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Strengthening Teaching and Learning of K-12 Mathematics through the Use of High Leverage Instructional Practices

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Title: Strengthening Teaching and Learning of K-12 Mathematics through the Use of High Leverage Instructional Practices


1
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_at_air.org

2
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?

3
  • 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.

4
The 5 Key Elements of Effective Mathematics
Teaching
  • Classroom management
  • The content
  • The pedagogy
  • The tools and resources
  • The evidence of learning

5
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 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 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.
6
  • And what should it look like in our classrooms?

7
Some data. What do you see?




40 4
10 2
30 4

8
Predict some additional data




40 4
10 2
30 4

9
How close were you?




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

45 4
25 3
15 2
40 4
10 2
30 4
20 3
11
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
12
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
13
  • At this point,
  • its almost anticlimactic!

14
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
15
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.

16
  • 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.

17
Why do you think I started with these tasks?
  • 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!!!

18
Ready, Set..
  • 5 (-9)

19
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

20
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

21
Major Theme of the Day
  • Multiple Representations!

22
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!

23
The Ice Cream Cone
  • You may or may not remember that the formula for
    the volume of a sphere is 4/3pr3 and that the
    volume of a cone is 1/3 pr2h.
  • Consider the Ben and Jerrys 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?

24
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28
Ergo A Vision by Example
  • Solve
  • Reason
  • Model
  • Explain
  • Critique
  • CCSSM Math Practices
  • (Construct viable arguments and critique the
    reasoning of others)

29
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.

30
My content agenda
  • Part 1 Putting our work in context
  • Part 2 Its instruction, silly
  • Part 3 Tying things together
  • Part 4 The Smarter Balanced opportunities
  • Part 5 Final thoughts on moving forward

31
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)

32
Part 1
  • Putting our work in context
  • (glimpses at the what, why and how of what we do)

33
  • 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.

34
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.

35
  • 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.

36
Accuracy, Speed and Memory
  • Tell the person sitting next to you what is the
    formula for the volume of a sphere.
  • V 4/3 p r3
  • 4/3 ? r? 3? p?

37
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?

38
  • 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..

39
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
40
  • Questions?
  • Yeah buts

41
  • Not convinced?

42
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43
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44
  • Envision the last test you gave your students.
  • Compare your test with the Subway Employment Test.

45
  • Lets see if we can be hired.

46
  • 10.00
  • - 4.59

47
  • If the customers order came to 6.22 and he
    gave you 20.25, what is the change?

48
  • A customer complained that he was short changed
    by you, receiving only 13 from his 2.00 instead
    of 31. What would you do?

49
  • So
  • Four overarching contextual perspectives that
    frame our work and our challenges

50
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 its
    instruction that really matters
  • A recognition that the U.S. doesnt have all the
    answers

51
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

52
3. 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.

53
4. 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.

54
  • Yes.
  • A lot to think about.
  • But if you think everything is hunky-dory, youre
    not going to change.

55
  • Ready?

56
Breakfast or dessert?
57
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58
NCTM Standards
Process Standards Content Standards
Problem Solving Reasoning and Proof Communication Connections Representations Number Measurement Geometry Algebra Data
59
All the standards rolled up into one
  • Problem Solving What is this? Whats 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

60
  • Compare that with..

61
  • Simplify
  • 45
  • v2 v7

62
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

63
So..
  • Its Instruction, silly

64
Join me in Teachers Room Chat
  • 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 THEY THEY BLAME BLAME BLAME
  • An achievement gap or an INSTRUCTION gap?

65
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.

66
So its 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 high-performing
    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
    representationsincluding pictures and concrete
    materialsas 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.

67
Accordingly Some Practical, Research-Affirmed
StrategiesforRaising Student Achievement
Through Better Instruction
68
My message today is simple We know what works!
  • Active classes
  • Questioning classes
  • Thinking classes
  • K-1
  • Reading
  • Gifted

69
Our job is to extract from these places and
experiences specific strategies that can be
employed broadly and regularly.
70
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?

71
Now consider mathematicsTAKE OUT YOUR HOMEWORK.
  • - 1 19
  • - 2 37.5
  • - 3 185
  • (No why? No how do you know? No who has a
    different answer?)

72
Strategy 1
  • Adapt from what we know about reading
  • (incorporate literal, inferential, and evaluative
    comprehension to develop stronger neural
    connections)

73
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?

74
Number from 1 to 6
  • 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?

75
Good morning Boys and GirlsNumber from 1 to 5
  • 1. What is the value of tan (p/4)?
  • 2. Sketch the graph of (x-3)2 (y2)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?

76
Strategy 2
  • Incorporate on-going cumulative review into
    instruction every day.

77
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.

78
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

79
Ready, set, picture..
  • three quarters

80
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

81
Ready, set, picture..
  • 20 centimeters

82
Ready, set, picture..y sin xy 2 sin xy
sin (2x)
83
Ready, set, picture..The tangent to the
circlex2 y2 25 at (-4, -3)
  • .

84
Strategy 3
  • Draw pictures/
  • Create mental images/
  • Foster visualization

85
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?

86
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

87
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
88
  • So lets look more deeply at alternative
    approaches and multiple representations

89
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
  • Whos right? Does it matter?

90
  • Multiplying Whole Numbers

91
Remember How
  • 213
  • X 4

92
Understand Why
  • 213 x 4
  • 213 213 213 213 852
  • 200 10
    3
  • 4 800 40
    12
  • 4 ( 200 10 3) 852


93
Which leads to
  • 4 threes
  • 4 tens
  • 4 two hundreds
  • 213
  • X 4
  • 12
  • 40
  • 800
  • 852

94
  • Multiplying Decimals

95
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.

96
Understand Why

4.2
gallons
4.39
Total
How many gallons? About how many? Max/min cost?
97
Understand Why

4.2
gallons
4.39
Total
183.38
Context makes ridiculous obvious, and breeds
sense-making
98
  • Solving Simple Linear Equations
  • 3x 7 22

99
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

100
3x 7 22
  • 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?

101
3x 7 22
  • Where did we start? What did we do?
  • x 5
  • x 3 3x 15 3
  • 7 3x 7 22 - 7

102
3x 7 22
  • X X X IIIIIII IIII
    IIII IIII IIII II
  • X X X IIIII
    IIIII IIIII

103
Tell me what you see.
  • 73
  • 63

104
Tell me what you see.
  • 2 1/4

105
Tell the person sitting next to you five things
you see.
106
Tell me what you see.
  • .

107
Tell me what you see.
  • f(x) x2 3x - 5

108
Strategy 4
  • Create a language rich classroom.
  • (Vocabulary, terms, answers, explanations)

109
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

110
And next
  • Look at the power of context

111
My Store
  • SALE
  • Pencils 3
  • Pens 4
  • Erasers 5
  • Limit of 3 of each!
  • SO?

112
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?

113

Pens 7 0 1 3 3 2 1 0 8
Pencils 8 0 1 3 5 7 0
Erasers 9 10 9 8 7 6 5 4 3 3
83 83
114
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.

115
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

116
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

117
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

118
You choose
  • 1.59 ) 10
  • vs.
  • You have 10.
  • Big Macs cost 1.59
  • SO?

119
That is.
  • The one right way to get the one right answer
    that no one cares about and isnt 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.

120
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.

121
  • 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

122
You Choose
  • Solve for x 16 x .75x lt 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?

123
Which class do YOUwant to be in?
124
Strategy 5
  • Embed the mathematics in contexts
  • Present the mathematics as problem situations.

125
Implementing Strategy 5
  • Heres 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
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

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?

129
Nex
130
Part 3Tying things together
  • Pancakes
  • Skin
  • Peas

131
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?

132
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?
  • Whats radius of single 3/8 thick pancake of
    same volume?
  • Draw a graph of Peters progress.

133
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

134
Putting it all together one way
  • Good morning class.
  • Todays objective Find the surface area of right
    circular cylinders.
  • Open to page 384-5.
  • 3
  • Example 1 S.A. 2prh 2
    pr2
  • 4

  • Find the surface area.
  • Page 385 1-19 odd

135
TIMSS Video Study 2
  • Teacher presents complex, thought-provoking
    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

136
Putting it all together another way
  • Overheard in the ER as the sirens blare
  • Oh my, look at this next one. Hes 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 were in trouble.

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  • 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.

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139
Valid or Invalid?Convince us.
  • Grapple
  • Formulate
  • Givens and Goals
  • Estimate
  • Measure
  • Reason
  • Justify
  • Solve

140
Your thoughts and reactions
  1. The one thing that Ive most agreed with today is
    _________
  2. The one thing Im 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 weve talked about
    today is __________

141
Part 4
  • And how will all of this be supported by Smarter
    Balanced??
  • http//sampleitems.smarterbalanced.org/itempreview
    /sbac/index.htm

142
  • Learn Zillion www.learnzillion.com
  • Inside Mathematics www.insidemathematics.org
  • Illustrative Mathematics www.illustrativemathemat
    ics.org
  • Conceptua Math www.conceptuamath.com
  • NCTM Illuminations http//illuminations.nctm.org
  • Balanced Assessment http//balancedassessment.con
    cord.org
  • Mathalicious http//www.mathalicious.com
  • Dan Meyers three act lessons https//docs.google
    .com/spreadsheet/ccc?key0AjIqyKM9d7ZYdEhtR3BJMmdB
    WnM2YWxWYVM1UWowTEE
  • 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

143
Part 5
  • Final thoughts on moving forward

144
Jo Boalers Work
Action Typical HS Railside HS
Lecture 21 4
Questioning 15 9
Individual Work Practicing 48
Group Work 72
Student Presenation 0.2 9
145
Jo Boalers Work
  • Typical Class
  • 2.5 minutes/problem
  • 24 problems/class
  • Railside HS class
  • 5.7 minutes/problem
  • 16 problems/90 minute period

146
Jo Boalers WorkMultidimensional 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.

147
Jo Boalers WorkMultidimensional 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.

148
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

149
  • 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.

150
Jo Boalers Work
  • Other characteristics at Railside
  • Teaching students to be responsible for each
    others learning
  • High cognitive demand
  • Effort over ability
  • Clear expectations and learning practices
  • Instruction Matters!

151
  • 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

152
What we know (but too often fail to act on)
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 leader, at its
core, is to help people envision, understand,
practice, receive feedback and collaborate.
153
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

154
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

155
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.

156
 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)

157
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

158
  • 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.

159
Finally lets 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 wont change practice in any
    tangible way, and your students wont learn any
    more math.

160
  • Prove me wrong
  • by
  • Sharing
  • Supporting
  • Taking Risks

161
Next steps SharingPractice-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
  •  

162
Next steps Supporting The mindsets with which
to start
  • Were all in this together
  • People cant do what they cant envision. People
    wont do what they dont understand. Therefore,
    colleagues help each other envision and
    understand.
  • Cant 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

163
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

164
  • 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.
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