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Bella and the Cubes: A Measurement Mystery

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Although they perceive length at an early age, children have difficulty measuring it. To do so, they need to learn several basic mathematical concepts, like the idea of a constant, conventional unit of measurement.This handout describes challenges children face as they learn to measure lengths using non-conventional units (in this example, connecting cubes), and those faced by interviewers in trying to determine what children understand about measurement.

by Colleen O. Oppenzato and Herbert P. Ginsburg

In everyday life, young children notice and focus on size and size differences. Terry might say that this building is “super big.” Tom may say he is littler than his dad. Taniesha might roll playdough into a snake, making it longer and longer, and proclaiming that it is the “hugest ever.” However, while perceiving and attending to such differences, children may have difficulty in quantifying and describing them accurately. Children need guidance to understand and communicate about precise measurement.

Many teachers introduce measurement by having children use non-conventional units such as connecting cubes, rather than conventional units such as inches. One reason is that rulers can be challenging for young children to manipulate and read accurately. Another reason is that conventional units are abstract and may not make immediate sense to children. There are no objects called inches in young children's worlds. Objects like cubes, on the other hand, are familiar and present in the daily lives of preschool-aged children and can be used to help them learn to measure. However, as you are about to see, use of manipulatives like connecting cubes is complex and does not guarantee successful learning.

Bella’s Length Measurement

Let’s watch Bella, who at age four years, nine months, has never measured with cubes.

The interviewer begins by saying that she will show Bella how to measure, and she places two connected cubes next to the green rod. (On the video the rod may seem to be black, not green.) The interviewer says, “I want to find out how many cubes long this green rod is. Is it two cubes long, or do I need another cube?” Bella answers, “You need another cube.” The interviewer then says, “Another, to make them the same length.”

Consider what is happening in this first brief segment. The interviewer’s intention was to have Bella understand that the task is to figure out the green rod’s length in terms of the number of cubes. But the interviewer says at the end of the exchange shown above, “Another, to make them the same length.” 

Given this statement, Bella then focuses on equivalence of lengths. She says that more than two cubes are needed to make the stack of cubes the same length as the green rod. However, when a third cube is added, she says, "That makes it not the same length because it's a little bit the same length but on this side [gesturing to the bump which connects the cubes] it has a different length." While her attempts at precision are interesting, the interviewer tries to get her to ignore the bump, a minor feature that is not important for the major task, namely to focus on the number of cube units. The interviewer says, “So it’s not exactly the same length, but it’s pretty close, right? It’s about three cubes long, do you think?” Bella agrees and even points out that using four cubes would result in a stack that is longer than the rod. 

In summary, Bella clearly interpreted the task as determining whether the length of the stick was the same as the length of the stack of cubes. Although the interviewer’s intention was to present a measurement task (determining how many cubes were needed to describe the length of the rod), she talked about comparing lengths a great deal. Even though the interviewer thought she was assessing and scaffolding Bella’s measurement method, the interviewer’s language unintentionally promoted a focus on equivalent lengths. When assessing what a child knows, it is important to pay attention not only to the child's responses, but also to how the interviewer phrases questions.

The interviewer then asked Bella to measure an orange stick.

Bella carefully places five cubes against the stick and says, "It's not exactly the same. Look." She then says she thinks she needs one more. She explicitly states that the outcome is uncertain: “…[the cube stack] might be bigger; it might be the same size.” After adding the sixth cube, she says, "I think it's now the same size."

Then something interesting happens when Bella is asked how many cubes long the stick is. She counts along the orange stick, "One, two, three, four." The interviewer wonders if Bella miscounted the cubes, so she asks, "Can you point and count? How many cubes do you have there?" Bella correctly counts six cubes. But when asked again if the stick is six cubes long, Bella responds by counting along the orange stick. "This is one, two, three." She knows how to count the cubes, but, to the interviewer's surprise, does not seem to connect the cube length to the length of the stick. It isn't clear what units she may be considering, if any, as she attempts to determine the length of the stick. She heard the interviewer's instructions and question and, as children often do, is trying her best to comply. She seems to understand that some sort of units are involved—that she is being asked how many somethings long the stick is. But she does not choose to use the cubes to measure the length and indeed may not even understand that the cubes are measurement units she could use to determine the length of the stick.

A little later, Bella is asked to measure the length of a longer, blue stick.

Bella continues to be very precise as she lines up the cubes. When she is finished, she announces, "I'm gonna count the cubes," and counts the stack of 10 cubes accurately. When asked, "So, how long is the blue?" Bella counts "one, two, three, four" along the blue stick. This is both intriguing and confusing. Why is she counting the flat sticks differently than the cube stacks? What unit, if any, is she using to determine the length of the flat sticks? 

The interviewer tries to delve into her thinking by asking, "What are you counting when you count one, two, three, four?" However, as is common among young children, Bella does not explain. She repeats the count sequence, and as she does so, segments the stick into four imaginary lengths. When she says “one” she points to the very end of the first imaginary segment. She says “two” at the end of the second segment, and so on. She seems to have a rough idea of a unit and believes that it is relevant for measurement.

When the interviewer asks another clarification question, "But you said this is ten cubes long?" Bella makes a distinction between the stack of cubes and the stick. Gesturing to the former, she says, "This is 10 cubes long" and gesturing to the latter, she says, "And this is four." So, there is a disconnect: despite the interviewer's efforts, Bella does not see how the stack of cubes offers measurement units that can be used to determine the length of the stick.

Eager to find out what units she may be using, the interviewer points to the blue stick and asks, "This is four what?" Bella replies, "The blue is four." When asked, "Four cubes?" she starts to agree, but then corrects herself, saying it is not four cubes. She holds up the blue stick and says, "This is four!" And then referring to the cube stack, she says. "This is ten." 

Next, the interviewer decides to ask her the same kind of questions about a smaller stick.  Sometimes children display more competence with small number problems than large.

When asked about the orange stick, Bella again counts in segments, and says it is "one, two, three." She then counts the cubes beside the orange stick and says that the stack is six cubes long. Even though Bella lines up the cubes very accurately, she does not see the cube stack as providing units that can be used to measure the stick’s length. So the orange stick is three mysterious units long, but the cube stack beside it is six. While it is unclear how she is measuring the blue and orange sticks, it is interesting that she says that the longer blue stick is four mysterious units long and the shorter orange stick is three mysterious units long. Her actions and descriptions suggest that she has some notion, however imperfect, that longer means more units and shorter means fewer.

Bella’s disconnect between the cubes and sticks still puzzled the interviewer, who therefore asked her to compare the sticks.

When asked, "So, which is longer, the blue or the orange?" Bella places all of the sticks and stacks of cubes together and says, "This one," pointing at the stack of 10 cubes, not at the blue stick. She says she knows this because "I measured it and this one [the stack of 10] is taller." She seemed to be comparing overall lengths, which in this case is easy to do without counting. Then, when asked if she could look at the number of cubes to compare them, she treats them as quantities, saying 10 cubes is longer than six cubes because the count sequence "one, two, three, four, five, six, seven, eight, nine, ten" is longer than "one, two, three, four, five, six." So she is able to count and understand quantities, but does not appear to use them in comparing the lengths.

Bella’s thinking about measuring the sticks still remains a mystery. But the interviewer uncovered some of her thinking. She can see the difference between the cube stacks. She can determine the number of cubes in each stack and knows which number is higher by referring to the relative length of the counting word sequence. But she does not use the cubes as units of measurement for the length of the sticks. 



Summarizing the disconnect

Bella’s behavior is interesting because she was successful in creating and counting the connecting cubes but did not use them as non-standard units to measure the length of the sticks. Bella worked with several different approaches: she could compare the lengths of sticks by looking; she could count the cubes; and she could count her mysterious length units. She seemed to understand that some kind of unit was needed to measure the sticks, but the nature of her unit is a bit of mystery. Further, it is clear that she did not make the connection between the use of cubes as a unit of measure and stick length. So she knows a good deal about length comparison, about counting objects, and about the need for measurement units, but does not appear to integrate what she knows into effective measurement with non-standard units.

What explains Bella’s approach? One possibility is that the interviewer biased or confused Bella by inadvertently suggesting, in the first episode, that Bella should focus on length comparisons, and not on using the cubes to measure length. The argument is that if the interviewer had not done this or had used a different task, Bella would have used the blocks as non-standard measurement units to measure accurately throughout the interview.

We think that this argument is implausible. The main reason is that after the first episode, the interviewer made several attempts to make clear the goal of using the cubes to determine the length of the sticks. Perhaps the interviewer could have done this more effectively, but watching all of the clips leads us to conclude that Bella’s approach did not simply result from misunderstanding of the instructions, but from failing to appreciate the cubes’ use as a measurement tool. 

But consider other possibilities. One is that if only different manipulatives were used, the child’s competence would be revealed. For example, maybe she would have succeeded if she had been given ordinary non-connecting blocks or small square tiles. Or maybe she would have succeeded if the instructions were clearer.  The interviewer should not give up on the child too easily because children often possess competencies of which adults are unaware and which require clever methods of assessment to uncover. Children are often smarter than we think, and we need to get smarter to appreciate their abilities.

At the same time, we must consider the real possibility that Bella actually does not understand the use of non-standard units for measurement. After all, the video shows Bella’s very first attempts at measurement with manipulatives. We have no doubt that, over time, with the help of her teacher, Bella can come to learn that a continuous quantity, like the length of a stick, can be described precisely—that is, measured—by a specific number of arbitrary units.


What children need to learn about measurement  

Measurement is more complex than it might initially appear. Children need to appreciate that the stick is not just little or big and it is not just longer or shorter than another stick. It is exactly five units long or exactly three units longer than another stick. Children also need to learn that the arbitrary units, in this case cubes, must be placed end to end, with no overlaps or spaces between them. The cubes must span the entire length of the object being measured. And most importantly, children must understand the idea that the total number of cubes represents the length of the object being measured. After children understand these ideas, they will be prepared to understand conventional measurement by an inch or metric ruler. They will realize that the inch or centimeter on the ruler is just like the cube: it is a unit by which to measure a length. 


The use of manipulatives

Many early childhood curricula suggest using common manipulatives such as connecting cubes to introduce measurement. Indeed, Bella demonstrated a great deal of competence with making cube stacks that were equal in length to the sticks. However, despite using the cubes to complete the tasks, Bella did not actually measure with them. As the videos show, it’s not enough to work with a manipulative; the manipulative must be used to help the child understand the need for, and role of, artificial units (like cubes or inches) in measuring length. By itself, lining up objects with cubes does not guarantee understanding of measurement.  Manipulatives can be a wonderful resource, but be careful to use them in a purposive, conceptual manner.

Your goal should be to use intentional teaching to connect use of the manipulatives to the ideas of measurement. Some children will figure out on their own how to use manipulatives as a measurement tool. Many will need help.

One strategy is to first introduce the cubes and ask children to play with them and perhaps use them to make “something interesting.” After they are comfortable with the cubes, present children with the task of measuring the length of books (or some other object). Organize children into small groups and give each group several books of different sizes. Some books will be both longer and wider than others. Let’s forget about depth for now and just deal with two dimensions. Ask each group to use the cubes to figure out the exact length and width of the books. Go around the room and help the children to place the cubes correctly along the sides of the books (starting at the origin), to count the cubes carefully, and to write down the number of cubes (plus a little left over, if any) they lined up along each side of the book. In this way, you can help the children understand, in ways Bella may not have, that the line of cubes provides information about the lengths and widths of the books, and that this process is measurement

You can go continue over a period of days. For example, you can have children compare measurements of the different books (how many cubes longer is José’s book than Jenny’s?), then talk about their measurements and explain their use of the cubes, then collect data on length and width, represent the data in graphs, read the graphs, and finally extend the measurement activity to other objects in the classroom. 


Additional Exercises

For additional exercises to accompany this thinking story, visit Using Unedited Videos in Your Courses.

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