Tuesday, July 13, 2010

Paper Puzzle

I've been off the grid for about ten days, but still have things to share from my math teacher circle workshop from before I left.  Here's a puzzle for you.  Make the following with one piece of normal paper and a pair of scissors.  Nothing else.  No tape, glue, or photoshop magic.


  1. What am I missing? This seems (a) completely obvious (b) almost nothing to do with teaching maths...

    I'm not trying to be obnoxious, but as someone who cares deeply about maths and the teaching of maths, I'm quite disturbed by what I see of your approach on this site: it seems to be an exemplar of a trend to favour trivial creativity over solving hard problems, and it seems likely to encourage teacher-pleasing rather than mathematical work. Did you ever read my comment on your May post about the minimally defined problem? Another thing that bothered me was the example you gave in your June 10th Call for Problems - you were apparently targeting it at middle schoolers, but my 6yo answered it instantly, and I'd certainly expect any 11yo with reasonable number sense to do so, if not distracted by wondering what teacher wanted today...

  2. Edit: Sorry, my previous comment was inadvertently posted before I finished.


    First of all, I appreciate you taking the time to read my blog. While tone is often difficult to convey or interpret on the interweb, it seems that you are finding this blog unsatisfying and I wonder why you continue to read it. I am all for pushback, but just like my classroom I expect it to be respectful and free of ego. Explicitly, let's refrain from the "this is easy/trivial" comments. One, not everyone will find this problem trivial and there's nothing that kills interest faster than being intrigued and surprised by a problem, trying to work through the visual disconnect, and then having someone imply that you're stupid because you haven't figured it out yet. Secondly, while you may value difficult problems I'm sure it's not the only thing you value in a problem. There must be problems that you think are interesting that you also find trivial. In this blog I'm trying to shift the paradigm away from just solving problems and moving on to reflecting on 1, how we solve the problem and 2, why we find the problem interesting or uninteresting. Anyway, I have not gotten a respectful or ego-free impression in any of your posts and encourage you to consider this if you choose to post further. As for your first response about minimally defined problems, I haven't responded yet because I feel that your response along with Ms V's response about lazy students necessitates a full blog post (and probably many full blog posts). In brief, though, here are some responses:

    Re minimally defined problems: This is a new approach and I have every expectation for there to be some pushback from students, teachers, parents, etc. I'm a little surprised, though, to get so much pushback from someone with a PhD in mathematics, someone who spent years in graduate school *creating* mathematics problems, someone who at some point in time was asked to approach math as a verb instead of a noun. And no, I don't find this approach trivial whatsoever and yes, I believe that creativity *should* play a more significant role in the mathematics classroom. As for the teacher pleasing aspect, this is actually something I've thought about a lot. I think so much of school is teacher pleasing, regardless of philosophy/pedagogy/curriculum and a significant part of my constructed theory has been around analyzing whether or not students will be more likely to find intrinsic value in this approach.

    Re call for problems: You may have misunderstood the intent of this post, because you talk about how this problem is "too easy." I was simply looking for math problems that people found interesting (You may value difficult problems. There's nothing wrong with this, but as I said earlier I'm sure this isn't the only thing you value. I'd love for you to contribute a few problems that you like and that you would find appropriate for middle schoolers.) I am in no way arguing that a six year old couldn't solve the problem I included (finding 2+4+6+...+20) and I'm uninterested in how long it takes them. What I'm curious about is *how* kids would solve this problem. Would they chug out the arithmetic? Would they reorder? Would they see the relationship between the answer I give them and the new problem? Kudos to your 6 year old for answering it instantly (although I really try to de-emphasize speed), but I'd be more interested in how he/she answered it, allowing me to then ask appropriate follow up questions (which might be along the lines of...if I start at a number n and count by m to add a total of x numbers, what's my sum in terms of n, m, and x?)or even better, allowing your six year old to ask follow-up questions of his/her own.

    Thanks, and I hope this reply clarifies both my intent of this blog and my expectations for a collegial environment.

  3. Perdita-
    I don't think this problem is obvious to everyone--it certainly wasn't obvious to me, and I have a BA in math and have been teaching math for the past 3.5 years. I think for people who have a natural affinity for spatial reasoning, for mentally visualizing objects, this problem may be trivial. That person isn't me, and it isn't for a lot of my students.

    I had a great conversation with one of my colleagues about how each of us solved this problem, which as Avery says, is one of the main foci of his work with this blog and ultimately on his dissertation. *How* do students solve these types of problems? I for one imagined picking up the flap that was sticking up and seeing how the other pieces would swing around and connect together. Then, while I understood how the figure worked, I still was a little uncertain how to physically make it. I thought about it some more and figured it out. I'm not saying I spent hours working on the problem--far from it--but it was a problem that I found interesting and non-trivial.

    I also believe that this problem pertains to the teaching of mathematics. The NCTM math standards, as well as the newly drafted Common Core Standards, contain both content and *process* standards. So again, the how of solving problems is important. Not sure what the UK standards look like, but I would guess that they have some sort of process standards as well. I could go on and on about why I think this is a good problem (and I probably will on my own blog), but this comment is getting too long already.

  4. Why am I reading? Principally because when I stumbled on your blog I was intrigued by the strength of my negative emotional reaction to your minimally defined problem post. Since you were asking how people would respond, I told you - by the way, I don't think your comment about "respect" is appropriate there; if you ask for reactions, you take what you get, even if you don't like it - and since then, I've come back periodically to see whether you've taken any notice, since I hate to think of you causing the same reaction in many children through a whole year, and would be reassured to think that you cared about the possibility and had some plan to mitigate it. (To check whether it was just me, I pointed a disparate group of friends whom I met through a non-mathematical connection at it, and asked them, without telling them what I thought myself. However unscientific such a poll, this at least confirmed that it wasn't just me. I offered "I'd have really enjoyed such a homework question", "Meh, OK" and "I'd have detested it" and more people voted for the third option then for the previous two put together. I do hope you have a better plan for the students who react that way than calling them lazy.)

    Objectively, the problem with this approach, and the reason why having a PhD enhances, rather than diminishes, my misgivings about it, is that the space of mathematical problems that are solvable with effort by a given person is very, very narrow by comparison with the space of all problems that that person can formulate. Preduction: if you ask children to make up their own problems, almost all of them will be either trivial, or insoluble. Neither promotes mathematical learning beyond, perhaps, the ability to pose problems. I'm not knocking the latter, but without also solving them it isn't the basis for a satisfying year's work. Indeed, the fact that it is hard to find problems that are interesting, but not insoluble, is why PhD students have supervisors whose job it is to direct them to areas where these may be found and to help them develop the skills of finding such problems for themselves. As I know from both sides, that is a very skilled role. Your suggestion of using such a problem as homework suggests that you do not appreciate this.

    I hope that my impression of you is mistaken, and I wish you well. I would be very interested to read about your plan for evaluating your year's work with your 6th graders.

  5. First, this doesn't look obvious to me at all. Second, I'm not sure if I'm seeing the problem correctly. Is the flap that's sticking up a single, connected of paper? Is there any paper cut off in this picture, i.e., is the flap just as as the missing segments are long?

  6. @ A mathematician improvises..
    Yes, the flap sticking up is a single, connected piece of paper. You can see all the cuts in the picture. So yes, the flap is the same length as the missing segments.

  7. By the way, to give credit where it's due, Avery stole this problem from me, and I stole it from a warm-up problem at a math circle for 7th graders at the MCCME in Russia. This piece of paper was sitting on the front table as the kids walked in, and they were asked to think about how it could have been made while they waited for everyone else to show up.

    I like having the photo instead of the physical piece of paper, though, to avoid the problem of kids being tempted to pick it up and fiddle! They have to figure out how to make their own. (My feeble attempts at getting around it by describing it as a "poison piece of paper" always seemed a bit silly to me.)

  8. Hey,

    I just replicated your picture. It was quite easy to do and that makes it all the more cool. It would never have occurred to me to do this in the first place. I agree with the poster above that this can be a great ice breaker with my little nieces and nephews.


  9. This is a really cool puzzle, I'm still trying to figure it out :)

  10. oh awesome! Just figured it :) that's quite clever :D

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