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Strengthening Teaching and Learning of K-12 Mathematics

through the Use of High Leverage Instructional Practices

Raleigh, North CarolinaFebruary 11, 2013

Steve LeinwandAmerican Institutes for Research

sleinwand@air.org

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.

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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 lesson’s objective was accomplished.

9. Effective teachers of mathematics demonstrate through the coherence of their instruction that their lessons – the tasks, the activities, the questions and the assessments – were carefully planned.

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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

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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,

it’s 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 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.

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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!!!

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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 Why5 + (-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!22

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The Ice Cream ConeYou 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?

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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: It’s 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 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?

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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

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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Let’s see if we can be hired.

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10.00

- 4.59

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If the customer’s 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 it’s

instruction that really matters• A recognition that the U.S. doesn’t have all the

answers 50

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

2. Where we live on the food chain

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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.

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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.

Yes.

A lot to think about.

But if you think everything is hunky-dory, you’re not

going to change.54

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? 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

Compare that with…..

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Simplify:

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√2 + √7

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…..

It’s Instruction, silly

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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?

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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 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.

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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 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 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

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Accordingly:

Some Practical, Research-Affirmed

Strategiesfor

Raising Student Achievement Through

Better Instruction

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My message today is simple:

We know what works!

• K-1

• Reading

• Gifted

• Active classes

• Questioning classes

• Thinking classes

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Our job is to extract from these places and

experiences specific strategies that can be employed broadly and

regularly.

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?)

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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?

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?

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Good morning Boys and GirlsNumber 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?

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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

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”

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.

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?

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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

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Or using a model:

11 kg

Rahim

Siti

29 kg

So let’s look more deeply at alternative

approaches and multiple

representations88

Ready, set,

8 + 9 =17 – know it cold 10 + 7 – decompose the 9 to get to 1018 – 1 – add 10 and adjust16 + 1 – double plus 120 – 3 – round up and adjust

Who’s 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 Why213 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 threes4 tens4 two hundreds

213 X 4 12 40 800

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Multiplying Decimals

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Remember How

4.39x 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.”

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Understand Why

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4.2 gallons

$ 4.39

Total

How many gallons? About how many? Max/min cost?

Understand Why

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4.2 gallons

$ 4.39

Total183.38

Context makes ridiculous obvious, and breeds sense-making

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

1. Tell me what you see: 3 x + 72. 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?

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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.

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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 #4Like 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 = coveringCos = bucket Cubic = SEllipse = locus of points with constant sum of

distances from 2 fociTan = sin/cos = y/x for all points on the unit circle

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?

Your turnPencils 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¢

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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 5+4=9

• 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 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.

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 difference121

You Choose:

Solve for x: 16 x .75x < 1Vs.

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

Here’s 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

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:1. Incorporate on-going cumulative review2. Parallel literal to inferential to evaluative

comprehension used in reading3. Create a language-rich classroom4. Draw pictures/create mental images5. Embed the math in contexts/problems

And always ask them “why?”

Nex

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Part 3:Tying 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?• What’s radius of single 3/8” thick pancake of

same volume?• Draw a graph of Peter’s progress.

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.

Today’s objective: Find the surface area of right circular cylinders.

Open to page 384-5.3

Example 1: S.A.= 2πrh + 2 πr2

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Find the surface area. Page 385 1-19 odd

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 problem135

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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. 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.

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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

• Solve139

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.smarterbalanced.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=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE

• 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 Boaler’s WorkAction 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 Boaler’s 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 Boaler’s 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 Boaler’s 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 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!

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“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”

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.

What we know(but too often fail to act on)

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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 1Potential 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 2Potential 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.

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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, won’t work,

already tried it, kids don’t care, they won’t let us)

Long Reach HSHoward 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 – 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.

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Prove me wrongby

Sharing

Supporting

Taking Risks

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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

 

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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

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Next steps: Taking Risks It all comes down to taking risksWhile “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.