us teach several things. Possibly the most important is that students know that their thinking is valued even if it takes us a while to catch on. Another is that students see that our focus is on the logic, the sense they were trying to make even if it did not match our intent, and that we are assuming that’s their focus, too. That sets logic, not an answer to a particular question, at the center of the mathematical game. It values clarity, and it shows that we, too, struggle to communicate clearly. It detoxifies errors without fanfare and without “celebrating mistakes,” which students recognize as school propaganda. (Nobody ever says “Woohoo! I made a mistake!”) It models asking questions when we get lost in communicating an idea. (After all, if the teacher does that, it must be a useful and respectable tool.) And it acknowledges that trying to express mathematical ideas in words is clumsy and difficult—the problem is often not the thinking, but the communication—and that’s why mathematics has special vocabulary, notation and conventions. It’s not because mathematicians like fancy words and symbols.
And when we can’t understand students’ logic, we can admit that, legitimizing “I don’t understand what you mean” by showing that that happens to us, too. Kids’ explanations, even when they are totally correct, are often elliptical or garbled, so there’s plenty of opportunity for us to say, “Wait, I don’t get it. Could you explain again?,” giving you a chance to understand and giving them a chance to clarify and perhaps even rethink.
Finally, what about that issue of teachers’ imprecision in language? Being routinely more precise takes a lot of thought, a lot of knowledge, and a kind of self-consciousness and control that is hard to achieve, but building good “mathematical hygiene” (I attribute that lovely term to Roger Howe) with appropriate use of mathematical vocabulary and correct use of notation is a certainly a thing for teachers to think about. On the other hand we must also recognize that there will remain times when conveying a rough idea of what we mean is the best we can do, times when communication, especially with a child, can’t achieve understandability and precision at the same time. Teaching must walk a fine line.
Mathematics is so much easier than English.