Sunday, October 20, 2013

Mastermind, Part 2: Refining our Notion of Equality

We had finished up our puzzles, but we still had a question: How many turns does a winning Mastermind Puzzle usually take?

I pooled data from the kids' Mastermind Puzzles and from the boards that Anna's 6th Grade class (totally coincidentally) happened to be making, and made sure that they knew what data means. (Turns out? Cell phone plans have forever changed kids' awareness of the word "data." Hmm.)

Then I asked them to organize all of the data any way that they saw fit. (ala TERC. They have kids doing this with counting the contents of boxes of raisins.)

This, frankly, was a bit open for their comfort and the kids weren't so into it. There wasn't a whole lot to talk about in the different ways they organized things, though I tried. And my attempts to spin a conversation about what constitutes the "typical" number from the data set were foiled by the widespread agreement that the typical number of turns is 5.


But I noticed two really interesting things from their work, which you can see in the pictures above:

  1. Almost all the kids wrote something along the lines of 7=1 or 6=3 in their work.
  2. Almost all the kids said that the average was 5, because it was the most common result.
I decided to poke at equality. I asked them what "equals" means. The kids offered two definitions, and we put them both on the board:

  1. "Equals means 'the answer is, duh duh duh DUH!'" The vast majority of the class agreed with this.
  2. "Equals means the same." There was only tepid acknowledgement of this meaning.
I tried to put pressure on them by pointing out that 2=1 or 7=1 is a weird thing to say in some contexts, but the defenders shrugged me off. So I upped my game, and I wrote 3 = 2 + 1 on the board. Does this mean "3, the answer is 2 + 1"? I made some kids nervous, but then B offered that this was really just asking what 2+1 was.

But 2 + 1 = 3 + 0 was much more problematic for them. How do you read this as offering an answer to some question? A few kids tried to suggest that these were answers to the question "Is 2 + 1 the same as 3 + 0?" but I didn't like that because that's not what they said "equals" meant.

I argued that equals means the same amount, not "the answer is...", and by the end of class the kids were vocally agreeing with me, so I guess that worked? I got a better sense of where we stood the next day, when (after looking at their homework) I realized that they didn't know what an equation was either.

This, of course, is super-related to what "equals" means. The idea that an equation is any problem is supported by the notion that equals just marks the answer. I'll spare you the details, but we sorted this out too. 

Next up, putting pressure on their notion of average.


  1. Hmm, I've found that arguing logically with my kids typically just gets them frustrated and more confused than before we started, but maybe it worked differently for your kids. (Even though that's how I prefer to convince my friends or co-workers...)

    How about using analogies or other uses of the words "equals" or "equality"? Ask them what "equality" means in the context of siblings, racial groups, or basic human rights, and draw out the idea of a balanced scale or some other metaphor for what you and I think of when we hear the word "equal".

    Just an idea. Good luck showing them about equality. (What grade/age level is this again?)

    1. Oh, I see the label "4th Grade", so that might be harder to do with them. Were some of them receptive to your arguments, or did they seem more confused overall?

  2. Great that you're pressuring them on what "equals" (or "=") means so early. The more common belief in your classroom is apparently a majority view even for much older students. They see this "=" as saying, "Here come THE ANSWER to my right!" Your examples do a nice job of problematizing that for them.

    I've hypothesized that part of the problem many students have with "=" may stem from seeing tons of 2 + 1 = 3 and very little of 3 = 2 + 1, at the elementary level, and then later encountering teachers who prefer to have answers like x = 3 and NOT 3 = x, so that students become committed to the notion that "=" only reads from left to right. And then when "<," ">," and their siblings arrive, there's still a resistance to reading from right to left, which at times can be a definite disadvantage.

    Your students may not realize it now, but what you're doing here will serve them enormously well in the future.

  3. Maybe there's also something happening here with cardinal and nominal (or ordinal?) numbers. That 7=1 is sorta standing for S = 1 where S is the number of 7s.

    Which is not to say that this isn't a great opportunity to push on what equals means, just that this kind of shortcut labeling of tables/lists is a lot like they want to say

    Number How many?
    5 -------------- 3
    6 -------------- 1
    7 -------------- 1

    Which is a little different from when they write 7 + 3 = 10 * 2 = 20 to try to explain what happens when 2 kids each get 7 plums from mom and 3 plums from Uncle Ray.

    1. Missed the first part of the issue here last night, as I came to this from the other blog and was more thinking about the meaning of average, then didn't scroll all the way to the beginning of this blog entry. Having looked at it now, I agree with @Max Ray that this is a shorthand notation for talking about frequency. Given that they have no previous experience with that in all likelihood, the just used what made sense to them, rather than writing f(5) = 3, etc.

      What's interesting is that when confronted, their misunderstanding about what "=" means in mathematics really let them down. They clearly would like it to be some sort of sign that magically means that whatever is on the right gives the "answer" to whatever you (the teacher? the reader? the student him/herself?) wanted to know about what is on the left.

      So mind-reading comes with this math notation for a lot of students. Kids say all the time, "But you KNEW what I meant!" They are legitimately puzzled at times that you don't. That's one reason that the stuff you're looking at here, Michael, and the things Christopher Danielson does with his kids, are so very powerful. Keep sharing!