Teaching for Mathematical Sense-Making

in the Spirit of the Common CoreAlan H. Schoenfeld

University of California

Berkeley, CA, USA

Alans@Berkeley.Edu

My Agenda1. Playing with some mathematics

2. What’s happening in California & the U.S–Common Core Standards,Smarter Balanced Assessments

3. What to look for in productive mathematics classrooms

4. A few thoughts on PD

5. Q&A

Which story matches the graph?

2. Thoughts about Mathematical Sense-Making:

Where the US is heading with the Common Core Standards

and Smarter Balanced Assessments and What it

Needs to do to Get Ready for It

Not Sense-Making:

How many two-foot boards can be cut from two five-foot boards?

Kurt Reusser asks 97 1st and 2nd graders:

There are 26 sheep and 10 goats on a ship.

How old is the captain?

76 students "solve" it, using the numbers.

H. Radatz gives non-problems such as:

Alan drove the 12 miles from his house in Berkeley to the Oakland

Zoo at 7 AM. On the way he picked up 2 friends.

Sense-Making

What happens when you add two odd numbers?

7 + 9

7 + 9

7 + 9

The Challenge:To make sense of:

- The (Common Core) Standards

- High Stakes Assessment and what it’s likely to mean in (half) the U.S.

- Formative Assessment as a mechanism for making good stuff happen in our classrooms.

- What to look for in productive classrooms.

Let’s start with context.

The Common Core State Standards in Mathematics (CCSSM) now exist.

But, what really defines what the Standards mean?

In today’s high stakes context, it’s the assessments.

In California, that has meant the CST. But that’s going to change, and it’s a challenge. Here are some CST release items.

They’re skill-oriented.

But the CCSSM demand more.

What, and what can be done?

That’s the rest of the conversation.

First, the Standards: Content and Practices

Content: Getting Richer

Practices: Much deeper and richer

The Practices in CCSS-M:• Make sense of problems and persevere

in solving them.• Reason abstractly and quantitatively.• Construct and critique viable arguments• Model with mathematics• Use appropriate tools strategically• Attend to Precision• Look for and make use of structure• Look for and express regularity in

repeated reasoning.

Let’s get serious about what matters.

Of course content counts. (Doh!)

BUT, the real action is in the practices.

And…

You can’t “check the practices off” if you do them once a month, or once per unit.

(We’ll return to this later, with videos.)

Real Mathematical Substance, as in CCSSM, hadn’t been

a focus of testing in California …

but it will be.

The reason: The Smarter Balanced Assessment

Consortium (SBAC)

http://www.k12.wa.us/smarter/

(Just google SBAC)

Half the Students across the US will be taking the new tests

devised by SBAC.

(The other half will take another test that should be equivalent. Time will tell.)

Here are some of the headlines.

Four Major Claims [Dimensions for Assessment] for the SMARTER Balanced Assessment

Consortium’s assessments of theCommon Core State Standards for Mathematics

Claim #1 - Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.

Claim #2 - Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.

Claim #3 - Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.

Claim #4 - Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.

Total Score for Mathematics

Content and Procedures

Score

40%

Problem Solving Score

20%

CommunicatingReasoning

Score

20%

MathematicalModeling

Score

20%

So: A large part of the exam will be devoted to things we

haven’t tested before.

Here’s an Example.

“Hurdles Race.”

Think of the Content involved:

• Interpreting distance-time graphs in a real-world context

• Realizing “to the left” is faster

• Understanding points of intersection in that context (they’re tied at the moment)

• Interpreting the horizontal line segment

• Putting all this together in an explanation

Think of the Practices involved:

• Make sense of problems and persevere in solving them.

• Reason abstractly and quantitatively.• Construct viable arguments…• Model with mathematics…

25% Sale, Part 1

In a sale, all the prices are reduced by 25%.

Julie sees a jacket that cost $32 before the sale. How much does it cost in the sale?

25% Sale, Part 2In the second week of the sale, the prices are reduced by 25% of the previous week’s price.

In the third week of the sale, the prices are again reduced by 25% of the previous week’s price.

In the fourth week of the sale, the prices are again reduced by 25% of the previous week’s price.

Alan says that after 4 weeks of these 25% discounts, everything will be free. Is he right? Explain your answer.

Again: Core content, central practices.

Want to see more?

Check out the SBAC specs;

look at

The Mathematics Assessment Project (google the name or go to

http://map.mathshell.org/materials/)

OK, you say,

But how does a difference in tests matter?

Can’t we keep doing what we’ve been doing?

From an SVMI study of 16,420 kids taking the MARS and SAT-9:

In other words, “business as usual” won’t work.

Even improving “business as usual” won’t work – the SBAC assessments will demand not only more, but different skills.

The SBAC tests will be in use in 2014.

This raises some questions about timing for California

schools to consider.

How do we prepare kids to do well on assessments like the

Smarter Balanced Assessments?

(I thought you’d never ask!)

There are resources on the web:

-Mathematics Assessment Project

-Silicon Valley Math Initiative

-Inside Mathematics-Math Forum

web sites

And, we can do more…

By way of formative assessment.

The purpose of formative assessments is not simply to show what students “know and can do” after instruction, but to reveal their current understandings so you can help them improve.

Important Background Issues

1. Formative assessment is not summative assessment given frequently!

2. Scoring formative assessments rather than or in addition to giving feedback destroys their utility (Black & Wiliam, 1998: “inside the black box”)

3. This is HARD to do. Tools help!

A Tool:

The formative assessment lesson, or FAL:

A rich “diagnostic” situation

and

Things to do when you see the results of the diagnosis.

A Challenge:

We know that students have many graphing misconceptions, e.g., confusing a picture of a story with a graph of the story in a distance-time graph.

Here’s one way to address the challenge.

Before the lesson devoted to this topic, we give a diagnostic problem as homework:

Describe what may have happened. Is the graph realistic? Explain.

We point to typical student misconceptions and offer suggestions about how to address them…

The lesson itself begins with a diagnostic task…

Students are given the chance to annotate and explain…

Follow-up Task: Card SortThe students make posters.

Students work on converting graphs to tables:

Tables are added to the card sort…

And the class compares solutions together.

The Mathematics Assessment Project’s goals are to:

• Help students grapple with core content and practices in CCSSM, and prepare them for the rich assessments they should (and it looks like, will) experience;

• Support formative assessment; and

• Do so in “curriculum-embeddable” ways.

We’re building 20 FALs at each grade from 6 through 10.

They’re FREE, athttp://map.mathshell.org/materials

To sum things up thus far:The Common Core Standards and their instantiation in the

Smarter Balanced Assessments offer a welcome

challenge.

California’s teaching force (and the nation’s) has to rise to meet that

challenge.

3. Thoughts about what to look for in productive

mathematics classrooms

What do you want to look for in a math classroom? What

counts?

I want to return to the issue of practices as a way of

“living” mathematics.

To illustrate this, let’s look at two videos.

Video 1: TIMMS geometry

Video 2: The Border Problem (from “Connecting Mathematical Ideas” by Jo Boaler and Cathy Humphries)

Key Questions for Math Classes:• Was there honest-to-goodness math in

what students and teacher did?

• Did students engage in “productive struggle,” or was the math dumbed down to the point where they didn’t?

• Who had the opportunity to engage? A select few, or everyone?

• Who had a voice? Did students get to say things, develop ownership?

• Did instruction find out what students know, and build on it?

Was there honest-to-goodness math in what students and teacher did?

Did students engage in “productive struggle,” or was the math dumbed down to the point where they didn’t?

Who had the opportunity to engage? A select few, or everyone?

Who had a voice? Did students get to say things, develop ownership?

Did instruction find out what students know, and build on it?

Put everything together:

and you have the dimensions of a framework for assessing lesson quality.

Develop rubrics tailored to different classroom activities:

• Whole Class discussions,

• Small Group work,

• Student Presentations,

• Individual work

And you get . . .

The Teaching for Robust Understanding of Math

(TRU Math) Scheme

4. Thoughts on

Professional Development

What do you need for successful PD?

• A (theoretically grounded) vision

• Systemic Coherence

• Tools

• Mechanisms for building community and supporting teachers

Our Plans for PD• Vision: rich teaching

• Coherence: everyone experiences math lessons the way they should be taught

• Tools: Formative Assessment Lessons, and the TRU Math Scheme

• Community Building: Lesson study using FALs and TRU Math.

Whew!

You’ve made it to part 5:

Q & A.