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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?

40 410 230 4

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Predict some additional data

40 410 230 4

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How close were you?

40 410 230 420 3

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All the numbers – so?

45 425 315 240 410 230 420 3

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A lot more information(where are you?)

Roller Coaster 45 4Ferris Wheel 25 3Bumper Cars 15 2Rocket Ride 40 4Merry-go-Round 10 2Water Slide 30 4Fun House 20 3

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Fill in the blanks

Ride ??? ???Roller Coaster 45 4Ferris Wheel 25 3Bumper Cars 15 2Rocket Ride 40 4Merry-go-Round 10 2Water Slide 30 4Fun 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 4Ferris Wheel 25 3Bumper Cars 15 2Rocket Ride 40 4Merry-go-Round 10 2Water Slide 30 4Fun House 20 3

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The Amusement ParkThe 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)28

My Goal TodayEngage 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 CountsVirtually 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

answers50

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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