This is a Math Thinking Story about a five-year-old preschool girl, who was interviewed towards the end of the school year in May. Here we explore her understanding of addition and mathematical symbols.

###
Charlotte: A Math Thinking Story

Like many other children her age, five-year-old Charlotte demonstrates a basic informal understanding of addition. But what is even more fascinating and perhaps unexpected is that she understands a good deal about mathematical symbols—written numbers, the plus and minus signs, and the equals sign. Charlotte’s story explores what it means to understand these mathematical symbols, how young children may indeed know both more and less about them than is ordinarily expected, and what all this implies for early math education.

Before going further, I want to note that the video on which the story is based is an example of a skilled and insightful clinical interview, conducted by Cassie Freeman (who simultaneously did the camera work as well, not an easy combination). You can learn a great deal about interviewing by studying the embedded video clips and the entire interview from which they were taken.

The story begins with Cassie writing down a simple number sentence.

###
3 + 1 =

Charlotte immediately said, “Three plus one,” and wrote down 4. She did not say “equals.”

Cassie followed up by asking, “What did you do?” and “Why?” These are the fundamental questions of the clinical interview: they attempt to reveal the underlying thinking that leads to the child’s response. Charlotte gave a very interesting set of responses.

Charlotte answered by saying that, “Three plus one is four.” Note that she was able to read the symbol + as “plus,” but did not say “equals” in response to the symbol = . Then she explained how she got the answer. At first, she mumbled something about “two plus two,” maybe just pointing out that two plus two also is four. But then she said, with a few stumbles, that she began with the number three, added one more to it, and then concluded that doing so adds to four.

This segment reveals some very interesting features of Charlotte’s understanding.

First, she can read the written symbols 3, 1, 4, and + with no difficulty but said nothing about the equals sign. This is not particularly unusual. Preschool aged children sometimes see these symbols in books, and teachers may display them in the classroom and even attempt to teach them. As a result, young children usually learn the small written numbers (say, up to ten) very easily. They may or may not learn the symbols + , - , and = .

What’s more impressive is that Charlotte was able to interpret the sentence accurately. She understood that it referred to adding one to an existing quantity in order to get the result. Moreover, she could provide an articulate explanation of her counting strategy.

All this is extremely important: for Charlotte, the symbols were meaningful, referring to a process of addition that she understood and could express in words. Indeed, this is the fundamental issue of teaching symbols in Early Math Education. All too often, symbols are taught without explanation so that the child’s only choice is to memorize them. Thus, a child may learn to say “plus” in response to the symbol + , but may have no idea about what it means.

###
2 + __ = 5

Next, Cassie gave Charlotte the *missing addend* problem 2 + _ = 5. The goal is to determine what number added to the two yields five, so that both sides of the sentence are balanced or equivalent. When I first saw the video, I thought that posing this problem was a big mistake because it would be too hard for her. But watch what happens.

Cassie kept her silence for about 40 seconds. To me it seemed much longer. Why did she do this when her lack of response may have put the child on the spot? The answer is that silence may be golden when it gives the child time to think. Silence may also signify to the child that the adult has confidence in the child’s ability to think. And as you saw, Charlotte was clearly thinking deeply and apparently using her fingers to solve the problem. She looked back and forth from the paper to her fingers. But at the end, Charlotte proclaimed that the problem was hard (as the research literature shows). She didn’t seem to know what to do.

But Cassie didn’t allow her to accept defeat. She asked Charlotte to explain what she was doing. This is not only a good interview strategy but also can be an effective teaching method.

Charlotte said that four might go with two to get five, but thought that four was wrong. Instead of providing the correct answer, Cassie encouraged Charlotte to think about how she might get it. Charlotte first used her fingers to determine that two and four is six and therefore four is not the needed missing addend. After that, Charlotte not only got the correct answer, but also explained her reasoning.

Charlotte said essentially that if four is too large a number to combine with two in order to get five, she needs to take away one from the four to get the correct missing addend. This is excellent mathematical reasoning, going far beyond simply memorizing the number facts. Charlotte not only knows the meaning of the symbols, but can also reason about the problems that they represent in order to get a sound answer.

###
What does the = mean?

Next, Cassie asks what the = sign means.

Like many children her age and even much older, Charlotte was not clear about its meaning. She said that “equals means plus,” moving her hands together as if to suggest combining sets. I think she meant that the = sign indicates that the addition operation should be carried out. Many children say that = means “makes” or “get the answer.”

###
1 + 2 = 2 + 1

Cassie went on to explore this “makes” interpretation of = . She wrote and said the problem, 1 + 2 = 2 + 1. We know that this number sentence simply describes the equivalence of the numbers on both sides of the equals sign. The idea of equivalence is very important to understand, especially as a foundation for doing algebra. Watch how Charlotte interprets this problem and then some others.

She said that the number statement is wrong because it has “two pluses,” and should instead be written as 1 + 2 = 3. She also rejected 3 = 2 + 1, because it is “backwards.” Then, presented with 1 = 1, Charlotte claimed at first that this was not true but then perhaps reluctantly agreed that it was, even though it lacked a + , thus implying that the statement was still not quite right.

It’s interesting to see and hear how Charlotte reacted to these non-canonical number statements; that is, number sentences that children seldom encounter. Usually, indeed almost exclusively, children see number sentences like 5 + 2 = ? and are asked to solve for the missing number. If so, then it makes considerable sense to think that + means add and = means get the answer. Although the result, 7, is equivalent to 5 + 2, the child’s main task is to compute the sum.

So I think that in this context, and others as well, = does legitimately mean “get the answer,” at least in part. Not everyone agrees. Many teachers think that they are teaching equivalence, and are surprised to discover that their students have instead learned “makes.” Professor Ginsboo untangles the issue in his remarkable paper, "Does = Equal makes?"

###
Conclusions

Here are some conclusions from the interview with Charlotte.

- First, she has a sound understanding of informal addition and uses sensible strategies and reasoning to get her solutions; she does not simply memorize the addition facts.
- Second, she understands some basic symbolism, particularly small written numbers (numerals like 3 or 6), as well as the + sign, which she knows indicates addition.
- Third, she sees her job as getting an answer—doing something—in response to a standard number sentence like 3 + 1 = ?. She gets the answer, 4, but is not concerned with the equivalence relationship between the addends (3 + 1) and the answer she obtains (4).
- Fourth, her knowledge of symbolism is imperfect: Charlotte has trouble with non-canonical statements like 1 + 2 = 2 + 1. But this is hardly remarkable. Most children, even in the elementary grades, fail to understand atypical symbolic statements like these, which they seldom encounter in the classroom.

Overall, we should be impressed by what Charlotte does know, not what she doesn’t. The bottom line is that she has some sound ideas about addition and some understanding of mathematical symbolism.

What does this imply for early math education? The main lesson is that teaching needs to connect symbolism with the child’s existing informal knowledge. The child should not just memorize how to say the word “plus” in response to the symbol + . That’s important, but the child also needs to know the ideas behind the written symbol. Effective and engaging teaching needs to help the child create a synthesis between personal, informal knowledge and the symbols of formal mathematics. This is what meaningful learning entails and what allows us to conclude that teaching symbols to children as early as in preschool can be developmentally and educationally appropriate.

Finally, a word about Charlotte. She was from an impoverished neighborhood in New York City. Many people, including educators, hold to the stereotype that “disadvantaged” children—that is, those lacking advantages considered necessary to academic success—are not capable of the kind of abstract thinking that Charlotte displays. This belief leads then either to no teaching of math or to teaching that presents few challenges. But Charlotte’s story suggests that this view may be myopic. No doubt, Charlotte is not typical of all disadvantaged preschoolers, but her example suggests that at least some of her peers are capable of her level of achievement. There are enormous individual differences among children within any age level and within any ethnic, socioeconomic, and racial group. Given these facts, we have reason to be optimistic about the potential of early math education for all children—if and only if our teaching is meaningful. We need to teach all young children as if they are Charlotte.

And finally, the total amount of video you saw was just a little less than five minutes, during which Cassie said little and Charlotte spent a lot of time thinking. You can learn a great deal from a short interview!