A Story of Units
Rigor Breakdown
A Three Part Series
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A Story of Units
Rigor Breakdown
2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
Rigor Breakdown
Part 1: Conceptual Understanding
Grades PK–2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
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.
4
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A Story of Units
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)
5
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A Story of Units
Conceptual Understanding Revisited
Reflection:
• What does conceptual
understanding look like / sound like
in the classroom?
6
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A Story of Units
Conceptual Understanding Revisited
Accessible through use of:
•
•
•
•
Concrete and pictorial models
Real-world contexts
Conceptual questioning
Speaking and writing about understanding
7
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A Story of Units
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.
8
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A Story of Units
Lesson Engagement – Number Towers
Reflection:
• What mathematical practice(s) do
you see being enacted in the
lesson?
9
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A Story of Units
GK—M5 Models and Contexts
Reflection:
• What mathematical practice(s) do
you see being enacted in the
lesson?
10
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
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.
11
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A Story of Units
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.
12
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A Story of Units
Lesson Engagement –
Selecting Models and Contexts
Review this standard:
Grade 1:
1.OA.3
Grade 2:
2.MD.2
13
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A Story of Units
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.
14
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A Story of Units
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.
15
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A Story of Units
Bridging Gaps in Prerequisite Knowledge
What prerequisite understanding is
important for success in this
lesson?
16
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A Story of Units
Bridging Gaps in Prerequisite Knowledge
How can my models and contexts be
used / modified / supplemented to
bridge the gaps?
17
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A Story of Units
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.
18
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
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.
19
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
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.
20
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A Story of Units
Video Clip: Counting Sticks
21
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
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.
22
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
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?”
23
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A Story of Units
Lesson Engagement –
Examples of Conceptual Questioning
Find other examples of conceptual
questioning from the lessons in GK—M5.
24
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A Story of Units
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.
25
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
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.
26
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
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.
27
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
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.
28
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A Story of Units
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.
29
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A Story of Units
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.
30
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
Key Points
• Conceptual understanding can be promoted in a
variety of ways – concrete and pictorial models, realworld 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.
31
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
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?
32
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