What is Mathematical Thinking and How Do We help All

©2021fosteringmathpractices www.fosteringmathpractices.com

What is Mathematical Thinking and How Do We help All Students

Develop It?

Grace Kelemanik

#FosteringMPs@AmyLucenta

@GraceKelemanik

Amy Lucenta

©2021fosteringmathpractices

Goals

▪ Learn about three avenues of mathematical thinking championed in the CCSS Standards for Mathematical Practice

▪ Learn the Connecting Representations reasoning routine designed to develop student capacity to think structurally

▪ Develop instructional practices that support all students (including students with learning disabilities and English language learners) to develop as math thinkers and problem solvers.

▪ Understand how reasoning routines can be integrated into the Eureka curriculum.


Agenda

▪ Intro, Goals, Agenda▪ What is Mathematical Thinking▪ Break (9:45)▪ Connecting Representations: teaching structural thinking▪ Break (11:50)▪ Planning and Preparing to Facilitate Connecting Representations▪ Lunch (1:00)▪ Next Steps▪ Adjourn (1:50)


What is Mathematical Thinking?


We are currently preparing students for jobs that don’t yet

exist . . . using technologies that haven’t yet been invented . . . in order to solve problems we don’t

even know are problems yet.

—The Jobs Revolution Richard Riley


Students need ways into and through prickly problems

We must teach them to think like mathematicians!


Standards for Mathematical Practice▪ MP1 Make Sense of Problems and Persevere in Solving Them▪ MP2 Reason Abstractly and Quantitatively▪ MP3 Construct Viable Arguments and

Critique the Reasoning of Others▪ MP4 Model with Mathematics▪ MP5 Use Appropriate of Tools Strategically▪ MP6 Precision in Mathematics▪ MP7 Look for and Make Use of Structure▪ MP8 Look for and Express Regularity in Repeated Reasoning


5TOOLS

8REPETITION 7

STRUCTURE

1PROBLEMSOLVING

3CONSTRUCT & CRITIQUE

4MODEL

2

QUANTITIES& RELATIONS

6PRECISION

Not All Math Math Practices Are Equal!


RepetitionQu

antit

ies

& Re

latio

nshi

ps

MP2 MP7 MP8

MP5Page 3

MP4

MP6MP3

MP5MP4

MP6MP3

Stru

ctur

eMP5

MP4 MP6

MP3


Avenues of Thinking

What do they look like?


Fish Tank Instructions:

Solve the task on your own.

Discuss your approach with a

partner.

A 20.5 gallon fish tank is 4/5 full. How many more gallons will it take to fill

the tank?


Avenues of Thinking

Quantitative Reasoning

Structural Thinking

Repeated Reasoning

Attend toAsk Yourself

Actions to Take

Chapter 3


Three Avenues of Thinking in Math

Chapter 3

Quantitative Reasoning Structural

Thinking

Repeated Reasoning


Quantitative Reasoning Avenue of Thinking

Attend to…

Quantities and

Relationships

Ask yourself…• What can I count or measure

in this problem situation?• How do the quantities relate

to each other?• How can I represent this

problem?• What does this (variable,

number, shaded region, etc.) represent in the problem context?


Students Typically Attend to Number

7 ½


When Students Only Attend to Numbers…


Students Typically Attend to Number

7 ½Does the number tell me

something about a quantity or is it describing

a relationship?


Quantity or Relationship?

▪ Grace has 7 cookies

▪ Amy has 7 more cookies

than Grace

▪ Grace ran ½ mile

▪ Amy ran ½ as far as

Grace


What is a Quantity?

A quantity is something you can count or measure:

▪ The number of…

▪ The amount of...

A quantity answers one of the questions:

▪ How many?▪ How much?

A quantity has 3 parts:

▪ Value (7, ½ , 2π, x, etc)

▪ Unit (cookies, miles, etc )

▪ Sign ( + / -)


What are the Quantities in Gina’s Garden?

Gina planted 24 flowers in her yard. Some of them were red and some of them were purple. There are twice as many purple flowers as red flowers.

The number of…

What can I count in this

situation?


QUANTITIES in Gina’s Garden

I can count…▪ The total number of flowers▪ The number of red flowers▪ The number of purple flowers▪ The number of flowers that are neither red nor purple▪ The number of colors of flowers


Common Core State Standards, HS Number and Quantity Conceptual Category, p58

Why Develop Quantitative Reasoning? in High School…


Entering the Fish Tank via the Quantitative Avenue of Thinking

Pay attention to…▪ The amount of water:

The tank holdsAlready in the tankAdded to the tank

▪ Relationships between the amount of water the tank holds and the amount already in the tank

A 20.5 gallon fish tank is 4/5 full. How many more gallons will it take to fill the tank?


Ask yourself...

Entering the Fish Tank via the Quantitative Avenue of Thinking


● Is the 20.5 a value for a quantity or does it describe a relationship?

● What quantity has a value of 20.5?

● Is the 4/5 describing a relationship or a quantity?

● What two quantities have a 4/5 relationship?

● Are there quantities here that don’t have a value given?

● How can I represent the quantities so I can see the relationship between them?


Quantitative Reasoning Avenue of Thinking


Quantitative Reasoning Actions

▪ Identify quantities explicitly mentioned in the problem statement▪ Surface hidden or implied quantities▪ Note relationships between quantities▪ Abstract problem situations▪ Use representations to see quantities and relationships▪ Recall and consider referents


Numbers

Key words

Quantities

Total amt. of H2O the tank holdsAmt. of H2O already in tank

Amt. of H2O to added

RelationshipsAmt. of H2O in tank is 4/5 of amt. of H2O tank holds

20.5

4/5is

more


Quantitative Reasoning Shifts

Look beyond the numbers and key words in a problem statement

To the quantities and relationships those

numbers and key words describe


How Could Quantitative Reasoning Support Students?

Number Grabbing Understanding Quantities

Blindly Operating Working with relationships


Quantitative Reasoning Supports ALL Students….Especially

▪ Students who don’t know where to begin to solve a word problem▪ Students who struggle with multi-step problems▪ Students who benefit from working within contexts▪ Students who benefit from drawing/using visual representations



Chapter 4


Thinking

Repeated Reasoning


Structural Reasoning Avenue of Thinking

Attend to…

Organization and Properties

Of

Number and Space

Ask yourself…• Is this behaving like

something else I know?• How can I use properties to

uncover structure?• How can I change the form to

make it easier to work with?• How can I “chunk” this to

make sense of it?• How can I connect this to

math I know?


Entering the Fish Tank via the Structural Avenue of Thinking

Pay attention to…▪ Types of numbers

▪ Composition of fractionsA 20.5 gallon fish tank is 4/5 full. How many more gallons will it take to fill the tank?



Ask yourself…▪ Is there another way I can

think about 4/5 full?▪ How can I change the form of

20.5 and 4/5 to make them easier to work with?





When Students Don’t Think Structurally


Structural Thinking Actions

▪ Chunk complicated objects▪ Connect math ideas and representations▪ Change the form of objects▪ Recall and use properties, rules of operations, and geometric

relationships▪ Shift perspective


Find the total number of circles quickly “in your head” (i.e. without counting each circle individually)

http://www.visualpatterns.org/

http://www.visualpatterns.org/








Structural Thinking Shifts

A collection ofunrelated

results and procedures

to know

A set of interconnected ideas that build on each other and make sense


Structural Thinking Supports ALL Students….Especially

▪ Students who lose track of their work and/or calculations

▪ Students who see the ‘big picture’

▪ Students who benefit from multiple representations



Chapter 5


Thinking

Repeated Reasoning


Repeated Reasoning Avenue of Thinking

Attend to…

Repetition inProcesses

CountingCalculating

Constructing

Ask yourself…▪ Do I keep doing the same

thing over and over again?

▪ What about the process is repeating?

▪ How can I generalize the repetition?

▪ Have I included every step?


Repeated Reasoning (MP8)

Actions You Take…• Count in an organized way• Draw or build several figures• Try several numbers and observe the process• Record and track calculations• Generalize the repetition• Simultaneously maintain oversight of the process while

attending to details

• Monitor & evaluate reasonableness of intermediate results


Decompressing Repeated Reasoning

▪ Pay attention to the process▪ Sense the regularity▪ “Shortcut” the process▪ Connect the process to an “input” value▪ Generalize the process to a rule



Repetition in Constructing

https://docs.google.com/file/d/0B-OZ54hUSHfCZ2Z3eW50dzhXOXM/preview


Repetition in Constructing

https://docs.google.com/file/d/1sDC52w-X3fHvu0U3Xbt2-w68CRLzimWD/preview



Attend to Repetition in Processes

CountingConstructingCalculating


Repeated Reasoning (MP8)Adam has a 20.5 gallon fish tank that is ⅘ full. How many gallons will it take to fill the tank?

GUESS

And

CHECK



Adam has a 20.5 gallon fish tank that is ⅘ full. How many gallons will it take to fill the tank?


When Students Focus on Number Pattern vs. Repetition in Process


WE HAVE TO PRIVILEGE THE PROCESS!


Repeated Reasoning Shifts

Patterns in

Numbers and

Results

The counting, calculating and

constructing processes that

generated those numbers and results


How Could Repeated Reasoning Support Students?

‘Magic’ RulesGeneralizations that

are rooted in concrete processes


Repeated Reasoning Supports ALL Students….Especially

▪ Students who benefit from multiple modalities

▪ Students who struggle to abstract and generalize

▪ Students who work in organized and/or systematic ways

▪ Students who benefit from seeing how rules are developed


Math Thinking and Reasoning 3 Key Messages

▪ Not all Math Practices are Equal

▪ Start with the Three Avenues of ThinkingQuantitative reasoning (MP2)Structural Thinking (MP7)Repeated Reasoning (MP8)

▪ Rather than being barriers, the math practices support struggling math learners.

Math Thinking

and Reasoning


How do we develop avenues of thinking in each and every student?


Instructional Routines: Repeatable & Predictable

"Designs for interaction

that organize classroom activities”

– Magdalene Lampert,NCSM 2015

Teachers

StudentsContent


Instructional Routines that Develop Mathematical Practices

▪ Capturing Quantities (MP2)▪ Connecting Representations (MP7)▪ Recognizing Repetition (MP8)▪ Three Reads (MP1)▪ Decide and Defend (MP3)▪ Contemplate then Calculate (MP7)

www.FosteringMathPractices.com

Connecting Representations

WHAT: Match visuals to expressions by chunking and connecting to math you know

WHY: To “think like mathematicians”, to use mathematical structure to match two different representations.


Think Make connections

Share & study connections

Reflect on

learning

Create representation

Think

Ask yourself…

What part of the visual will help me connect to a chunk of the expression?

What about the expression will help me connect to the visual?

“I noticed… so I looked for...”

“... connects to ... because...”

Make Connections

We noticed… so we ...We knew... so we...

They noticed… so they ...They knew... so they...

Share and Study Connections

Create a Representation

THINKAsk yourself…

What do I notice about this expression?How can I chunk this expression

into pieces I can represent with a visual?


PAIR

Share your interpretations of the expression.Together, create a matching visual


SHARE

They noticed…..so theyWhen they saw….it made them think of….so they…

Meta-Reflection

A. When interpreting a visual / expression, I learned to pay attention to…

B. When connecting representations, I learned to ask myself…

C. A new mathematical connection I made was…

Reflect

I notice…

I wonder…

5 Essential Instructional Strategiesthat keep the focus on mathematical thinking while

providing access for ALL learners…routinelyAsk-yourself questions

Annotation

Sentence frames and starters

The Four Rs – repeat, rephrase, reword, record

Turn-and-Talkwww.fosteringmathpractices.com

?

What are Ask-Yourself Questions?

▪ A prompt for students to consider and eventually internalize

▪ Orient student thinking, without taking it over

?

What are the important

quantities in this situation?

Can I change the form of this to

make it easier to work with?

What about this process is

repeating?

Why Use Ask-Yourself Questions?

▪ Combat learned helplessness

▪ Promote student agency

?

What is Annotation?

▪ A representation of student thinking

Why Use Annotation?

Annotation Connects the verbal to the visual

What are the Four Rs?

Repeat Rephrase Reword Record

Why Use the Four Rs?

Math Ideas

& Language

Process

Develop

Refine

Articulate

What are Sentence Frames and Starters?

We noticed…..so we….

They knew….so they….

A question I learned to ask myself is…..

Why Use Sentence Frames and Starters?

▪ Orient student thinking

▪ Help students organize their ideas

▪ Support language production

▪ Promote pair and full group shares

What are Turn and Talks?

▪ An opportunity for students to work out mathematical ideas and language together

Why use Turn and Talks?

▪ So that every student has an opportunity to speak, develop language and thinking

▪ So that the teacher can hear the range of thinking in the room

A. 3(x+2)

B. 3x+2

C. 3(x+6)

1.

2.

Connecting Representations in the Classroom

Harris MintzMaria Ambroselli▪ Mather H.S., NYC▪ Grade 9, Algebra CTC▪ Students - 1st time▪ Teachers - 2nd time

This material is based upon work supported by the National Science Foundation under Grant No. 1238157

How do these strategies support all students to think and communicate mathematically?

Ask Yourself

QuestionsAnnotation

Sentence Frames

and Starters

Four Rs Turn and Talks

?

https://docs.google.com/file/d/1AuSzuoklw9ptbeWWaAIh4jI23BbGZnoJ/preview


Ask Yourself

QuestionsAnnotation

Sentence Frames

and Starters


?

https://docs.google.com/file/d/1DC3UJtt9dIAUeeJMqmD0MB2sDa6BCGfS/preview


Ask Yourself

QuestionsAnnotation

Sentence Frames

and Starters


?

https://docs.google.com/file/d/1vzbsoHzbJKGtgszgyMyzu93FEGCYoUiZ/preview

Stop and Jot

How does Connecting Representations support learners?

What about the designs for interaction orient students to structural thinking? How?

What about the designs for interaction engage students? How?



What makes a good task set?



Planning and Preparing to Facilitate


Connecting RepresentationsPlanning

www.fosteringmathpractices.com


#ConnectingReps Planning Part 1

▪ Consider the representations Key features Annotate connections

▪ Anticipate student thinking What will students notice? How will they use what they notice to connect reps? What incorrect connections might they make? Why?

▪ Consider your students What representation type will support/challenge students? What representation will you omit for students to create? Why?


#ConnectingReps Planning Part 2

▪ Work through pages 2 and 3 of the planner▪ Respond to prompts▪ Prepare the slides-- ‘taskify’ the prompts.▪ Full Group Rehearsal


WHAT: Match visuals to expressions by chunking and connecting to math you know

WHY: To “think like mathematicians”, to use mathematical structure to match two different representations.


ThinkMake

connections

Share & study

connections

Reflect on

learning

Create representatio

n

Think

Ask yourself…

What part of the visual will help me connect to a chunk of the expression?

What about the expression will help me connect to the visual?

“I noticed… so I looked for...”

“... connects to ... because...”

Make Connections

We noticed… so we ...We knew... so we...

They noticed… so they ...They knew... so they...

Share and Study Connections


THINKAsk yourself…

What do I notice about this expression?How can I chunk this expression

into pieces I can represent with a visual?


PAIR

Share your interpretations of the expression.Together, create a matching visual


SHARE

They noticed…..so theyWhen they saw….it made them think of….so they…

Meta-Reflection

A. When interpreting a visual / expression, I learned to pay attention to…

B. When connecting representations, I learned to ask myself…

C. A new mathematical connection I made was…

©2020fosteringmathpractices www.fosteringmathpractices.com

What we knowRepresentation

Type 11.2.3.

Representation Type 2

A.B.



Instructional Routine


Selecting Connecting Representation Tasks

Ask yourself:▪ Can you make accurate ‘matches’ for non-mathematical reasons?

(Are there superficial connections?). NO!▪ Does the task set contain enough similarities to cause students to

discern mathematical structure and surface important mathematics? YES!

▪ Is it annotatable? YES!▪ Do students have to change the form? NO!

www.fosteringmathpractices.com


▪ Offer two or more representations (e.g. a visual model, diagram, graph, etc.)

▪ Connect representations (e.g. graphs to words, visuals to numbers or expressions, etc.)

▪ Cause students to discern structural nuances

Look for tasks in your curriculum that…


Wrap Up

▪ Check out Resources on www.fosteringmathpractices.com▪ Read Routines for Reasoning chapter 4 and parts of chapter 7▪ Enact Connecting Representations in the classroom 3 or more

times.

Documents

What is Mathematical Thinking and How Do We help All