Children need to understand some basic mathematical ideas in order to measure accurately. First, they need to know what Goldilocks knows (as revealed by my extensive clinical interview with her), namely that Baby Bear is smaller than Momma Bear and Momma Bear is smaller than Poppa bear; and also that Momma is at the same time smaller than Poppa Bear and larger than Baby Bear. Children also need to learn what Goldilocks does not yet know, namely how to measure the bears exactly. How much smaller is Baby Bear than Momma Bear? And what is this smallness anyway? Is Baby Bear shorter, or less voluminous (skinnier), or lighter, and by how much exactly?
This thinking story describes how children’s basic ideas of small, larger, and largest gradually evolve into sound understanding of measurement. We begin with a rather old video, shot many years ago and not up to current production standards. But this cinéma vérité is very revealing.
Lizbeth and the Staircase
The interviewer gave Lizbeth, who was about three years of age, the seriation problem, developed by Jean Piaget many years ago, which involves putting a collection of wooden sticks (or “rods”) into a simple series, that is, an ordering from smallest to largest (or largest to smallest).
The interviewer begins with, “Watch this.” He then places on the table several red “sticks,” and tells a story about building a staircase for a man who wants to climb up so as to go higher and higher. Note that, although saying nothing, Lizbeth is paying close attention both to the story and to what the interviewer did with the sticks.
At the outset, the interviewer takes care to line up the first three sticks so that the bottoms are on the same level (the base) and the tops increase evenly from left to right. Given this, the fictitious man is able to ascend the steps one at a time, in a very even progression, to get to the highest step. The interviewer then asks Lizbeth to put out the next stick, the bottom of which she does not line up properly next to the bottoms of the other sticks. In response, the interviewer says, very explicitly, “Make it just like this,” as he lines up the stick on the base, and then asks her to continue. “Can you make the staircase higher and higher?”
Note that the interviewer modeled the construction of the initial three steps of the series, and also explicitly described the need to line up sticks at the same base and demonstrated how to do so. Given all this, will Lizbeth be able to construct the remainder of the series?
Clip 2 shows that she experiences some difficulty.
She adds sticks of different sizes, and mostly, they do rise higher and higher, although this is accomplished by ignoring the base. Piaget interpreted this as centration, which refers to focusing or centering on only one aspect of the situation (the heights of the tops of the sticks, ignoring the bases) because it is too hard for a young child to keep both in mind (in working memory) at the same time. Also, at the end, even her limited (and presumably easier) focus on the one aspect breaks down as she puts the tops of the last two sticks below the top of the highest stick preceding them. (Advice for interviewers: Don’t put the objects so far away from the child so that she has to practically climb over the table to get them!)
But don’t jump to conclusions. Watch what happens when, after Lizbeth flounders a bit more, the interviewer does some very direct instruction intended to clarify the need to focus on both base and height.
Lizbeth gets very excited, and in fact utters her first words during this interview, something like “I’ll find it.” She then carefully lines up each stick on the bottom as she puts it out. She clearly learned something from the instruction. But notice something quite remarkable. Each time, she selects the correct stick in the sequence, the one that is just a little higher than the one before it. To accomplish this, she does not have to try out various sticks by placing them on the base. Her method was simple and logical: when reaching out for the next stick, she chooses the one that is shorter than all those remaining and therefore larger than the last stick already in the series.
Next, the interviewer introduces a new problem, namely to insert a new stick into the already completed series. Piaget found this to be a challenging problem for preschool-age children because solving it requires breaking up the overall configuration that took much thought to produce in the first place. Watch how Lizbeth handles this situation.
Lizbeth does not have a clear strategy at the outset. She puts the stick at the beginning of the series and then at the end of it. Then, even after being reminded to consider the bottoms of the sticks, she breaks up the series here and there to no good effect. After the interviewer wipes her nose (I am unmasked: what interviewer other than a father would wipe an interviewee’s nose?) and also suggests that the stick had to go somewhere in the middle, she breaks up the series, going one by one from long to short, and opens a space between adjacent sticks. As she is doing this, she does not look at the stick to be inserted. When she gets to what she thinks is the right place, she inserts the missing stick. How did she know where to put it? Did she use an image of the stick in her head? I’m not sure.
This interview suggests some important lessons about seriation and about interviewing. First, thinking about increases in size, whatever attribute is involved (length, height, weight, volume, and so on), is more complex than it may initially appear. It requires what we might call “fair comparisons.” The child must learn, for example, that he cannot stand on a chair to be considered taller than his sister (just as he has to make sure that the sticks need to have a common base). It also requires understanding these increases in size are orderly, getting bigger and bigger, just as in a growing pattern. The child may also understand, as Lizbeth did, how the smallest in one collection may be the largest when placed in another. And finally, the child needs the flexibility of thought required to examine what she has created and to modify it as necessary. That’s a lot to know, even when the child seems only to be creating a staircase with sticks. And you should see the reasoning that goes into the child’s block play and construction. But that’s another topic!
The episode also teaches some assessment lessons. First, don’t assume that a child’s initial failure is conclusive. Modifying the problem, or even doing a bit of direct instruction, may reveal unsuspected competence. Of course, if you have just taught the child something, you have to take care to determine whether the child’s correct response is a mere copying of what you did or whether the direct instruction essentially helped the child to understand what the task and question were all about. (“Oh, I get it; he wants me to make a staircase that is flat on the ground, as if I made the sticks stand up.”) Second, you can learn a lot about a child’s knowledge by having the child work with objects in response to your questions, even if the child says very little. This is not naturalistic observation (observation of everyday behavior), bur rather observation of behavior in a test-like situation that forms the basis of the clinical interview. Here, Lizbeth “answered” the clinical interview questions by arranging sticks in various ways.
Practice just for you: Maya the Elder
Let’s continue with four-year-six-month-old Maya the Elder (my granddaughter) who is beginning to build a staircase.
That is all I am going to reveal. This video gives you an opportunity to practice what you have learned and reflect on it. Analyze and interpret what Maya the Elder does. Does she understand the role of the base? Can she insert a stick in the right place? Can she fix mistakes? Can she select the smallest of all those remaining in order to find the largest stick of the staircase already constructed? Can she explain what she has done and why? You figure it out.
Maya the Younger
When you see the interesting thinking of a child at the age of about four, you might forget about its origins just a year or so earlier. Here’s 21-month-old Maya the Younger’s first staircase—sort of.
She didn’t do so well, and her only language was “OK!”
Ethan’s unappreciated measurement and negativity
At age four-years-four-months, Ethan (Maya’s brother) builds a tower with three-dimensional sticks. When you look closely at them, you can see dividing lines suggesting that each stick is separated into what appear to be individual cubes attached to one another.
(In math talk, we might say that the rectangular prism shows square faces adjacent to one another on the four surfaces. This arrangement gives the appearance of cubes connected to one another.)
At the outset of this episode, Ethan has already built a staircase with six sticks.
I then ask Ethan to find the next stick without counting. He selects the red stick and looks at it very intensely, obviously trying to count. He asks whether it is OK to count and is told that it is not. Then he takes the red stick and taps the squares in sequence, again obviously counting, which he denies!
After doing the interview, my first reaction was that I had failed to get Ethan to engage in the problem I was interested in, namely whether he could use methods other than counting to solve the problem. I wanted him to do the Piaget seriation task. But on reflection, I realized that he was instead telling me that he was interested in precise measurement drawing upon the standard units (the square faces) provided by the sticks. He could have solved the problem more easily by visual comparison, for example just by holding up the red stick next to the orange. Even so, he was interested in exact measurement, even though I was not. (Advice to interviewers: Avoid imposing your perspective on the child!) His refusal to play my game turns out to be a blessing in disguise because now I can show in this paper how non-counting seriation can lead to and be replaced by exact measurement.
Watch next the extent to which Ethan’s numerical measurement approach is extremely sophisticated. At the outset of the video, Ethan has constructed a very neat tower and I ask what comes next. He counts the number of cubes, gets nine for the black and eight for the red. He places the eight at the taller end of the staircase. Then I ask him which of the two remaining sticks, black or blue, comes next, after the red. Let’s follow his reasoning.
He chooses the black this time without having to count the squares. The most likely interpretation is that he immediately sees from the way the sticks are lined up (with a common base) that the black is shorter than the blue and therefore should come immediately after the red. (This is the argument that Lizbeth implicitly uses to the effect that the smallest of the sticks remaining will be the largest of the sticks in the staircase.)
But then, when I push for a reason, Ethan engages in an explicit analysis of the lengths and their significance. He begins by saying that he knows that the blue stick is 10. (How he knew this is not clear but is not important for this analysis.) Then he holds up the black stick and says that it has nine because it has one less than the 10 and that if it had one more it would be 10. Then he uses some remarkable language to describe the block of nine: “It has a less square so it’s nine…” In effect, he is saying that the nine would equal the 10 if not for its missing square. Further, “a less square” might be taken to indicate that he is using a negative number, saying that 9 – (-1) = 10. You might conclude that this is quite a remarkable use of the double negative. And then he added: “Pretty easy.”
So Ethan solved the problems by both non-numerical means and by exact measurement. Not so easy as he suggests! And his explicit description of his thinking is not so easy either and should be encouraged in young children.
We have traced children’s journeys in developing informal ideas and methods for dealing with series’ of objects, and then formal ideas and methods for exact measurement as well. Enthusiastic Maya the Younger is followed by Lizbeth’s struggles with seriation and her eventual, although mostly non-verbal, grasp of some basic seriation ideas. Then Maya the Elder provides verbal elaboration on seriation and also demonstrates many competencies that you are urged to discover in the video of her performance. Finally, Ethan shows how exact measurement emerges (despite the interviewer’s efforts) to provide a happy ending to the Thinking Story.