Spatial Relations

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Betsy and the Three Mathematical Bears

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This 40-second video shows a young child’s understanding of spatial concepts such as in front of and behind. The video demonstrates that taking the child’s point of view can lead the adult to appreciate how an apparently wrong answer can result from sound ideas about spatial relations. The teacher needs to dig beneath the surface behavior of right and wrong answers to undercover the child’s interesting ideas. 

by Herbert P. Ginsburg

Betsy is three years and nine months old. Here she is, charming, and perhaps with golden locks. About one minute from the beginning of the interaction, the interviewer, Mia Almeda, begins with this statement, “Since these bears are in a line …” as she twice moves her finger left to right, from the blue bear to the yellow bear. Mia then asks, "Which one do you think is the bear at the end of the line?” Betsy points to the yellow bear. Next, when asked, “Which one is the bear at the beginning of the line?” Betsy points to the blue bear.

Asked to explain, Betsy changes her answer. She says that the blue bear is at the “back,” the orange bear in the “middle" (even though she was not asked about the orange bear), and the yellow bear is in the “front.” Perhaps surprised, Mia ends the questioning, after which Betsy demonstrates her abiding affection for the hard-working bears.

This very short excerpt raises several interesting issues.

At first, Betsy’s two answers seem contradictory. The initial answer was that the blue bear is at the beginning and the yellow bear at the end. The second answer was that the blue bear is in the back and the yellow bear in front.

What can we make of this? One possibility is that Betsy realized that her initial answer was wrong and then corrected it. Perhaps that’s the case, although I didn’t see any direct evidence from her expression or language that she saw a contradiction or was attempting to correct it. 

Here is another possibility: Perhaps both answers are sensible. At the outset, as she said, “Since these bears are in a line,” Mia pointed at the blue bear and then slid her finger to the yellow bear. In fact, she did this twice. Her question referred to the bear “at the end of the line.” If Betsy interpreted the question as referring to temporal order, then the correct answer is the yellow bear: it was pointed to last, “at the end.”  But if the question was about spatial order, then the correct answer is also the yellow bear because it ends up in the front of the line while the blue bear is in the back, as Betsy explicitly says. 

It then follows that if the two questions are different, then the two different answers are not necessarily contradictory. The yellow bear can be both “at the end” and “in front,” and the blue bear could be both “at the beginning” and also “in back.” 

But what would have happened if Betsy were asked to explain how this could be true? Did she see a contradiction in her descriptions? The answer is that we will never know. Sometimes we don’t have enough evidence to judge the plausibility of an interpretation. Still, there are several interesting lessons that we can draw from the episode. 


The first lesson is that the child may interpret a question differently from the adult who asked it. 

This happens more frequently than you might suppose. After all, one of Piaget’s (who was arguably the founder of cognitive developmental psychology) oldest and most important insights is that the adult may see the world differently from the child.

Teachers are no exception to the rule. One example concerns the symbol we call the “equals sign" (=). The teacher asks what the symbol means, pointing to an equals sign (=). The teacher thinks it means equivalence and thinks that’s what the child has learned. But the child says that it means, “makes,” as in two plus three makes five. For the child, it is not at all obvious that the teacher’s idea (two plus three is “equal to” or “the same as” five), makes sense.

So when you interview or interact with a child, as teacher, parent, or person, you need to try to take the child’s perspective. It may or may not agree with yours or be correct. If you don’t try to understand the child’s thinking, you may assume that she is wrong, even though her response may make sense from her perspective, and your response may not! Another way of saying this is that you need to get beyond your own egocentrism to understand the child’s.

A second lesson is that the language of questions matters a great deal

The interviewer can use many different words to state the problem. Which bear is

  • first in line?
  • third in line?
  • in front of the line?
  • in back of the line?
  • at the beginning of the line?
  • at the end of the line?

The interviewer’s job is to determine how the child interprets the specific words used in the question so as to rule out the possibility that the child is responding to a question that the interviewer did not intend to ask. To do this, you should consider alternative wordings for questions and also be prepared to ask the child to elucidate the meaning of a particular word.

A third lesson is that there is a lot of mathematics that can be found in the modest line of the Three Bears

Here are only some examples:

  • The bears are right side up.
  • They are on the same horizontal plane. 
  • They are all the same size, but each has a distinctive color.
  • The yellow bear is first, orange bear second, and blue bear third.
  • The yellow bear is first, the orange bear next, and the blue bear is last.
  • The yellow bear is in front, the orange bear is in the middle, and the blue bear is at the end.
  • The yellow bear is facing forward and so are the others.
  • The yellow bear is facing forward, the orange bear is facing the yellow bear, and the blue bear is facing the orange bear.
  • From our perspective, the yellow bear is on the left, the orange bear in the middle, and the blue bear on the right.
  • From the perspective of someone looking from directly opposite us, the blue bear is on the left, the orange bear in the middle, and the yellow bear on the right.
  • The yellow bear is one bear, the orange bear is one bear, and the blue bear is one bear.
  • The yellow bear and the orange bear are two, the yellow bear and the blue bear are two, and the orange bear and the blue bear are two.
  • And finally the yellow bear and the orange bear and the blue bear are three bears altogether.

Can you come up with more mathematics in our simple line?

Isn’t it interesting that that a short segment is so rich? Think of the discussions that teachers and children can have about this small sleuth (look it up!) of Three Bears.

Finally, although this segment is about 30 seconds long (without the emotional farewell), it took me much, much longer to figure out what was happening and to write this short thinking story about the Three Mathematical Bears.

To celebrate the occasion of the story’s completion, here is a haiku:

Yellow Bear’s Existential Ambiguity
Betsy loves sweet bears

Yellow bear confused: first, last?

Two realities….

Final advice to Yellow Bear:  Go drown your sorrows in some porridge.

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