Monday, September 28, 2015

Does "center" always mean "half"?

I'm spending the next couple months trying out some new non-dance, moving- and body-scale math lessons for my new book.  There's an activity in Mathematics Their Way (from the '70s) that I want to try out.

The lesson is ostensibly about using your whole, moving body to find the "middle" or the "center" of a space that you can't fold. I was thinking I'd start my version of the lesson by asking first and second graders "What is half?" I planned to give them a piece of 8 1/2" x 11" paper and see what they come up with.

Except, as I was getting the materials ready today I noticed some paper cut into big triangles. I took one up to my 10yo who was doing her homework. I asked her to fold it in half. This is what she did.

 Then I asked her to fold it into half again. The image below illustrates what she was doing for a split second before she said...""

At which point she said, "I can't fold it into half that way. But I can fold it into fourths."

I tried to get her to tell me how she knew when she had folded it in half, but she wasn't interested in talking. But it did make me wonder how I might fold this triangle into eighths.

Then I thought: Wait a minute! We've folded this triangle into eight one-eighths and still haven't found its center/middle. Or, well, maybe we did on the first fold ... but maybe there's another center we haven't yet found?

I love it when what I think I know is challenged. We need to do this for our students on a regular basis.

And, I'm rethinking my lesson for tomorrow. I'll focus on the language of "middle" or "center" and use rectangular and triangular paper. But "half"?  Half I'll save for another time.

Tuesday, September 22, 2015

Swimming toward Specificity: Understanding Pattern Properties

Today's work was to make a four-beat Pattern A. I structure each day/"lesson" as an activity progression with planned pauses to discuss certain key points of the math/dance making.  Today I recorded two similar moments, in two different classes, in the process of making Pattern A. When I transcribed the conversations this afternoon what struck me was how differently they unfolded. 

LESSON SUMMARY: My goal on the second day of my residency is to end the class with every team having a Pattern A they can dance the same way with their partner. We do this by watching individual teams demonstrate their patterns and then talking about what we see/notice about the pattern. This ideally means talking about the pattern properties that reside within the three different categories that build our patterns (movement, foot position and direction). Yesterday's notices made me think that the categories made sense to most kids. At the beginning of each class today we reviewed "what do we use to make our patterns" but when we started looking at the Pattern As I realized that though both groups were making nice patterns, both groups still didn't quite "get" what they were looking at in others' work.

The first story illustrates what it looks/feels like to work with a class that has a sense of what dancing "the same" might mean but is very far away from saying exactly what is the same (maybe the equivalent to van Hiele Level 1?)

The second story illustrates a class that is still learning how to interact with the movement variables but has what seems like a more flexible understanding the relationship between the categories and the properties within those categories (van Hiele Level 1.5?)

NOTE: In the two stories below my words are italicized, and the kids' responses are bolded. In both conversations many children contributed, I just haven't specifically noted change of speaker here.


CLASS #1: Swimming Upstream

Let’s say what we noticed about that pattern (that the demo team just danced).  Kid:They’re both doing the same thing and they’re not off beat.

Okay, so by “same” do you mean same tempo? Did you mean same in any other way? (To class): [First kid] was seeing same in the tempo, the same speed. What else did you notice about their pattern? Let’s watch it again. That they’ve got a really good rhythm, like how they do it.

So we’ve talked about the tempo. What else did you notice? That they were doing it together they weren’t like one person wasn’t doing different things they were all together.

Okay, so “all together”  What does it mean? People were saying they were dancing together. So what does it mean to be "all together"? What words will help us describe a pattern that is "all together"? What kinds of things can we identify?  [Kids start picking words kinda randomly from our word board] Ooh! Symmetry. Interesting. But what have we been talking about how the patterns have been made? What kinds of things are we using to build our patterns that we could also use to say if a pattern was being danced the same or not?

Ummm. TransformNo…we haven’t actually talked about that yet. Reflect or…

You know, I've been spending most of my time when I’m up at the front of the classroom, near these posters.  This is the information that I’d like you to think about. What kind of words on these posters might help you describe these patterns? 
[Long pause] Movements? 

Yeah! [talking to the demo team] Did you guys do mostly jumps? Were the jumps the same on every beat.?[bringing the demo team back up to the ] Let’s say jump for each beat while they dance…Now there was another movement in there.  Look up at the list. The category of movement has a lot of movement words. Turn?

Turn! Right, let’s say jump, jump, jump, jump/turn. And how far did they turn? If they went from front to back? 180!

Now, let’s look at their foot positions. Let’s do it slowly. We can also describe the sameness of their dance by looking at their foot positions.  Right now they’re together. Do the first beat and stop. Is that split, together, right, left or cross? Split.

So let’s see their second beat. Ready, go. What are their feet doing now? Split. Where’s their third beat? Split. 

Good! You’re staying in this category of foot position…And where’s the fourth beat? Split! Did they do 4 splits?  My goodness. Let’s watch this pattern one more time and let’s talk about their direction.  So their beat zero is where? Center.  So first beat the direction is what? Diagonal.

Thank you! Is it a right diagonal or a left diagonal do you think? Right. Now let’s look at the next direction. They went to the…side. 

The right side!  And now the third direction? It looks like they went to a different kind of side with their feet to the right and left. Give them a round of applause! So what I’d like you to do right now is go back to your work with your partner and we need to make sure that everyone has the same pattern as their partner. You’re working together. 


CLASS #2: Swimming Downstream
So what do you notice about this pattern? Let’s watch it one more time and see if you can come up with some words that can help us understand what you see in the pattern. What did you notice about that?  [inaudible] So you noticed a type of foot position which wasa cross. Okay let’s watch again and notice other foot positions.  

Tell me more about what you noticed? They really liked corners. 

(giggles from all the kids) What else could we be noticing? 

Okay before we watch this tell me what else we could be noticing? We noticed their starting position, we noticed that their feet were crossed at one point. What else could we notice? Feet?  Their left. Well, my right. 

So right now we’ve been talking about feet? What other CATEGORY could we notice while they’re moving?  

Step? They stepped backward? Yeah, but which category is that? Movement?

So let’s look at their movements...impromptu chorus of kid voices: Step, cross…

Well , what category does cross belong to?  Feet.

While we’re looking at the pattern let’s find words that describe what it is we’re seeing? They start with feet together.  Where are they?  Center center.

So if we’re looking at movements let’s see if we can figure that out. Step, step, jump, jump/cross…

We don’t say cross because we’re only looking at one category at a time. But this is challenging, I know that. So let’s say those movement words again. Step step jump jump.

Now let’s look at directions. They start in center and let’s do the first beat and stop. The first beat is where? Back left corner.

Where’s the second beat? Back right corner.

And where’s the third beat? Center.

And where’s the last beat? Front corners…

Me trying to talk over them: front, front…just front. Front corners.

Well, if they’re in both corners we’ll just say they’re at the front.  And what would be the foot position word that would describe the fact that their feet are in both corners?  Split.

So can you see how all three of those would…(to dancers) show us what your feet would look like if they were together in front.  So we would call that a together front. Now show us a split back…and a together back.  Do you see how that works? You need more than one piece of information to describe any one beat.

Monday, September 21, 2015

Pulling Language Forward: Notice & Wonder Day 6

After experimenting with Notice & Wonder last week during a Math in Your Feet residency with three classes of fourth graders I:

  1. ...feel like I had more feedback about student thinking than ever before. I've always seen how children think with their bodies, but was never happy with their written reflections, which didn't really reflect what I saw them do while they were making their patterns. I have also not been happy with how hard it's been to help kids use and apply the terminology of the Movement Variables. I found Notice & Wonder to be a great tool at pulling the language forward in a way that was natural and meaningful for the students themselves.
  2. ...noticed that each of my three classes last week moved from general noticings to much more specific noticings as the week progressed. Not only that, but they noticed more and more, and in great detail their own making process and the process of reflection; the math/dance pattern properties and structure (movement, direction and foot position); the differences between a reflected pattern and a non reflected pattern; and the process of unitizing two patterns into one larger one.
  3. ...finally felt like students owned everything about the week -- their actions and the language they used to describe their own and others' patterns.
  4. ...was grateful for the classroom teachers who made the decision to support the residency by making their normal math time a chance to explore the math we were using in the math/dance making. This was particularly important in terms of the angles/rotations and the idea of congruence. 
This week I'm back at the same school with three more 4th grade classes. Their Notices & Wonders today were really nice but, like last week, fairly general. I'm excited to see if they follow the same course as last week's students in becoming more and more specific, precise and descriptive with their noticings and pattern analysis.

And today I had a new Wonder: What if I bring a few notices and a few wonders forward each day (collected from all three classes) and put them in front of the students. They won't be identified by class or student.  Right now I'm thinking that it won't hurt to try and I'm curious what might happen.  Here are the student responses that I'm choosing to pull forward for tomorrow:
  • I noticed all the different words you can use.
  • I noticed all the movements, feet and directions we did today.
  • I noticed you can combine different movements.
  • I noticed we were doing the same pattern over and over again and I think we should choose something different.
  • I noticed dance isn't just movement; it's feet, movement AND direction.
  • I wonder what our pattern is going to be?
  • I wonder how many 4-beat patterns we can make with all the things ups there [on the movement variables poster.]
  • I wonder if I'll have better ideas tomorrow.
My original three wonders can be found here: INTRO/DAY 1
The rest of last week can be found here and, like I mentioned, document the progression of general noticing to much more specific and precise language: DAY 2 | DAY 3 | DAY 4 | DAY 5

Friday, September 18, 2015

Reflected or Not Reflected? Notice & Wonder Day 5

Wonderful, goofy, creative, thoughtful fourth graders! :-)
Today was the last day of the five-day residency with three classes of 4th graders. The first four days I experimented with Notice & Wonder. Today we had to finalize 8-beat patterns and had some reflection games to play to end the week. So, we were busy, and I felt that I would probably see interesting things in their final reflections in their residency journals. I decided to make sure that I followed up on yesterday's reflection lesson. And, instead of ending the final class with N&W we ended by dancing our dance patterns to music!

Before we practiced for the reflection games (doing our 4- or 8-beat patterns both congruently and reflected) I held conversations with each class about what it meant to reflect a dance pattern across a line of reflection. Here is a conversation that illustrates really well the kinds of things kids needed to consider, both in language and in their bodies:

Me: So tell me, what it is that has to change in the dancing? Give me specific examples of what has to change in the pattern if you are the reflection:

Kid 1: That your feet are doing the opposite.
Kid 2: It has to be congruent. Well, it doesn't have to be the exact same but it has to be very similar.
Me: So what is the thing that is different about the reflection when they do the dance?
Kid 3: If the reflection does its right, the other person has to do its left.
Me: So who is the reflection reacting to? Themselves or the original?
Kids: The original.
Me: So if the original puts his foot in the outside corner (away from the line), the reflection's foot is...
Kid 4: on the inside.
Me: Really?
Kids: Yeah.
Me: Hmmm. Why don't you guys stand up so we can watch this. So let's say that you're the original. Put your foot in the outside (away from the line) upper right corner. So the reflection puts his foot where?
Kids: On the outside...
Me: On the outside...?
Kids: Left corner.

Me: So what else needs to be different? Is it just feet? Raise your hand and tell me what else it could be.
Kid 5: Um, your like, body movement?
Me: Tell me more.
Kid 5: Like, so, say, this person twists this way (gesturing to the right) and the reflection turns to the left.

Me: Is there anything else that needs to be different?
Kid 6: For like diagonals, then they have to be facing each other? [positioning herself in the square]
Me: Oh! I see. So if you're the original, when you're doing a left diagonal [with feet split apart] so the reflection would have to do the...
Kids: Right diagonal.
Me: And the body is involved because it's far? How far has it been turned from front?
Kids; 45 (degrees).


Me: Who wants to play "Reflected or NOT Reflected?!"

Pattern 1:
Mr C & 4th grader dance, when they finish there are audible gasps from the audience, like they've figured it out!
Okay...let's watch AGAIN! Was it reflected or not reflected?
Kid: It was reflected because when they turned their backs were facing each other.
Me: So if they're backs are facing each other, that has to mean what?
Kids: They're facing in the opposite direction.

Pattern 2:
Me: So, what was their intention, even though they didn't do it exactly the way they wanted, what was it?
Kid: She was reflecting because when she was diagonal, she did a right diagonal and the other person did a left diagonal.
Me: Was there anything else that you saw that told you it was a reflection?
Another kid: Um, the steps they were using when they stepped. So when C. stepped to the right, G. stepped to the left.

Pattern 3: 
Me: Why did you not think it was reflected.
Kid: Because when they did a right diagonal, the other person did a right diagonal.
Me: What other evidence did you have to support your claim that it was not reflected?
Another kid: When they turned they turned the same way.

Pattern 4:
Me: Do you have a reason for why it was not reflected?
Kid: Because they were going the same way and when they, like, moved their feet together they did the same crosses.
Me: Oh, okay. So by "same cross" what makes the cross the same?
Kid: They used the same legs.

Pattern 5
Me: Oooh. Surprise ending! What did you see?
Kid: When they, like, turned, they landed in the same place.
Me: They landed in the same place, but what did you notice about the turn?
Kid: They turned different.
Me: They turned in different...what?
Kids: Ways.
Me: Ways?
Kids: They turned IN.
Me: Did one slide and the other jump? Or was it more like they turned in different directions?
Kids: Different directions.
Me: Did you see that pattern, you almost couldn't tell it was reflected? It was SNEAKY! And you didn't really know until the end! Give them some applause for the sneaky-ness!

Me: Now, who wants to play "Whoooooo's the reflection?!"

Pattern 6
Kid: When they did the cross T. did the opposite way.
Me, to class: Any other ways to describe how T. was different?
Other kid: When D. turned to the right, I think, T. turned to his left.

Pattern 7
Me: So who do you think the reflection is?
Kid: S., why do you think it was S? What did you see or what did you notice about what he did?
Kid: When M. turned he faced that way and went to that corner and S went to that corner.
Me: So, they went to opposite corners.

Pattern 8:
Me; Okay. Who changed the pattern?
Kid: G.
Me: And why do you think it was G.?
Kid: Because after he did a pattern and (his partner) A. followed...
Me: You saw him following along? Ohhh...
Another kid: Grant changed his body while A changed his body the other way so the bodies were the opposite.
Me: But how did you know it was G. specifically?
Third Kid: Because, for the last time I was glad you said it was the last time they would do it congruently because I didn't get to look at their diagonals. So, their diagonals...A's was a right diagonal I think, and G did a right diagonal.
Me: How many people noticed his first jump?

Pattern 9
Me: Who do you think the reflection was?
Kid: H. because N turned the right direction and when H. saw that he turned the left direction.
Me: How many people agree (hands). Was there any other information?
Kid: In the beginning it was, when they did it congruently, it was right diagonal, left diagonal? And then H. went left diagonal, right diagonal.

Pattern 10
Me: What do you think?
Kid: It was H.
Me: Who wants to give me a reason for why?
Kid 2: I think it was H. because on the second beat her feet went...when they did it congruently I memorized their feet positions and which corner their feet went and then I was watching that corner when they did the other one then that person would be the reflection.

A fun last day, and a fun week of learning with 4th graders! I will be puzzling over my "data" from this week. If you're interested in reading more about the Notice & Wonder experiment I've been running this week you can read each day's account here:

INTRO/DAY 1  |  DAY 2  |  DAY 3  |  DAY 4

Thursday, September 17, 2015

Equivalence Relations & Spatial Reasoning: Notice & Wonder Day 4

Two girls in the process of reflecting their math/dance pattern. Different year, different school, different kids. Same tape. 

Today I took the noticing and wondering of my 4th graders from yesterday and used that information to restructure the frequently stressful last two days of my five day residency.  I will be going into more detail about how N/W has impacted my teaching after I finish the week. Today, however, it was clear that all three classes were noticing the math in their math/dance making more and more specifically.

1. Students practiced/reviewed their A & B patterns. 

2. A body-based lesson on reflection symmetry and how to apply that idea to their foot-based dance patterns.  This included more discussion about how we were evaluating sameness. The classroom teachers have moved their geometry unit to this week so they can make specific connections to the MiYF work. I checked in with each class and asked them if we could use the word congruent to describe how they dance their patterns exactly the same as their partner. One boy said that our dances couldn't be congruent because all the 4th graders are different sizes. This led to a discussion about the relevant properties with which we assess sameness/congruence (all properties the same). FANTASTIC.

3. A chance to combine their A & B patterns into a new 8-beat pattern, otherwise known as unitizing.  All you have to do is read their notices below and you'll see how cool this is.

The process of reflecting a 4 beat pattern across a line of reflection

  • My pattern even looks good in a mirror
  • Doing the reflection was pretty easy because it was like just doing the opposite foot. His foot was on the outside [away from the line of reflection] and I used my outside foot too. 
  • The whole mirror thing was kind of tricky. I had to do the complete opposite.
  • It was easier for my partner to be the reflection because normally when we'd do it she'd do the opposite anyhow. [So, she was well cast in the role of reflection?! lol] 
  • It was hard for us to mirror each other and do the same thing, like opposite 
  • When I was pretending I was a reflection I had to do the feet the opposite and it messed me up
  • I noticed how people were doing the same thing being con-gruent (sic) 
  • When you were doing the reflection not everybody was doing the exact same but they were pretty similar. 
  • It was harder doing  reflection because [my partner] was trying to do the same as me like we did before and she is now doing the opposite 
Unitizing two 4-beat patterns into a new 8-beat pattern C [As you read, also put yourself in a spatial orientation/reasoning frame of mind]
  • I noticed how we combined two patterns into one big one
  • That we have a C pattern with eight beats
  • That you have to take away one beat to make it Pattern C...[did you take away one beat or did you take away your starting position?] Starting position. 
  • When we did B+B it was very hard because we kept getting confused with our moves and what to do because of our 180 turn. We were used to starting our pattern facing backward, and it was confusing doing the pattern [again] facing forward. 
  • It was harder for us to do A to B because in our A we land in a split but In our B we start at a zero in the corner so it was harder to get there
  • It's harder when you do same pattern twice, like A+A together because I finish over HERE and I have to jump to over HERE. It was freaky to put A+A together.
  • When we did [our pattern] forward and then we turned and tried to do it the other way it was harder. We were facing this way the whole time and then when we tried to start from the other side and look this [other] way it was harder.
Properties & Structure
  • I noticed lots of diagonals and turns
  • When we were doing it congruently I noticed they [a nearby team] has a similar pattern. 
  • I noticed there's yellow tape
Learning Process & Miscellaneous
  • During the reflection it took like three goes and then we got it down 
  • A lot of people were tired
  • We did really good even though it was difficult.
  • That everyone did Pattern A better than B...because we knew A better. 
  • We can get through A+B a medium speed 
  • [When they were working on reflecting their pattern] one person turned left the other person turned left and then the 15th time one person turned right and the other turned left
Heading into our final day I am finally seeing the process of math/dance making from the inside of the process. There is so much to notice from the outside in observing children's bodies thinking, moving and learning all at the same time. But having Notice & Wonder this week has taken my understanding of their learning to a whole new level. 


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