Thursday, September 5, 2013

Day 8: An Unstable Chair

We started the guided notes today in geometry.  The students had an odd mixture of disappointment and relief.  I think the relief came from the fact that we were going to a format with which they are familiar.  Students have been conditioned to expect lecture and note-taking so when things go differently, they are often confused.

Students sitting in lecture remind me of people returning to family who are not nice.  There is discomfort and awkwardness, but there is a level of security because of how familiar the situation is.  You KNOW that your aunt Ethel is going to ask when you're finally going to find someone and settle down.  You KNOW that your uncle Jeff is going to be disappointed that you didn't bring home any hot friends.  Students know that they are expected to sit quietly, listen to the lecture and write down ...whatever.

I hope that by the end of this year, my students will have forgotten these expectations for lecture out of disuse.  I want the "sit and take notes" muscle to atrophy.

So, even with the guided notes packet, I tried to keep the conversation lively and make it an actual conversation.  I've been thinking very much about the idea that I should never say anything that a student can say.  I ask questions that direct students in specific ways and I'm not satisfied with answers until I get the one I want, or one like it.

We talked about points, line and planes and I had students go to the board to answer the questions in their note packets.  I ended with a critical thinking question:

What makes a chair wobble?
I happen to have a chair in my room that wobbles, so I put it on the desk and rocked it back and forth.
I received a ton of answers that were true, but weren't what I was going for.  "The legs are uneven." "The screws came out." "It's unstable!"

"So the legs are uneven.  Why does that make it wobble?"
"One leg is shorter."
"Yes, but why does that make it wobble?  Why does it matter than one leg is shorter, or longer?"

Finally, I got what I wanted.  "It means the feet of the chair are on different planes."

We talked a bit about this too and I asked about the two legs that don't leave the floor when I rock it.  One kid actually said "they are in both planes!"

Me: "Are there other points that are in both planes?"
S: "Not that we can see."
Me: "What do you mean?"
S: "Well, all of the points between those two would be also, right?"
Me: "What do you mean 'all the points'?"
S: "Like, if we drew a line between those two points, everything on that line."

So I ran a string between the two legs, as he suggested.
Me: "Like this?"
S: "Yeah! Everything on that line would be in both planes."
Me: "So what we can tell about where two planes meet?"
Class: "It's forms a line!!"

I was VERY pleased.

I also had my first real discipline issue in that class.  For the past 7 days, one of the students has been doing almost nothing.  He comes in, puts his head down, refuses to do most of the tasks and refuses to engage with me when I talk to him.  He sat through the whole class with his arms pulled into his sleeves.  Today, after asking him multiple times to take out a pencil and get to work and he refusing to do so, I took him into the hallway.  I explained to him that this was a high level academic class and his behavior was not acceptable.  He stared at me blankly.  I asked him what I could do to help him get back on track, that I wanted him to be successful and that I would help him however I could.  He stared at me blankly.

So I called his dad, who was very supportive.  I explained that I wasn't trying to get him punished, but wanted to make sure we were all on the same page with the expectations of behavior and academics.  I think it went well, but we'll have to wait to see if it produces any results.

Pre-algebra is falling back into the habits that I sadly expect of those students.  Today, about 70% did their homework, which is much better than last year, but not as good as last week (85%.)  I quickly did my lesson, which was dull, then had students work independently or in pairs for about 15 minutes and then put their answers up on the board.  They then got time to work on their homework, almost all of whom finished and got their answers checked.

This is really a great group of kids.  I often struggle to determine what is age-appropriate behavior.  How much of the acting out is because they don't know how to behave, or because they are 13.  In high school, I was able to treat them like adults, but here, they are on the cusp.  Some are over it and some won't be for a few years.  It's hard to be fair and treat everyone justly when they are at such different places.

Perhaps that will be the topic of next week's #MSMathChat.

Warning: The follow content is about content! Read at your own risk!

I've been thinking about procedures.  A few comments from Kate Nowak on Twitter today got me thinking about why we ask kids to do things in a certain way.  Sometimes it's because it's the only way to do it correctly, like Order of Operations, but sometimes it's because it's easier for us to give them steps and see if they can follow them.  But by the time they get to middle or high school, is there still educational value in that?  If they haven't learned it by then, maybe there's something else going on.

Also, I want to draw a distinction between not wanting to follow directions and not being able to.

I think there is a third reason for having do something procedurally.  This came up today when I asked my students to follow the steps I gave, not because I care how they solve a problem, but because I've been teaching long enough that I know where the common mistakes are and the steps help to avoid those.

I'm not opposed to making mistakes and there is great value in learning from them, but when the notation the students are using leads directly to the confusion, I think it's important to at least provide them with a road map until they can find their own path.  This is the example that has been coming up recently.

Evaluate 3b + c for b = 1, c = 4  (Number 1 in the picture)

Clearly the student knew what he was doing and got the right answer.  My concern is the string of equal signs.  As the problem gets more complicated, this type of notation will start to trip him up, and it did 5 problems later.  I don't want to impose my own procedural preferences on them, but I also don't want them forming habits that will hurt them later.

This is a very fine line...


  1. 1. I love your wobbly chair example and plan to use it!

    2. I don't think that's a fine line at all. The student wrote 3+1=4 and it doesn't. There are two options to give the kid: 3*1+c=3+c or 3*1=3 (new line) 3+c. This isn't imposing your own steps so much as making sure their reasoning is valid.

    1. I agree with you, but my concern was that even though I think his notation is ugly, his answer was right and HE knew what he was talking about. I made him explain it to me and he was able to. I wanted to check with other teachers because I wanted to make sure my making him write stuff on a new line was good pedagogy and not just imposing my own sense of order on his mental process.

  2. Agree with Tina.
    If he *always* gets everything *perfectly* right then it will be hard to justify, but if not, then doing the "new line" thing will make it possible to debug his work.
    It *did* trip him up. Your job is to help 'em not get tripped up, eh?

  3. There's mathematical convention, too. Just like languages have conventions, math does as well. I think it's good to show students the structure that mathematicians use and encourage them to do the same. We don't all need to have the same ideas, or follow the same steps, but we should be able to follow the flow of work that someone else does.

    1. I understand that and I certainly require that some things be done in a specific way, like nomenclature. I suppose the idea of letting the students make decisions is very new to me and I don't always know how to classify my preferences versus tried-and-true mathematical convention.

  4. Justin,
    I feel your pain on this. I once sat down and really looked at how I personally used the equals sign in lessons. I realized that I had used it 5 different ways in two days.
    The term for when you use the same symbol to mean different things is "code shifting" because we are shifting the encoded meaning from one thing to a different thing. We know what we mean, but the learner only knows one or two of the codes so they have to decipher and come up with the new meaning on their own.

    If math teachers do this, how can we expect learners not to do this as well? I have been much more mindful of my use this last year and this year, and I have noticed learners not doing this as much. I also have been talking to my department about being careful with equal signs and not code shift on the learners.

    As I learn more about teaching, I realize those "formalities" of math that I thought were not really that important 4 or 5 years ago are very important to learning.

    That is a step in my growth as a teacher.

    1. You're absolutely right. If I don't use the equal sign correctly, how could I expect them to. I've been very careful not to use an equals sign when we're talking about evaluating expressions because I want to draw a distinction between expressions and equations. Also, I've been very careful to put each thing on a new line.

      Ex: Evaluate (5+2)-4(2+1)

      (5 + 2) - 4( 2 + 1)
      (7) - 4 (3)
      7 - 12

      Later, when we talk about solving equations, I have the discussion with them about the equals sign being a balance where each side of the equation has the same value. Even this, I think, isn't really what I want to be doing, but my own ingrained ideas are hard to shrug off.

      It's something that I really to work on.

  5. Please, please, please have a short conversation with your class about the question, "What is the meaning of the equal sign?"

    There are two common conceptions of the equal sign's meaning. (1) "Here comes the answer" and (2) "The things on either side have the same value".

    This work suggests a student who is operating with conception 1, which is extremely common but totally unproductive.

    If a student thinks that the equal sign signals the place to put the answer, then the mathematical convention will seem arbitrary: "My teacher insists that I put each answer on a new line; seems like a huge waste of paper to me." This is very different from, "My teacher insists that I only write true equations."

    Those students who submit to our will to write each calculation on a new line are often only conforming to what they see as an arbitrary convention, rather than changing their understanding of the meaning of the equal sign. But then all the algebraic moves we have them make are again about following arbitrary rules rather than about maintaining equality. One sure sign is a student who is uncomfortable with 3=x being the last line in an algebraic computation. "x isn't the answer," the student thinks, "3 is!"

    A good way to launch this conversation with a class is by having them write what goes in the box in the equation 8+4=box+5. And then have them write answers that they think OTHER people might write. Poll the class and you'll surely get some kids who want to put 12 in the box, and possibly a couple who want to put 17 in there. That gets us to talking about what the equal sign MEANS. (for the record, the majority of my future elementary teachers typically write 12 the first time through this task; high school graduates who placed into college level math, or who got there through the developmental math sequence)

    1. Your last point speaks directly to my confusion. I agree with you that it's something we need to drum out, but if people who placed into college math using this bad/wrong practice, then doesn't that validate their methods?

      I know that in mathematics education we just as much, if not more, about the process than about the result, but if this process is producing the desired results, then how do we determine which processes we should be enforcing and which we should be allowing the students to develop on their own?

      This is leading my mind in other directions. In the early and middle levels, especially with students who are math-averse, I think it's important to help build confidence. With a student who works very hard on a problem using the method described above, gets the right answer after a long struggle with the calculation, that student would have serious difficulty understanding why their answer is right, but the method was wrong, especially if that method works consistently.

      It's putting me in mind of cooking as well, which may be a bad analogy. Two people bake a cake, One uses the recipe and cleans as they go and one doesn't and cleans everything at the end. Both cakes come out looking the same and tasting the same. Now I have to tell one person that the way they made the cake is wrong.

      I'm not saying I don't want to. I think the way he completed this problem is ugly and will lead to trouble down the line, whether or not he becomes a mathematician. I'm just trying to see it from the student perspective so that I can do a better job of trying to explain why his method is wrong.

    2. Do this...In every class on Monday, have your students do the 8+4=BOX+5 problem. Each kid gets a Post It, they write their answer on the Post It. No names. You give no further instructions beyond "on your Post It write the number that should go in the box". Collect the Post Its. Remember that it's a box, not an x or other variable. Report back. If anything close to 100% of those Post Its have a 7 on them, I will eat my hat (which, fortunately, is made of Tootsie Rolls).

      See, I am not arguing that the run on equal signs need to be forbidden. No, I am arguing that the run on equal signs are a symptom of a deep problem, which is that people often do not understand the meaning of the equal sign. And this misunderstanding has serious consequences mathematically.

      But we can have that conversation later. For now, do this little action research project for me (or rather, for you!), OK?

    3. Absolutely! I may do it in my geometry class as well with a different color of post-it to see if that same issue persists there.

      I'll put it in my Monday post!

      Also, does that hat come in rice crispy treat? I''ll take 4

    4. Cool! Think maybe I'll do it too Monday (or Tuesday/Wednesday "block day") ... This was a fun read!

  6. has a lesson on it ... I like the box idea.

  7. Ok so the basic gist is over 50% of kids will tell you that the equal signs means "the answer". Your next response will be it means “they are equal”. Truth is, ask them a little more and they don’t even know what the equal sign means. But are these answers wrong? Not necessarily. But are students aware that this symbol means a lot more than what they think it does...ABSOLUTELY NOT. In order for students to be successful with linear equations or any other mathematical relationships it is extremely important for them to have conversations around the question: “What does the equal sign mean?” (then eventually less than, greater than) That is why last year I started my linear equation unit with exactly that question. see here→

    This conversation did not just happen in one day. We talked and did some activities for about 3 days. Now you might be thinking “YEAH RIGHT!? I don’t have that kind of time.” If you want to keep your sanity and you really want to help students be successful in mathematics class you will take this time to have these discussions. It will help you (and them) for the rest of any relational symbol stuff (systems of linears, inequalities, abs value, etc.) I have a little bit of what I talked about with them seen here → and here →

    I would love to chat more about what questions/lesson I had planned for them on the Same and Different day. The question in the picture is not the only one I asked them to do . I asked them a variety of questions that included money, pictures, tangrams, etc. I extended it beyond just numbers. I wanted to push their thinking. We also used equal mats printed on paper :)

    1. I love this! I have plenty of time to do this and I will! I completely agree with you that having them gain a better understanding of mathematical relationships and how they apply to the greater concepts in math.

      As I say in the post I put up today, I would rather take more time to solidify basic foundational skills that try to force complex concepts later without them.

      I now have two great things to do with the kids! I'm excited about this. Thank you for the input and the links!!

  8. Hey, sorry I'm late to the party. I assure you I'm /very/ passionate about appropriate uses of equal signs.

    I just wanted to clarify what I was agonizing about on Twitter. I was thinking of a very specific thing I used to do, teaching Algebra 2, which was "come up with fool-proof procedures for factoring quadratics and show them to kids and practice them to death so they remember all the steps." I generalized this in my Tweet to "kids have to be making decisions" but there's a distinction. With the factoring thing, by creating a procedure that didn't require any decision-making, I was robbing them of the opportunity to understand structure and evaluate equivalence, and, you know, DO MATH.

    I wasn't trying to say, "let kids do whatever as long as they get the right answer."

    What you're talking about here is something wholly other, I think. You recognize that their way of working through a problem illuminates a misconception about what the = notation means. It's important to address the misconception, and there's utility in insisting on they write their work a certain way. You're not trying to do the thinking for them, you're insisting they use notation correctly and communicate clearly.

    I hope that makes sense.

    1. Absolutely! In other cases where students are using faulty procedures to end up the right answer, I make up an example where their operation doesn't work and ask them to try it again. The one that comes to mind is with absolute value. Students DESPERATELY want to just change the sign so I show them some cases where it doesn't work. I follow that up with a talk about distance and how you can't "unwalk" even if you go backwards. I "make" them ask "how far is that number from zero?" and it seems to work.

      As I said in a previous comment, I was never taught how to find the line between what I should I explain to them and what they should discover themselves. Or even that that line was in a drastically different place for every student.

      This job is tough!


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