1. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Rigor Breakdown A Three Part Series

2. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Rigor Breakdown 2

3. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Rigor Breakdown Part 1: Conceptual Understanding Grades PK–2

4. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Session Objectives 4 Examine the conceptual understanding component of rigor in GK—M5. Explore conceptual understanding for select content from grades 1 and 2. Explore how cross-grade coherence is accessible through conceptual understanding. Recognize opportunities to emphasize the Standards of Mathematical Practice during activities that promote conceptual understanding.

5. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Conceptual Understanding Revisited 5 “Teachers teach more than ‘how to get the answer’ and instead support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations as well as writing and speaking about their understanding.” (excerpt from the Shifts)

6. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Conceptual Understanding Revisited 6 Reflection: What does conceptual understanding look like / sound like in the classroom?

7. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Conceptual Understanding Revisited 7 Accessible through use of: Concrete and pictorial models Real-world contexts Conceptual questioning Speaking and writing about understanding

8. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 8

9. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Lesson Engagement – Number Towers 9 Reflection: What mathematical practice(s) do you see being enacted in the lesson?

10. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units GK—M5 Models and Contexts 10 Reflection: What mathematical practice(s) do you see being enacted in the lesson?

11. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 11

12. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Lesson Engagement – Selecting Models and Contexts 12 Select a grade level: 1 or 2 Find a partner to work with. Have your standards available. Have your progressions documents from the binder available. Have the A Story of Units Curriculum Overview available.

13. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Lesson Engagement – Selecting Models and Contexts 13 Review this standard: Grade 1:Grade 2: 1.OA.3 2.MD.2

14. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Lesson Engagement – Selecting Models and Contexts 14 Decide on the model(s) – concrete and/or pictorial. Decide on contextual situations. Create a sample vignette to introduce the main concept.

15. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 15

16. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Bridging Gaps in Prerequisite Knowledge 16 What prerequisite understanding is important for success in this lesson?

17. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Bridging Gaps in Prerequisite Knowledge 17 How can my models and contexts be used / modified / supplemented to bridge the gaps?

18. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Coherence Across the Grades 18 Reflections What can you share with your colleagues to promote coherence relative to models and contexts across grades PK–5? Recall that A Story of Units recommends a finite number of concrete and pictorial models used coherently across the grades.

19. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 19

20. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip – Conceptual Questioning 20 How do these questions prepare students for the first grade work of adding and subtracting within 20? How do you know there are 20? Is it easy to count them when they are all jumbled up in a big pile? Do we need to go back and count them over and over? Explain how you counted them.

21. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip: Counting Sticks 21

22. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Video Clip – Conceptual Questioning 22 How do these questions prepare students for the first grade work of adding and subtracting within 20? How do you know there are 20? Is it easy to count them when they are all jumbled up in a big pile? Do we need to go back and count them over and over? Explain how you counted them.

23. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Conceptual Questioning – Key Points 23 Goes beyond getting the right answer Goes beyond Yes/No questions Encourages recognition of subtleties and exposes current level of student understanding “Can you think of a case where that would not work?” “Someone else says the answer is this. Can you prove that they are right/wrong?” “When we get a like unit for these two fractions, will the like unit be bigger or smaller than the units we have?” “Can you think of a number between 1/4 and 1/5?”

24. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Lesson Engagement – Examples of Conceptual Questioning 24 Find other examples of conceptual questioning from the lessons in GK—M5.

25. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 25

26. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Lesson Engagement – Selecting Conceptual Questions 26 Select conceptual questions to add to your lesson for Grade 1 or Grade 2 that assess and remediate gaps in prerequisite knowledge.

27. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 27

28. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Conceptual Questioning 28 Reflection: Think of a time when you were asked a question and were surprised to find out that you really did know the answer to that question – you just had to think about it.

29. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Bridging Gaps in Prerequisite Knowledge 29 Design lesson opening questions geared to uncover current understanding of prerequisite knowledge. Example – assess conceptual understanding of subtraction before learning to use subtraction in comparison situations. Write a word problem that requires me to subtract 3 from 7 to get the answer.

30. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Bridging Gaps in Prerequisite Knowledge 30 Work with your partner to design a conceptual question that assesses and bridges gaps in prerequisite knowledge for the lesson you’ve created.

31. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Key Points 31 Conceptual understanding can be promoted in a variety of ways – concrete and pictorial models, real- world contexts, conceptual questioning, and writing/speaking about understanding. Each of these ways can be used to coherently bridge gaps in prerequisite knowledge. Content knowledge directed by the standards and the progressions informs coherent and balanced instruction.

32. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Next Steps 32 How can you increase students’ deep understanding of the concepts you will be covering when you return to your schools? How can you increase coherence in your school’s delivery of concepts? What can you share with your colleagues about bridging gaps in prerequisite knowledge through conceptual models, contexts and questioning?