Monday, June 23, 2014

Two Textbooks' Handshake Problems

Like so many classic math problems, the handshake problem is easy to state: "If everybody here has to shake everyone else's hand, how many handshakes do we need?" Because it's such a well-known problem, the way that a teacher or a textbook uses it can offer an interesting window into their perspective. The problem is common, so how you choose to use it speaks a great deal.

I'm going to share how my two favorite geometry texts -- CME Geometry and Discovering Geometry -- use the handshake problem. The question is: what do their approaches to the problem say about their takes on geometry?

Discovering Geometry's Handshake Problem:

CME Geometry's Handshake Problem:

Some Observations:
  • Discovering Geometry breaks down the modeling process into eight steps, starting with completing a handshake table. CME doesn't offer any of those steps or supports.
  • Discovering Geometry asks the student to model the handshake process with polygons and segments. CME presents the handshake problem and the diagonals problem separately, and then asks students to consider connections between the two questions.
  • This isn't in the pictures, but the handshake problem is the first problem that appears in CME Geometry, while the handshake problem appears as subsection 2.4 in the Discovering text, titled "Mathematical Modeling."

I'm very curious to know what you folks out there think about these two presentations. My suspicion is that these two presentations speak to two different assumptions about how students learn to reason in mathematics. One assumption is that students need lots of informal experience that, with feedback and opportunity, will slowly shape their reasoning habits. The other assumption is that students need an explicit model of the reasoning process, either in addition to or at the beginning of their opportunities to reason on their own.

Your thoughts and criticisms are always appreciated, but I'll amplify the invitation for this post. I also know that authors of both of these texts hang out on the internet, and they could likely school us all a bit on their work.

Comments from the Bullpen:

Fawn and mrdardy want to see Discovering Geometry take away all that support for the kids, instead put it into a teacher's guide. They'd like to see CME get rid of the explicit connection between diagonals and handshakes, preferring texts that offer radically little upfront support for students.

l hodge finds CME's explicit call for a connection between diagonals and handshakes patronizing. Is there another way to push kids towards making that connection?

fivetwelvethirteen connects this with the CCSS standards of mathematical practice. How do you help kids get better at big skills like solving hard problems or mathematical modeling? My take: let's look to other fields, like literacy, because "problem solving" is as big a skill as "reading" or "writing."


  1. If I were to put the handshake problem into a textbook, I'd leave out entirely the connection between it and the diagonals of a polygon. Instead, the connection would be made in the teacher's edition. Because by putting it right there (in 8 steps) or as 2 separate problems (and back-to-back), the textbook just made the connections for the kids without giving them a chance to do so on their own.

    We, as teachers, can always make this connection for them if it does not come up eventually. Let kids find their owns ways to record, illustrate, explain their thinking. They get to see that some other group's recording may be more efficient, that doing a simpler problem is a solid strategy, that oh-I-see-what-you-did-there. It's too ironic that Discovering Geometry puts this problem under the the label "Investigation" and leaves little for kids to really investigate aside from following their set of instructions. The messiness of kids' initial trials and reasoning should be allowed. This problem was made it too tidy; real-world problems and investigations just don't happen like this.

    Another option is to pose the two problems some chapters apart. Let the kids see if one reminds them of the other.

    I'm not familiar with CME Geometry, but I am with Discovering Geometry, and I like it! (I like Discovering Algebra too.) So, this is just specific to the problem you post here.

    When I see too much information is given or too much scaffolding happening in the textbook, I simply just pose the problem to the kids without their books around.

    All this is to say our student textbooks should be a lot skinnier. Make the teachers' edition nice and thick. :)

  2. I agree with Fawn completely that the support work should be there for the teacher, not the student. Let the teacher have a variety of ideas about how to guide/steer the student conversation. Let the students approach the problem in a more organic way. We are working on our own Geometry text for use in our school for the fall and this problem is our opening day one. However, the problem is not in the book at all. Our teachers will introduce it and let the conversation flow. I happen to think that a diagram and some conversation about diagonals would be natural and I hope that it unfolds that way. We'll revisit this problem at a number of points during the year.

  3. We'll add our 2¢. The below link is to a snippet of a non-US curriculum's treatment of the same problem.

    Nothing novel there.

    Why do we comment? It's a fifth grade textbook.

  4. One fundamental idea here is how to scaffold two key Standards for Mathematical Practice (or at least that's how I'm interpreting it): MP.1, Make sense of problems and persevere in solving them, and MP.4, Model with mathematics. Something I'm working on is an articulation of key SMPs to put up in my classroom and refer students to, to make them more central and meaningful in my classroom. Here's a draft of those standards for that poster--

    I persevere when solving hard problems:
    I find out everything I can about a problem, even if I don’t know how to get to the answer
    I draw a picture or try a different method if I get stuck
    I estimate or make a prediction if I can’t find a precise answer

    I use mathematics to solve real-world problems:
    I choose appropriate mathematical tools to solve problems
    I use estimations and predictions to check if my answer makes sense
    I consider when a model might not give an exact answer, and adjust my work accordingly

    These are imperfect, but what I want to do is to give students some general scaffolds toward problem solving. I don't want to scaffold individual tasks (my ideal task looks more like the CME version), but to give students thinking tools to fall back on so that when they are struggling, they have an idea of where they might go next.

    Less scaffolding is great at increasing the quality of student thinking -- once students have cleared a certain bar of background knowledge and understanding. Too many times, I ask a class to attempt a problem, and for a few kids at the bottom end of the skill/mindset/knowledge spectrum, they try, fail, and reinforce a belief that they are fundamentally bad at math.

  5. Since the CME problem is at the very beginning of the book, I would guess that it is at least partially intended to establish certain expectations of the reader. I like that question #1 involves a specific case (however many students happen to be in the class), and question #2 is generalized. Not a huge fan of question #3 - seems patronizing. Overall, I would say that these questions have a reasonably good chance of easing the reader into the sort of thinking the book is presumably looking for.

    I often wonder about the intended purpose for presentations like that in discovering geometry.

    Is the purpose to introduce a formula for the number of diagonals? That is not really a central idea or particularly interesting result. Hardly seems worth it if getting there requires chopping it into an 8 step process.

    Is the purpose for the students to develop their ability to use the “technique” of building a table to analyze a problem? If so, what subsequent questions check whether the student has internalized this technique?

    Is the purpose of going “step by step” to make it more likely for the task to go “smoothly”?

    1. Our hope is that #3 shouldn't be necessary for most students, which is why there is a significant gap (in vertical space) between #2 and #3. We found in field testing CME Geometry that many students (on the first day) did these problems without thought toward how they might be connected, and #3 sits there mostly to say "Hey you, look deeper."

      These sorts of "hey you" prompts appear less and less frequently throughout the books.

      (Sorry for the horribly late reply, just reading now!)

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  7. First of all , this problem can be solved in many different ways.

    - geometrically by drawing the diagonals in a polygon and getting an algorithmic idea on how many lines have to be drawn (I do not like the figures in the textbooks, however),

    - in an algorithmic fashion by finding a way to systematically list all handshakes,

    - by counting the number of subsets of size 2 in a set,

    - recursively by asking how many handshakes an additional person would add.

    There are not many questions with such a broad scope of solutions pointing into various important topics in math.