Rigor Breakdown

1. Rigor Breakdown A Three Part Series

2. Rigor Breakdown

3. Rigor Breakdown Part 1: Conceptual UnderstandingGrades PK–2

4. Session Objectives • 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. Conceptual Understanding Revisited “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. Conceptual Understanding Revisited • Reflection: • What does conceptual understanding look like / sound like in the classroom?

7. Conceptual Understanding Revisited • Accessible through use of: • Concrete and pictorial models • Real-world contexts • Conceptual questioning • Speaking and writing about understanding

8. AGENDA • Promoting Understanding through Models and Contexts • Questioning, Writing and Speaking about Understanding • Examine examples from GK—M5. • Select strategies for Grades 1 and 2. • Consider ways to bridge gaps in prerequisite knowledge.

9. Lesson Engagement – Number Towers • Reflection: • What mathematical practice(s) do you see being enacted in the lesson?

10. GK—M5 Models and Contexts • Reflection: • What mathematical practice(s) do you see being enacted in the lesson?

11. AGENDA • Promoting Understanding through Models and Contexts • Questioning, Writing and Speaking about Understanding • Examine examples from GK—M5. • Select strategies for Grades 1 and 2. • Consider ways to bridge gaps in prerequisite knowledge.

12. Lesson Engagement – Selecting Models and Contexts • 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. Lesson Engagement – Selecting Models and Contexts Review this standard: Grade 1: Grade 2: 1.OA.3 2.MD.2

14. Lesson Engagement – Selecting Models and Contexts • Decide on the model(s) – concrete and/or pictorial. • Decide on contextual situations. • Create a sample vignette to introduce the main concept.

15. AGENDA • Promoting Understanding through Models and Contexts • Questioning, Writing and Speaking about Understanding • Examine examples from GK—M5. • Select strategies for Grades 1 and 2. • Consider ways to bridge gaps in prerequisite knowledge.

16. Bridging Gaps in Prerequisite Knowledge What prerequisite understanding is important for success in this lesson?

17. Bridging Gaps in Prerequisite Knowledge How can my models and contexts be used / modified / supplemented to bridge the gaps?

18. Coherence Across the Grades • 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. AGENDA • Promoting Understanding through Models and Contexts • Questioning, Writing and Speaking about Understanding • Examine examples from GK—M5. • Select strategies for Grades 1 and 2. • Consider ways to bridge gaps in prerequisite knowledge.

20. Video Clip – Conceptual Questioning • 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. Video Clip: Counting Sticks

22. Video Clip – Conceptual Questioning • 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. Conceptual Questioning – Key Points • 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. Lesson Engagement – Examples of Conceptual Questioning Find other examples of conceptual questioning from the lessons in GK—M5.

25. AGENDA • Promoting Understanding through Models and Contexts • Questioning, Writing and Speaking about Understanding • Examine examples from GK—M5. • Select strategies for Grades 1 and 2. • Consider ways to bridge gaps in prerequisite knowledge.

26. Lesson Engagement – Selecting Conceptual Questions Select conceptual questions to add to your lesson for Grade 1 or Grade 2 that assess and remediate gaps in prerequisite knowledge.

27. AGENDA • Promoting Understanding through Models and Contexts • Questioning, Writing and Speaking about Understanding • Examine examples from GK—M5. • Select strategies for Grades 1 and 2. • Consider ways to bridge gaps in prerequisite knowledge.

28. Conceptual Questioning • 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. Bridging Gaps in Prerequisite Knowledge • 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. Bridging Gaps in Prerequisite Knowledge Work with your partner to design a conceptual question that assesses and bridges gaps in prerequisite knowledge for the lesson you’ve created.

31. Key Points • 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. Next Steps • 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?