Rigor Breakdown

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Units

Rigor BreakdownA Three Part Series


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

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Units

Rigor BreakdownPart 1: Conceptual Understanding

Grades 3–5



Session Objectives

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• Examine the conceptual understanding component of rigor in G3—M5.

• Explore conceptual understanding for select content from grades 4 and 5.

• 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.



Conceptual Understanding Revisited

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“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)




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Reflection:• What does conceptual

understanding look like / sound like in the classroom?




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



AGENDA

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Promoting Understanding through Models and Contexts

Questioning, Writing and Speaking about Understanding

Examine examples from G3—M5. Select strategies for Grades 4 and 5. Consider ways to bridge gaps in

prerequisite knowledge.



Lesson Engagement – The Number Line

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Reflection:• What mathematical practice(s) do

you see being enacted in the lesson?



G3—M5 Models and Contexts

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Reflection:• What mathematical practice(s) do

you see being enacted in the lesson?



AGENDA

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Lesson Engagement – Selecting Models and Contexts

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• Select a grade level: 4 or 5• Find a partner to work with.• Have your standards available.• Have your Number and Operations

– Fractions progression available.• Have the A Story of Units

Curriculum Overview available.




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Review this standard:

Grade 4: Grade 5: 4.NF.4 5.NF.3




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• Decide on the model(s) – concrete and/or pictorial • Decide on contextual situations • Create a sample vignette to introduce the main

concept



AGENDA

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Bridging Gaps in Prerequisite Knowledge

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What prerequisite understanding is important for success in this lesson?




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How can my models and contexts be used / modified / supplemented to bridge the gaps?



Coherence Across the Grades

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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.



AGENDA

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Video Clip – Conceptual Questioning

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How do these questions prepare students for thinking about fractional units?• Which is a larger unit, an inch or a

centimeter?• Therefore which would yield a

greater number when measuring the book, inches or centimeters?



Video Clip: Inches and Centimeters

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Video Clip – Conceptual Questioning

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How do these questions prepare students for thinking about fractional units?• Which is a larger unit, an inch or a

centimeter?• Therefore which would yield a

greater number when measuring the book, inches or centimeters?



Conceptual Questioning – Key Points

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• 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?”



Lesson Engagement – Examples of Conceptual Questioning

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Find other examples of conceptual questioning from the lessons in G3—M5.



AGENDA

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Lesson Engagement – Selecting Conceptual Questions

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Select conceptual questions to add to your lesson for Grade 4 or Grade 5 that assess and remediate gaps in prerequisite knowledge.



AGENDA

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Conceptual Questioning

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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.




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• Design lesson opening questions geared to uncover current understanding of prerequisite knowledge.

• Example – assess conceptual understanding of multiplication before learning to multiply with fractions: • Write a word problem that requires me to multiply 3 x 7

to get the answer.




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Work with your partner to design a conceptual question that assesses and bridges gaps in prerequisite knowledge for the lesson you’ve created.



Key Points

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• 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.



Next Steps

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• 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?

Rigor Breakdown

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