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

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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

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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?

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• 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.

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The 5 Key Elements of Effective Mathematics
Teaching

• Classroom management
• The content
• The pedagogy
• The tools and resources
• The evidence of learning

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

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Some data. What do you see?




40 4
10 2
30 4

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Predict some additional data




40 4
10 2
30 4

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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
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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
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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
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• At this point,
• its almost anticlimactic!

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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
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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.

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• 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.

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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!!!

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Ready, Set..

• 5 (-9)

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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

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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

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Major Theme of the Day

• Multiple Representations!

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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!

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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?

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Ergo A Vision by Example

• Solve
• Reason
• Model
• Explain
• Critique
• CCSSM Math Practices
• (Construct viable arguments and critique the
  reasoning of others)

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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.

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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

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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)

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Part 1

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

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• 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.

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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.

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• 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.

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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?

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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?

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• 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..

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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
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• Questions?
• Yeah buts

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• Not convinced?

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

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• Lets see if we can be hired.

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• 10.00
• - 4.59

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

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

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

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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

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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

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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.

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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.

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

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• Ready?

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Breakfast or dessert?
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NCTM Standards
Process Standards Content Standards
Problem Solving Reasoning and Proof Communication Connections Representations Number Measurement Geometry Algebra Data
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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

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• Compare that with..

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• Simplify
• 45
• v2 v7

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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

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So..

• Its Instruction, silly

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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?

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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.

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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.

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Accordingly Some Practical, Research-Affirmed
StrategiesforRaising Student Achievement
Through Better Instruction
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My message today is simple We know what works!

• Active classes
• Questioning classes
• Thinking classes

• K-1
• Reading
• Gifted

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

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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?)

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Strategy 1

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

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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?

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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?

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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?

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Strategy 2

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

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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.

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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

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Ready, set, picture..

• three quarters

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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

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Ready, set, picture..

• 20 centimeters

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

• .

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Strategy 3

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

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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?

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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

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

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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?

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• Multiplying Whole Numbers

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Remember How

• 213
• X 4

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Understand Why

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

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Which leads to

• 4 threes
• 4 tens
• 4 two hundreds
• 213
• X 4
• 12
• 40
• 800
• 852

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• Multiplying Decimals

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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.

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Understand Why

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

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

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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

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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?

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3x 7 22

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

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3x 7 22

• X X X IIIIIII IIII
  IIII IIII IIII II
• X X X IIIII
  IIIII IIIII

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Tell me what you see.

• 73
• 63

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Tell me what you see.

• 2 1/4

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

• .

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Tell me what you see.

• f(x) x2 3x - 5

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Strategy 4

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

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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

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And next

• Look at the power of context

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My Store

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

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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?

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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
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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.

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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

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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

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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

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You choose

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

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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.

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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.

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• 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

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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?

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Which class do YOUwant to be in?
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Strategy 5

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

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Implementing Strategy 5

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

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Last and most powerfully

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

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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

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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?

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Nex
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Part 3Tying things together

• Pancakes
• Skin
• Peas

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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?

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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.

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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

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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

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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

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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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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 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

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Part 5

• Final thoughts on moving forward

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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
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Jo Boalers Work

• Typical Class
• 2.5 minutes/problem
• 24 problems/class
• Railside HS class
• 5.7 minutes/problem
• 16 problems/90 minute period

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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.

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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.

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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

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• 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.

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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

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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.
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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

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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

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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.

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 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)

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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

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• 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.

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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.

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

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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
•  

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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

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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

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• 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.