Math Coaches and the Struggling Learner Valerie Faulkner Coordinating Teacher SpEd Services Wake County Public School System [email protected] 858-1599.

Math Coaches and the

Struggling Learner

Valerie Faulkner

Coordinating Teacher SpEd Services

Wake County Public School System

[email protected]

858-1599

Agenda

The Math – Working on our own skills: How can I improve my own mathematical thinking?

The Culture – How is it different to work with a population of students versus a population of teachers?

The Instruction – What specific changes can I help teachers implement?

The Questions

Best practices – How do I get teachers to use them?

Number Sense – How do I get teachers to embed it in their instruction?

Differentiation – How do I support stronger differentiation in classrooms?

Assessment – How do we assess and use it for instruction?

The Math

Elementary Math is Hard!

Are you seeing the

STRUCTURES of the arithmetic

or still looking for Answers?

The Culture

Remember – We have a Cultural problem.

We tend to think about getting the answers in this lesson, rather than SYSTEMATICALLY working on mathematical structures.

Analyzing Student Work:Question posed to me by a colleague

names changed to protect the excellent!

I was in a 3rd grade math class and the students were working on this problem:

Jack had picked 12 apples.

Syreeta picked 5 fewer apples than Jack.  

How many apples did Jack and Syreeta pick altogether?

BRAINSTORM:

How is this problem difficult for students?

SMALL GROUP:

How would you support teacher?

The word fewer confuses them

Research Based Approach # 1Use number line to ground student in

magnitude and quantity. Process the vocabulary through the

number line.Develop “diagram literacy” grounded in

magnitudeMath Talk!

Subtext taught: Vocabulary grounded in the number line.Numbers ‘live’ on the number line.

Mathematical Problem

At Food Lion, butter costs 65 cents per stick.This amount is 2 cents less per stick, than butter at

Lowes.If you need to buy 4 sticks of butter, how much will

you pay at Lowes?

(adapted from Hegarty, Mayer, Green, 1992)

Converting a sentence into a mental representation

At Food Lion, butter costs 65 cents per stick.

So --- Food Lion = 0.65

65

Assignment Sentence: Easy - peasy

Relational Sentences:Uh – oh…

This amount is 2 cents less per stick, than butter at Lowes.

Say What?

Cultural Problem

The problem is that almost all of the strategies we learned in math pretend that life is filled with

nothing but assignment sentences!

The only thing that can really help is to Deeply Develop and Understand

Arithmetic Relationships –VISUALLY AND STRUCTURALLY.

This allows us to make sense of the relational sentences.

• Building a mental model of the problem situation– The hard part of this problem is NOT the

multiplication. It is figuring out the cost of butter at the two stores!

FOOD LION LOWES?

65 LOWES?

Younger than, fewer than, fewer, less, less than, more than, help!

“More than means add”

“Less than means subtract”

“For less than take the two numbers and subtract but switch the two numbers”

5

4

3

2

1

10

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

1

10

9

8

7

6

Number Worlds

Vocabulary Development with Meaning - Not “disembodied”

Develop Vocabulary in CONTEXT

The number line IS the context

The Otter lesson

Integrating

• Yolanda is four years younger than Zelda

Y ZZ?

Z – 4 = YY + 4 = ZZ – Y = 4

?

5

4

3

2

1

10

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

1

10

9

8

7

6

Comparison bars and math mountains don't work very well.  

Develop diagram literacy grounded in magnitudeDevelop diagram literacy grounded in the

structures of the arithmetic operationsDevelop student understanding that numbers are

made of other numbers

Subtext taught: Math takes timeRespect the depth of Arithmetic: “Profound Understanding of Fundamental

Mathematics”

Research Based Approach # 2

Doug Clements, Julie Sarama

Subitizing

This is a critical skill and may lay underneath early math number sense difficulties with addition and subtraction.

Number Sense and Instructional Choices

Number Sense and Instructional Choices

Math Mountains

Total/Sum

Jack14

Syreeta 5

Syreeta

Jack Syreeta

?12

Jack 12

Syreeta5

Jack – 5 = Syreeta12 – 5 = Syreeta12 – 5 = _____

Jack + Syreeta = _____12 + ______ = ______

Comparison Bars

Structure addition

Join and Part-Part Whole

– There is something, and you get more of it?

– There are two kinds, how many all together?

Start Unknown

– Some are given away, some are left, how many were there to start?

Compare--total unknown

- I know one amount and I have some amount more than that--how many do I have?

?

?

?

Addition types Adapted from Carpenter, Fennema, Franke, Levi and Empson, 1999, p. 12 in Adding it Up, NRC 2001.

Taken Left

What didI start with?

How many altogether?

How many do I have?

Structure3 types of subtraction

Ask yourself if a problem is a subtraction problem—

Does it fit one of these three types?:

– The Classic “Take away”

(how many left?)

– Comparison

(difference between? who has more?)

– Deficit/Missing amount

(what’s missing?)

You’ve got some amount and “take away” from it: What’s left?

You compare to see: Who has more or less?

You need some more to get where you want to be:What is the missing amount?

?

?

?

Subtraction types adapted from Ma and Kessel, Knowing Mathematics, 2003

If they have had Number Worlds maybe they would know to count backward from 12 five times.

Research Based Approach # 3

Teach through Base-TenMath Facts ARE important

- but, perhaps, for a different reason than we think!

Conceptual Repetitions

Subtext taught: Our number system is based on tenI can derive answers using ten as my cornerstone

Math Facts

8 + 7

Fact Families?Flash Cards?

Rewards?

Problem? We see this as a one-step building block

Cognitively, it is not.

Makes 10 Left over

________ + _________ = __________

Makes 10 Left over

____8____ + ____5_____ = _________

Makes 10 Left over

9 + 7

9 + ( ____ + ____ )

( 9 + ____ ) + ____

(____) + _____

_________

Demonstrate the Associative Property of Addition

Makes 10 Left over

The Instruction

• How might The Math affect how you will support instruction?

NumerationNumeration

Quantity/Quantity/

MagnitudeMagnitude

Base TenBase Ten

EqualityEqualityForm of aForm of aNumberNumber

ProportionalProportional

ReasoningReasoning

Algebraic andGeometricThinking

The Big Ideas© 2007 Cain/Doggett/Faulkner/Hale/NCDPI

Language

The Components of Number Sense

Embedding Number Sense in Instruction

How many Therapists does it take to change a lightbulb?

How many educators does it take to change a lesson plan?

So how do you deal with the CULTURAL issues to get to the INSTRUCTIONAL

issues?

Staff Development Mini-Modules

Create

Objective

and Measurable ‘challenges’

for your staff to grapple with as colleagues

Equality

8 + 5 = ___ + 3

When the entire class blows it…

The Math Coach Swoops In!

Two trucks = one elephant

Are two trucks “the same thing” as one elephant?

How might Subitizing support instruction in Equality?

Video: Does 6 = 6?

Remember!

Part of Assessing is the

Victorious Post-Test!

Give the problems again!

fluency

Can you use a fluency challenge to sneak base-ten into the school?

Is it possible that ASSESSMENT

can help us with this?

Makes 10 Left over

Makes 10 Left over

9 + 7

9 + ( ____ + ____ )

( 9 + ____ ) + ____

(____) + _____

_________

Demonstrate the Associative Property of Addition

Stations, Interventions, Activities to support Meaningful Fluency?

Discuss in Small Group

Group Discussion

Discuss One target issue in your small (2-4 people) group

Develop a plan to:

AssessIntervene

Affect

So you are working Together with your teachers!

Assess Intervene Affect!

Knowledge is the Engine

Suggested Books and Articles for Group Study

Ma, Liping. Knowing and Teaching Elementary Mathematics

Carpenter, Franke, Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School

Burns, Marilyn. About Teaching Mathematics: A K-8 Resource

Carpenter, Hiebert, Fennema, Fuson. Making Sense: Teaching and Learning Mathematics with Understanding

Ball, Deborah: Select journal articles by topic (late elem) at: http://www-personal.umich.edu/~dball/articles/inex.html (articles posted on-line!)

Griffin, Sharon Select journal articles by topic (late elem) at: http://www.clark.edu/academicCatalog/facultybio.cfm?id=39 (you can also

email me—here articles are not posted on-line, just the titles) Clements, Doug Subitizing: What is it? Pdf on Blackboard

Faulkner, Valerie and Cain, Chris Components of Number Sense and How it affected one teacher. Pdf on blackboard