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PPT – Making Math Work for Special Education Students Phoenix, AZ February 7, 2014 Steve Leinwand SLeinwand@air.org www.steveleinwand.com PowerPoint presentation | free to download - id: 69e58f-ZmVhO

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Making Math Work for Special Education

Students Phoenix, AZ February 7, 2014 Steve

Leinwand SLeinwand_at_air.org www.steveleinwand.com

- And what message do far too many of our students

get? - (even those in Namibia!)

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A Simple Agenda for the Day

- The Math
- Special
- Instruction
- Access
- Culture of Collaboration

- An introduction to the MATH

Sothe problem is

- If we continue to do what weve always done.
- Well continue to get what weve always gotten.

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- 7. Add and subtract within 1000, using concrete

models or drawings and strategies based on place

value, properties of operations, and/or the

relationship between addition and subtraction

relate the strategy to a written method.

Understand that in adding or subtracting

three-digit numbers, one adds or subtracts

hundreds and hundreds, tens and tens, ones and

ones and sometimes it is necessary to compose or

decompose tens or hundreds.

Ready, set..

- 10.00
- - 4.59

Find the difference

- Who did it the right way??
- 910.91010
- - 4. 5 9
- How did you get 5.41 if you didnt do it this way?

So what have we gotten?

- Mountains of math anxiety
- Tons of mathematical illiteracy
- Mediocre test scores
- HS programs that barely work for more than half

of the kids - Gobs of remediation and intervention
- A slew of criticism
- Not a pretty picture!

If however..

- What weve always done is no longer acceptable,

then - We have no choice but to change some of what we

do and some of how we do it.

- But what does change mean?
- And what is relevant, rigorous math for all?

Some data. What do you see?

40 4

10 2

30 4

Predict some additional data

40 4

10 2

30 4

How close were you?

40 4

10 2

30 4

20 3

All the numbers so?

45 4

25 3

15 2

40 4

10 2

30 4

20 3

A lot more information (where are you?)

Roller Coaster 45 4

Ferris Wheel 25 3

Bumper Cars 15 2

Rocket Ride 40 4

Merry-go-Round 10 2

Water Slide 30 4

Fun House 20 3

Fill in the blanks

Ride ??? ???

Roller Coaster 45 4

Ferris Wheel 25 3

Bumper Cars 15 2

Rocket Ride 40 4

Merry-go-Round 10 2

Water Slide 30 4

Fun House 20 3

- At this point,
- its almost anticlimactic!

The amusement park

Ride Time Tickets

Roller Coaster 45 4

Ferris Wheel 25 3

Bumper Cars 15 2

Rocket Ride 40 4

Merry-go-Round 10 2

Water Slide 30 4

Fun House 20 3

The Amusement Park

- The 4th and 2nd graders in your school are going

on a trip to the Amusement Park. Each 4th grader

is going to be a buddy to a 2nd grader. - Your buddy for the trip has never been to an

amusement park before. Your buddy want to go on

as many different rides as possible. However,

there may not be enough time to go on every ride

and you may not have enough tickets to go on

every ride.

- The bus will drop you off at 1000 a.m. and pick

you up at 100 p.m. Each student will get 20

tickets for rides. - Use the information in the chart to write a

letter to your buddy and create a plan for a fun

day at the amusement park for you and your buddy.

Why do you think I started with this task?

- Standards dont teach, teachers teach
- Its the translation of the words into tasks and

instruction and assessments that really matter - Processes are as important as content
- We need to give kids (and ourselves) a reason to

care - Difficult, unlikely, to do alone!!!

Lets be clear

- Were being asked to do what has never been done

before - Make math work for nearly ALL kids and get

nearly ALL kids ready for college. - There is no existence proof, no road map, and

its not widely believed to be possible.

Lets be even clearer

- Ergo, because there is no other way to serve a

much broader proportion of students - Were therefore being asked to teach in

distinctly different ways. - Again, there is no existence proof, we dont

agree on what different mean, nor how we bring

it to scale.

- An introduction to SPECIAL

SPECIAL EDUCATION

- Students with disabilities are a heterogeneous

group with one common characteristic the

presence of disabling conditions that

significantly hinder their abilities to benefit

from general education (IDEA 34 300.39, 2004).

More practically

- SPECIAL
- Different
- Better
- More individualized, but still collaborative and

socially mediated - Differentiated

But How?

- Mindless, individual worksheet,

one-size-fits-all, in-one-ear-out-the-other

practice is NOT Special!

For SwD to meet standards and demonstrate

learning

- High-quality, evidence-based instruction
- Accessible instructional materials
- Embedded supports
- Universal Design for Learning
- Appropriate accommodations
- Assistive technology

SwD in general education curricula

- Instructional strategies
- Universally designed units/lessons
- Individualized accommodations/modifications
- Positive behavior supports
- Service delivery options
- Co-teaching approaches
- Paraeducator supports

Learner variability is the norm!

- Learners vary
- in the ways they take in information
- in their abilities and approaches
- across their development
- Learning changes by situation and context

Two resource slides

- http//www.udlcenter.org/sites/udlcenter.org/files

/updateguidelines2_0.pdf - But compare Amusement Park (teaching by engaging)

to Networks and UDL (teaching by showing and

telling) and notice that these are summaries for

nerds.

3 Networks 3 UDL Principles

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- So what is a more teacher-friendly way to say all

of this?

Join me in Teachers Chat Room

- They forget
- They dont see it my way
- They approach it differently
- They dont follow directions
- They give ridiculous answers
- They dont remember the vocabulary
- They keep asking why are we learning this
- THEY THEY THEY BLAME BLAME BLAME
- An achievement gap or an INSTRUCTION gap?

Well..if..

- They forget so we need to more deliberately

review - They see it differently so we need to

accommodate multiple representations - They approach it differently so we need to

elicit, value and celebrate alternative

approaches - They give ridiculous answers so we need to

focus on number sense and estimation - They dont understand the vocabulary so we need

to build language rich classrooms - They ask why do we need to know this so we need

to embed the math in contexts.

Pause..

- Questions???
- Most intriguing/Aha point?
- Most confusing/Hmmm point?

- Soan introduction to Instruction

My message today is simple We know what

works. We know how to make math more accessible

to our students Its instruction silly!

- Active classes
- Questioning classes
- Thinking classes

- K-1
- Reading
- Gifted

9 Research-affirmed Practices

- Effective teachers of mathematics respond to most

student answers with why?, how do you know

that?, or can you explain your thinking? - Effective teachers of mathematics conduct daily

cumulative review of critical and prerequisite

skills and concepts at the beginning of every

lesson. - Effective teachers of mathematics elicit, value,

and celebrate alternative approaches to solving

mathematics problems so that students are taught

that mathematics is a sense-making process for

understanding why and not memorizing the right

procedure to get the one right answer.

- Effective teachers of mathematics provide

multiple representations for example,

models, diagrams, number lines, tables and

graphs, as well as symbols of all mathematical

work to support the visualization of skills and

concepts. - Effective teachers of mathematics create

language-rich classrooms that emphasize

terminology, vocabulary, explanations and

solutions. - Effective teachers of mathematics take every

opportunity to develop number sense by asking

for, and justifying, estimates, mental

calculations and equivalent forms of numbers.

- 7. Effective teachers of mathematics embed the

mathematical content they are teaching in

contexts to connect the mathematics to the real

world. - 8. Effective teachers of mathematics devote the

last five minutes of every lesson to some form of

formative assessments, for example, an exit slip,

to assess the degree to which the lessons

objective was accomplished. - 9. Effective teachers of mathematics demonstrate

through the coherence of their instruction that

their lessons the tasks, the activities, the

questions and the assessments were carefully

planned.

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- Yes
- But how?
- OR
- Making Math Work for ALL (including SwD)

Number from 1 to 6

- 1. What is 6 x 7?
- 2. What number is 1000 less than 18,294?
- 3. About how much is 32 and 29?
- 4. What is 1/10 of 450?
- 5. Draw a picture of 1 2/3
- 6. About how much do I weight in kg?

Strategy 1

- Incorporate on-going cumulative review into

instruction every day.

Implementing Strategy 1

- Almost no one masters something new after one or

two lessons and one or two homework assignments.

That is why one of the most effective strategies

for fostering mastery and retention of critical

skills is daily, cumulative review at the

beginning of every lesson.

On the way to school

- A fact of the day
- A term of the day
- A picture of the day
- An estimate of the day
- A skill of the day
- A measurement of the day
- A word problem of the day

Or in 2nd grade

- How much bigger is 9 than 5?
- What number is the same as 5 tens and 7 ones?
- What number is 10 less than 83?
- Draw a four-sided figure and all of its

diagonals. - About how long is this pen in centimeters?

Consider how we teach reading JANE WENT TO THE

STORE.

- Who went to the store?
- Where did Jane go?
- Why do you think Jane went to the store?
- Do you think it made sense for Jane to go to the

store?

Now consider mathematics TAKE OUT YOUR HOMEWORK.

- 1 19
- 2 37.5
- 3 185 (No why? No how do you know? No who

has a different answer?)

Strategy 2

- Adapt from what we know about reading
- (incorporate literal, inferential, and evaluative

comprehension to develop stronger neural

connections)

Tell me what you see.

Tell me what you see.

- 73
- 63

Tell me what you see.

- 2 1/4

Strategy 3

- Create a language rich classroom.
- (Vocabulary, terms, answers, explanations)

Implementing Strategy 3

- Like all languages, mathematics must be

encountered orally and in writing. Like all

vocabulary, mathematical terms must be used again

and again in context and linked to more familiar

words until they become internalized. - Area covering

Quotient sharing - Perimeter border Mg

grain of sand

Ready, set, picture.. three quarters Picture

it a different way.

Why does this make a difference? Consider the

different ways of thinking about the same

mathematics

- 2 ½ 1 ¾
- 2.50 1.75
- 2 ½ 1 ¾

Ready, set, show me. about 20 cms How do you

know?

Strategy 4

- Draw pictures/
- Create mental images/
- Foster visualization

The power of models and representations

- Siti packs her clothes into a suitcase and it

weighs 29 kg. - Rahim packs his clothes into an identical

suitcase and it weighs 11 kg. - Sitis clothes are three times as heavy as

Rahims. - What is the mass of Rahims clothes?
- What is the mass of the suitcase?

The old (only) way

- Let S the weight of Sitis clothes
- Let R the weight of Rahims clothes
- Let X the weight of the suitcase
- S 3R S X 29 R X 11
- so by substitution 3R X 29
- and by subtraction 2R 18
- so R 9 and X 2

Or using a model

11 kg 11 kg 11 kg 11 kg 11 kg 11 kg

Rahim

Siti

29 kg 29 kg 29 kg 29 kg 29 kg 29 kg

A Tale of Two Mindsets (and the alternate

approaches they generate)

- Remember How
- vs.
- Understand Why

Mathematics

- A set of rules to be learned and memorized to

find answers to exercises that have limited real

world value - OR
- A set of competencies and understanding driven by

sense-making and used to get solutions to

problems that have real world value

- Number facts

Ready??

- What is 8 9?
- Bing Bang Done!
- Vs.
- Convince me that 9 8 17.
- Hmmmm.

- 8 9
- 17 know it cold
- 10 7 add 1 to 9, subtract 1 from 8
- 7 1 9 decompose the 8 into 7 and 1
- 18 1 add 10 and adjust
- 16 1 double plus 1
- 20 3 round up and adjust
- Whos right? Does it matter?

- 4 29
- How did you do it?
- How did you do it?
- Who did it differently?

- Adding and Subtracting Integers

Remember How

- 5 (-9)
- To find the difference of two integers, subtract

the absolute value of the two integers and then

assign the sign of the integer with the greatest

absolute value

Understand Why

- 5 (-9)
- Have 5, lost 9
- Gained 5 yards, lost 9
- 5 degrees above zero, gets 9 degrees colder
- Decompose 5 (-5 -4)
- Zero pairs x x x x x O O O O O O O O

O - - On number line, start at 5 and move 9 to the

left

Lets laugh at the absurdity of the standard

algorithm and the one right way to multiply

- 58
- x 47

- 3 5
- 58
- x 47
- 406
- 232_
- 2726

- How nice if we wish to continue using math to

sort our students!

- So whats the alternative?

Multiplication

- What is 3 x 4? How do you know?
- What is 3 x 40? How do you know?
- What is 3 x 47? How do you know?
- What is 13 x 40? How do you know?
- What is 13 x 47? How do you know?
- What is 58 x 47? How do you know?

3 x 4

- Convince me that 3 x 4 is 12.
- 4 4 4
- 3 3 3 3
- Three threes are nine and three more for the

fourth - 3
- 4

12

3 x 40

- 3 x 4 x 10 (properties)
- 40 40 40
- 12 with a 0 appended
- 3
- 40

120

3 x 47

- 3 (40 7) 3 40s 3 7s
- 47 47 47 or 120 21
- 3
- 40 7

120

21

58 x 47

- 40 7
- 50
- 6

- 58
- x 47
- 56
- 350
- 320
- 2000
- 2726

- Multiplying Decimals

Remember How

- 4.39
- x 4.2
- We dont line them up here.
- We count decimals.
- Remember, I told you that youre not allowed

to that that like girls cant go into boys

bathrooms. - Let me say it again The rule is count the

decimal places.

- But why?
- How can this make sense?
- How about a context?

Understand Why

So? What do you see?

Understand Why

gallons

Total

Where are we?

Understand Why

4.2

gallons

Total

How many gallons? About how many?

Understand Why

4.2

gallons

4.39

Total

About how much? Maximum?? Minimum??

Understand Why

4.2

gallons

4.39

Total

184.38

Context makes ridiculous obvious, and breeds

sense-making. Actual cost? So how do we

multiply decimals sensibly?

- Solving Simple Linear Equations

3x 7 22

- How do we solve equations
- Subtract 7 3 x 7 22
- - 7 - 7
- 3 x 15
- Divide by 3 3

3 - Voila x 5

3x 7

- Tell me what you see 3 x 7
- Suppose x 0, 1, 2, 3..
- Lets record that
- x 3x 7
- 0 7
- 1 10
- 2 13
- 4. How do we get 22?

3x 7 22

- Where did we start? What did we do?
- x 5
- x 3 3x

15 3 - 7 3x 7 22 - 7

3x 7 22

- X X X IIIIIII IIII IIII

IIII IIII II - X X X

IIIII IIIII IIIII

Lets look at a silly problem

- Sandra is interested in buying party favors for

the friends she is inviting to her birthday party.

Lets look at a silly problem

- Sandra is interested in buying party favors for

the friends she is inviting to her birthday

party. The price of the fancy straws she wants

is 12 cents for 20 straws.

Lets look at a silly problem

- Sandra is interested in buying party favors for

the friends she is inviting to her birthday

party. The price of the fancy straws she wants

is 12 cents for 20 straws. The storekeeper is

willing to split a bundle of straws for her.

Lets look at a silly problem

- Sandra is interested in buying party favors for

the friends she is inviting to her birthday

party. The price of the fancy straws she wants

is 12 cents for 20 straws. The storekeeper is

willing to split a bundle of straws for her. She

wants 35 straws.

Lets look at a silly problem

- Sandra is interested in buying party favors for

the friends she is inviting to her birthday

party. The price of the fancy straws she wants

is 12 cents for 20 straws. The storekeeper is

willing to split a bundle of straws for her. She

wants 35 straws. How much will they cost?

So?

- Your turn. How much?
- How did you get your answer?

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Pulling it all together

- Or
- Escaping the deadliness and uselessness of

worksheets

You choose

- 3 4 10 - 3 2 x 4 etc.
- Vs.
- SALE
- Pencils 3
- Pens 4
- Limit of 2 of each!

OOPS Wrong store

- SALE
- Pencils 3
- Pens 4
- Erasers 5
- Limit of 3 of each!
- SO?

Your turn

- Pencils 7
- Pens 8
- Erasers 9
- Limit of 10 of each.
- I just spent 83 (no tax) in this store.
- What did I purchase?

Single-digit number facts

- More important than ever, BUT
- - facts with contexts
- - facts with materials, even
- fingers
- - facts through connections and families
- - facts through strategies and
- - facts in their right time.

Deep dark secrets

- 7 x 8, 5 6 7 8
- 9 x 6, 54 56 54 since 549
- 8 9 18 1 no, 16 1
- 63 7 7 x ___ 63

You choose

- 85
- - 47
- vs.
- Ive got 85. Youve got 47.
- SO?

You choose

- 1.59 ) 10
- vs.
- You have 10. Big Macs cost 1.59
- SO?

You choose.

- The one right way to get the one right answer

that no one cares about and isnt even asked on

the state tests - vs.
- Where am I? (the McDonalds context)
- Ten? Convince me.
- About how many? How do you know?
- Exactly how many? How do you know?
- Oops On sale for 1.29 and I have 20.

You choose

- Given F 4 (S 65) 10
- Find F when S 81.
- Vs.
- The speeding fine in Vermont is 4 for every

mile per hour over the 65 mph limit plus 10

handling fee.

Which class do YOU want to be in?

Strategy 5

- Embed the mathematics in contexts
- Present the mathematics as problem situations.

Implementing Strategy 5

- Heres the math I need to teach.
- When and where do normal human beings encounter

this math?

Last and most powerfully

- Make why?
- how do you know?
- convince me
- explain that please
- your classroom mantras

To recapitulate

- Incorporate on-going cumulative review
- Parallel literal to inferential to evaluative

comprehension used in reading - 3. Create a language-rich classroom
- Draw pictures/create mental images
- Embed the math in contexts/problems
- And always ask them why?
- For copies SLeinwand_at_air.org
- See also Accessible Math by Heinemann

Nex

Processing Questions

- What are the two most significant things youve

heard in this presentation? - What is the one most troubling or confusing thing

youve heard in this presentation? - What are the two next steps you would support and

work on to make necessary changes?

Next Steps

People wont do what they cant envision, People

cant do what they dont understand, People cant

do well what isnt practiced, But practice

without feedback results in little change,

and Work without collaboration is not

sustaining. Ergo Our job, as professionals, at

its core, is to help people envision, understand,

practice, receive feedback and collaborate.

To collaborate, we need time and structures

- Structured and focused department meetings
- Before school breakfast sessions
- Common planning time by grade and by department
- Pizza and beer/wine after school sessions
- Released time 1 p.m. to 4 p.m. sessions
- Hiring substitutes to release teachers for

classroom visits - Coach or principal teaching one or more classes

to free up teacher to visit colleagues - After school sessions with teacher who visited,

teacher who was visited and the principal and/or

coach to debrief - Summer workshops
- Department seminars

To collaborate, we need strategies 1

- Potential Strategies for developing professional

learning communities - Classroom visits one teacher visits a colleague

and the they debrief - Demonstration classes by teachers or coaches with

follow-up debriefing - Co-teaching opportunities with one class or by

joining two classes for a period - Common readings assigned, with a discussion focus

on - To what degree are we already addressing the

issue or issues raised in this article? - In what ways are we not addressing all or part of

this issue? - What are the reasons that we are not addressing

this issue? - What steps can we take to make improvements and

narrow the gap between what we are currently

doing and what we should be doing? - Technology demonstrations (graphing calculators,

SMART boards, document readers, etc.) - Collaborative lesson development

To collaborate, we need strategies 2

- Potential Strategies for developing professional

learning communities - Video analysis of lessons
- Analysis of student work
- Development and review of common finals and unit

assessments - Whats the data tell us sessions based on state

and local assessments - Whats not working sessions
- Principal expectations for collaboration are

clear and tangibly supported - Policy analysis discussions, e.g. grading,

placement, requirements, promotion, grouping

practices, course options, etc.

The obstacles to change

- Fear of change
- Unwillingness to change
- Fear of failure
- Lack of confidence
- Insufficient time
- Lack of leadership
- Lack of support
- Yeah, but. (no money, too hard, wont work,

already tried it, kids dont care, they wont let

us)

Finally lets be honest

- Sadly, there is no evidence that a session like

today makes one iota of difference. - You came, you sat, you were taught.
- I entertained, I informed, I stimulated.
- But It is most likely that your knowledge base

has not grown, you wont change practice in any

tangible way, and your students wont learn any

more math. - And this is what we call PD.

- Prove me wrong
- by
- Sharing
- Supporting
- Taking Risks

Next steps Taking Risks It all comes down to

taking risks

- While nothing ventured, nothing gained is an

apt aphorism for so much of life, nothing

risked, nothing failed is a much more apt

descriptor of what we do in school. - Follow in the footsteps of the heroes about whom

we so proudly teach, and TAKE SOME RISKS

- Thank you!

Appendix Slides

The Basics an incomplete list

- Knowing and Using
- , -, x, facts
- x/ by 10, 100, 1000
- 10, 100, 1000,., .1, .01more/less
- ordering numbers
- estimating sums, differences, products,

quotients, percents, answers, solutions - operations when and why to , -, x,
- appropriate measure, approximate measurement,

everyday conversions - fraction/decimal equivalents, pictures, relative

size

The Basics (continued)

- percents estimates, relative size
- 2- and 3-dimensional shapes attributes,

transformations - read, construct, draw conclusions from tables and

graphs - the number line and coordinate plane
- evaluating formulas
- So that people can
- Solve everyday problems
- Communicate their understanding
- Represent and use mathematical entities

Some Big Ideas

- Number uses and representations
- Equivalent representations
- Operation meanings and interrelationships
- Estimation and reasonableness
- Proportionality
- Sample
- Likelihood
- Recursion and iteration

- Pattern
- Variable
- Function
- Change as a rate
- Shape
- Transformation
- The coordinate plane
- Measure attribute, unit, dimension
- Scale
- Central tendency

Questions that big ideas answer

- How much? How many?
- What size? What shape?
- How much more or less?
- How has it changed?
- Is it close? Is it reasonable?
- Whats the pattern? What can I predict?
- How likely? How reliable?
- Whats the relationship?
- How do you know? Why is that?