Making Math Work for
Special Education Students
Phoenix, AZ
February 7, 2014
Steve Leinwand
SLeinwand@air.org
www.steveleinwand.com
1
And what message do far too
many of our students get?
(even those in Namibia!)
2
3
A Simple Agenda for the Day
•
•
•
•
•
The Math
“Special”
Instruction
Access
Culture of Collaboration
4
An introduction to the MATH
5
So…the problem is:
If we continue to do what we’ve
always done….
We’ll continue to get what we’ve
always gotten.
6
7
8
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.
9
Ready, set…..
10.00
- 4.59
10
Find the difference:
Who did it the right way??
910.91010
- 4. 5 9
How did you get 5.41 if you didn’t
do it this way?
11
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!
12
If however…..
What we’ve 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.
13
But what does change mean?
And what is relevant,
rigorous math for all?
14
Some data. What do you see?
40
4
10
30
2
4
15
Predict some additional data
40
4
10
30
2
4
16
How close were you?
40
10
30
4
2
4
20
3
17
All the numbers – so?
45
25
15
40
10
30
4
3
2
4
2
4
20
3
18
A lot more information
(where are you?)
Roller Coaster
Ferris Wheel
Bumper Cars
Rocket Ride
45
25
15
40
4
3
2
4
Merry-go-Round
Water Slide
10
30
2
4
Fun House
20
3
19
Fill in the blanks
Ride
???
???
Roller Coaster
Ferris Wheel
Bumper Cars
Rocket Ride
Merry-go-Round
Water Slide
45
25
15
40
10
30
4
3
2
4
2
4
Fun House
20
3
20
At this point,
it’s almost anticlimactic!
21
The amusement park
Ride
Time Tickets
Roller Coaster
Ferris Wheel
Bumper Cars
Rocket Ride
Merry-go-Round
Water Slide
45
25
15
40
10
30
4
3
2
4
2
4
Fun House
20
3
22
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.
23
The bus will drop you off at 10:00 a.m. and
pick you up at 1:00 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.
24
Why do you think I started with
this task?
- Standards don’t teach, teachers teach
- It’s 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!!!
25
Let’s be clear:
We’re 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 it’s not widely believed to be possible.
26
Let’s be even clearer:
Ergo, because there is no other way to
serve a much broader proportion of
students:
We’re therefore being asked to teach
in distinctly different ways.
Again, there is no existence proof, we
don’t agree on what “different” mean,
nor how we bring it to scale.
27
An introduction to SPECIAL
28
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).
29
More practically:
SPECIAL:
- Different
- Better
- More individualized, but still
collaborative and socially
mediated
- Differentiated
30
But How?
Mindless, individual worksheet,
one-size-fits-all, in-one-ear-outthe-other practice is NOT
Special!
31
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
32
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
33
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
34
Two resource slides:
http://www.udlcenter.org/sites/udlcenter.or
g/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.
35
3 Networks = 3 UDL Principles
36
37
So what is a more
teacher-friendly way to
say all of this?
38
Join me in Teachers’ Chat Room
•
•
•
•
•
•
•
They forget
They don’t see it my way
They approach it differently
They don’t follow directions
They give ridiculous answers
They don’t remember the vocabulary
They keep asking why are we learning this
THEY THEY THEY BLAME BLAME BLAME
An achievement gap or an INSTRUCTION gap?
39
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 don’t 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.
40
Pause…..
Questions???
- Most intriguing/Aha point?
- Most confusing/Hmmm point?
41
So……an
introduction to
Instruction
42
My message today is simple:
We know what works.
We know how to make math more
accessible to our students
It’s instruction silly!
• K-1
• Reading
• Gifted
• Active classes
• Questioning classes
• Thinking classes
43
9 Research-affirmed Practices
1. Effective teachers of mathematics respond to most
student answers with “why?”, “how do you know
that?”, or “can you explain your thinking?”
2. Effective teachers of mathematics conduct daily
cumulative review of critical and prerequisite skills
and concepts at the beginning of every lesson.
3. 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.
4. 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.
5. Effective teachers of mathematics create languagerich classrooms that emphasize terminology,
vocabulary, explanations and solutions.
6. Effective teachers of mathematics take every
opportunity to develop number sense by asking for,
and justifying, estimates, mental calculations and
equivalent forms of numbers.
45
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 lesson’s 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.
46
47
Yes
But how?
OR:
Making Math Work for
ALL (including SwD)
48
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?
49
Strategy #1
Incorporate on-going
cumulative review into
instruction every day.
50
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.
51
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
52
Or in
nd
2
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?
53
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?
54
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?)
55
Strategy #2
Adapt from what we know about
reading
(incorporate literal, inferential,
and evaluative comprehension to
develop stronger neural
connections)
56
Tell me what you see.
57
Tell me what you see.
73
63
58
Tell me what you see.
2 1/4
59
Strategy #3
Create a language rich
classroom.
(Vocabulary, terms, answers, explanations)
60
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
Perimeter = border
Quotient = sharing
Mg = grain of sand
61
Ready, set, picture…..
“three quarters”
Picture it a different way.
62
Why does this make a difference?
Consider the different ways of
thinking about the same
mathematics:
•2½+1¾
• $2.50 + $1.75
• 2 ½” + 1 ¾”
63
Ready, set, show me….
“about 20 cms”
How do you know?
64
Strategy #4
Draw pictures/
Create mental images/
Foster visualization
65
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.
Siti’s clothes are three times as heavy as
Rahims.
What is the mass of Rahim’s clothes?
What is the mass of the suitcase?
66
The old (only) way:
Let S = the weight of Siti’s clothes
Let R = the weight of Rahim’s 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
67
Or using a model:
11 kg
Rahim
Siti
29 kg
68
A Tale of Two Mindsets
(and the alternate approaches they
generate)
Remember How
vs.
Understand Why
69
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
71
Ready??
What is 8 + 9?
17 Bing Bang Done!
Vs.
Convince me that 9 + 8 = 17.
Hmmmm….
72
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
Who’s right? Does it matter?
73
4 + 29 =
How did you do it?
How did you do it?
Who did it differently?
74
Adding and
Subtracting Integers
75
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”
76
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
77
Let’s laugh at the absurdity of “the
standard algorithm” and the one
right way to multiply
58
x 47
78
3 5
58
x 47
406
232_
2726
79
How nice if we wish to
continue using math to
sort our students!
80
So what’s the
alternative?
81
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?
82
3x4
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
12
4
83
3 x 40
• 3 x 4 x 10 (properties)
• 40 + 40 + 40
• 12 with a 0 appended
3
120
40
84
3 x 47
• 3 (40 + 7) = 3 40s + 3 7s
• 47 + 47 + 47 or 120 + 21
•
3
120
40
21
7
85
58 x 47
40
50
6
7
58
x 47
56
350
320
2000
2726
86
Multiplying Decimals
87
Remember How
4.39
x 4.2
“We don’t line them up here.”
“We count decimals.”
“Remember, I told you that you’re not
allowed to that that – like girls can’t go into boys
bathrooms.”
“Let me say it again: The rule is count the
decimal places.”
88
But why?
How can this make sense?
How about a context?
89
Understand Why
So? What do you see?
90
Understand Why
gallons
Total
Where are we?
91
Understand Why
4.2
gallons
$
Total
How many gallons? About how many?
92
Understand Why
4.2
gallons
$ 4.39
Total
About how much? Maximum?? Minimum??
93
Understand Why
4.2
gallons
$ 4.39
184.38
Total
Context makes ridiculous obvious, and breeds sense-making.
94
Actual cost? So how do we multiply decimals sensibly?
Solving Simple Linear
Equations
95
3x + 7 = 22
How do we solve equations:
Subtract 7
Divide by 3
Voila:
3 x + 7 = 22
-7
-7
3x
= 15
3
3
x
=
5
96
3x + 7
1. Tell me what you see: 3 x + 7
2. Suppose x = 0, 1, 2, 3…..
3. Let’s record that:
x
3x + 7
0
7
1
10
2
13
4. How do we get 22?
97
3x + 7 = 22
Where did we start? What did we do?
x
5
x3
3x
15
÷3
+7
3x + 7
22
-7
98
3x + 7 = 22
X X X IIIIIII
XXX
IIII IIII IIII IIII II
IIIII IIIII IIIII
99
Let’s look at a silly problem
Sandra is interested in buying party favors
for the friends she is inviting to her
birthday party.
100
Let’s 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.
101
Let’s 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.
102
Let’s 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.
103
Let’s 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?
104
So?
Your turn. How much?
How did you get your answer?
105
106
107
108
109
110
111
112
113
Pulling it all together
Or
Escaping the deadliness and
uselessness of worksheets
114
You choose:
3+4=
10 - 3 =
Vs.
2 x 4 = etc.
SALE
Pencils 3¢
Pens 4¢
Limit of 2 of each!
115
OOPS – Wrong store
SALE
Pencils 3¢
Pens 4¢
Erasers 5¢
Limit of 3 of each!
SO?
116
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?
117
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.
118
Deep dark secrets
• 7 x 8, 5 6 7 8
• 9 x 6, 54 56 54 since 5+4=9
• 8 + 9 …… 18 – 1 no, 16 + 1
• 63 ÷ 7 =
7 x ___ = 63
119
You choose:
85
- 47
vs.
I’ve got $85. You’ve got $47.
SO?
120
You choose:
1.59 ) 10
vs.
You have $10. Big Macs cost $1.59
SO?
121
You choose….
• The one right way to get the one right answer
that no one cares about and isn’t 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.
122
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.
123
Which class do YOU
want to be in?
124
Strategy #5
Embed the mathematics in
contexts;
Present the mathematics as
problem situations.
125
Implementing Strategy #5
Here’s the math I need to
teach.
When and where do normal
human beings encounter this
math?
126
Last and most
powerfully:
Make “why?”
“how do you know?”
“convince me”
“explain that please”
your classroom mantras
127
To recapitulate:
1. Incorporate on-going cumulative review
2. Parallel literal to inferential to evaluative
comprehension used in reading
3. Create a language-rich classroom
4. Draw pictures/create mental images
5. Embed the math in contexts/problems
And always ask them “why?”
For copies: SLeinwand@air.org
See also: “Accessible Math” by Heinemann
128
Nex
129
Processing Questions
• What are the two most significant things
you’ve heard in this presentation?
• What is the one most troubling or confusing
thing you’ve heard in this presentation?
• What are the two next steps you would
support and work on to make necessary
changes?
130
Next Steps
People won’t do what they can’t envision,
People can’t do what they don’t understand,
People can’t do well what isn’t 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.
131
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
132
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
133
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
• What’s the data tell us sessions based on state and local assessments
• “What’s 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.
134
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, won’t work,
already tried it, kids don’t care, they won’t let us)
135
Finally – let’s 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 won’t change practice in
any tangible way, and your students won’t
learn any more math.
And this is what we call PD.
136
Prove me wrong
by
Sharing
Supporting
Taking Risks
137
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
138
Thank you!
139
Appendix Slides
140
The Basics – an incomplete list
Knowing and Using:
• +, -, x, ÷ facts
• x/ ÷ by 10, 100, 1000
• 10, 100, 1000,…., .1, .01…more/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
141
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
142
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
143
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?
What’s the pattern? What can I predict?
How likely? How reliable?
What’s the relationship?
How do you know? Why is that?
144