Yesterday’s lesson began with me drawing a rectangle, a square, and a rhombus.

Me: A rectangle is a figure with 4 sides and 4 right angles. Is a square also a rectangle?

Girls unanimously: No!

Me: Does a square have 4 sides? (Yes.) Does it have 4 right angles? (Yes.) Then why is it not a rectangle?

Varya: It doesn’t have two long sides and two short sides. (This was accompanied with drawing long and short sides in the air.)

Me: But was that in the definition of a rectangle? (Again a unanimous yes.)

We then moved on to the definition of a rhombus. I define a rhombus as a figure with 4 equal sides and ask whether a square is also a rhombus. Of course not! This time Katie volunteers the explanation, “Because its angles are not pointy.” I then take the square and rotate it by 45 degrees. “Is it now a rhombus?” Yes. “But it was not one before?” No. I decided not to insist on anything just yet. I did come back to the questions of whether a square was a rectangle or a rhombus a few more times, each time making sure to reiterate the definitions; the girls’ answers, however, did not change.

After failed attempts at classifying the figures, we started playing with their symmetries. I had prepared some cut-out figures that I folded along various lines and opened back up. We drew the rectangle’s two lines of symmetry and verified them with folding. I then asked whether they thought the rectangle had any more lines of symmetry. Katie said yes and drew the diagonals. We then folded a rectangle along the diagonal and I had the girls verify that the two parts did not land exactly on each other; we agreed that the diagonals were not lines of symmetry for a rectangle. We then proceeded to find the rhombus’ two lines of symmetry. Here again Katie initially suggested that the line that bisects opposite sides was another line of symmetry and we verified through folding that it was not. Finally, we discovered that the square has 4 lines of symmetry: two that bisect opposite sides like a rectangle’s and two that bisect opposite angles like a rhombus’.

So now the question is how to proceed. I think that the next time that I decide to come back to this topic I will just insist that a square is both a rectangle and a rhombus because it satisfies all of the conditions. I believe that the main reason why they didn’t see it this time was because of preconceived notions about the figures: everything that they had previously seen called a rectangle had one pair of sides longer than the other, and so a square must not be a rectangle. I would love to hear any other opinions that exist out there.

In my opinion, you need to play it as a game again. Sit them 45 (actually 135) degrees to each other, give one of them a square and ask each one is it a square or not. Then play around their answers. As for the rectangle/square dillema, take a rectangle that is visuallyclose to a square, wait for their answer that it is a square, and show, that it is not. Than discuss your square/not square question again, cut a little strip to make it a square etc. Even better, if you can make a rubber square or rectangle and play with it, asking them when, on their opinion, a rectangle stops to be a rectangle and becomes a square.

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Here are the few posts from Zhenya’s blog that I think are fantastic in their approach

http://janemouse.livejournal.com/733041.html

http://janemouse.livejournal.com/654503.html

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I think the issue is of the same sort as what they call ‘scalar implicatures’ in linguistics – they used a non-literal, ‘enriched’ meaning of what you give them as definitions. you say “a figure with 4 sides and 4 right angles”, but they enrich it with “NOT with equal sides”, because there’s a more narrow, more ‘informative’ notion of square that is “4 sides, 4 right angles, equal sides”. the same principle is at work when we, in natural language, use quantifiers like ‘some’ or ‘all’ – when we say “i ate some of the cookies”, logically, it’s just existential quantification and is compatible with the situation where there are no cookies left (i.e. i ate all of them). but what we in fact infer from this sentence is “i ate some BUT NOT ALL of the cookies” – an enriched meaning. linguists say that this inference is there because there is a conversational reasoning involved in our ordinary language that makes us think something like “if a stronger alternative (‘all’) was true, the speaker would have used it, but she didn’t, so it’s not true”, and so the negation of the stronger alternative is added to the meaning. this is a feature of ordinary, non-scientific, or non-terminological communication. and it seems to me that the girls are in this ordinary mode of communication while doing math with you, they just haven’t built this ‘stronger’ ‘terminological’ mode yet, where the literal meanings play a crucial role. at least the features of not having built this distinction yet strike me when i do ‘logic’ with my 5-year-old. does it sound intuitively right?

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Lisa, your explanation makes perfect sense. There is no reason to call something just a rectangle when you know that it is a square. On the other hand, if they ever encounter a question about a rectangle, they should understand that the answer might be a square. Maybe the thing to do is give them such a question; any ideas?

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Rather than giving them a definition let them develop the characteristics – similarities and differences – for each of the objects and write these down under each object. Then pose the question starting with: “Is a rectangle a square?” They will more than likely answer “NO!” then reverse the question: “Is a square a rectangle?” They will most probably answer, as in your experience, “No.” then with some Socratic questioning you can lead them to the conclusion that a square is a ‘special’ rectangle.

I’ve done this with middle-school kids with considerable success.

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Wally, this certainly sounds like an interesting approach, but my fear would be that with 5 year olds they’d get bored before you reach the desired destination. What sort of questions do you discuss with your middle school kids in this way?

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Wow, Lisa, this is a great explanation! I think their brains function like this very often.

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Wally, I think that at this age it is simpler to use deductive way. We can draw a few rectangles closer and closer to the square and show them one by one, finally coming to the square, but still being in a “rectangle mood”.

Lisa, I really like your explanation. I believe you are absolutely right, these two examples have the same psychological base.

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