Today I went into my daughter’s first-grade classroom, full of six-year-old girls, and gave a presentation about Möbius bands.
We cut strips of paper and at first curled them into simple bands, cylinders, which we proved had two sides by coloring them one color on the outside and another color on the inside. Next, we cut strips and curled them around, but added a twist, to make a true Möbius band.
These, of course, have only one side, a fact that the children proved by coloring it one color all the way around. And we observed that a Möbius band has only one edge.
We explored what happens with two twists, or more twists, and also what happens when you cut a Möbius band down the center, all the way around.
It is very interesting to cut a Möbius band on a line that is one-third of the way in from an edge, all the way around. What happens? Make your prediction before unraveling the pieces–how many pieces will there be? Will they be all the same size? How many twists will they have?
Overall, the whole presentation was a lot of fun. The girls were extremely curious about everything, and experimented with additional twists and additional ways of cutting. It seemed to be just the right amount of mathematical thinking, cutting and coloring for a first-grade class. To be sure, without prompting the girls made various Möbius earrings, headbands and bracelets, which I had to admit were fairly cool. One girl asked, “is this really mathematics?”
It seems I may be back in the first-grade classroom this spring, and I have in mind to teach them all how to beat their parents at Nim.
Moebius strips are always fun, even if you’re all grown up. It’s nice to see that children can grasp and tackle such concepts as well.
Maybe if more people would do similar things the next generation won’t grow up thinking that mathematics is all about solving equations of sines, cosines, and integrals…
I agree with Asaf. We need more of this sort of thing happening in schools.
I agree, Dana and Asaf. I hope to get back into the classroom soon.
Wow! This post made `Editors’ Picks’ at mathblogging.org!
I plead guilty 😉 I think it’s not only an important topic but especially important to have top researchers do this kind of outreach.
Also, I had planned to put (part of) your instructions on the image slider — but I found a bug in our underlying software.
On a more philosophical note, the Moebius strips are prime examples of objects that have properties that are at once both mathematical and physical. Do the existence of Moebius strips prove mathematical Platonism (indeed, Moebius strips are examples of objects that are both mathematical and physical….)?
Hmmmmn…My view is that actual physical existence, in the sense that rocks and trees and chairs and electrons exist, is far more mysterious and opaque than mathematical existence. We seem to be able to give a fairly clear account of what it means for the set to exist whose only member is the empty set, but what does it really mean, at bottom, for something to exist in the physical universe? What does it mean for this desk in front of me to exist? I have no idea, and I find it utterly bewildering. I can’t imagine anyone giving a clear account of this without making some appeal to our first-person experience of the physical world, which seems to me to beg the question entirely.
Would you call this approach a phenomenological account of reality (this first-person experience of the physical world)? For me, at least, I am satisfied that by giving a strip of paper a half-twist and joining the ends together, I have constructed what Neil de Beaudrap called on philosophystackexchange “a finite non-orientable 2-manifold of a certain well-known kind which we call [a] ‘Mobius strip’ “. By constructing such a ‘ finite non-orientable 2-manifold’ it obviously must exist , since it exhibits the properties of such a manifold (though technically it is really a 3-manifold since a strip of paper has width) in the real world.
The phrase “though technically it is really a 3-manifold since a strip of paper has width” should read “though technically it is really a 3-manifold since a strip of paper has thickness”–sorry. Also, since cutting lengthwise at different locations on the strip has different physical outcomes (eg. cutting lengthwise one-third the way in from an edge has a different physical outcome than cutting lengthwise down the center of the strip), can these locations be deemed ‘physical properties’ of the object (that is the ‘finite non-orientable 2-manifold’ known as a Mobius strip). If so, then the distinction between mathematical and physical existence may be tenuous at best….
Reposted by Zaine Ridling https://plus.google.com/108871877301789098084/posts/FYuwQy42mfJ