Thursday, April 5, 2012

Non-Linear Thinkers in a Linear World

I'm going to re-write the lyrics to Madonna's Material Girl: "We are living in a linear world - but I am not a linear girl!" It makes sense in my head. That's all that matters and is a perfect example of where this piece is coming from.

I was reading an article (and I really wish I could find it again to reference it - I promise to update this if I do find it) that was talking about assessments and classroom procedures for non-linear thinkers. In a day that it seems common sense is out the window, students can't think for themselves and we have an awful time figuring out what we are trying to say, it is just amazing to me that technology has advanced as far as it has.

Math itself is very linear. At least, the traditional notion of math makes it linear. Computers, codes, everything we have is built from that linear aspect. In fact, most of it involves LINEAR ALGEBRA, a class I took twice in college (as an intro course and an advanced course, not because I failed it). Matrices, patterns, kernals - I barely remember what those are. Maybe I should have passed. I knew what they were then so that counts for something. I also know it has something to do with an identity. But all of this is neither here nor there.

 I know how the NL thinkers work and I know how frustrating it is to not be able to share your thought process. I, Brynn Cody, am a non-linear thinker. I switch between conversations as if you could hear what I was saying in my head, so it would make sense for me to follow up out loud. Today L and I were in the car talking about the Micheal's cake course I'm taking, but we pulled up to the light near the gym as conversation lagged. I started thinking about how I am sad I didn't sign up for the last round of classes. He asked me when I was going to find out about the next classes and I told him I had a gym date later this week. He was still talking about the cake classes - he didn't switch just because I looked at the gym. And that doesn't totally make sense to me. Isn't it obvious that we should talk about the gym simply because we are in front of it?

So I wonder, how can we take our non-linear thinkers and prepare them for a linear world? I think the answer is tons of practicing and a handful of scaffolding. We need to practice circumventing the traditional processes to let our children discover algorithms on their own. We have to help them verbalize their thoughts so they can find patterns in what they're already doing.

THIS CAN HAPPEN IN ANY CLASSROOM. I'm not just talking about student-led activities and open discover classes where they create every math rule, or science definition, or whatever! Even during traditional lectures, you can be there to support the non-linear thinkers. I let a lot of students explain what they are thinking and find ways to connect it to where we are going. For example, take the problem I gave my students as a warm-up one day: You are saving up to buy a new flat screen. You have saved $35 so far and plan to save $10 a week until you have enough money. The TV you want is $456. How many weeks would you have to save before you can get the TV?

My clearly linear thinkers went to 10x+35=456 and solved for x. Okay, good, they get the concept of writing equations.

Some, let's say parabolic thinkers did 10(40)=400+35=435+10(2) so it was 42 weeks. Apparently they know some stores where $455 is enough to buy a TV that costs $456. Of course they also came up with answers like "you can borrow a dollar from a friend" or "you can get it on sale." Touche, you outsmarted me. Why didn't I think of that?

Of course, some students gave up because they started 35+10=45+10=55+10=65.... and decided it was going to take too long.

Then there's my non-linear thinkers. The ones who worked backwards and said they needed $456 but already had $35 so they only really needed $421. Then they saved $10 per week so they divided by 10 and came up with 42.1 weeks. Of course, they didn't say it so clearly because that is too linear. But they did DO it, I just had to pull the word-part out of them. So in this particular problem, I compared what they did to how the linear kids would solve their equation. Was it bad that they couldn't come up with a formula for it? Are the other kids smarter? Of course not! They just thought about it in a different way. We talked about this problem for a lot longer than I had meant to, but I was happy we could. We also got to compare the repeated addition of 10s to the multiplication some others did. All-in-all, it was a really valid conversation that all of the kids could benefit from.

We help the non-linear thinkers by encouraging them. Most teachers probably tell them they are wrong, they need to do better, or they need to do it 'this way.' We need to be the types of adults - not just as teachers but as leaders of this world - who are patient with kids, let them work through the frustrations and confusion, and watch for the final product to be created. Besides, there will always be linear thinkers out there. But without the dreamers, where would we be?


  1. We did a warm-up problem pretty much exactly like that today!

    1. Did you get a really diverse range of responses? When we did it, it really reminded me of that "a farmer has chicken and rabbits in the yard with 30 heads and 50 feet" or whatever that problem was. What I love about math is that, even though there's a very traditional route to it, you can also approach problems in a variety of ways. It's awesome!

  2. I love different ways to solve problems. Several years ago I had a student who was very bright. I truly could not follow his way of thinking, but we always got to the same place in the end. Keep up the good work, Brynn.

  3. I would have pinned you for a linear thinker...probably because you teach math and have always been good at it. Shows my linear thinking at work!

    Your kids are lucky to have you though. How important it is to have someone who can think through a problem a zillion different ways and also understand and appreciate the kids who don't take the traditional route.

  4. I love this story, and believe this is exactly what teaching SHOULD be - helping students identify and cultivate their innate (and clearly very unique) cognitive styles. How do we help NL Thinkers better communicate their methods and thought processes to their Linear Thinking peers, though?