Sunday, March 17, 2013

Some Equal Signs Are More Equal Than Others

Here's a challenge.

INTRODUCTION:
I'm introducing concepts of Algebra to my students. That is to say, if a student asked, "Mr.Rowinsky, what are we doing today in class?" I would likely answer, "Algebra."

We're playing with the idea of equations. Balance. Left Side. Right Side. Equal sign in the middle. Some great discussion about this new mathematical animal takes place in the classroom. It's a challenge. It sometimes involves working backwards. It sometimes involves taking tiles and trying to match it with an algebraic abstraction. But mostly it's just abstract. There are moments of "I don't get it" and moments of "Oh ya, that makes sense!" It's frustrating in so many ways to some students. Likely more ways than I can imagine, and definitely more ways than I can teach. But this is the challenge.

MAIN:
A certain student has no difficulty solving x - 7 = 18. They will even momentarily suspend the trivial nature of the equation1 and indulge in my suggestion that adding 7 to both sides might be helpful. x - 7 + 7 = 18 + 7 and yes, as you already knew x = 25.
However, when the question is flipped, 18 = x - 7, some interesting things happen.

"Do I have to subtract 7 because it's reversed.
"How do I do this, it's all backwards?"
"I subtracted 18 from both sides but it didn't work."

These are not the loudest voices in the class. Just a certain few. The few that I'm trying to highlight here because I believe I know why they are having difficulty with this equation. (Outside of the fact that

INITIAL THOUGHTS:
Students have been convinced to believe the this symbol "=" actually means, "Put answer here".
That is math to them. Find the answer! And if they don't know the answer, they are not good at math. Or, they can grab a calculator and that solves everything.

FURTHER:
I am now collecting some like terms. Again, sometimes tiles are used. Some seemingly concrete ideas to explain the abstract. And, they are following and having fun with it.

If I ask, what is 3x + 2x, they say 5x.
How about 7a - 3a, they say 4a.
And b + 8b - b...tough one, but yes, 8b works.

But again, a certain student might step back a bit and then ask me, "But what's the answer?"
They are frustrated that there is no equal sign, and that they still don't know the value of x, or, a, or b?
"How is this helping me?" they'll ask.

They want an equal sign. And they want a blank space for an answer.
They don't want to simplify.
They don't want to evaluate.
They don't want to factor.
They don't want to represent this equation with tiles.

They want an equal sign and they want the answer!

And by they, I mean certain students. And that is my challenge.
Oh, I'm up for it, but it's challenging nonetheless.

HELP ME:
I'd love to hear what works for you. Please share what you've seen, and what you've heard in your classes when you first introduce Algebra.

1. [This alone is a point to consider. A triumph in my eyes.]





19 comments:

  1. For sure, "18 = x - 7" gives kids the hives. I'm really trying hard this year to have kids ask the question of "Is my answer reasonable?" after every problem. I fall off my chair when I see answers boxed up neatly that make absolutely no sense. It's a uphill battle, but I'm willing to spend the time and energy because number sense is so key to everything.

    I'm thinking of the visual patterns that I have my kids do. They come up with an equation for the number of tiles for that pattern, but they know there is no "answer" for it because this equation is waiting for further instruction -- the step number.

    I'd direct the question back to them, like ask, "What the value of 5x? How much is it worth?" Since they shouldn't have an answer, they might just stop asking! :)

    Send them home. Thanks, Nico.

    ReplyDelete
  2. The visual patterns you are doing are great. It's a gold mine for sure. I find that students are making those connections between an algebraic expression and a visual pattern or at least they are up for the challenge and enjoy it. Your visual patterns projects helps with this challenge.

    Here it seems that SOLVING an algebraic expression takes their minds somewhere else. Seeing 3x - 1 = 5 does not connect (yet) with the visual pattern that goes 2 , 5 , 8 , 11. It's a new thing to them. If I throw in a 3x + 2x on top of that, we're on a different world completely. Fascinating in the teacher sense of the word.

    ReplyDelete
  3. I have started to remove the word 'equals' from my daily vocabulary in the math class and I replace it with "is the same as". Hence the equation reads "18 is the SAME AS x - 7". I find that this helps them to undetstand what the true meaning of the equal sign is. Usually a simple (or complicated) example like 3+ 9 -5 = 2 x 3 + 1. I'll ask the class to discuss and decide if it's true. My goal is to let them get to the point that it is true because 3+9-5 is the SAME AS 2x3+1.

    ReplyDelete
  4. Interesting post. I just wrote something about how research shows that problems with equality start in early elementary school: http://nicoraplaca.com/two-different-worlds/. Lots of research supports what you are saying: students think of the equal sign as a direction to "do something." In fact, there are quite a few people who offer suggestions of how to address this misconception in the early grades. Now, obviously you can't go back to elementary school and change how they were taught to think about the equal sign. However, you may want to think about how to get them to start thinking about the equal sign as a symbol that represents two equivalent quantities or amounts.

    ReplyDelete
    Replies
    1. Thanks Nicora.

      It's such a tiny detail but it lasts their entire math career.

      BigT's comment seems to agree with yours, as he suggested to remove the word 'equals' from his daily vocabulary in an attempt to correct the misconception. I'm sure there are lots of things we can do each step of the way.

      Elementary, middle and high school...there's something here for everyone to consider.

      Delete
  5. When we have to go back to explaining the balancing of equations, I try and do just that--use a physical balance (balance scale with unit weight manipulatives) to demonstrate that the sides need to be the same for the equation to balance. Grams become x's; 2 gram units become y's etc. depending on the problem at hand.
    The concrete form of balancing helps some students remember the concept a little better.

    ReplyDelete
  6. A few weeks ago while I was discussing this topic with a student, she mentioned, "Hey, it's like a balance." I explained to her that people sometimes use an actual balance to explain this sometimes. The concrete concept came to her even though we were discussing it in the abstract. Thanks for dropping by LL.

    ReplyDelete
  7. I am 100% convinced (tons of success with all types of learners) that the way to teach Algebraic equations is through the use of manipulatives (red/yellow chips and cups). In the coming months I am starting my own teacher math blog and Ill post an actual couples lessons me teaching it with manipulatives. The biggest problem I have seen with this method is the teacher is no comfortable enough with it to actually give students a chance to really explore with it. We have got to get away from teaching a bunch of rules and teach common sense. I wish I had the time to really go into it right now but manipualitives I am convinced can teach any child how to do algebra and it will make total sense to them. The problem is we move to the abstract because that is where we are comfortable instead of lingering in places that make sense to our kids. When you actually move to the algebra you have to make sure to always relate it back to the manipulative and the students will get it. I teach a math enrichment class at my Junior High and using manipulatives within 5 days students are solving multi- step algebra problems, and moving into systems of equations using manipulatives. When I move the concrete they have no problem because in their mind they are seeing the manipulative. We don't need a lot of rules because it just makes sense.

    ReplyDelete
  8. Hi tgillespey,

    I love seeing your passion for this issue. I agree that some teachers are uncomfortable with certain methods and that results in students suffering even more. I can also see that you have had success with the use of manipulatives when it comes to teaching Algebra. I look forward to seeing your activities when you share it on a blog. Please stay in touch and follow up by email if necessary.

    Thanks for dropping by and sharing.

    ReplyDelete
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