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Making Math Work for Special Education Students Phoenix, AZ February 7, 2014 Steve Leinwand SLeinwand@air.org www.steveleinwand.com

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Title: Making Math Work for Special Education Students Phoenix, AZ February 7, 2014 Steve Leinwand SLeinwand@air.org www.steveleinwand.com

1
Making Math Work for Special Education
Students Phoenix, AZ February 7, 2014 Steve
Leinwand SLeinwand_at_air.org www.steveleinwand.com
2
• And what message do far too many of our students
get?
• (even those in Namibia!)

3
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4
A Simple Agenda for the Day
• The Math
• Special
• Instruction
• Access
• Culture of Collaboration

5
• An introduction to the MATH

6
Sothe problem is
• If we continue to do what weve always done.
• Well continue to get what weve always gotten.

7
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8
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9
• 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.

10
• 10.00
• - 4.59

11
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?

12
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!

13
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.

14
• But what does change mean?
• And what is relevant, rigorous math for all?

15
Some data. What do you see?

40 4
10 2
30 4

16
Predict some additional data

40 4
10 2
30 4

17
How close were you?

40 4
10 2
30 4
20 3
18
All the numbers so?

45 4
25 3
15 2
40 4
10 2
30 4
20 3
19
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
20
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
21
• At this point,
• its almost anticlimactic!

22
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
23
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.

24
• 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.

25
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!!!

26
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.

27
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.

28
• An introduction to SPECIAL

29
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).

30
More practically
• SPECIAL
• Different
• Better
• More individualized, but still collaborative and
socially mediated
• Differentiated

31
But How?
• Mindless, individual worksheet,
one-size-fits-all, in-one-ear-out-the-other
practice is NOT Special!

32
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

33
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

34
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

35
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.

36
3 Networks 3 UDL Principles
37
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38
• So what is a more teacher-friendly way to say all
of this?

39
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?

40
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.

41
Pause..
• Questions???
• Most intriguing/Aha point?
• Most confusing/Hmmm point?

42
• Soan introduction to Instruction

43
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

44
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.

45
• 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.

46
• 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.

47
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48
• Yes
• But how?
• OR
• Making Math Work for ALL (including SwD)

49
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?

50
Strategy 1
• Incorporate on-going cumulative review into
instruction every day.

51
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.

52
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

53
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?

54
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?

55
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?)

56
Strategy 2
• (incorporate literal, inferential, and evaluative
comprehension to develop stronger neural
connections)

57
Tell me what you see.

58
Tell me what you see.
• 73
• 63

59
Tell me what you see.
• 2 1/4

60
Strategy 3
• Create a language rich classroom.
• (Vocabulary, terms, answers, explanations)

61
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

62
Ready, set, picture.. three quarters Picture
it a different way.
63
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

64
Ready, set, show me. about 20 cms How do you
know?
65
Strategy 4
• Draw pictures/
• Create mental images/
• Foster visualization

66
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?

67
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

68
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
69
A Tale of Two Mindsets (and the alternate
approaches they generate)
• Remember How
• vs.
• Understand Why

70
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

71
• Number facts

72
• What is 8 9?
• Bing Bang Done!
• Vs.
• Convince me that 9 8 17.
• Hmmmm.

73
• 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?

74
• 4 29
• How did you do it?
• How did you do it?
• Who did it differently?

75
• Adding and Subtracting Integers

76
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

77
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

78
Lets laugh at the absurdity of the standard
algorithm and the one right way to multiply
• 58
• x 47

79
• 3 5
• 58
• x 47
• 406
• 232_
• 2726

80
• How nice if we wish to continue using math to
sort our students!

81
• So whats the alternative?

82
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?

83
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
84
3 x 40
• 3 x 4 x 10 (properties)
• 40 40 40
• 12 with a 0 appended
• 3
• 40

120
85
3 x 47
• 3 (40 7) 3 40s 3 7s
• 47 47 47 or 120 21
• 3
• 40 7

120
21
86
58 x 47
• 40 7
• 50
• 6
• 58
• x 47
• 56
• 350
• 320
• 2000
• 2726

87
• Multiplying Decimals

88
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.

89
• But why?
• How can this make sense?
• How about a context?

90
Understand Why

So? What do you see?
91
Understand Why

gallons
Total
Where are we?
92
Understand Why

4.2
gallons

Total
How many gallons? About how many?
93
Understand Why

4.2
gallons
4.39
Total
About how much? Maximum?? Minimum??
94
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?
95
• Solving Simple Linear Equations

96
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

97
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?

98
3x 7 22
• Where did we start? What did we do?
• x 5
• x 3 3x
15 3
• 7 3x 7 22 - 7

99
3x 7 22
• X X X IIIIIII IIII IIII
IIII IIII II
• X X X
IIIII IIIII IIIII

100
Lets look at a silly problem
• Sandra is interested in buying party favors for
the friends she is inviting to her birthday party.

101
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.

102
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.

103
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.

104
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?

105
So?
• Your turn. How much?
• How did you get your answer?

106
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107
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108
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109
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110
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111
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112
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113
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114
Pulling it all together
• Or
• Escaping the deadliness and uselessness of
worksheets

115
You choose
• 3 4 10 - 3 2 x 4 etc.
• Vs.
• SALE
• Pencils 3
• Pens 4
• Limit of 2 of each!

116
OOPS Wrong store
• SALE
• Pencils 3
• Pens 4
• Erasers 5
• Limit of 3 of each!
• SO?

117
• Pencils 7
• Pens 8
• Erasers 9
• Limit of 10 of each.
• I just spent 83 (no tax) in this store.
• What did I purchase?

118
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.

119
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

120
You choose
• 85
• - 47
• vs.
• Ive got 85. Youve got 47.
• SO?

121
You choose
• 1.59 ) 10
• vs.
• You have 10. Big Macs cost 1.59
• SO?

122
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.

123
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.

124
Which class do YOU want to be in?
125
Strategy 5
• Embed the mathematics in contexts
• Present the mathematics as problem situations.

126
Implementing Strategy 5
• Heres the math I need to teach.
• When and where do normal human beings encounter
this math?

127
Last and most powerfully
• Make why?
• how do you know?
• convince me
• explain that please
• your classroom mantras

128
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

129
Nex
130
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?

131
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.
132
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

133
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

134
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.

135
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)

136
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.

137
• Prove me wrong
• by
• Sharing
• Supporting
• Taking Risks

138
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

139
• Thank you!

140
Appendix Slides
141
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

142
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

143
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

144
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?