HOW MIGHT WE SUPPORT TEACHERS AS THEY DEEPEN THEIR OWN UNDERSTANDING AND EXPLORE STUDENT THINKING RELATED TO THE K-6 GEOMETRY PROGRESSION? GINI STIMPSON

GOAL  What? We will analyze the K-6 Geometry Progression  How? By learning about its structure, developing a shared understanding of the vocabulary, and then reading the progressions document  Why? So that we can anticipate experiences and resources that teachers may need in order to implement the Common Core State Standards in the K-6 Geometry strand.

KEY FOCUS FOR WORK WITH TEACHERS AND STUDENTS From the Mathematics Teaching Practices Support productive struggle in learning mathematics. From the TDG Math Habits of Mind I explore MISTAKES and STUCK POINTS to start new lines of reasoning and new math learning. From the Common Core Standards for Mathematical Practice Math Practice 3: Construct viable arguments and critique the reasoning of others.

BACKGROUND  Developing Mathematical Ideas (DMI): Measuring Space in 1, 2, & 3 Dimensions & Examining Features of Shape  K-8 Work Teams that focused on mistakes and stuck points related to multiplication  Working with representative teachers from each elementary school in a large school district who worked to make sense of each K-5 progression and related work from the Illustrative Mathematics website and then identified necessary work that all teachers at their grade level would need to prepare them for addressing the Common Core Content and Practice Standards.

READ THE IMPORTANCE OF SPATIAL STRUCTURING IN GEOMETRIC REASONING Be prepared to share:  How would you describe “spatial structuring” in your own words?  What are you curious about in terms of how students’ you work with would do some of these problems?  How might teachers use ideas from this article to find more about their own students thinking?

ANTICIPATING STUDENTS CONFUSIONS Read Case 19: A Big Clump of Cake from DMI: MS123.  What does Kalil understand at the end of this case that he did not understand when his teacher first conferred with him?  How does reading this case impact how you are thinking about introducing and working with arrays or supporting students who do not recognize why arrays are a model for multiplication?

PREPARE TO READ THE K-6 GEOMETRY PROGRESSION The Geometry Progression discusses the most important goals for elementary geometry according to three categories.  Geometric shapes, their components (e.g., sides, angles, faces), their properties, and their categorization based on those properties  Composing and decomposing geometric shapes  Spatial relations and spatial structuring

LEVELS OF GEOMETRIC THINKING (SEE NOTE IN PAGE 3 MARGIN)  Visual. Students recognize shapes, e.g., a rectangle “looks like a door.”  Descriptive. Students perceive properties of shapes, e.g., a rectangle has four sides, all its sides are straight, and opposite sides have equal length.  Analytic. Students characterize shapes by their properties, e.g., a rectangle has opposite sides of equal length and four right angles.  Abstract. Students understand that a rectangle is a parallelogram because it has all the properties of parallelograms

IDENTIFY THE GRADE LEVEL THAT WILL BE YOUR FOCUS Within your group, identify a person that will responsible for providing a detailed description of the work at a particular grade level, K, 1, 2, 3, 4, 5, or 6. Read the details for the three grade levels, the one immediately before your focus grade, your focus grade, and the one following your focus grade. Summarize the three or four key points that you want to share with the whole group. As time permits, read pages 2-5 and page 20.

QUIET READING TIME  What ideas and/or words are you puzzling about so you want to discuss with others in your table group?  What seems new or different?  What connections are made between the geometry strand and other mathematics at a particular grade level?

SHARE INSIGHTS AND QUESTIONS WITH YOUR GROUP  Each person should share their key points including what you noticed in terms of what may be new or different including connections with other strands within a grade level.  Identify ideas and/or words that are puzzling.

WHY LITTLE GEOMETRY MAY BE TAUGHT K Additional Clusters  Describe and compare measurable atributes  Identify and decribe shapes  Analyze, compare, create, and compare shapes Grades 1 and 2 Additional Cluster  Reason with shapes and their attributes Grade 3 Supporting Cluster  Reason with shapes and their attributes

Grade 4 Additional Clusters  Geometric measurement: understand concepts of angle and angle measure  Draw and identify lines, and angles, and classify shapes by properties of their lines and angles Grade 5 Major Cluster  Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition Grade 5 Additional Cluster  Graph points on the coordinate plane to solve real-world and mathematical problems Grade 6 Supporting Cluster  Solve real-world problems involving area, surface area, and volume.

QUICK IMAGES FROM DMI: MS123 Gather interlocking cubes. After seeing an image for 3 seconds, you will build it. You will be given a second 3-second look at the image so that you can revise your structure, if needed.

SHARE YOUR THINKING  What aspects of the figure did you note as you worked to hold an image of it in your mind?  In other words, how did you remember the structure?

HOW DID YOU REVISE YOUR APPROACH? Talk with your elbow partner about how you changed the way you looked at the shapes after hearing from other participants or reflecting on your own thinking.

DO SOME GEOMETRY ASSESSMENT TASKS FROM ILLUSTRATIVE MATHEMATICS What additional questions or insights do you have as a result of working through these assessment tasks?

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