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: ...is 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 turned...how far? How far has it been turned from front?
Kids; 45 (degrees).

TIME FOR GAMES!!

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.
Me...so, 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

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.

ACTIVITY OVERVIEW
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.

4th GRADERS NOTICED
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. 

On Process & Properties: Notice & Wonder, Day 3

Kids practicing their math/dance patterns on my dance board before we started class this morning.

The room was hot all day. The kids were overly concerned about things being fair in terms of who got to show their work. They were also a bit squirrel-y (and my focus could have been better) but we all worked hard on the process of creating a second 4-beat pattern, Pattern B.

The math involved in today's activities included: 
- observing and comparing moving patterns and identifying relevant properties
- increased focus on using precise language to describe the math/dance steps
- increased emphasis on focusing/identifying one relevant property at a time in the dancing (movement, foot position, direction)
- continued growth in understanding and noticing sameness in the dancing (unison/congruence) including rights and lefts for foot placement and for direction of turns

My takeaway on today:
Compared to other days, this day was a huge challenge (in a good way). I have always known that Pattern B is challenging, even without hearing what the kids noticed. But their notices were so full of details today, way more than the first few days, about the process of making a new pattern as different as possible from their first pattern.

My wonder from today: 
I'm still pondering what to do with these noticings. It's already a great tool for assessing what and how they're thinking about the material. But I keep thinking there has to be something specific I can *do* with this information. And then I think that maybe that the explicit process of reflecting like this is helpful in itself for my learners, but I can't quite put my finger on why that might be exactly. So, for now, I will be content with what I have and look forward to the final two days of this residency.

In case you're interested, my original wonderings/goals re: adding Notice/Wonder to how I facilitate Math in Your Feet can be found here.  
________________________________________

WHAT THEY NOTICED....

Observations on process
- We had to do two patterns so people had to know Pattern A and Pattern B and it's confusing...they have to know which one.
- [Me and my partner] would share our ideas and then we would do them all and then figure out which one was the easiest and the one we could do all together.
- It was hard because of the turns [So you gave your self a challenging pattern?]  Yep.
- Pattern B took more time [to make] because we had to come up with different ideas and if we had two or more turns we had to do less
- The first time I did it wrong and the second time I did it right.
- It's hard to do the same thing as your partner.
- We got confused a lot and we stepped on teach others feet and we got confused. [Why?] Because we kept on changing it and sometimes we'd remember A and mess up Pattern B.
- It doesn't take long to decide on a Pattern B if you talk and give ideas out to your partner.
- It was really loud in here when we were creating Pattern B
- Our Pattern B was very easy [to dance]
- We had to change our Pattern B because our first one was too hard. [What made it hard for you?]  We did all this tricky stuff and turned around a lot and crossed our feet a lot and it got hard so we changed it.
- Our slide had good sound but we need to work on the sound for the step
- People were making mistakes which is a good thing because if you make a mistake it's a FAIL which is a "first attempt in learning.
- I heard people being respectful to each other
- People were messing around

Observations on properties and equivalence (sameness and difference)
- Some people had jumps in Pattern A but in Pattern B they had turns and slides
- There was a lot of 90 degree turns
- We couldn't do the same pattern on Pattern B. We had to be different
- A lot of people came up with really different B patterns [compared to Pattern A]
- If people had lots of spins in Pattern A they did not do a lot of spins in Pattern B
- Everybody jumped at least once in Pattern B
- We did the same beats and the same feet for this pattern
- It took us a long time to get our B fully finished

WONDERS (similar to yesterday; I'm including the ones that stood out to me)
- I wonder what the rules are for [making] Pattern C?
- If every partner can agree with the other partner. [So, were you noticing that you were having some difficulties?]  Ummm...a little. [Did you guys come up with a Pattern B?]  Yeah.
- If everybody's A and B patterns will go good together?

You Never Know Until You Ask: Notice & Wonder, Day 2

SYNOPSIS

1. I'm engaged in a week-long experiment with Notice & Wonder as a way to help kids in Math in Your Feet reflect on their body based learning and experiences. My Day One post is here if you want to get a sense of what my goals and wonders are.

2. For Day 2 I added a Notice into our observation of the math/dance patterns in progress, which led to increased student engagement and pattern analysis.

3. Make sure to read the Notices and Wonders at the end of this post. They are lovely and philosophical and insightful as to the creative process. They are also interesting evidence of children's processing around turns, degrees of turns, and combinations of turns. Why hello Trigonometry!

EPIPHANY: MORE NOTICE, LESS TELL

Today we started making a four-beat Pattern A. As part of this process I initiated a conversation to help kids focus more closely on the categories of foot position, movement and direction as resources for our pattern building and for noticing the structure of these temporal, moving patterns. Working on Pattern A also requires an emphasis on "sameness" -- making sure both dancers are dancing the same pattern and dancing it the same way.

Part of the work of making a math/dance pattern is about stepping out of the "actor" role (as the one involved in making the pattern) to observe others' work. This kind of stepping back can help you gain insight and perspective on your own work.  Today I tried something new during this observation time and had an epiphany:

Instead of directly guiding conversation by asking "What kind of movement (or direction, or foot position) did they use in that pattern...?" I decided (on a whim, let it be known!) a Notice. After a team of dancers would show us their pattern I would first ask something like: "What did you notice about their pattern?"  For the most part, children in all three classes noticed at least one relevant attribute without me prompting. And, even better, these initial noticings started interesting conversations during which I could contribute/introduce new ideas or observations and deepen the conversations.

The kids were more engaged in these discussions than I've seen before. Ever.

I'm beyond thrilled.

YOU NEVER KNOW UNTIL YOU ASK 
Since I only have five days max with any one class, I never have enough time to really figure out exactly what makes each group tick. Imagine having the first week of school over, and over, and over again. But, with the Notice/Wonder (first done in conversation with a friend and then shared out to the whole group) I am getting much more information from these 4th graders about how they're thinking about their work than I ever did before. 

Here are their Notice & Wonders from Day 2, but not all of them. I've combined all three classes' responses -- some of them were repeated in each class. Refresher: today's mathematical themes were sameness, problem solving, categories, and identifying attributes/properties as a way to make sense of pattern and structure.

I noticed...
- It was hard for us to do the same thing [move] on the same beats
- It was hard to make a decision on what we should do [while making their pattern]
- It was hard for us to go at the same rate.
- We did good when no one was watching and when people were watching we didn't do it right
- One of us was (crossing our feet) right over left and the other was going left over right
- That sometimes when we did it we'd get confused, or get messed up, and sometimes we had success
- Every group we saw had a pattern with jumps in it
- Every group had a pattern with a split [two feet apart] in it
- Everybody was doing the same dance/pattern as the other partner
- When people started in the middle [of the square]  and they jumped to the front part of the square they usually jumped out of the square so we thought they should start at the bottom part of the square and jump to the top. [Spatial reasoning]
- Every single dance pattern was different [meaning each team had a different pattern]

I wonder...
- Can I clog at home?
- If we can add some more moves
- If you could change a 360 degree turn into 314 degrees or you could choose your number from zero all the way up to 360?
- If we combined all the patterns in the classroom and then see what it looks like
- If you could do a 460 degree turn
- If everyone will remember their dance moves
- If we we'll actually be able to do the dance we made. [Me: So you're not feeling like you can do it right now?] Well...it's tough. We're still working on it. [Me, to class: How many other people feel like they're still working on their pattern, that they don't exactly have it the way they want it? (lots of hands). So, you're not alone. This is part of the process]
- If we're going to learn other people's patterns
- If our pattern will work out? [Is it not working out?] Well it is; we've wondered from the beginning if it would work out. [Okay, what do you think so far? Thumbs up?! Okay!]
- What's the difference between the A & B pattern is [They'll find out tomorrow!]
- Why your body and your mind control the pace you go in...why only you can control your pace.
- How we come up with the things (pattern) we came up with. [Do you have any thoughts about that?]  Well I think it's about how much you're creative and how much you do that thing. Like if you like to turn and that's your favorite thing and you do it all the time you probably would like to add that in to your pattern.

Bonus: 4th grade Teacher Wonder! ...What it would be like to make a math problem out of our turns...if we were adding and subtracting the angles as we turned, what the extent of that math problem would look like.

I can't wait to see what happens tomorrow.

Vignette: Noticing, exploring, and discovering space with moving bodies

Today was my second of five days with three classes of 4th graders. Today Mr. C, whose class needs to be with me at 9am, smoothed out his morning routine and told them to come straight to the LGI room. Kids trickled in from buses, drop-off, breakfast, etc. In the absence of any particular activity the kids immediately started exploring the empty space. 

The activity mostly consisted of jumping from box to box and eventually led to discussion about which leap was the furthest and most difficult. 

I am a big fan of self-initiated explorations of space like this but after a while there were too many kids in the room to do this safely. Luckily one kid wanted to try out my dance board which led to a wonderful round robin of dancers, all being cheered on by their friends. So cute!

If you change a familiar space with tape kids will notice. When they notice they will explore this new structure with their bodies through movement. When they explore you will see how they are thinking. In this case their exploration was spatial and also mathematical in terms of magnitude and direction.