On the Magnitude of Numbers: Rubik’s Fail
I’m not sure if this is an admission of guilt or an indictment of myself as a beta-nerd, but I just don’t enjoy Rubik’s Cubes.
In fact, if we’re being frank–which raises pronoun-related metaphysical issues–I think Rubik’s Cubes are an excellent analogy for mediocre teaching and learning.
If you’re to be “good” at Rubik’s cube, you are to memorize a set of steps, or choose from a small cadre of these sets based on trivial inputs and variable states.
You’d think this sounds like good math, but it isn’t. Yeah, some of you are super-dweebs and are about to write me a comment about group theory. Look, if you’re teaching group theory to high schoolers, then you can have an account on this blog and write all about it; I promise.
Identifying math in situations where it is the best tool for the job is central to the success of a student in any STEM1 related situation. Sadly, the Rubik’s cube, and its myriad tens of procedures superficially look like math, but the actual process of math is dirtier, and less, well, primary colored.
Why am I ripping on the Rubik’s cube? Mostly because of this literature they just sent me in the mail on using them in the classroom:
Yup. 14,000 billion years old. Holy shit, Rubik knows something we don’t.
That makes the cosmic microwave background radiation, um, not microwaves.
That means that all the dark energy that physicists have their collective panties wadded up over isn’t actually a significant part of the matter-energy in the universe.
Finally, that means that the Rubik’s cube people should probably stop mailing me garbage about its toys and start collecting its Nobel.
Sadly, in the end, and this is important, the value of a giant number doesn’t matter. Students have no better way to conceptualize 14 trillion vs 14 billion years. It just falls in the folder marked “Old Balls.”
Here’s a proportional comparison:
Footnotes:
1. I’m sorry for using the STEM acronym. Here’s an explanatory list from my favorite twitter people:
- Salty Trollops Eating Mangoes – Me
- Stymied Teachers in Educational Malaise – TracePickering
- Savvy Teachers Escape Mania – FourKatie
- saintly triceratops evade all mallcops – TieAndJeans
- sometimes those emoticons matter :@ – maxmathforum
- Sit Then Eat Meat – achmorrison
- Students Try Education Memes – bvancil
Garden Wheat Presented Without Comment: Kayaks
When I first started playing around with Rubik’s Cubes, I was extremely frustrated at all of the solutions that just amounted to a prescribed set of moves that you had to memorize. It was quick, but very unsatisfying.
Then I ran across a website that tried to teach some basic theory about commutators in terms of the Cube. Basically, there is a simple set of rules that you can follow to switch the positions of any three pieces of the same type. If you understand that basic concept, you can solve the whole Cube. It will just take a lot longer. The advantage of this is that it can be applied to a Rubik’s Cube of any size. I have done it myself on a 3x3x3, a 4x4x4, and a 5x5x5 Rubik’s Cube. I have no doubt in my mind that I can do the 6x6x6 and 7x7x7 as well.
I actually think this is a great way to explain to students the power of using fundamental principles. They don’t need a “box on a ramp tied to another box that is hanging over the edge” equation if they understand the basics of Newton’s Laws.
Amen. My high school friends and I were nerdy enough to enjoy memorizing the standard solutions… but it *really* got fun when we found Philip Marshall’s “Ultimate Solution” site and realized you could apply it to weird shapes too, like the dodecahedron, We emailed him and he even gave us a shoutout on the site:
http://helm.lu/cube/MarshallPhilipp/Megaminx.htm
I agree the usual solutions are plug-and-chug memorization, but it could be a respectable math class puzzle to figure out what changes and what doesn’t as you generalize Marshall’s approach to shapes with extra sides, more layers, etc. (even without going into the group theory).
Also, someone else has documented this approach for a slew of cube-like puzzles:
http://rubiksultimatesolution.blogspot.com/
The welcome post is dated from 2014. I assume that’s the year when the rest of us discover the universe is really 14,000 billion years old. Ugh, shame about that.
Thank you for giving me permission to be a beta-nerd. I, too, have not been very excited by the old RC, and have always felt a little guilty. And I even have taught some group theory to high-schoolers, although we stuck with rotations of objects that have no internal joints…
This demonstrates your contention, Shawn, that to make something appear valid all you need to do is ascribe a number to it. Remember: 89.7% of all statistics are made up on the spot.
I totally agree about the fail that the Rubiks cube is for motivating pertinent math but I think it could be useful to introduce kids (in an extra credit assignment or on a down day) to math culture.
Mr. Thistlethwaite is an interesting character. His son is a mathematician. Have students (or maybe just the interested ones) explore the Math Geneology project page for Mr. Thistlethwaite. Let them see how advisors and their students work on similar branches of mathematics. Illustrate the small world phenomena by showing that if you go back far enough you get to famous mathematicians (2 steps to Veblen, 3 to EH Moore, a couple more to Gibbs and Darboux). Almost no one knows about the culture of mathematics and their opinions on where famous mathematicians get their ideas from seems more based on Good Will Hunting than than on the concept of academic lineage.
My 2 cents.
>Look, if you’re teaching group theory to high schoolers, then you can have an account on this blog and write all about it; I promise.
I’m pretty sure Paul Solomon does this.
OMG, of course he does. Well, if wants to post it here, the offer stands.
Hahaha! YES, I do! (Thanks, Michael.)
I’ve been doing an Algebra 2: Functions and Abstract Algebra class for the last couple years that includes groups, rings, and fields.
Even better, Anna Weltman and I did a one-semester Modern Algebra elective last year that got way further into the group theory. Pretty awesome.
Maybe we’ll take you up on that offer after all..
As for the Rubiks Cube, I never liked it much at all. UNTIL I read Metamagical Themas by Douglas Hofstadter. He has some fantastic sections on the Rubiks Cube there that show how you could easily piece together a solution, and it made me feel way more connected to the whole of it. Totally worth checking out!
Yes, we sure did teach a semester-long elective on group theory to high school students! It was a total blast. I’ve also taught that Algebra 2 class that Paul describes, and I incorporate activities that involve group theory ideas into most all of my classes. I’d write about it. What do you want to know?
How about starting by teaching the rest of us what group theory actually is, and how it can trickle down into our lowlier classes? Shoot me an email shawn.thinkthankthunk@gmail.com, let’s talk about an article or seven! Thanks!