© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Units
Rigor BreakdownA Three Part Series
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Rigor Breakdown
2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Units
Rigor BreakdownPart 1: Conceptual Understanding
Grades PK–2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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)
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Conceptual Understanding Revisited
6
Reflection:• What does conceptual
understanding look like / sound like in the classroom?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
AGENDA
8
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Lesson Engagement – Number Towers
9
Reflection:• What mathematical practice(s) do
you see being enacted in the lesson?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
GK—M5 Models and Contexts
10
Reflection:• What mathematical practice(s) do
you see being enacted in the lesson?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
AGENDA
11
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Lesson Engagement – Selecting Models and Contexts
13
Review this standard:
Grade 1: Grade 2: 1.OA.3 2.MD.2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
AGENDA
15
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Bridging Gaps in Prerequisite Knowledge
16
What prerequisite understanding is important for success in this lesson?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Bridging Gaps in Prerequisite Knowledge
17
How can my models and contexts be used / modified / supplemented to bridge the gaps?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
AGENDA
19
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Video Clip: Counting Sticks
21
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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?”
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Lesson Engagement – Examples of Conceptual Questioning
24
Find other examples of conceptual questioning from the lessons in GK—M5.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
AGENDA
25
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
AGENDA
27
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M 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?