Puzzle Presentation

I led a short workshop yesterday about logic puzzles. I think it went pretty well. People seemed to have a good time and there were some really good discussions about the kinds of thinking participants were using when they worked on the puzzles. Without me explaining it, people figured out my goal: to get students to develop language and reasoning skills which lead to proof.

I based my workshop on the methodology that Jeffrey J. Wanko laid out in his article Deductive Puzzling which was published in the May 2010 issue of Mathematics Teaching in the Middle School. Sadly, I don’t think that you can access the article unless you have a subscription.[1] In a nutshell, I had participants look at a solved puzzle and try to figure out what the goal of the puzzle was. Then they worked on an unsolved puzzle with a partner or small group and I asked them to think about strategies for good starting points while they worked. After most groups had solved the first puzzle, I called the group together and we solved a similar puzzle together on the computer[2]. I had people tell me what they wanted me to do and also why they knew that step had to be true. We did this process for three different puzzles: Shikaku and Hashiwokakero, which are in the MTMS article, and also Slitherlink, which I worked on almost obsessively last school year.

Anyways, I just wanted to jot down some observations so that I remember them the next time I give this workshop. So, I thought it went well enough that I’d like to do it again. That’s something!

I thought the order in which I introduced the puzzles was good. The one I started with was definitely the right one—it was accessible and easy to figure out what the rules were. I want to experiment with flipping the other two, doing Slitherlink second and Hashi third. I was surprised by the seemingly common belief in the room that Slitherlink was much harder than the other two puzzles. I want to play around with that and see if that remains the case with the order switching. I conjecture that part of this difficulty arose from it being very different game-play than the preceding puzzle. And I think it is a little more similar to the first puzzle’s game play.

I want to highlight something that arose naturally, the idea of developing a common language to talk about the puzzle-solving. People made up some really good terminology—“hallways”, “end-stops”, “corridors”, “double-bridge”, etc.—which is something that is extremely helpful when solving a puzzle together. I’ve usually only solved puzzles by myself, in which case there is no need for this. I also loved that it happened organically, that it was participant created language, not terminology I forced them to adopt.

I would like to have a little more time to discuss the overall “how might we use this in our classrooms?” question at the end. I know we started late, so I didn’t have my full 90 minutes, and maybe doing three puzzles in that amount of time was a little ambitious in the first place. We did have plenty of time to do all the puzzles, but more discussion time would have been nice.

I want to stick firm to my “no answers provided” policy. I might even want to make this explicit to people who want the answers. Part of what I am trying to accomplish is the idea of proving your ideas. If you know what the solution is and you check your work against it, I think this undermines that. You have to first prove things to yourself in order to convince anyone else.

And one last little side note. There was a comment that was voiced a few times in this workshop that just irritated me. If you’ve read this far, I’d love your feedback on ways to respond to this. Someone said: “There’s no way my students could handle this” (or words to that effect). I get an internal reaction similar to the sound of fingernails scraping down a chalkboard when I hear someone say this. And it’s not like I’ve never thought this myself. I completely understand where this comes from and I empathize with the feeling. However, I have learned to re-frame this reaction in my own mind and state it more along the lines of “I’m not sure how to make this idea accessible to my students yet”. The “yet” is really important, as is the “I”. Ultimately, I’d love to have a non-preachy response locked and loaded that I can say when someone says this in a workshop.


[1] If you would like to read the article, give me your email address and I’d be happy to send you a copy.

[2] I used the sample puzzles on Nikoli.com, which have really awesome game play, including the ability to easily undo multiple steps, and a “try” feature which outlines moves in a different color.

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9 thoughts on “Puzzle Presentation

  1. Oh I love these. Please send me the article. Thanks. You get my email in the blog comment notification right?

    As for your question… I’m the wrong person to ask. My inability to play nice with such teachers is a well-documented fact. I felt my insides get all twisted up just from reading that last paragraph.

    The biggest problem is that this teacher is right of course. If you’ve setup your climate such that you’ve got “can do” and “can’t do” already segregated in your head, and you never try to push, well then yeah, that teacher’s class is going to fail horribly at anything the students are uncomfortable with. Then this teacher will be confirmed in his/her beliefs.

    I think you handled it well though. “How can you adapt it?” being a good start. Also, some teachers are just too concrete with this kind of stuff. They only see the exact puzzles you used rather than the whole problem solving set up. Perhaps they need a preface before hand that you’re going to give them some specific puzzles to do but these specific ones are irrelevant to what we’re learning about.

    • I just sent you the article. So let me know if you don’t get it.

      I didn’t really handle this in the workshop. I just let it slide right by. Which is part of the reason that I wanted to ask about this. I want to have something to say so that I can feel comfortable challenging that belief in the future. I didn’t feel like I had something, or could come up with something on the fly at that moment.

      I like your question “how can you adapt it?”. I thought of a related question: “What would you need to do to get your students ready for this kind of activity?”

      I think questions are the best way to handle this type of statement. It’s harder to make someone defensive by asking a genuine question than by telling them that the way they are thinking is wrong. Pointed questions that aren’t really questions are another matter entirely.

  2. As one of the actual teachers present at this workshop who made one of the offending comments (and who saw how it made your jaw lock up), I would like to offer some insight into what was going on for me when I responded the way I did.

    My hope in doing so is to offer some perspective from a different (and perhaps unconsidered) vantage point and also to gain some insight into other ways I might deal with what I’ve got in my classroom.

    First of all, I loved your presentation framework and the idea for using puzzles as solving tools and entry points into explorations of communication about mathematical logic and reasoning. Thank you for a very thoughtful presentation.

    Now about my comment. You should know that about 80% of my students (all public school) are deeply discouraged math learners. Actually, “traumatized” would be a fairer characterization. They come into the classroom hunched over in a defensive psychological crouch, praying that their cloak of invisibility will hold all year if they don’t make any sudden moves.

    They got shoved into Algebra 1 in eight grade, before they were ready, then they failed it (great for adolescent self-regard, let me tell you!), and now they have to take it again in high school — only now, they have to be there twice as long every week.

    These students are stuck in the “freeze” response, characteristic of traumatized people.

    My first job (which tends to take all fall semester) is just to get them to breathe, relax, and believe that if they stay in the classroom and do not act out in ways that get them thrown out of the classroom (or into jail), something non-horrifying might happen… and it will probably involve them doing math.

    But we’re way too early in the trust-building process, and to cross that first bridge, they need to get some wins.

    One criterion I have found that works really well in the early stages of working with these students is something I learned from my study of information design with Edward Tufte. I use a lot of his principles and work in designing information graphics for my students that invite them to relax and explore.

    These designs need to be clear, compelling, and low-stakes. The same criteria I would use for designing a user interface for a piece of application software.

    The Shikaku and Hashiwokakero puzzles met this criterion of mine for a warm-up acivity for the population I work with. The cleanness and simplicity of their graphic design makes them very inviting, regardless of one’s experience with puzzles, math, or logic.

    The Slitherlink puzzles did not. They had a much less intuitive information design, with no clear way to start the puzzle without reading the tiny, buried instructions at the bottom of the page.

    And once I *did* read those instructions, I encountered this little gem which stopped me dead in my own tracks:

    “There is one unique solution, and you should be able to
    find it without guessing.”

    Oh good grief.

    I’ve spent 8 weeks convincing my students that hazarding a guess is a low-stakes activity they can use to begin working with problems. Some of these kids are very close to believing that the hand of God will not reach out of the sky and smite them dead if they toss out a wrong guess. So you want me to hand them a sheet of paper that tells them they’re a failure if they can’t find the solution without guessing?

    Thanks, but no thanks for that piece of advice.

    At this point in the semester, I’m this close to winning them over to the idea of giving in and just trying it. So the idea of handing them this particular confusing sheet of paper with shaming instructions did not seem like the better part of pedagogical wisdom.

    Does this give you any different responses or constructive suggestions for me?

    • Elizabeth, I am once more reminded of what a little world the math ed community is. I apologize for “calling you out” like this. I didn’t want to offend you or anyone else. My intent in writing about this was solely to gain insight for myself in how to react to similar situations, and I strongly regret not having the confidence to have had this conversation with you during the workshop. I wish that I had been able to talk to you about this then instead of doing an e-conversation after the fact.

      That being said, I would like to respond to your comment in detail, since you raise some really good points. Firstly, I know that you are aware that I teach at an independent school, with a very different demographic than your current school. What you may not know is that for the past four years I have taught at several schools with the exact same clientele that you have described. Students who have been beaten down again and again by the system, and by math in particular, students whose home lives are heartbreaking and the definition of unstable.

      I love the criteria you have developed for determining what makes a good problem for your students. You are right on in figuring out that your students (and all students) need problems to be accessible. Or to have “multiple entry points” if I were using edu-jargon.

      I’m really glad that you found the Shikaku and the Hashi puzzles helpful. Part of the reason that I picked these puzzles was because of the fact that they are “language independent” (as Wanko put it). In other words, once you know the rules there isn’t any confusing language that might prevent you from solving them.

      What seems to be your main issue with the Slitherlink puzzles is the wording of the instructions at the bottom of the page. I definitely don’t want you “to hand them a sheet of paper that tells them they’re a failure if they can’t find the solution without guessing”. Hell no! I was not majorly concerned with having those instructions at the bottom of the page because I knew that the group I was working with were mathematically trained adults—not students. If I were to give those problems to students I would make a LOT of changes to the format. For you guys I just printed off a couple of pages from the interweb and called it good enough. My suggestion, should you (or anyone else) ever decide you want to explore using something like this in the classroom, is to think about how you present it to students. Which would look significantly different than how I presented it in the workshop. The questions that Jason and I were throwing around might be helpful: “How can I adjust it?” and “What could I do to get them ready for this?”

      Obviously, one of the first things you would do would be to re-write the directions. That would be my instinct as well. I would probably also put fewer puzzles on the page to start with.

      The thing that I found really interesting was that I think Slitherlink is actually more intuitive than Hashi, but that didn’t seem to be the case in how you and the other participants approached the puzzles. I found that fascinating, and hope that I will get the chance to explore different presentation frameworks to address that in the future.

  3. Bree- Not to worry, and no offense taken. I came to the workshop to learn and to reflect and to improve my own practice. It was obvious to me that your intentions were sincere that that you were grappling with something genuine.

    That’s why I felt OK about naming the cognitive dissonance and not just letting it fester. I wanted to close the gap between what I was trying to say and what you heard when I was trying to say it. My meditation practice has taught me that “knots” don’t come up unless there is something powerful going on below the surface. And that’s usually where the payoff of learning is hiding.

    For whatever reason, I found the Shikaku and Hashi puzzles much more intuitive — and addictive — than the Slitherlink ones, which is really all I was trying to say. I tried to give the Slitherlinks a fresh try again today, but all the dots and numerals just made my eyes glaze over. Different strokes, I guess.

    Your presentation gave me a giant insight into how I could use logic puzzles as a “hook” to engage my particular bunch of kids and help them learn how to persevere in their problem-solving. What a great alternative tool for giving them a chance to experience the satisfaction that come with hanging in there and struggling through to a solution.

    Thank you for sharing these tools with us and for engaging with me.

  4. Pingback: Puzzle Presentation Part II « The Space Between the Numbers

  5. @Elizabeth I don’t think kids would be intimidated by this because it doesn’t “look” like math. My kids are pretty open to whatever as long as it doesn’t look like school-like. Of course, if you were to drop a grade on it and say “Solve this for 100 points and I’ll mark you off for each wrong box” then yeah..problem.

    Also, sounds like you teach at the high school we feed into. We put all our kids in Algebra regardless (and 1 geometry class) and last year had a new record of 15% of the kids pass the CST. So …yeah….sorry about that.

    And thank you for clarifying your comment. When I read “My kids can’t handle that” it immediately makes me think of teachers underestimating the abilities of their students. Whenever I hear that I think that what they really mean is “I don’t know how to teach it in a way they’ll understand it so I’m going to blame the kids” and I flip out.

  6. David Maloney and Mark Masters of IPFW do something similar with game logs of abstract strategy board games. They give the students a board and a sets of move lists including who wins and the students must deduce the basic rules of the game.

    The activity is meant to show how scientific knowledge is generated based on observations and hypothesis testing. And how we can never prove a hypothesis (or game rule) but can instead only disprove them. You can put together a scientific theory (set of game rules) that accounts for everything observed so far, but there is room for new observations to contradict some of the existing rules without having to overhaul the entire set of rules (scientific theory).

    Homepage for game of science: http://users.ipfw.edu/maloney/game_of_science.htm

    Paper: http://dx.doi.org/10.1119/1.3274353

  7. Pingback: Puzzle Presentation: An Update « The Space Between the Numbers

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