Thoughts about Mathematical Sense-Making Where we re heading with

Thoughts about Mathematical Sense-Making: Where we’re heading with the Common Core Standards and Smarter Balanced Assessments and How to Get Ready for It Alan H. Schoenfeld University of California Berkeley, CA, USA [email protected] Edu

Today’s Activities 1. Playing with some mathematics 2. What’s coming down the pipe – Common Core Standards, Smarter Balanced Assessments 3. Thoughts about what to look for in productive mathematics classrooms 4. Q&A at any time 5. Lunch

1. Let’s Play!

David says, If you draw in the two diagonals of a quadrilateral, you divide the quadrilateral into four equal areas. The question: Is David’s claim (a)Always right? (b)Sometimes right? (c) Never right?

Always, Sometimes, or Never True?

Always, Sometimes, or Never True?

Always, Sometimes, or Never True?

Always, Sometimes, or Never True?

Always, Sometimes, or Never True?

Always, Sometimes, or Never True?

David says, If you draw in the two diagonals of a quadrilateral, you divide the quadrilateral into four equal areas. The question: Is David’s claim (a)Always right? (b)Sometimes right? (c) Never right?

Student Work 1 If you draw in the two diagonals of a quadrilateral, you divide the quadrilateral into four equal areas. 12

Student Work 2 If you draw in the two diagonals of a quadrilateral, you divide the quadrilateral into four equal areas. 13

Some Questions. Did You Have To • 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?

More Questions. • Was there honest-to-goodness math in what we did? • Did you engage in “productive struggle, ” or did I dumb it down to where you didn’t? • Who had the opportunity to engage? A select few, or everyone? • Who had a voice? Did people get to say things, develop ownership? • Did instruction find out what you know, build on it?

2. Thoughts about Mathematical Sense-Making: Where we’re heading with the Common Core Standards and Smarter Balanced Assessments and How to Get Ready for It

Not Sense-Making: How many two-foot boards can be cut from two five-foot boards?

National Assessment of Educational Progress, 1983: An army bus holds 36 soldiers. If 1128 soldiers are being bussed to their training site, how many buses are needed? 29% 31 R 12 18% 31 23% 32 30% other

Kurt Reusser asks 97 1 st and 2 nd 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 Tilden Early Childhood Center at 3 PM. 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 California - Formative Assessment as a mechanism for making good stuff happen in our classrooms.

Let’s start with context. The Common Core State Standards in Mathematics (CCSSM) now exist.

But what do they mean? Huh? What do you mean, what do they mean? The words are there on the page…

Remember Alice and Humpty Dumpty?

Here’s WC Fields as Humpty Dumpty in the 1933 film “Alice in Wonderland”

“When I use a word, ” Humpty Dumpty said in rather a scornful tone, “it means just what I choose it to mean – neither more nor less. ” And so it is with Standards (Common Core or otherwise)

What defines the Standards? In today’s high stakes context, it’s the assessments. And in California, that’s meant the CST.

Why is this such a problem? WYTIWYG

But, the CST is going away… So things will change. How, and what might we do? That’s the rest of the conversation.

First, the Standards: Content and Practices

(Alan’s Biased Predictions) Content: Getting Richer Practices: A BIG Opportunity

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.

Remember the “processes” in the ‘ 89 NCTM Standards: Mathematics as Problem Solving Mathematics as Communicating Mathematics as Reasoning Mathematics as Connections

Remember the goals of the 1992 CA Mathematics Framework: Mathematical Power Mathematical Performance Large Assignments Complete Work

Remember NCTM’s (2000) Principles and Standards: Five Content Standards: Number & Operations Algebra Geometry Measurement Data Analysis and Probability

Remember NCTM’s (2000) Principles and Standards: And Five Process Standards: Problem Solving Reasoning and Proof Communication Connections Representation

It’s no exaggeration to say that all of these things “count” in the Common Core Standards. But will they count in California?

It’s looking like the answer is YES And the reason is … Assessment

Specifically, the Smarter Balanced Assessment Consortium (SBAC) http: //www. k 12. wa. us/smarter/ (Just google SBAC)

Here are some of the headlines.

Four Major Claims [Dimensions for Assessment] for the SMARTER Balanced Assessment Consortium’s assessments of the Common 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 Problem Solving Score 40% 20% Communicating Mathematical Reasoning Modeling Score 20%

So, OK… but, what do the tasks look like?

“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 2 In 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 The Mathematics Assessment Project (google the name or go to http: //map. mathshell. org/materials/)

OK, you say, But does a difference in tests really matter?

YES!

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

So, 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. We zipped through one. Here’s another.

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 Sort The 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, at http: //map. mathshell. org/materials and I hope they help!

To sum up this part of our conversation: The Common Core Standards and their instantiation in the Smarter Balanced Assessments offer a welcome challenge. Let’s roll up our sleeves and work together toward meeting it.

3. Thoughts about what to look for in productive mathematics classrooms

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

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? Algebra Teaching Study, UC Berkeley/MSU, Alan Schoenfeld and Bob Floden, PIs; http: //ats. berkeley. edu

“TRU Math” (Teaching for Robust Understanding of Mathematics) Essential Dimensions of Productive Mathematics Classrooms Algebra Teaching Study, UC Berkeley/MSU, Alan Schoenfeld and Bob Floden, PIs; http: //ats. berkeley. edu

Two honest questions: Do these matter? Can it help to think about them?

You’ve made it to parts 4 and 5: 4. Q & A, for as long as you can stand it; 5. Lunch!