Friday, January 16, 2015

Moving Negative

Wow - I missed a blogging month. And I had so much to say about it... we did Math in Your Feet, some excellent student projects, lots of new lessons, assessment thoughts... So I was thinking about resolving to blog less big, more frequently. Then Sue Van Hattum blogs her #edustory, and I think challenges me and a few others...

I've been thinking about doing more microblogging - and maybe I'll try it. I get stopped by "nobody wants to read that" which makes me forget that I'm writing out my own understanding, so that shouldn't matter. I'm not an author who's trying to please a fan base, I'm a teacher trying to work my way to understanding.

Actual Post
But what's really on my mind is embodied cognition. Last summer I got to try a session with Malke Rosenfeld and Christopher Danielson at TMC14 on Embodied Cognition.  (My account.) Outside of their session, Malke worked with Michael Pershan and Max Ray and others on doing a life size complex plane and number line. I wasn't even a part of that but it got me wondering. Malke and Max have continued to work on the idea, and there is an MTMS article in the works. They were willing to share their writing on that so I could try things in class.

The course is for preservice middle school teachers. I start off with negative numbers (and probably end with negative numbers too if you know what I mean. Where's my Dangerfield font?) because it is a good setting for talking about operations as story and action (exposing them to the CGI structures), and rolling in some content from our preservice elementary classes on fact families and operation strategies, models and landscapes of learning.

For the Cognitively Guided Instruction stories we watched the Kindergartener uses Direct Modeling video from the Heinemann site. Then sorted these stories.  Usually students sort them by operation needed to solve them, but the video was a great focus, because they really did a great job discerning actions. The idea is that young students encountering stories before direct operation instruction classify stories by what's happening in them. Are amounts increasing or decreasing? Are we comparing separate amounts or looking at static groups of different types? They then model and invent strategies that fit the contexts. For example, students who are taught addition first but then have to use it in a decreasing context often have difficulty solving the story problems. (James gave away 3 pencils but still had 5 left. With how many did he start?)

We followed that by brainstorming contexts  for negative numbers: money, debt, bills, weights/balloons, depth/sea level, golf, football yardage, temperature… the usual suspects but a good variety. When we tried writing stories for them, it was challenging to ask the questions in a way that the answer was negative. I nudged them towards the idea that one of the strongest contexts for negative numbers is when the numbers are describing change rather than direct quantities.

Some of the questions from Max and Malke:
  • where she was compared to where she started
  • tell us how far away from someone they were, and in what direction
  • a plan for how everyone could, in a coordinated way, get from their home position to their spot that was the same distance away but in the opposite direction
  • identify if there was someone who was the same distance away from Shane, but in the opposite direction
  • give students a target result and ask them to come up with a series of moves that resulted in the given displacement

We just used stickies to make the numberline. I marked a square as 0 and asked the PSTs to place stickies for 5, 10, 15, -5, 10, -15. First discussion: are we using the squares or the edges in between? (Squares, because of placement of zero.)

Students moved to various numbers on the line, called out by the teacher. Discussion: left right, direction big part of idea of negative. Distance talk, however, is naturally positive.

Then we started modeling change. If a student C walks from A to B: how far did they go? (Positive.) What is the change in their position? (Signed.) We did several iterations of moving in both directions. Discussion: PSTs started noticing how zero figures into the strategies. Frequently found change by b to 0, 0 to c.

PSTs challenged in groups to come up with a question that could be modeled on the line.
  • 1st group: stood at 8, 5 and -3. Class brought up person at 8 could be change: how big a change when walking from -3 to 5. Another said -3 could be how big a change from 8 to 5? Discussion: no way for 5 to be the change in that situation. 
  • Next group of 2 stood at 14 and -7. Their story was: Samantha climbed a 14 foot hill and jumped in the water, sinking to 7 feet below the surface. How far did she dive? Someone brought up 14, and dove 21 feet, where is she? (“Dead”)
Shared Max and Malke’s challenge to come up with a combination that resulted in a net difference. Students proposed 3, 4 and 5 move challenges to get the goal, took the challenge to mean literally standing on the line. Brought up how the challenge could lead to better strategies than counting one space at a time.

End of day 1, informal assessment: was this worth their time? All 4s and 5s on a 0 to 5 finger scale.

Homework: asked them to read one or more of the following: 
Day 2 we set the number line back up and started thinking about how to explicitly model addition and subtraction. The first group shared the idea to face positive: if addition of positive move forward, if subtraction of positive turn back, if negative turn... I raised my opposition to things that just feel like more rules and not connected to ideas. In discussion, the idea was raised to face neutral by default, turn positive to add and negative to sutract, walk forward for positive, walk backward for negative. I brought up how the class deciding this idea for themselves is probably the most valuable part.

Once we had decided on the representation, we got to some exciting stuff. We had students do a walk on the number line. (Start, turn, walk, stop) and then we wrote it down. 5 + -4 = 1. We discussed how non-threatening it was to walk for something like this, and it was a place where you were really free to experiment. When someone brought up different options, 7 - 13 or 7 + -13, we talked about how you can tell and was there really a difference. Then we hit on the idea that you could walk out equivalences. Are these two things equal? Let's try them! Someone had the idea to try commutativity. What would associativity look like? (Hard to walk.) There was an interesting side effect: some common student errors are impossible to walk. They just don't make sense in embodied cognition land

I brought a game idea, of course. Take a deck of cards, remove everything but A to 7, red is negative and black is positive. 

End of the Line (game)
Shuffle, and deal two decks, like for War. Both players start at zero, facing each other. Flip the top card of your deck. Player with the smaller magnitude number goes first. You can add your number to yourself or subtract it from your opponent. The first player's move decides whether they are going positive or negative, and the 2nd player is going the other way. The goal is to get off the number line. (Ours went to +15 and -15.)

Sample turn:
  • player Positive is at 2 and flips red 6. Negative is at -3 and flips red 2. 
  • Smaller magnitude goes first, so Negative adds -2 to their position. 
  • Faces positive and walks back two squares to -5. Positive player doesn't want to add -6, so makes Negative subtract. Negative faces negative, then walks backward 6 squares to 1.
In the picture there are two different games going on. Other students tried the game or modeling stories with a chip model. People liked the game pretty well with the numberline, and got them a lot of practice thinking about adding and subtracting positive and negative, as well as use on the numberline.

Day 3 involved no embodied cognition. We discussed fact families and addition and subtraction strategies, and then discussed how to show a variety of strategies for integer addition and subtraction in symbolic records and in number lines. The number lines really showed the benefit of the previous classes' movement as they felt the directions really made sense.

All in all, great start to the year. I've only become more convinced of the need for more opportunities to embody mathematics, and the value of the intuition that this helps build through experience. And, of course, I'm interested in your stories about this, or ideas for what else you might have tried.

EDIT/postscript: several of the preservice teachers are blogging about this.
  • Sam - clearer explanation of the game.
  • Brittany - thinking about the human number line in terms of variety
  • Dakoda - trying it out in her tutoring
  • Kevin - finding some online number line games

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