Welcome 

Do Now (T‐P‐S)What math practice(s) were you 

using when working on DiophantusTask

A Mathematical Thinking Avenue into and through Non‐routine Problems

Grace KelemanikClinical Teacher EducatorBoston Teacher ResidencyGKelemanik@Comcast.net

Reasoning Quantitatively

And NOW

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Instructions:

1. Solve the textbook task.

2. Analyze the student work.

3. Talk with a tablemate about how the student was thinking.

Session Goals

Understand what it means to “reason abstractly and quantitatively”  (MP2). You will know your learning is on track if you can:

o Describe the key ideas of MP2o Identify when you or a colleague is reasoning 

abstractly and quantitatively

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Session Goals

Learn how the instructional strategies, designed to foster MP2, also embodies NCTM Teaching Principles. You will know your learning is on track if you can:

o Identify when and how  Math Goals that focus Learning are Established

o Describe how the strategies Pose Purposeful Questions

o Identify how the strategies Support Productive Struggle in Learning Mathematics

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Agenda

1. Opening, Framing, Goals2. Unpack Abstract and Quantitative Reasoning3. Do Math: Gina’s Garden4. Name and Reflect on Practice‐Prompting 

Instructional Strategies5. Do Math: Three‐Reads6. Reflection:  NCTM Teaching Principles

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What’s an Avenue of Thinking?

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Standards for Mathematical Practice 

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning

of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

Structure

MP2

MP4 MP5

MP6MP3

MP7

MP4 MP5

MP6MP3

MP8

MP4 MP5

MP6MP3

© Kelemanik, Lucenta, and Creighton

Three Avenues of Thinking into and through a math problem

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MP 2

MP 7

MP 8

What does it mean to reason abstractly and quantitatively?

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Two Key Ideas of MP2

Decontextualize & Contextualize

Attend to Quantities and Relationships  

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DECONTEXTUALIZE

CONTEXTUALIZE

MP2 Key Idea #1

MP2 Key Idea #2 Attend to Quantities and Relationships

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“Mathematically proficient students make sense of the quantities and their

relationships in problem situations.”CCSS-M

What’s a Quantity?

List the quantities you see.

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What Quantities

Do You See?

Quantity

Something you can count or measure

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Gina’s Garden

Gina planted 24 flowers in her yard.  Some of them were red and some of them were purple.  

• Ask yourself: what can I count or measure?

• The number of…• The amount of…

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Gina’s Garden

Gina planted 24 flowers in her yard.  Some of them were red and some of them were purple.  

• Ask yourself: How are the quantities related?

• Think:  Quantity – “Relator” ‐‐ Quantity

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There were twice as many purple flowers as red flowers.

Stop and Jot

How was your attention oriented to quantities and relationships?

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Strategies to Support the Development of Quantitative Reasoning

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#3 Sentence Starters/Frames

• The number of…• The amount of…

• Quantity – “Relator” ‐‐Quantity

#1 Problem Stem• No question• Breaks rush to problem solve

#2 Ask Yourself Questions

• What can I count or measure in this situation? 

• How are the quantities related? 

Entering a Problem

The Three Reads Strategy

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Read the Problem 3 Times

1st Read   What is the problem about?

2nd Read  What is the question?

3rd Read  What information is important?

Reading for Quantities and Relationships

Adapting the Three Reads to explicitly target reasoning abstractly 

and quantitatively (MP2)

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Lesson Goal

Learn to “read like a mathematician”.  Pay attention to quantities and relationships in the 

problem statement. 

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Read the Problem 3 Times

1st Read   What is the problem about?

2nd Read  What quantity or quantities are we trying to find?

3rd Read What quantities and relationships are important for answering the question?

What is the problem about?

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1st Read

What quantity are we trying to find?

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2nd Read

A prince picked a basketful of golden applesin the Enchanted Orchard.  On his way home,the prince was stopped by a troll who guarded the orchard.  The troll demanded payment of one‐half of the apples plus two more.  The prince gave him the apples and set off again.  A little further on, he was stopped by a second troll guard.  This troll demanded payment of one‐half of the apples the prince now had plus two more.  The prince paid him, and set off once more.  Just before leaving the Enchanted Orchard, a third troll stopped him and demanded one‐half of his remaining apples plus two more.  The prince paid him and sadly went home.  He had only two golden apples left.  How many apples had he picked?

What quantities and relationships are important for 

answering the question?

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3rd Read

A prince picked a basketful of golden apples in the Enchanted Orchard.  On his way home, the prince was stopped by a troll who guarded the orchard.  The troll demanded payment of one‐half of the 

apples plus two more.  The prince gave him the apples and set off again.  A little 

further on, he was stopped by a second troll guard.  This troll demanded payment of one‐half of the apples the prince now had plus two more.  The prince paid him, and set off once more.  Just before leaving 

the Enchanted Orchard, a third troll stopped him and demanded one‐half of his 

remaining apples plus two more.  The prince paid him and sadly went home.  He had only two golden apples left.  How 

many apples had he picked?

Ask Yourself:

What can I count or measure in this situation?

How are the quantities related?

I can count the number of…

I can measure the amount of…

Partner Work

How many apples did the prince pick?

Developing Students’ Capacity to Reason Abstractly and 

QuantitativelyHow did the Three Reads process 

focus student attention on quantities and relationships?

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NCTM Math Teaching Practices

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Identify when and how teachers Pose Purposeful Questions

Describe how MP2 Three‐Reads Support Productive Struggle in Learning Mathematics

Identify how MP2 Three‐Reads promotes and makes use of Establish Mathematics Goals to Focus Learning

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Student work

A Shift in Focus

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Pre‐CCSS‐ Focus on 

number‐ Focus on key 

words‐ Make sure answers 

had labels

CCSS‐ Focus on quantities‐ Focus on relationships 

through visual representations

‐ Build students’ capacity to move in and out of contexts

HS Conceptual Category  Number and Quantity

Quantities. In real world problems, the answers are usually not numbers but quantities: numbers with units, which involves measurement. In their work in measurement up through Grade 8, students primarily measure commonly used attributes such as length, area, and volume. In high school, students encounter a wider variety of units in modeling, e.g., acceleration, currency conversions, derived quantities such as person‐hours .... They also encounter novel situations in which they themselves must conceive the attributes of interest. For example, to find a good measure of overall highway safety, they might propose measures such as fatalities per year, fatalities per year per driver, or fatalities per vehicle‐mile traveled. Such a conceptual process is sometimes called quantification. Quantification is important for science, as when surface area suddenly “stands out” as an important variable in evaporation. Quantification is also important for companies, which must conceptualize relevant attributes and create or choose suitable measures for them.

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DisclaimerThe National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

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