Devlin's Angle

May 2011

The Keyring Problem: NCTM fumbles the ball

Last month, this tweet from NCTM caught my eye:

"What a nice little problem to form the basis of a class discussion," I thought. Okay, it is perhaps a bit contrived. How often has anyone needed to know how many different arrangements there are? But that aside, it has a number of features that give it great pedagogic potential:

  • it is simple to state
  • it is easy to visualize
  • a teacher could get the class to each bring in a key ring and some keys, and begin the investigation experimentally, first with one key, then two, etc.
  • the question evidently has multiple solutions, depending on how you interpret what is being asked, making it a super exercise in mathematical modeling of a real-life situation
  • those different solutions are in some cases orders of magnitude different
  • the simplicity means that students should be able to formulate their own models and put forward clear explanations of why they chose the particular model they did.
  • I should add that one solution seemed to me to be the best, but I suspect that others would argue for an alternative. (My answer is 3,840. I'll come back to it in just a moment. Before you read on, you might want to see what answer you get.)

    From what I have said so far, it is clear that I follow the NCTM's Twitter feed, and on that basis you may infer, correctly, that I have an interest in K-12 mathematics teaching. You may even suspect, also correctly as it turns out, that I am a member of NCTM. Indeed, the week before that tweet was published I attended their annual conference in Indianapolis. Not as a math teacher, for I am not qualified to teach mathematics at K-12 level and have no experience doing so; rather as a mathematician interested in K-12 teaching and believing that my knowledge of mathematics may be of assistance to those who do teach in our nation's schools, to those who train them, and to those who provide them with educational materials.

    ASIDE: Why do I make this disclaimer? Because our society appears to be riddled with individuals who, on the basis of having been taught K-12 math as children, seem to think they know how math should be taught. (In some cases, those individuals freely admit that they were not taught well and never really got it, yet that does not prevent them advocating one method or another, even the one that did not work on them.) In stark contrast, I do not find many people who, on the basis of having flown in an airplane, think they could fly one, or indeed have anything useful to tell a pilot. Nor do I find people who think their experience being treated by a physician makes them qualified to operate on a patient or pontificate on medical practice.

    Teaching, like many things in life, might look simple from the outside, but it assuredly is not. Now, it is true that the likely outcome of a non-pilot taking control of a jet airliner or a non-physician treating a sick patient would be dramatic and tragic, in that people could die. In contrast, if someone not adequately educated in mathematics and untrained in mathematics pedagogy teaches a math class, the children are not likely to die. The worst that could happen are one or more of the following:

  • the children learn little or no math
  • the children learn some wrong mathematic
  • the children learn to fear mathematics
  • the children come to believe they cannot do mathematics
  • the children come to believe that mathematics is a collection of arbitrary, disjointed rules and procedures that have to be learned
  • the children give up mathematics at the earliest possible opportunity
  • the children come to believe that mathematics is ugly, illogical, pointless, and useless.
  • Yup, the outcome is not dramatic, and no one gets killed. But nonetheless tragic, don't you agree?

    What makes it particularly hard to appreciate the importance of good teaching is that, unlike flying a plane or performing heart surgery, it is very difficult - though not impossible - to measure the outcomes of mathematics teaching. (Standardized tests are about as useful in determining whether an individual can think mathematically as measuring a tennis player's bicep to see if she or he has a good serve.) Thus, the lack of benefit - or worse, negative outcomes - of inadequate or bad math teaching tend to remain hidden. This may go a long way to explain why so many people think that teaching math is not a skillful profession the same way that being an airline pilot or a doctor are. (Though the large number of adults who claim they cannot do math, and even hate the subject, should give us a BIG CLUE that something, somewhere is not right in many of our nation's classrooms, and has not been for at least three generations, and probably longer.)

    But I digress. Back to that NCTM keyring problem. I had just one nagging worry when I read that tweet. Was the person who posed it falling into the familiar trap of defaulting to one particular interpretation, and posing the problem as one that (wait for it) has a "right answer." Given the wealth of materials that NCTM publishes and promotes on good mathematics teaching, the odds were against this, but I had sufficient worry to tweet a reply:

    Sadly, my fear proved to be correct. Later that day, the NCTM tweeter (or is the appropriate actor a "Twitterer"?) published the answer that was expected:

    Of all organizations to make such a blunder, the NCTM is surely one of the worst. No, I am not going to cancel my membership of NCTM, which does a lot of good work. (Besides, some of my best friends are card-carrying NCTM members.) The fact is, the organization is made up of people, and people make mistakes. All the time. Moreoever, instant media like Twitter (and an MAA column like mine) increase the risk of something going out that a bit of reflection would have prevented. There is absolutely nothing wrong with getting something wrong. That is how we learn. Only two things are indefensible about getting something wrong: not learning from the mistake and not trying again. We Americans and British (I'm both) typically find it hard to admit an error. (Though a crucial factor in the continued entrepreneurial success of Silicon Valley, where I live, is that not only is failure accepted, if someone does not repeatedly fail they are viewed as not trying hard enough.) We need to overcome that fear of failure. (My friends in Japan tell me it is much worse there.) The fact is, with the keyring problem going out into the Twittersphere under the logo of the NCTM, we have a wonderful news hook on which to hang a valuable reminder of the power of what the blogging math teacher (and now Stanford doctoral student) Dan Meyer refers to as WCYDWT teaching. ("What Can You Do With This?")

    The keyring problem is, as I indicated at the start, a great problem for investigation. There is no single right answer. It all depends on how you interpret the question. As occurs so often with applications of mathematics, you have to start by turning an underspecified question into something precise. With answers ranging from 12 to 3,840 (Any advance on those limits, anyone?), it is clear that how you introduce the precision can make an enormous difference.

    The NCTM tweeter presumably modeled the situation as: "How many ways are there to order five points on a non-oriented circle having no distinguished point?" I'm not going to give any of the calculations, by the way. I may not have K-12 classroom experience, but I've done enough math teaching in my life to know that good teaching is a matter of helping students figure it out for themselves. This, incidentally, is why Kahn Academy (which I have endorsed elsewhere and will continue to do so) does not teach anyone how to do math (i.e., how to think mathematically), any more than a dictionary, thesaurus, and grammar book can teach someone how to write a novel. What Salman Kahn does is provide an excellent instructional resource for some of the tools you need to do math. A particularly gifted and motivated individual could likely use it to learn mathematics on their own, particularly if they had access to a mathematician who could assist them when they are stuck. Thank you for doing that, Sal. Your site is valuable, and you deserve all the attention it has garnered. The danger I see in all the media coverage of late is if people think that the provision of a tool that can play a part in improving the nation's mathematical abilities is actually the entire solution. (As far as I am aware - and I have met Sal a few times and we have talked about his material - he himself has not claimed that, though others seem to have come close.)

    The NCTM tweeter's interpretation of the keyring scenario is an easy one for experienced math teachers to slip into without realizing it, because it is the one that leads to the easiest computation. But as I pointed out (at length, as is my wont) in my May 2010 column, interpreting in simplistic ways problems formulated in real-world terms is tricky and not without significant dangers. Unfortunately, formulating abstract mathematical and logical puzzles in seemingly r eal-world terms (word problems) is a tradition that goes back to the very beginnings of mathematics several thousand years ago, and has been used ever since. As I noted in that earlier essay, this is fine for those of us who enjoy puzzles and learn to interpret the problems in the intended way - to "read the code" - but is problematic for everyone else. And in this case the everyone elses constitute the majority of people!

    Here is how I arrived at my answer. In a drawer at home, I have several key rings I have acquired over the years. So many that I suspect that, from that one source alone, a future historian might be able to retrace many places I have visited and institutions I have interacted with. Those keyrings all have one feature in common, namely a topologically circular ring, and the majority share a second feature, namely to the ring is attached some kind of object, often a badge or shield, sometimes a ball-shaped object, an occasional pen or flashlight, etc. I'll add that some of those badge-like attachments have two distinct faces and are rigidly attached to the ring, some have two distinct faces but the entire badge can be rotated on an axle, and some are symmetrical so that the entire keyring looks the same from both sides. ("What difference can these features make?" one or two of your students might ask. If they don't, you should ask them.)

    I also have an even larger collection of keys. (Why do we keep old keys, by the way? I have no idea what the majority of them lock, and for sure they no longer fit anything I still have access to.) A very small number of those keys are symmetrical: when I turn them over, they look identical. (Again, is this significant to how we model the problem?) But the vast majority do not have such symmetry.

    When I modeled the keyring probem, I based my model on my notalgia-based collection of real keyrings and my irrational collection of old keys. In other words, I did what I think any sane, sensible person would do outside the math class: I took the problem at face value, as asking about real keys on real keyrings. I based my solution on what I most commonly saw in my drawer.

    I then had to decide what constitutes an "arrangement". Again, I interpreted this word (I almost said "key word", but decided that would be too tacky, though puns are their own reword) in the way I thought most natural. The problem asked about different arrangements of keys, not about permuting them around the ring. In particular, if I change the orientation of one of the (asymmetrical keys), I change the arrangement. Again, there is a story to be told, and a decision to make, when symmetrical keys are involved. And I wondered for a while what to do about having two or more identical keys on the same ring, when the mathematical model you choose could depend on why the question was asked in the first place.

    In the end, I came out with 120 x 32, namely 3,840. A far cry from the NCTM's 12. Which of us is right? Neither. There is no right answer for the problem as posed. It's a modeling task - using mathematics to analyze something we encounter in the world. For sure, I think my answer is more in accord with real keys on real keyrings, and hence more realistic, than the NCTM's. But the value of the problem is that it has so many different answers, and shows that building a mathematical model is a tricky thing.

    Solving a math problem that arises from a situation in the world can be hard. But nothing like as hard as formulating the issue mathematically in the first place. And as I argued in another earlier column in July 2010, in the U.S., our economic future and our security depend on our excelling at the latter kind of activity. Put frankly, a crucial economic reason why it is important for as many Americans as possible to know how to solve math problems is so they can think mathematically about real world problems.

    Yes, the keyring puzzle is a great problem, and definitely one that NCTM should be promoting. Devlin's Angle is updated at the beginning of each month. Find more columns here. Follow Keith Devlin on Twitter at @nprmathguy.


    Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition. His most recent book is Mathematics Education for a New Era: Video Games as a Medium for Learning, published by AK Peters/CRC Press.