making mathematical thinking processes 4/26/2016 visible · 2020-04-02 · students need only to...

Making Mathematical Thinking Processes Visible

4/26/2016

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Looking at Skill Sets

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Making Mathematical Thinking

Processes Visible

Tuesdays for Teachers

April 26, 2016


Welcome!

• Daphne Atkinson, GED Testing Service

• Debi Faucette, GED Testing Service

• Bonnie Goonen, Consultant to GEDTS

• Susan Pittman, Consultant to GEDTS

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4/26/2016



In this session, we will:

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Session Objectives

• Discuss why visible thinking

routines are useful

• Identify different thinking routines

• Discuss the impact of integrating

thinking routines into

mathematical reasoning

• Share resources and ideas

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Let’s get

started by

taking a

quick poll.


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Is this your first-time

attending a Tuesdays for

Teachers webinar?

Yes

No

Staying

Informed


Have you used strategies/materials from a

Tuesdays for Teachers’ webinar in your

classroom/program?

Yes

No

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registered for the GED

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newsletter – In Session?

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Staying

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4/26/2016



Have you accessed the

revised Assessment Guide

for Educators (March 2016)

and the Performance Level

Descriptors?

Yes

No

10

Staying

Informed


Did you know that you are

invited to attend the GED

Testing Service® Conference

in Westin, Virginia on July

27-29?

Yes

No

11


Identifying Thinking

• Recall a lesson or activity

you’ve seen that you feel

really engaged students in

developing understanding.

• What kinds of thinking did

you observe the students

engaging in during that

activity or lesson?

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4/26/2016



Notice, Name, and Highlight Thinking

What kind of thinking do we want our students

to do?

– Make connections

– Reason with evidence

– Observe closely and describe

– Consider different viewpoints

– Capture the heart and form conclusions

– Build explanations and interpretations

– Solve problems in different ways

– ? ? ?

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How Do We Teach

Thinking Skills? Research and Support

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“Students often enter a

math class with about as

much enthusiasm as one

brings to a root canal

procedure.”

— Gary Stogsdill, Mathematics Professor

Prescott College, Arizona


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Paradigm Shift

Traditional Approach – Teacher delivers the prescribed

curriculum to the students. AKA trying to get what is in

our heads into our students heads.

Teaching for Understanding– “Trying to get what is in the

students’ heads into our own so that we can provide

responsive instruction that will advance learning.”

Making Thinking Visible, p.35

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Effective Teaching and Learning. (2014). In Principles to Actions : Ensuring

mathematical success for all (p. 11). Reston, VA: NCTM.

Unproductive Beliefs Productive Beliefs

Students can learn to apply

mathematics only after they have

mastered the basic skills.

Students can learn mathematics through

exploring and solving contextual and

mathematical problems.

The role of the student is to memorize

information that is presented and then

use it to solve routine problems on

homework, quizzes, and tests.

The role of the student is to be actively

involved in making sense of mathematics

tasks by using varied strategies and

representations, justifying solutions,

making connections to prior knowledge or

familiar contexts and experiences, and

considering the reasoning of others.

An effective teacher makes the

mathematics easy for students by

guiding them step by step through

problem solving to ensure that they are

not frustrated or confused.

An effective teacher provides students

with appropriate challenges, encourages

perseverance in solving problems, and

supports productive struggle in learning

mathematics.

Productive vs. Non-Productive Beliefs


Productive vs. Non-Productive Beliefs

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Unproductive Beliefs Productive Beliefs

Mathematics learning should focus on

practicing procedures and memorizing

basic number combinations.

Mathematics learning should focus on

developing understanding of concepts and

procedures through problem solving,

reasoning, and discourse.

Students need only to learn and use

the same standard computational

algorithms and the same prescribed

methods to solve algebraic problems.

All students need to have a range of

strategies and approaches from which to

choose in solving problems, including, but

not limited to, general methods, standard

algorithms, and procedures.

The role of the teacher is to tell

students exactly what definitions,

formulas, and rules they should know

and demonstrate how to use this

information to solve

mathematics problems.

The role of the teacher is to engage

students in tasks that promote reasoning

and problem solving and facilitate

discourse that moves students toward

shared understanding of mathematics.

Effective Teaching and Learning. (2014). In Principles to Actions : Ensuring

mathematical success for all (p. 11). Reston, VA: NCTM.


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The Making Thinking Visible Project

• Harvard Graduate School of

Education Project Zero

• Investigated the development of

learning processes since 1967

• Researched-based teaching and

learning strategies (routines)

• Strategies designed to move

student learning beyond teaching

and a test to thinking and

understanding.

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What is a thinking routine?

• Simple structures and tools that can be used

across levels and content areas

• A way to advance understanding and provide

ways to make thinking

• Patterns of behavior to help us use our minds in

new situations

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Tools for the Teachers – Habits for the Students


“At the core of Visible Thinking are

practices that help make thinking

visible. Thinking Routines loosely

guide learners’ thought processes and

encourage active processing. They

are short, easy-to-learn mini-

strategies that extend and deepen

students’ thinking and become part of

the fabric of everyday classroom life.”

— Pzweb.harvard.edu


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Thinking Routines

• Easy to remember

• Named and can be identified as a common

practice

• Goal-oriented; used for a specific purpose of

directing or scaffolding thinking

• Work across a variety of contexts and levels

• Encourage students to actively engage with a

topic by asking them to think beyond facts

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Samples of Thinking Routines

• What Makes You Say That? (interpretation with justification)

• Think Puzzle Explore (sets the stage for further inquiry)

• I used to think…Now I think (reflecting on how and why our thinking

changed)

• See Think Wonder (exploration into what is seen)

• Connect Extend Challenge (making connections, identifying new

information, and posing questions)

• GSCE (Generate, Sort, Connect, Elaborate – a routine for

organizing one’s understanding of a topic through concept mapping)

• Question Starts (A routine for creating thought-provoking questions)

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But there are more . . .

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Remember, it’s

not about a

specific routine,

but rather it is a

tool to develop

and advance

higher-order

thinking skills.


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Strategies for the Classroom

Using Modeling & Scaffolding

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Objective

Consider how two classroom strategies…

– Modeling (making thinking processes visible)

– Scaffolding (meeting students where they

are and taking them where they need to go)

can support incorporating visual thinking in

the classroom.

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Why Focus Here?

• Helping students learn how to learn is critical to

aiding the development of higher-order thinking

• Knowledge gained from higher-order thinking

processes

– Is more easily transferrable

– Lasts longer

– Becomes accessible for use in solving new

problems

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MTPV in Action

Applying to Mathematical Problem Solving

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Where do you start?

Start by concisely

describing for

students what you

and they will be

doing.

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Doing the Pre-Work

• Complete the task analysis of the thinking

skill to be learned

• Identify sample problems, examples, &

explanations

• Develop activity-dependent questions

(e.g. how, why, and how well)

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Making Thinking Processes Visible

(MTPV) 1. Select a skill to teach

2. Select an activity that requires successful

application of the skill

3. Work out the exact steps needed to

complete the task

4. Plan your script—you’ll play the

struggling student

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During Modeling…

1. Focus on explaining the “why”

2. Stay in character: YOU are in the learner role

3. Be sure that your process is visible

4. Provide time to debrief

5. Post the steps for future reference

6. Provide practice

7. Engage students: Ask/discuss: Where else can

you use this process?

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4/26/2016



“A problem is a

chance for you to

do your best.”

Duke Ellington


Why Word Problems?

• Require higher-order thinking skills—even in

their simplest forms

• Practice in multi-step processing

• Require close reading in a mathematical

context

• Provide a way to develop skills in determining

data sufficiency

• Exercise mathematical reasoning skills

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Simplified Task Analysis

• Demonstrate how to READ the problem

using the Three Read Process

• Understand the problem to solve

• Organize the data

• Identify the math needed to solve the

problem

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First Read: Read for Understanding

Second Read: Identify a Problem-Solving Process

Third Read: Solve the Problem and Check for Reasonableness

Reading and Reasoning Process

Miller, P. and Koesling, D. “Mathematics Teaching for Understanding: Reasoning, Reading, and

Formative Assessment. Danvers, MA


First Read: Read for Understanding

• Read through the problem aloud – noting your

reactions to what you’re reading?

• What vocabulary do I not know?

• What’s the real-world context of the problem?

• Is there a picture that can help you visualize the

problem?

• What questions are being asked? Miller, P. and Koesling, D. “Mathematics Teaching for Understanding: Reasoning, Reading, and

Formative Assessment.” Danvers, MA


Algebraic Sugar Cane

10 factories produce sugar cane.

The second produced twice as much as

the first. The third and fourth each

produced 80 more than the first.

The fifth produced twice as much as the second. The sixth

produced 40 more than the fifth. The seventh and eighth

each produced 40 less than the fifth. The ninth produced

80 more than the second. The tenth produced nothing due

to drought in Australia. If the sum of the production equaled

11,700, how much sugar cane did the first factory produce?


4/26/2016



Ten factories

40

#1 #2 #3 #4 #5

#6 #7 #8 #9 #10


Organize the information

• What is known? (10 factories with total

production of 11,700)

• What does the problem ask? (The output of

the first factory—discuss where what is to be

solved for is found)

• What are the important cues?

– How is the output of each factory described?

– What is the pattern I see?

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• How will I translate the words into mathematical

language?

• What is the pertinent information in this problem? Is

everything I need supplied?

• What problem-solving strategies can I use to solve

the problem?

• Which of those problem-solving strategies is best

suited for this problem?

Second Read: Identify a

Problem-Solving Process




4/26/2016



From words to “math”

Words

1. 1st

2. 2nd produced twice as much as the 1st

3. 3rd produced 80 more than the 1st

4. 4th produced 80 more than the 1st

5. 5th produced twice as much as the 2nd

6. 6th produced 40 more than the 5th

7. 7th produced 40 less than the 5th

8. 8th produced 40 less than the 5th

9. 9th produced 80 more than the 2nd

10. 10th produced nothing

Math

1. x

2. 2x

3. x+80

4. x+80

5. 4x

6. 4x+40

7. 4x-40

8. 4x-40

9. 2x+80

10. 0x

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Ten factories

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x 2x x+80 x+80 4x

4x+40 4x-40 4x-40 2x+80 0x


• Now that I understand the problem’s content,

how can I best use my math skills to solve

the problem?

• Am I answering the right question?

• How should the answer to the question be

expressed?

• In the case of our example, it is the output of the first

factory (not any of the others)

Third Read: Solve the Problem and Check for Reasonableness




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Answer

First write down what each factory produced in relation to

the first factory (whose production is x):

1. x

2. 2x

3. x + 80

4. x + 80

5. 4x

6. 4x + 40

7. 4x - 40

8. 4x - 40

9. 2x + 80

10. 0x

Add them all up and set equal to 11,700 to get the equation: 23x + 200 = 11,700

Solution is: x = 500


Routines for Problem Solving

Applying to Mathematical Problem Solving in the Classroom

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Claim, Support, Question

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• What do you see?

• What do you think about that?

• What do you wonder?


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Problem Solving Begins with…

• Allows all students to

participate

• Work independently or

in groups

• What is stated in the

problem

• What are the “givens”

of the problem

• Is the planning part

• Talk about strategies to

use

• Restate the problem

• Pose questions about

what they noticed

• Allows students to slow

down and think

• Brainstorm, list, and

discuss ideas

Noticing Wondering


Cranberry Craving

On Thanksgiving Day, Carissa ate some cranberries. The next day she couldn’t stop thinking about how

good the cranberries were and ate seven more cranberries than she had eaten on Thursday.

Each day after that she ate seven more cranberries than the day before. By the following Wednesday

night, she had eaten a total of 161 cranberries for the whole week.

What do you notice? What do you wonder?

Think About It!


Don’t forget to

teach multiple

ways of solving

through heuristics.

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4/26/2016



Must-Have

Strategies

for Problem

Solving

What are heuristics?


Two Questioning Routines for

Problem Solving

“Anyone who has

never made a

mistake has never

tried anything new.”

- Albert Einstein


Let’s SOLVE a Math Problem

Even Albert Einstein said:

“Do not worry about your difficulties in

Mathematics. I can assure you mine

are still greater.”


4/26/2016



S tudy the problem (What am I trying to find?)

O rganize the facts (What do I know?)

L ine up a plan (What steps will I take?)

V erify your plan with action (How will I carry out my

plan?)

E xamine the results (Does my answer make sense?

If not, rework.)

Always double check!

SOLVE a Problem (A thinking routine)


S = Study the problem

What is the problem

asking me to do?

Find the question.

Each week, Bob gets paid

$20 per hour for his first 40

hours of work, plus $30 per

hour for every hour worked

over 40 hours. Last month,

Bob made an additional $240

in overtime wages. If Bob

works 55 hours this week,

how much will he earn?

We are going to practice

SOLVE with this one!


O = Organize the Facts

• Identify each fact.

• Eliminate

unnecessary facts.

• List all necessary

facts.

What facts are provided in order

for you to solve the problem?

Each week, Bob gets paid $20

per hour for his first 40 hours of

work, plus $30 per hour for

every hour worked over 40

hours. Last month, Bob made

an additional $240 in overtime

wages. If Bob works 55 hours

this week, how much will he

earn?


4/26/2016



• Select the operations to use.

• State the plan/strategy that you will use in words.

Remember, it’s about making your thinking visible.

L = Line Up a Plan

I will use a multi-step approach. First, I will multiply

the number of regular work hours by the regular

hourly rate. Next, I will multiply the number of hours

of overtime by the overtime rate. To obtain Bob’s total

weekly salary, I will add the total amount earned for

his regular salary plus his overtime salary.


V = Verify Your Plan

$20.00

x 40

$800.00

$30.00

x 15

$450.00

Regular Wages $ 800.00

+ 450.00

$1250.00


E = Examine the Results

(Is it reasonable? Does it make

sense? Is it accurate?)

$1250.00 IS reasonable because it is

more than Bob’s average weekly salary.

Also, the answer is a whole number

because all of the facts were whole

numbers ending in zeros. Therefore, Bob

made $1250.00 in salary for the week.


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Active Reading and Thinking Routine for Math

• Read problem closely

• Identify the goal – the task(s) to be completed

– Paraphrase what author wants to be done

– Write in own words

• Identify the givens – information relevant to

solving the task

– Look for key terms

Goals and Givens


• Create a t-chart

Goals Givens

• Develop a plan to reach the goal

• Implement the plan and find the solution

• Check the reasonableness of the solution

Goals and Givens


Goals and Givens Template


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Posing Purposeful Questions

Regardless of the Routine – Questions Are Important

64


Posing Purposeful Questions

Four Types of Questions

1. Gathering Information

2. Probing Thinking

3. Making the mathematics visible

4. Encouraging reflection and justification

65

Effective teaching of mathematics uses purposeful questions to

assess and advance students’ reasoning and sense making about

important mathematical ideas and relationships.

Effective Teaching and Learning. (2014). In Principles to Actions : Ensuring mathematical success for all

(p. 36). Reston, VA: NCTM.


Purposeful Questions

66

Question type Description Examples

Gathering

information

Students recall facts,

definitions, or procedures.

• When you write an equation,

what does the equal sign tell

you?

• What is the formula for finding

the area of a rectangle?

Probing thinking Students explain,

elaborate, or clarify their

thinking, including

articulating the

steps in solution methods

or the completion of a task.

• As you drew that number line,

what decisions did you make

so that you could represent 7

fourths on it?

• Can you show and explain

more about how you used a

table to find the answer to the

Smartphone Plans task?




4/26/2016



Purposeful Questions

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Question type Description Examples

Making the

mathematics

visible

Students discuss

mathematical structures

and make connections

among mathematical ideas

and relationships.

What does your equation have

to do with the band concert

situation?

How does that array relate to

multiplication and division?

Encouraging

reflection

and justification

Students reveal deeper

understanding of their

reasoning and actions,

including making an

argument for the validity of

their work.

• How might you prove that 51 is

the solution?

• How do you know that the sum

of two odd numbers will always

be even?




Thinking routines are . . .

• One of the best ways to

develop higher-order

thinking skills

• Enhance knowledge for

longer term use

• Flexibility across content

areas and problem types

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Let’s Look at Few More Resources

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• MTPV Information

• Graphics Organizer

• Websites

• More . . .


4/26/2016



http://www.gedtestingservice.com/

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Questions

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Thank you! [email protected]

[email protected]

[email protected]

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