Patient Problem Solving – Scaffold Problem Solving, Not Problems!

We want students to do challenging things in mathematics, but we also want them to be successful.  Its a constant battle between scaffolding too much and not scaffolding enough. I’m always worried about not being clear in what I want, and asking students to do very confusing tasks, but I also don’t want to take them through things step by step, because then they’re not thinking about the math themselves.

What does not help?

Telling students what to do.  Giving students exact directions on what to do next.  If we have to do this on a daily basis, then we haven’t taught students how to actually think.  If we have to do this, then we have to scaffold each problem for our students, because each problem is different.  This obviously doesn’t help long term, because eventually we hope that students can do things without us. This involves instructions like:

  • create an equation for this sentence
  • measure the slope of the line

What does help?

Scaffolding thinking.  Using things like the Mathematical Processes to help our students think about big picture mathematics.  Having students do things like reflect on the reasonableness of their answer is useful in every problem.  Having students understand that we can represent mathematical models in various ways also helps in a variety of problems. Another way to use scaffolding is to scaffold in time during the lesson to have students discuss their own next steps.  Perhaps its a few minutes to talk about what should be done next.  Perhaps its some time soft students to look up vocabulary words that will help them solve a problem.  Here is a great summary of a few of these strategies.

Patient Problem Solving

In the end, its simple.  Help students think so that they can solve their own problems.  Don’t help them solve specific problems.  Long term, that’s not helping. If we scaffold each step, they will never be patient enough to think through something on their own.  On the other hand, if we scaffold thinking strategies, they will take the time to reason through their thoughts and work with others to make sense of the mathematics that we’re trying to teach.

The Ferris Wheel Problem – What info do you need vs. What info can you get???

I wanted to get my students to model periodic behaviour using Trigonometry.  This lesson did that, but it got us to talk about something much more mathematically important: What information is needed, but also what information is attainable?  For example, my students wanted to google the “Central Axis of the Ferris wheel at Chicago Navy Pier”.  This information is necessary to solve our problem, but unattainable in these words… I included a video with this discussion, and had some really great conversations with students about what information we can actually expect to find out there, and what math we can use in order to manipulate the information we find, into what we need.

Here are the slides I used.

Act 1: I started with just this clip of the Navy Pier Ferris Wheel from Chicago and asked the students what questions they had:

Not the most effective Act 1 I’ve ever had, but it got them to think about things like:

  • How long are you on the ride?
  • How tall is it?
  • How high off the ground do you get on?

But the question that got my attention, and that drove the rest of the class was: How far can you see from up there? So we made up the question “You can see US Cellular Field from a height of 100m.  How long do you have to take pictures of it?”

Act 2: We brainstormed for a long time the information needed.  I’ve included a video of the discussion here, but I think the coolest thing that came out of it is that we learned to translate math language into Lehman’s terms.  For example, my students wanted the “central axis of the Ferris Wheel” and I asked them if we could google it.  They all agreed with a “no” and told me I was ridiculous.  I agreed, so we tried to find what it was that we could google about the wheel.

Also, towards the end, a student asked if we could google the answer to “You can see US Cellular Field from a height of 100m.  How long do you have to take pictures of it?”, and I really appreciated that comment. Lots of good discussion came from it.

I finally gave them very little information that I could scrounge up, and it was this little slide:

Ferris info

Students moaned and groaned a bit, because in the video you may have noticed that they wanted things like: Radius, diameter, min heights, vertical shift etc… and none of that is given… they had to use math to interpret all of this information to make it usable.

Act 3: Use your model, and interpret whether it makes sense or not.

So that was it from me, the rest was on them.  They actually came up with very innovative ways to determine the second time when you’d reach a height of 100ft.  Here’s some of the work they produced:

IMG_2271 IMG_2270 IMG_2269 IMG_2268

I actually tried to interrupt them a few times…but to no avail…they wanted to figure it out themselves.  They make me want to keep teaching…

Let Students Impress You

I just had a conversation with a student in my class that made me really happy.  I got a student to believe that her figuring out information, made her a lot happier than me giving it to her.  I got her to impress herself, and to believe that she can impress me.

Completing the square is part of our curriculum, and we teach it as just a way to change standard form to vertex form when it comes to a quadratic relationship.  So after doing a lot of fun tasks around modelling situations that end up being quadratic, we have to … complete the square.

Bottom line is that I could easily tell my students that in order to complete the square in x^2+bx+c , you take half of the “b” value and square it, but what’s the point???

The Task

My students have worked with algebra tiles before, so I just ask them to take x^2+4x and arrange it as close as possible to a square. So for a while, their desk just looks like…

tiles1

After a bit of thought and some talking and arguing…it eventually turns into…

tiles2

And then I just ask them…”what’s missing to make it a square?” and they’ll very quickly show me this:

tiles3

 The Side Conversation

So while this is going on, one of my students who, before this semester, had a tough time with math class, starts a conversation with me:

Student: (Quietly) Sir…is it four?

Me: What do you think?

Student: (Louder) I think it has to be four…

Me: Could it be anything else?

Student: (Confident) Nope! It has to be four.

Me: Ok then…

Student: (With a big grin) I’m so smart!

The Cynicism

So into the second question I asked them, another student at her group says…”I’m pretty sure that its just “b” divided by two squared”, and the girl asks “Sir…why wouldn’t you just tell us that???”

The Teachable Moment

Me: How did you feel when you figured out that the answer is 4?

Student: Awesome!

Me: So what kind of person would I be if I took that away from you?

Student: Awwww… Thank you sir!

The Bottom Line

We have many answers… answers that can impress students… make them say ooooohhh and aaaaaahhhh, but why??? Why take those moments away from them? If we give them the chance, they can make their own moments, impress themselves, and most definitely impress us.

Cheers.

Anxiety in Math Class – Patricia Heaton on “Who Wants to Be a Millionaire?”

So Dan Meyer called me out for only having one post so far, but he did mention me which makes me feel oh so special! Today I’ll write my first post in the “What do I think?” category, where I’ll share some of my thoughts about education in general, and some things that I’ve used to get my message across.  I’ll start with this clip which outlines some effects of anxiety in math.

So think clip is a nugget of gold if you’re looking to do some PD around how traditional math class has done at least one person terribly wrong.  I’ve used this clip, and in fact have pared down a version which you can access at the bottom of the post.

How do I use the clip:

I usually play the clip without much warning, and I give people about 30 seconds to chat.  I then start asking some pointed questions:

  • What was her initial reaction to the question?
  • Why do you think that is?
  • Could she do the math that was required to complete this problem?
  • What do you think her first mathematical thought was, and why did it happen at that instance?
  • Why did she choose to call specifically her husband?
  • How did she feel when her husband left and she thought she lost?
  • How did she feel immediately after Regis said that she didn’t lose?
  • How did she feel after it sunk in that she didn’t lose?
  • How does this parallel the emotional experience of our students in many math classrooms?

There are many more questions we can ask, and I invite you to share in the comments section so that we can improve how we use this video.

Where has the discussion gone in the past?

The bottom line is that discussion usually leads down the path of:

  • She knew the math (did it all at the end)
  • Stress caused her to shut down and not start thinking
  • She called her husband because he was French (Euros…)
  • First mathematical thought didn’t happen until about 6 minutes into the clip – fight or flight probably kicked in before that
  • When she found out she didn’t lose, she was very excited!
  • This quickly went away again and turned to… more fear.

So what does it mean?

I think that what this does is outline the fact that people want to be able to do math… Scratch that, they want to be successful, it doesn’t matter what the task is.  For some reason, at some point, they started to believe that they can’t do math.  This is usually because in many math classrooms, if you can’t do calculations QUICKLY, you’re not good at math.  So let’s make sure that we don’t create students who shut down at the sight of the number $1.50, because that worries me…a lot.  We have to have an environment where students can think critically about math, and where their opinion can be valued.

Thanks for reading!

Oh yea… and here’s the shortened 3 minute version.