Strengthening Teaching and
Learning of K-12 Mathematics
through the Use of High
Leverage Instructional Practices
Raleigh, North Carolina
February 11, 2013
Steve Leinwand
American Institutes for Research
sleinwand@air.org
1
Ready? Set!
There are 310 million people in
the U.S. There are 13,000
McDonalds in the U.S.
There is a point somewhere in
the lower 48 that is farther from a
McDonalds than any other point.
What state and how far?
There are 310 million people in the U.S.
There are 13,000 McDonalds in the U.S.
McDonalds claims that 12% of all
Americans eat at McDonalds each day.
VALID? INVALID? SURE? NO WAY?
Make the case that this claim is valid or
invalid.
3
The 5 Key Elements of Effective
Mathematics Teaching
•
•
•
•
•
Classroom management
The content
The pedagogy
The tools and resources
The evidence of learning
4
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 sensemaking 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 language-rich 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.
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 5
– were carefully planned.
And what should it
look like in our
classrooms?
6
Some data. What do you
see?
40
4
10
30
2
4
7
Predict some additional data
40
4
10
30
2
4
8
How close were you?
40
4
10
30
20
2
4
3
9
All the numbers – so?
45
25
15
40
10
4
3
2
4
2
30
20
4
3
10
A lot more information
(where are you?)
Roller Coaster
Ferris Wheel
Bumper Cars
45
25
15
4
3
2
Rocket Ride
Merry-go-Round
Water Slide
40
10
30
4
2
4
Fun House
20
3
11
Fill in the blanks
Ride
???
???
Roller Coaster
Ferris Wheel
Bumper Cars
Rocket Ride
Merry-go-Round
45
25
15
40
10
4
3
2
4
2
Water Slide
Fun House
30
20
4
3
12
At this point,
it’s almost anticlimactic!
13
The amusement park
Ride
Time Tickets
Roller Coaster
Ferris Wheel
Bumper Cars
Rocket Ride
Merry-go-Round
45
25
15
40
10
4
3
2
4
2
Water Slide
Fun House
30
20
4
3
14
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.
15
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.
16
Why do you think I started
with these tasks?
- 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!!!
17
Ready, Set…..
5 + (-9)
18
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”
19
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:
xxxxx
OOOOOOOOO
On number line, start at 5 and move 9 to
the left
20
Major Theme of the Day
Multiple
Representations!
21
So look at what you have:
• Visual – the displayed slides
• Aural – my voice and passion
• Hard copy – the handout
Multiple representations to
maximize the opportunity to learn!
22
The Ice Cream Cone
You may or may not remember that the formula
for the volume of a sphere is 4/3πr3 and that
the volume of a cone is 1/3 πr2h.
Consider the Ben and Jerry’s ice cream sugar
cone, 8 cm in diameter and 12 cm high,
capped with an 8 cm in diameter sphere of
deep, luscious, decadent, rich triple
chocolate ice cream.
If the ice cream melts completely, will the cone
overflow or not? How do you know?
23
24
25
26
27
Ergo: A Vision by Example
•
•
•
•
•
Solve
Reason
Model
Explain
Critique
CCSSM Math Practices
(Construct viable arguments and critique
the reasoning of others)
28
My Goal Today
Engage you in thinking about (and then being
willing and able to act on) the issues of what
we teach, how we teach, and how much they
learn by:
• validating your concerns,
• examining standard operating procedures,
• giving you some tools and ideas for making
math more accessible to our students,
• empowering you to collectively take risks.
29
My content agenda
•
•
•
•
Part 1: Putting our work in context
Part 2: It’s instruction, silly
Part 3: Tying things together
Part 4: The Smarter Balanced
opportunities
• Part 5: Final thoughts on moving forward
30
My Process Agenda
(modeling good instruction)
•
•
•
•
Inform (lots of ideas and food for thought)
Engage (focused individual and group tasks)
Stimulate (excite your sense of professionalism)
Challenge (urge you to move from words to action)
31
Part 1
Putting our work in
context
(glimpses at the what, why and
how of what we do)
32
There is no valid psychological or logical
reason to limit students of lesser
academic ability or aptitude to practice
with paper and pencil procedures.
On the contrary, there is ample evidence
to suggest that such an approach is
often counter-productive, resulting in
little improvement in procedural skills
and increasingly negative attitudes.
33
from Everybody Counts
Virtually all young children like
mathematics. They do
mathematics naturally, discovering
patterns and making conjectures
based on observation. Natural
curiosity is a powerful teacher,
especially for mathematics….
34
Unfortunately, as children become
socialized by school and society, they
begin to view mathematics as a rigid
system of externally dictated rules
governed by standards of accuracy,
speed, and memory. Their view of
mathematics shifts gradually from
enthusiasm to apprehension, from
confidence to fear. Eventually, most
students leave mathematics under
duress, convinced that only geniuses
can learn it.
35
Accuracy, Speed and Memory
Tell the person sitting next to
you what is the formula for
the volume of a sphere.
V = 4/3 π r3
4/3 ? r? 3? π?
36
Sucking intelligence out…
Late one night a shepherd was guarding
his flock of 20 sheep when all of a
sudden 4 wolves came over the hill.
Boys and girls, how old was the
shepherd?
37
“The kind of learning that will be required of
teachers has been described as
transformative (involving sweeping
changes in deeply held beliefs,
knowledge, and habits of practice) as
opposed to additive (involving the
addition of new skills to an existing
repertoire). Teachers of mathematics
cannot successfully develop their
students’ reasoning and communication
skills in ways called for by the new
reforms simply by using manipulatives in
their classrooms, by putting four students
together at a table, or by asking a few
additional open-ended questions…..
38
Rather, they must thoroughly overhaul their
thinking about what it means to know and
understand mathematics, the kinds of tasks
in which their students should be engaged,
and finally, their own role in the classroom.”
NCTM – Practice-Based Professional
Development for Teachers of Mathematics
39
Questions?
Yeah buts…
40
Not convinced?
41
42
43
Envision the last test you
gave your students.
Compare your test with the
Subway Employment
Test.
44
Let’s see if we can be hired.
45
10.00
- 4.59
46
If the customer’s order
came to $6.22 and he
gave you $20.25, what is
the change?
47
A customer complained
that he was short changed
by you, receiving only 13¢
from his $2.00 instead of
31¢. What would you do?
48
So:
Four overarching contextual
perspectives that frame our
work and our challenges
49
1. What a great time to be convening
as teachers of mathematics!
• Common Core State Standards adopted by 46
states
• Quality K-8 instructional materials
• More access to material and ideas via the web
than ever
• A president who believes in science and data
• The beginning of the end to Algebra II as the
killer
• A long overdue understanding that it’s
instruction that really matters
• A recognition that the U.S. doesn’t have all the
answers
50
2. Where we live on the food
chain
Economic security and social well-being



Innovation and productivity



Human capital and equity of opportunity



High quality education
(literacy, MATH, science)



Daily classroom math instruction
51
3. 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.
52
4. 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.
53
Yes.
A lot to think about.
But if you think everything is
hunky-dory, you’re not
going to change.
54
Ready?
55
Breakfast or dessert?
56
57
NCTM Standards
Process
Standards
• Problem Solving
• Reasoning and Proof
• Communication
• Connections
• Representations
Content
Standards
• Number
• Measurement
• Geometry
• Algebra
• Data
58
All the standards rolled up
into one:
• Problem Solving: What is this? What’s
that white thing?
• Communication: Tell the person sitting
next to you.
• Reasoning: How do you know?
• Connections: A real rip-off ad.
• Representations: A picture
59
Compare that with…..
60
Simplify:
45
√2 + √7
61
So Why Bother?
Look around. Our critics are not all wrong.
• Mountains of math anxiety
• Tons of mathematical illiteracy
• Mediocre test scores
• HS programs that barely work for half the
kids
• Gobs of remediation
• A slew of criticism
Not a pretty picture and hard to dismiss
62
So…..
It’s Instruction, silly
63
Join me in Teachers’ Room
Chat
•
•
•
•
•
•
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 THEY THEY BLAME BLAME BLAME
An achievement gap or an INSTRUCTION gap?
64
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.
65
So it’s instruction, silly!
Research, classroom observations and common sense
provide a great deal of guidance about instructional
practices that make significant differences in student
achievement. These practices can be found in highperforming classrooms and schools at all levels and all
across the country. Effective teachers make the question
“Why?” a classroom mantra to support a culture of
reasoning and justification. Teachers incorporate daily,
cumulative review of skills and concepts into instruction.
Lessons are deliberately planned and skillfully employ
alternative approaches and multiple representations—
including pictures and concrete materials—as part of
explanations and answers. Teachers rely on relevant
contexts to engage their students’ interest and use
questions to stimulate thinking and to create language-rich
mathematics classrooms.
66
Accordingly:
Some Practical,
Research-Affirmed
Strategies
for
Raising Student
Achievement Through
Better Instruction
67
My message today is
simple:
We know what works!
• K-1
• Reading
• Gifted
• Active classes
• Questioning
classes
• Thinking classes
68
Our job is to extract from
these places and
experiences specific
strategies that can be
employed broadly and
regularly.
69
But look at what else this example shows us:
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?
70
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?)
71
Strategy #1
Adapt from what we know
about reading
(incorporate literal, inferential,
and evaluative
comprehension to develop
stronger neural connections)
72
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?
73
Number from 1 to 6
1. How much bigger is 9 than 5?
2. What number is the same as 5 tens
and 7 ones?
3. What number is 10 less than 83?
4. Draw a four-sided figure and all of its
diagonals.
5. About how long is this pen in
centimeters?
74
Good morning Boys and Girls
Number from 1 to 5
1. What is the value of tan (π/4)?
2. Sketch the graph of (x-3)2 + (y+2)2 = 16
3. What are the equations of the
asymptotes of f(x) = (x-3)/(x-2)?
4. If log2x = -4, what is the value of x?
5. About how much do I weight in kg?
75
Strategy #2
Incorporate on-going
cumulative review into
instruction every day.
76
Implementing Strategy #2
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.
77
On the way to school:
•
•
•
•
•
•
A term of the day
A picture of the day
An estimate of the day
A skill of the day
A graph of the day
A word problem of the day
78
Ready, set, picture…..
“three quarters”
79
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”
80
Ready, set, picture…..
20 centimeters
81
Ready, set, picture…..
y = sin x
y = 2 sin x
y = sin (2x)
82
Ready, set, picture…..
The tangent to the circle
x2 + y2 = 25 at (-4, -3)
.
83
Strategy #3
Draw pictures/
Create mental images/
Foster visualization
84
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?
85
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
86
Or using a model:
11 kg
Rahim
Siti
29 kg
87
So let’s look more
deeply at alternative
approaches and
multiple
representations
88
Ready, set,
8+9=
17 – know it cold
10 + 7 – decompose the 9 to get to 10
18 – 1 – add 10 and adjust
16 + 1 – double plus 1
20 – 3 – round up and adjust
Who’s right? Does it matter?
89
Multiplying Whole
Numbers
90
Remember How
213
X
4
91
Understand Why
213 x 4
213 + 213 + 213 + 213 = 852
4
200
800
10 3
40 12
4 ( 200 + 10 + 3) = 852
92
Which leads to:
4 threes
4 tens
4 two hundreds
213
X
4
12
40
800
852
93
Multiplying Decimals
94
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.”
95
Understand Why
4.2
gallons
$ 4.39
Total
How many gallons? About how many? Max/min cost? 96
Understand Why
4.2
gallons
$ 4.39
183.38
Total
Context makes ridiculous obvious, and breeds sense-making
97
Solving Simple Linear
Equations
3x + 7 = 22
98
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
99
3x + 7 = 22
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?
100
3x + 7 = 22
Where did we start? What did we do?
x3
+7
x
5
3x
15
÷3
22
-7
3x + 7
101
3x + 7 = 22
X X X IIIIIII
IIII IIII IIII IIII II
XXX
IIIII IIIII IIIII
102
Tell me what you see.
73
63
103
Tell me what you see.
2 1/4
104
Tell the person sitting next to
you five things you see.
105
Tell me what you see.
.
106
Tell me what you see.
f(x) =
2
x
+ 3x - 5
107
Strategy #4
Create a language rich
classroom.
(Vocabulary, terms, answers,
explanations)
108
Implementing Strategy #4
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.
Perimeter = border
Area = covering
Cos = bucket
Cubic = S
Ellipse = locus of points with constant sum of
distances from 2 foci
Tan = sin/cos = y/x for all points on the unit circle
109
And next:
Look at the power of
context
110
My Store
SALE
Pencils 3¢
Pens 4¢
Erasers 5¢
Limit of 3 of each!
SO?
111
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?
112
Pens 7¢ 0
1 3 3 2 1 0
Pencil 8¢ 0
1 3 5 7
s
Eraser 9¢ 1 9 8 7 6 5 4 3
s
0
83
¢
8
0
3
113
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.
114
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
115
Dear sirs:
“I am in Mrs. Eaves Pre-algebra class at the Burn
Middle School. We have been studying the area
of shapes such as squares and circles. A girl in
my class suggested that we compare the square
and round pizzas sold by your store. So on April
16 Mrs. Eaves ordered one round and one
square pizza from your store for us to measure,
compare and…
116
The search for sense-making/future
leaders
“What is the reason for the difference in the price
per square inch of these two pizzas? Is it harder
to cook a round pizza? Does it take longer to
cook? Because if 3.35 cents per square inch is
acceptable for the square pizza, then the same
price per square inch should be used for the
round pizza, making the price $10.31 instead of
$10.99.
Thanks for the tasty lesson in pizza values.”
Sincerely,
Chris Collier
117
You choose:
1.59 ) 10
vs.
You have $10.
Big Macs cost $1.59
SO?
118
That is….
• The one right way to get the one right
answer that no one cares about and isn’t
even asked on the state test
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.
119
You Choose:
F = 4 (S – 65) + 10
Find F when S = 81
Vs.
First I saw the blinking lights… then the
officer informed me that:
The speeding fine here in Vermont is $4
for every mile per hour over the 65 mph
limit plus a $10 handling fee.
120
Connecticut: F = 10 ( S – 55) + 40
Maximum speeding fine: $350
• Describe the fine in words
• At what speed does it no longer
matter?
• At 80 mph how much better off would
you be in VT than in CT?
• Use a graph to show this difference
121
You Choose:
Solve for x: 16 x .75x < 1
Vs.
You ingest 16 mg of a controlled
substance at 8 a.m. Your body
metabolizes 25% of the substance
every hour. Will you pass a 4 p.m.
drug test that requires a level of less
than 1 mg? At what time could you
first pass the test?
122
Which class do YOU
want to be in?
123
Strategy #5
Embed the mathematics in
contexts;
Present the mathematics as
problem situations.
124
Implementing Strategy #5
Here’s the math I need to
teach.
When and where do normal
human beings encounter
this math?
125
Last and most
powerfully:
Make “why?”
“how do you know?”
“convince me”
“explain that please”
your classroom mantras
126
Powerful Teaching
• Provides students with better access
to the mathematics:
–
–
–
–
Context
Technology
Materials
Collaboration
• Enhances understanding of the
mathematics:
– Alternative approaches
– Multiple representations
– Effective questioning
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?”
128
Nex
129
Part 3:
Tying things together:
Pancakes
Skin
Peas
130
Peter Dowdeswell of London,
England holds the world record
for pancake consumption!
62
6” in diameter,
3/8” thick pancakes,
with butter and syrup
in 6 minutes 58.5 seconds!
SO?
131
So?
•
•
•
•
•
About how high a stack? Show and explain
Exactly how high?
How fast?
How much?
Could it be, considering the size of the
stomach?
• What’s radius of single 3/8” thick pancake of
same volume?
• Draw a graph of Peter’s progress.
132
TIMSS Video Study 1
• Teacher instructs students in a concept
or skill.
• Teacher solves example problems with
class.
• Students practice on their own while
the teacher assists.
• In other words……
133
Putting it all together one way
Good morning class.
Today’s objective: Find the surface area of right
circular cylinders.
Open to page 384-5.
3
S.A.= 2πrh + 2 πr2
Example 1:
4
Find the surface area.
Page 385 1-19 odd
134
TIMSS Video Study 2
• Teacher presents complex, thoughtprovoking problem
• Students struggle with the problem
individually and in groups
• Student present their work
• Teacher summarizes solutions and
extracts important understandings
• Students work on a similar problem
135
Putting it all together another
way
Overheard in the ER as the sirens blare:
“Oh my, look at this next one. He’s completely
burned from head to toe.”
“Not a problem, just order up 1000 square
inches of skin from the graft bank.”
You have two possible responses:
- Oh good – that will be enough.
OR
- Oh god – we’re in trouble.
136
• Which response, “oh good” or “oh
god” is more appropriate?
• Explain your thinking.
• Assuming you are the patient, how
much skin would you hope they
ordered up?
• Show how you arrived at your answer
and be prepared to defend it to the
class.
137
138
Valid or Invalid?
Convince us.
•
•
•
•
•
•
•
•
Grapple
Formulate
Givens and Goals
Estimate
Measure
Reason
Justify
Solve
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Your thoughts and reactions
1. The one thing that I’ve most agreed with
today is _________
2. The one thing I’m most aggravated about
so far today is ____________
3. The biggest question I have about doing
these things in my class is __________
4. My biggest concern about what we’ve
talked about today is __________
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Part 4
And how will all of this be
supported by Smarter
Balanced??
http://sampleitems.smarterbal
anced.org/itempreview/sbac/
index.htm
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•
•
•
•
•
•
•
•
•
•
•
•
•
Learn Zillion: www.learnzillion.com
Inside Mathematics: www.insidemathematics.org
Illustrative Mathematics: www.illustrativemathematics.org
Conceptua Math: www.conceptuamath.com
NCTM Illuminations: http://illuminations.nctm.org
Balanced Assessment: http://balancedassessment.concord.org
Mathalicious: http://www.mathalicious.com
Dan Meyer’s three act lessons:
https://docs.google.com/spreadsheet/ccc?key=0AjIqyKM9d7ZYdEhtR3BJMmdBW
nM2YWxWYVM1UWowTEE
Thinking blocks: http://www.thinkingblocks.com
Decimal squares: http://www.decimalsquares.com
Math Assessment Project: http://map.mathshell.org/materials/index.php
Yummy Math: www.yummymath.com
National Library of Virtual Manipulatives:
http://nlvm.usu.edu/en/nav/vlibrary.html
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Part 5
Final thoughts on moving
forward
143
Jo Boaler’s Work
Action
Typical HS
Railside HS
Lecture
21%
4%
Questioning
15%
9%
Individual Work
Practicing
48%
Group Work
Student
Presenation
72%
0.2%
9%
144
Jo Boaler’s Work
• Typical Class:
– 2.5 minutes/problem
– 24 problems/class
• Railside HS class:
– 5.7 minutes/problem
– 16 problems/90 minute period
145
Jo Boaler’s Work
Multidimensional classes
“In many classrooms there is one practice that is
valued above all others – that of executing
procedures (correctly and quickly). The
narrowness by which success is judged means
that some students rise to the top of classes,
gaining good grades and teacher praise, while
other sink to the bottom with most students
knowing where they are in the hierarchy created.
Such classrooms are unidimensional.”
146
Jo Boaler’s Work
Multidimensional classes
“At Railside the teachers created
multidimensional classes by valuing many
dimensions of mathematical work. This
was achieved, in part, by having more
open problems that students could solve in
different ways. The teachers valued
different methods and solution paths and
this enabled more students to contribute
ideas and feel valued.”
147
When there are many ways to be
successful, many more students are
successful.
“When we interviewed the students and
asked them “what does it take to be
successful in mathematics class?” they
offered many different practices such as:
asking good questions, rephrasing
problems, explaining well, being logical,
justifying work, considering answers…
148
When we asked students in “traditional”
classes what they needed to do in order to
be successful they talked in much more
narrow ways, usually saying that they
needed to concentrate, and pay careful
attention.”
149
Jo Boaler’s Work
Other characteristics at Railside:
• Teaching students to be responsible for
each other’s learning;
• High cognitive demand;
• Effort over ability
• Clear expectations and learning practices
Instruction Matters!
150
“Most teachers practice their craft behind
closed doors, minimally aware of what
their colleagues are doing, usually
unobserved and under supported. Far too
often, teachers’ frames of reference are
how they were taught, not how their
colleagues are teaching. Common
problems are too often solved individually
rather than by seeking cooperative and
collaborative solutions to shared
concerns.”
- Leinwand – “Sensible Mathematics”
151
What we know
(but too often fail to act on)
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 leader, at its core, is to
help people envision, understand,
practice, receive feedback and
collaborate.
152
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
153
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
154
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.
155
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)
156
Long Reach HS
Howard County (MD) recognized that there
were a significant number of 9th graders who
were not being successful in Algebra 1. To
address this problem, the county designed
Algebra Seminar for approximately 20% of
the 9th grade class in each high school.
These are students who are deemed unlikely
to be able to pass the state test if they are
enrolled in a typical one-period Algebra I
class. Algebra Seminar classes are:
157
•
•
•
•
•
•
•
•
•
Team-taught with a math and a special education teacher;
Systematically planned as a back-to-back double period;
Capped at 18 students;
Supported with a common planning period made possible by Algebra
Seminar teachers limited to four teaching periods;
Supported with focused professional development;
Using Holt Algebra I, Carnegie Algebra Tutor, and a broad array of
other print and non-print resources;
Notable for the variety of materials and resources used (including
Smart Board, graphing calculators, laptop computers, response
clickers, Versatiles, etc.);
Enriched by a wide variety of highly effectively instructional practices
(including effective questioning, asking for explanations, focusing of
different representations and multiple approaches); and
Supported by county-wide on-line lesson plans that teachers use to
initiate their planning.
158
Finally – let’s be honest:
Sadly, there is no evidence that a day 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.
159
Prove me wrong
by
Sharing
Supporting
Taking Risks
160
Next steps: Sharing
“Practice-based professional interaction”
• Professional development/interaction that is situated
in practice and built around “samples of authentic
practice.”
• Professional development/interaction that employs
materials taken from real classrooms and provide
opportunities for critique, inquiry, and investigation.
• Professional development/interaction that focuses
on the “work of teaching” and is drawn from:
- mathematical tasks
- episodes of teaching
- illuminations of students’ thinking
161
Next steps: Supporting
The mindsets with which to start
• We’re all in this together
• People can’t do what they can’t envision.
People won’t do what they don’t understand.
Therefore, colleagues help each other envision
and understand.
• Can’t know it all – need differentiation and
team-work
• Professional sharing is part of my job.
• Professional growth (admitting we need to
grow) is a core aspect of being a professional
162
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
163
Thank you.
Now go forth and start shifting YOUR
school culture toward greater
collegial interaction and collective
growth that results in better
instruction and even higher levels of
student achievement.
164