Basic question about definition of a rectangle or any shape?

A

apple2357

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My son ( 9 year old) was doing this question below for homework ( its from a challenge competition).
The confusion is my son said the answer is B because he said rectangles cannot have lines in them? As presumably these are the only types of rectangles he is used to seeing. But when do we clarify with young children the definition of a rectangle and what is inside doesn't matter? Is it just assumed?

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apple2357 said:
But when do we clarify with young children the definition of a rectangle and what is inside doesn't matter?
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The sooner the better. If your son disagrees ask what shape he thinks it is. Or take a rectangular piece of paper and draw a line on one side. Ask him how it can change shape depending on which side is visible.
 
Interesting question! This may just be one of those things that you have to learn by experience, just as many aspects of English usage are learned by hearing the language used, which are very hard to state clearly.

I would say that a rectangle is a figure consisting of four line segments at right angles; what we don't explicitly state is that it is only those lines that matter, not what other lines might be present inside or around them. The fact that students first see geometric figures in isolation (or filled in so that the name seems to refer to the whole figure including its interior) is meant to make it easier to learn, but can be an oversimplification.

It's not uncommon for a student who has never seen a question like this not to see, until they are pointed out, that there are several overlapping rectangles. If I were introducing this to children, I would probably start with an example, marking one of the more subtle rectangles in red or shading its interior to call attention to it.

So, when do we clarify this? Probably whenever we first show such a problem! There's no shame in not realizing it on that first exposure.
 
apple2357 said:
My son ( 9 year old) was doing this question below for homework ( its from a challenge competition).
The confusion is my son said the answer is B because he said rectangles cannot have lines in them? As presumably these are the only types of rectangles he is used to seeing. But when do we clarify with young children the definition of a rectangle and what is inside doesn't matter? Is it just assumed?

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The question is not at all ambiguous.
Tell your son to look at the left most vertical line segment.
How many rectangles to its right is it in? (3)
Move to the next vertical line, it in (2).
Can he finish?
 
apple2357 said:
How should the question be phrased to be less ambiguous?
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It has to refer to a definition of a rectangle. If the definition is based on a closed figure with a perimeter consisting of four straight lines meeting at right angles, then the correct answer is 6. If the definition is based on a closed figure consisting of exactly four lines, then the correct answer is 3.

It is an ambiguous question because no definition is provided. That exactly is what the child is rightfully complaining about.
 
apple2357 said:
How should the question be phrased to be less ambiguous?
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I think an example, like I suggested, would be the best way to make the problem clear; apart from that, children just have to chalk it up to experience -- there are a lot of things I misunderstood the first time, but then I knew what was meant. This is a classic kind of puzzle question that you get used to.
 
lev888 said:
Why does it have to? Including all definitions with each problem would waste quite a bit of paper/electrons.
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Because when you ask how many X's are there, there must be an agreement on what X means. Otherwise you cannot agree on a common answer. Promoting the idea that math does not depend on rigorous definitions is bad math. It is perfectly reasonable to say that a rectangle is a closed figure made up of four straight lines and four right angles. If we were to ask 100 adults who are not mathematicians if that is a correct definition, I suspect 100% would say "yes." If we were then to show those same 100 adults this problem, they might say, "Maybe that definition I just said was correct does not apply to this problem." Adults recognize that outside of math definitions may have fuzzy borders. That is a valuable lesson for children to learn, but math is not where to teach it.

Back in the 70's there was an interesting psychological experiment about the difference between adults and children in finding a vocabulary for things that did not fit well into the vocabulary of English and using that vocabulary to communicate with others of like age. The result of the experiment was that both children and adults had no trouble creating a vocabulary on the spot, but that young children neglected to come to agreement on the meaning of that vocabulary. Adults on the other hand explicitly came to an agreement on what brand-new words referred to what before trying to actually use the word. (I am misdescribing a bit what the adults did. They said things like "Do you see the thing that sort of looks like a sail boat" and, once they had agreed that there was something that had some similarity to a sail boat, they just called it the "sail boat.") The point is that young children do not have an adult's experience with the trickiness of communication and tend to assume that words have exact fields of meaning.
 
If you label the "outside" vertices A, B, C and D, then the shape ABCD is a rectangle no matter what else is drawn in or around it. An analogy: a plate with food on it is still a plate.
 
I can see both sides of the argument here and you have provided some good analogies to help explain. I like the plate in the food one! Thanks all.
I think its a reflection on what kind of rectangles children get used to seeing- so yes this might be the first time the discussion around the definition of a rectangle gets explored. You could argue the question is designed to do exactly that!
 
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