Fascinating article from today's New York Times. In 2-dimensions we talk about tessellating objects to fill the plane. Circles of course will always leave gaps. In 3-dimensions, equal spheres will also leave gaps when packed as closely as possible, but the question then becomes, "How do we arrange the spheres which would result in the densest packing. Turns out that the grocer's method of stacking oranges solves that problem! Equal cubes can be packed together without any gaps, so we can say that the densest packing for cubes is 100%, that is, identical cubes can be packed so that they use 100% of the available space.
But packing regular tetrahedrons (a pyramid whose 4 faces are congruent equilateral triangles) as tightly as possible has defied the best logical mathematical minds, including Aristotle's, for nearly 2000 years. Recently, significant strides have been made, not only by the best mathematical and scientific minds, but also by graduate students, like Ann Chen from U. of Mich.,, who have taken dozens of tetrahedral dice from Dungeons and Dragons games and are using a hands-on approach to build various configurations and then computing the density of the packing. For the past several months teams from different schools have published their latest and best attempts, but, as of this moment, Ann has found the densest packing at 85.63%. That's right, she's outdone the best theoretical mathematicians and scientists in this quest for the new Holy Grail of packing problems. Aristotle mistakenly asserted that there exists a perfect 100% packing for tetrahedrons, but this was shown to be false. Now it appears that the percent is far more than we thought. Professor Nash, we need you!
I strongly believe that there is a place for this kind of discussion in our math classes from the earliest years on. Let students know that solving mathematical problems often involves hands-on experimentation as in science! Besides, who's to say that there isn't some middle schooler out there right now who might sit in her room playing with these dice who will arrive at 86%!!
"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)
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8 comments:
Hi, I liked your blog, check this one as well, very interesting http://online-math-education.blogspot.com
Thank you for blogging about the hands-on empirical approach to mathematics. As an applied mathematician, I do various numerical "experiments" to test out my conjectures. Experimental mathematics should be more emphasized in middle and high school, as well as in early college math courses. For example, simulations are easy to do in Excel and a lot of conceptual insight can be gained from such an experience.
Lisa,
Thanks for the support. I checked out your new blog. Interesting and informative. Best wishes...
Reva,
You have to appreciate the irony here. I've always regarded myself as a pure mathematician! My doctoral work was in Algebraic Number Theory. However, my perspective about how children learn mathematics has evolved over decades in the classroom.
On this blog I've tried to represent a 'balanced' view between the two extreme positions in the Math Wars.
I absolutely believe that our understanding of concepts deepens with hands-on experimentation. Using tools like Excel or Geometer Sketchpad is invaluable in this process. I believe that algebraic relations form in the mind through concrete numerical patterning and how we represent the data. Did e = mc^2 occur to Albert in a vacuum!
On the other hand, I also believe strongly that students need to develop facility with the computational and algebraic tools in order to be able to recognize and manipulate observed patterns. Mathematical discovery is often based on the ability to represent relationships in different forms. This requires strong skills.
Finally, I always wanted my students to understand the difference between "truth" in a scientific "law" and a mathematical assertion. It's in the methods of proof that distinguish one from the other.
You might want to look for Doris Schattschneider's article in The Mathematical Gardner on pentagon tilings of the plane. She describes, among other things, the discovery of four families of tilings by an amateur mathematician, Marjorie Rice, who had had no mathematical training since high school.
Currently, 14 families of tilings are known, but more may exist.
Eric,
It doesn't surprise me anymore that someone blessed with ingenuity, spatial and mathematical insight can solve problems without formal training. Someone by the name of Srinivasa R comes to mind!
Actually, I will always remember two students of mine, John K and Ken B whose raw talent and genius left me in awe...
As a former math professor, maybe you can answer this burning question of mine: Is there a difference between teaching geometry based on its spatial aspects vs. teaching the subject based on its methodology? Appreciate any insight/perspective!
I have a burning question on geometry teaching--is there really a difference between teaching the subject based on its spatial aspects vs. teaching it based on its methodology? Appreciate any insight.
Jeff,
In my opinion, the formalism of proof should never precede sense-making of a theorem, principle, etc.
Young children can "discover" the Triangle Inequality using sticks of varying lengths long before they need to consider a deductive argument, but that seems self-evident, doesn't it?
I don't believe all students need a formal treatment of proof but certainly we want all of our students to undserstand the nature of mathematical truth. If they can explain in a logical manner why the diagonals of a rhombus are perpendicular bisectors of each other AND CONVERSELY, I'd be very pleased (ok, amazed!).
Some learners seem to actually prefer a structured argument such as 2-column proofs, others are more right-brained. As a teacher, I needed to address both hemispheres and that's not easy b/c each of us has a preferred dominant side. I see myself as more left-brained.
Finally, Jeff, as you may have figured out, I believe in BALANCE. I tried to balance an intuitive approach to axioms and theorems with a deductive approach. This begins with challenging my students to explain why something must be true, not just because it appeared to be or to produce a counterexample. For example, I would draw a quadrilateral with a pair of parallel sides (marked) and the other pair congruent. I would intentionally make it appear to be a parallelogram and challenge students, working in their "learning pairs", to prove or disprove that it was in fact a parallelogram!
You have asked the essential question here. I don't believe it's an "either/or" issue.
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