Series: Coaching Towards Collaborative Learning

Math.Practice.MP7

Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP7:  Look for and make use of structure.

    Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property.



    In the expression x2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective.


    They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Download Common Core State Standards (PDF 1.2 MB)

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Math.3.OA.B.5

Common core State Standards

  • Math:  Math
  • 3:  Grade 3
  • OA:  Operations & Algebraic Thinking
  • B:  Understand properties of multiplication and the relationship between multiplication and division
  • 5: 
    Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Download Common Core State Standards (PDF 1.2 MB)

Creating a Culture of Collaborative Learning
Lesson Objective: Engage in a learning lab cycle
All Grades / Math / Professional Learning
Math.Practice.MP7 | Math.3.OA.B.5

Thought starters

  1. How does Ms. Gray create a collaborative culture that allows for authentic professional learning?
  2. How does the learning lab cycle help teachers deepen their understanding of the mathematical content?
  3. How do the teachers support each other? How do they push each other's thinking?
34 Comments
Great stuff
Recommended (1)
Thanks so much Gregory!
Recommended (0)
Hey Kristin--what a powerful learning practice! I want to share it with coaches in our district. I am curious how you're able to schedule this time. I heard there's a weekly 45 minute PLC time. But it seems like it would be hard to fit the full cycle into 45 minutes. What does the time frame/schedule usually look like?
Recommended (1)
Thanks Sean! I talked about the timing and organization of the Learning Labs more here in my latest blog post: https://www.teachingchannel.org/blog/2016/02/11/planning-a-learning-lab-gbt/ Let me know if that helps and if you have any other questions!
Recommended (2)
Can't wait to share this at next week's PLC. Thanks!
Recommended (1)

Transcripts

  • Creating a Culture of Collaborative Learning Transcript

    Kristen Gray: My name is [Kristen Gray 00:00:07]. I am a K5 math specialist

    Creating a Culture of Collaborative Learning Transcript

    Kristen Gray: My name is [Kristen Gray 00:00:07]. I am a K5 math specialist and Teaching Channel Laureate at Richard A. Shields Elementary School in Lewes, Delaware. My Getting Better Together project this year is creating a culture around collaborative learning, and it's really thinking deeply about so many pieces of how we learn as adults and also embedding in that how students learn and thinking about both of those in relation to one another.
    Today I'm really focusing on my planning and questioning in creating this safe culture. I would really love to see how creating the space allows teachers to be open and authentic in their learning while pushing each others' ideas. I also would really like to think about how the mathematics, putting that at the forefront, helps them better understand their students' thinking and where they're going from here in their lessons.
    What you're going to see today is an example of a learning lab. The first phase, the teachers come together and plan an activity around a specific mathematical goal.
    We had picked our goal as that math practice of applying the different properties of operations.
    The math lesson that we will see today is what's called a quick image. We want to see if we can design an image that may bring out the different properties, the associative and distributive, really just bringing them to the forefront and having them think about what's happening in the picture of dots to match what is happening in their equation.
    This feels like the one?
    Female: I think so.
    Kristen Gray: So we'll move forward with that one? Okay.
    We then go into the second phase of the cycle, which is in the classroom.
    Female: Okay. I want you to think about how many dots you see and how you see them.
    Kristen Gray: We have a teacher who volunteers to lead the lesson while the other teachers sit amongst the students in the classroom.
    Student: If you put them in different rows, it would still equal 32.
    Kristen Gray: During that time, any of the teachers can pause the teaching, so we may call what's called a teacher timeout.
    Female: Could you show what she did?
    Kristen Gray: So it's really learning in public.
    Female: There's so many different equations that make 32. Maybe they could think about that.
    Kristen Gray: After the lesson, we come back together in the third phase of the cycle and reflect on what we just saw.
    Female: I think this is a good opportunity now next time go back and model that.
    Kristen Gray: We also think about the next moves from here.
    In the next step of this, everybody will go back and try this image. Is there something different that you may change based on this?
    This is the first team where I truly grounded our work in research and a planning sheet that was completed ahead of time before they came in. So a lot of the work today was almost a formative for me also to say 'Okay is this what I had in mind for the planning? Is this working?'
    We are going to start our meeting today by quickly referencing our norms that we've established.
    So in creating a culture where teachers are collaboratively planning and being authentic and open in their practice, I thought it was really important to start our day together talking about the norms that they had established.
    We can add, revise so the next time we meet those revisions will be in place.
    Those were things they had come up with so I thought that was a great way to tie that all back in.
    As we go through the anticipated responses today, it's something really important to think about: How are we recording? How does our recording impact student thinking?
    One thing that I'm trying to be very thoughtful of is asking them to expand on their thinking and not be the one that's saying why I think they're doing it which is something that I tend to do. Wait time is not my thing!
    So I'm wondering what you think is very important that you're doing and recording to move these students-
    Female: As far as recording, I was trying to do-
    Kristen Gray: Really I want to ask questions that bring to the forefront these intricate things that they don't really think about that they're doing in the classroom and bring them out.
    Female: I can tell that it's starting to develop.
    Kristen Gray: I didn't want them to feel disconnected from the work they were doing. I really wanted them to truly reflect.
    Female: She circled it in different colors and then recorded it in both ways.
    Kristen Gray: That's a really strong visual for seeing why that commutative property is working.
    Female: We're not at the point of distributive property or anything like that so I think it would be really helpful to see that in a classroom...
    Kristen Gray: I also try to bring it back to the mathematics when I ask questions. So how do we think these properties are showing up? How are we going to connect them? How do we want students to connect them?
    Female: I think once somebody's taken their way when we're sharing they'll try to think of a different way to match what they had to their equation. Then it gets a little different.
    Kristen Gray: I think appreciating the ones that play around with it, because I love that creativity and that really becomes that math practice 7 where they have the structure of numbers-
    I think just overall listening to everyone contributing is something that says a lot about the culture. I think in the past the PLCs, people could've gotten by with nodding. That sounds like a great idea and I'll try it.
    Female: Are all of our students seeing the commutative property?
    Kristen Gray: I think today they were really thinking about how their students would think about it. What questions they would ask.
    Female: How do we solve for 4 groups of 8? Would that be a question?
    Female: Yes. I think that if mine just say 4 groups of 8 then I would ask, 'How did you solve that?'
    Kristen Gray: Typically the math specialist was running the show. Now it's nice to see the questions that I had planned are coming from them.
    Female: I did switch my wording to 'How many dots did you see and how did you see them?' From, 'What did you notice?' When I did 'what did you notice?' They said, 'I see a line of symmetry' verses actually focusing on the math solutions.
    Kristen Gray: If you're comfortable recording up there?
    Female: Sure!
    Kristen Gray: When we first started learning labs, I started to get the vibe that a lot of teachers thought the reflection was going to be about the teacher that was leading the lesson or the activity.
    Female: The most basic, I think, would be 8+8+8+8.
    Kristen Gray: I really tried to make it a goal of mine to let them know that we're really looking at the math and we're looking at the student thinking around the math.
    Female: So what question would you ask to make that clear because they may not have seen it that way as a double?
    Female: Usually what I would ask if then say that I doubled it, I ask them what did you double?
    Female: I think you can ask, 'How do you see it as an equation?' Before I take the jump and start recording.
    Female: Right, because I think the distributive property will come out, but if the student does like [Jen 00:06:35]says and said 'I did 4x4' and they say 'I doubled it' what does doubling look like to you, multiplying by 2, that now could later bring out the associative property.
    Kristen Gray: I think at this point, it would make the most sense to talk about which image we want to now focus all of our time on, before we get in with the students.
    Female: I think the first image pulled out a lot.
    Female: I think we spent the most time going over it up there, and with what [Jen 00:06:56] said, as it is and see if they can break apart those groups of 8 into groups of 4.
    Kristen Gray: Teacher time out, real quick. Turn and talk when we think an opportunity presents itself. Do we need any type of signal? Are you good with just talking?
    Female: Talking is fine.
    Female: I think they're more comfortable [inaudible 00:07:11].
    Female: You first saw it as 4 groups of 8, you said. Can you say the second part again?
    Student: Those two columns had to be 4x2.
    Female: I heard her say she saw it in columns, which is not what the other student did. Can you show what she did?
    Kristen Gray: In the past, the next time we were available to meet was the following week. Today was the first time we got to do it and then reflect right after and it was a world of difference coming fresh out of the experience.
    Female: I can't remember if it was rows or columns and someone had pointed out that-
    Female: She was in the back corner and she said 'there's 4 columns' that's why I was like 'she's looking at it totally differently in an array sense'.
    Kristen Gray: When we're going in, looking to see how they decompose those 2 groups of 8 and see if they doubled that and really see if that associative property was going to come out because that was important to their work they were going to be doing with arrays. I thought it's something that then we all have a common eye for.
    Student: Then I did 12+12, then I saw that there was 8 more left.
    Female: I was wondering why he was more comfortable with 6 than breaking the 8 into 4.
    Kristen Gray: It's almost like a little bit of excitement around going in there, so now we're just going in there to watch, but we're now going in there invested in this curiosity that we have.
    Female: Earlier with the 6+6 and the 6+6, they didn't seem as confused.
    Female: When you're just adding the extra on.
    Female: It's a different operation. [crosstalk 00:08:44] It's like they see it as a remainder.
    Daylen: I have a different way. I saw 2,4,6,8,10,12,14,16...
    Female: And then what did you do with that 16? I'm wondering if he kept doing that all the way down the next one, or used it to help with the next one. [Daylen 00:09:05], what did you do with that?
    Daylen: Doubled the 16.
    Female: So how would I record that as a multiplication equation?
    Daylen: 16x2=32
    Female: I had put I noticed a lot of addition, and I think we had anticipated maybe that would happen. I liked how you pushed them 'How would we write that as multiplication?' And then trying to re-write them-
    Female: I felt like somebody over here was going to time it out with I didn't [crosstalk 00:09:37]
    Female: Something I'm less surprised about is I didn't hear skip counting. Even though we did addition, I did not hear the skip counting, which I had anticipated.
    Female: That's a really nice movement from that, when we read those learning progressions, now they're getting into this level 2, level 3 type of thinking where it's more multiplicative.
    Kristen Gray: So when we shove this together as one array, which you guys will do with the arranging chairs activity, do you see any connection?
    Next with this group, I really just want to focus on taking what they learned from those journal entries and learning to somehow to embed that in our next moves.
    Female: What if they split it into rows? [crosstalk 00:10:17] that could be a different way of-
    Female: Then you get that language out there too that-
    Kristen Gray: I think today was really a testimony to these teachers really take that content seriously. They want to learn it better, and just making the time and space for them to do that is something that is invaluable in every single classroom.
    Female: Before when we have PLCs and be kind of in the back or maybe not involved, this is the most involved and most excited I've seen teachers. I think it's the most relevant piece of learning that we've done.
    Female: I think the reciprocal nature of it is nice. Being able to open your door, that makes you more comfortable to go into someone else's room.
    Female: It's really nice for me to take a minute and slow down and go 'okay, we made some tweaks and it's going better this time' and the teachers are appreciating it.
    Female: You didn't make it feel like when you were coming in or someone else was coming in that it was something we needed to improve upon, it was a journey were were taking together.
    Female: Is our curiosity around this commutative property.
    Kristen Gray: I think a lot of the changes I've made for this round are nice to now go try with different grade-levels that haven't had their second math learning lab experience.
    Student: 8x4 is equal to 4x8?
    Female: How do you know?
    Female: In the end our students really benefit from this culture that we're creating because we're showing them that we're learners, just as much as they are.

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School Details

Shields (Richard A.) Elementary School
910 Shields Avenue
Lewes DE 19958
Population: 707

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Kristin Gray
Math / Kindergarten 1 2 3 4 5 / Teacher

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