One of these things is not like the others

When you read the title of this post, did you think Sesame Street? Foo Fighters? Or, like me, both?

Recently, Geoff shared seven (sneaky) activities to get students talking mathematically. One activity, ‘odd one out’, involves having students pick the one mathematical thing that doesn’t belong. This reminds me of one strategy used by Dr. Marian Small to create open questions – asking for similarities and differences.

Here’s my ‘odd one out’ question:

Which of the following quadratic functions doesn’t belong? (Dr. Small might ask “Which of these four functions are most alike?”)
y=2\left( x-1\right) ^{2}+3
y=\dfrac {1} {2}\left( x-3\right) ^{2}-5
y=3\left( x+2\right) ^{2}-4
y=-\dfrac {3} {2}\left( x-4\right) ^{2}+6

Students might say,
y=2\left( x-1\right) ^{2}+3 because it does not cross the x-axis
y=\dfrac {1} {2}\left( x-3\right) ^{2}-5 because it is a vertical compression of y = x²
y=3\left( x+2\right) ^{2}-4 because it is a horizontal translation to the left
y=-\dfrac {3} {2}\left( x-4\right) ^{2}+6 because it opens down

Do the graphs of these functions strengthen your choice or make you change your mind?

I carefully chose the values of ap, and q in y = a(x – p)² + q so that students could reasonably argue that any one of the functions could be picked as the odd one out. Because I am not looking for one particular answer, each student should be able to confidently answer the question and contribute to a mathematical discussion. Planning disagreement is key; it means students will have to justify their mathematical thinking.


7 Replies to “One of these things is not like the others”

  1. I’ve had a lot of trouble selling kids on the Marian Small/Malcolm Swan “odd man out” questions. It must be in the way that I’m presenting it to them, because kids always seem to just shrug and say, “It could be any of them. You want me to choose? Fine, the third equation.” Or something like that.

    Any advice on how to pull of those questions in the classroom? Have you found better and worse ways of selling kids on them?

  2. Michael, I can see students responding in this way. Unlike other activities I have blogged about, I haven’t tried this strategy – I learned about it after becoming a helping teacher. I no longer have a class of my own. One teacher I work with told me that it took time for her to begin pulling this off – you’re fighting the mindset that in math there is one correct answer.

    I see there being a couple of places for Marian Small’s open questions. First, to get students talking about math at the beginning/end of class. Second, to assess understanding. Students figure out what we think is important by what’s on the test – and what we say we value is often missing from these tests. I’d include open questions on my quizzes/tests. Maybe this gives these questions credibility in the classroom? But this doesn’t quite sit well with the ed reformer in me. ‘It’s going to be on the test’ goes against my beliefs about motivation. Still, I think asking “Which of these functions are most alike?” is an improvement on (only) having questions like “Which of these parabolas opens down?” So there you have it – a better and worse way of selling kids on them all wrapped up in one.

    Someone recently blogged about polling (Nat Banting maybe? I’ll see if I can find it) and I remember thinking at the time that this might be useful with these type of questions. Would polling before/after presenting motivate students to make a convincing argument in front of their classmates?

    In my example, a case can be made for any pair of functions being most alike. Maybe this is too much. If a case could not be made for some pairs, then maybe this avoids the ‘Fine, I choose A and C’ thing.

  3. I really like your idea to include it on a quiz or a test. Every once in a while I throw in an “explain this”, or a “which technique would you prefer” question on a quiz, but I haven’t done it systematically. I think that I’d like to, and when I do it I’d like to put a very open question like this one on it.

    I think, in general, it’s good to start experimenting with things such as this on quizzes. There’s a sort of mutiny that can happen when I try to push students past their limits in the whole-group setting. It might go like this:

    Me: Can anyone convince me why this would this be true?
    Kids: [Silence]
    Kid 1: Who cares why it’s true?
    Kid 2: Why did you make it so complicated?
    Kids 3 – 4: Yeah! Why did you make it so annoying?
    Me: You really need to understand this.
    Kids: O.M.G.

    The issue is that in the whole-group the kids can coordinate to lower the expectations. But I *never* have that issue when I throw an “explain this” question on a quiz or a test. So I like the idea of undermining their cynicism by asking them deeper questions in individual, rather than whole-group time.

    One of the things I struggle with is how to keep track of these sorts of questions in my more traditional, closed-question grading. That’s something that I need to think about, probably over the summer.

    I think another thing that would help me pull this off is being clear that there are a lot of bad answers to open questions. It’s not like no matter what you say, you’re right. I could start with one of these open questions and then ask them to give me a bunch of terrible answers before we dive into some good ones. Actually, that sounds like a lot of fun.

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