Does CMP Do It All?

Re-envisioning the work of mathematics teachers in a standards-based classroom.

Valerie L. Mills Northwest Regional Conference October 13, 2007

What is the role of a teacher?

“I used to think I was a good teacher because I could come up with interesting problems for students to work on. But now that we use CMP with lots of interesting tasks, what is my job?” “Now that we have CMP and know the materials fairly well, how can we help more students learn?”

The BIFOCAL Project

Project Directors Valerie Mills & Edward Silver Project Team Alison Castro, Charalambos Charalambous, Gerri Devine, Hala Ghousseini, Melissa Gilbert, Dana Gosen, Lawrence Clark, Kathy Morris, Jenny Sealy, Beatriz Strawhun, & Gabriel Stylianides Project Supported by: Oakland Schools The Mathematics Education Endowment Fund Michigan State University The National Science Foundation (via CPTM) University of Michigan

Beyond Implementation: Focus On Challenge and Learning BIFOCAL Goals  Understand  Assist

Opening BIFOCAL Session

 MS teachers from 5 districts all using standards-based instructional materials  Discussing a written case study titled: The Case of Catherine Evans and David Young* * Smith, M.S., Silver, E.A., Stein, M.K., Boston, M., Henningsen, M.A., and Hillen, A.F. (2005).

and functions Cases of mathematics instruction to enhance teaching Volume 2: Algebra as the study of patterns

. New York: Teachers College Press.

What issues are teachers exploring as they discuss the case study?

QuickTime™ and a decompressor are needed to see this picture.

Turn and talk with a neighbor:

 What issues are teachers exploring?

 How would you characterize these questions?

Instructional Issues

Teachers seemed to be grappling with questions about  instructional decision making, rather than  questions related to the particular task or the instructional materials Issues raised included:  How and why do you work with multiple student solution strategies in the classroom?

 How do you manage classroom conversations effectively?

Other instructional dilemmas discussed:

 Identifying and working with the trajectory of mathematical goals over many lessons;  Making instructional decisions based on mathematical goals;  Assessing student understanding during instructional activities;  Anticipating and planning for possible student solutions and/or misconceptions; and  Judging when and how to allow students to grapple with tasks, while keeping them engaged in learning.

What is the role of a teacher?

Working with student solutions

Teachers must decide “what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge”.

NCTM, 2000, p.19

Working with multiple solution paths

Melinda’s Problem

Tree Circum.

Giant Sequoia Coast Redwood White Oak 83.2 feet 79.2 feet 31.8 feet

Height

275 feet 321 feet 96 feet

Spread

107 feet 80 feet 119 feet Problem 1.3

A. Which of the following statements accurately compare the largest trees to each other?

1. The spread of the largest white oak is greater than that of the largest Coast Redwood by a ratio of about 3 to 2.

Melinda sharing her lesson with colleagues:  Melinda ask students to create posters of their work that described how they were able to make sense of the various comparison statements and how they checked their accuracy.

 While the students worked, she walked around and ranked student’s solutions according to their degree of sophistication from lowest (1) to highest (4).

What instructional choices did Melinda make in order to have a successful lesson?

Turn and talk with a neighbor:

 What do you think Melinda may have thought about or planned for as she got ready to teach this lesson?

Lesson Planning Template Adapted from Thinking Through A Lesson Protocol by Margaret Schwan Smith

1.

Identify the mathematical goals of the lesson both short term and long term. 2.

Identify the all the ways in which the task can be solved.

o o o Which of these methods do you think your students will use? What misconceptions mig ht students have ? How will you help students correct these misconceptions ? What errors might students make ? How will you help students recognize and correct their errors?

3.

Launch: How will you introduce students to the activity so as not to reduce the demands of the task? What will you hear that lets you know students understand the task? 4.

Which solution paths do you want to have shared during the class discussion in order to accomplish the goals for the lesson?

o o Which will be shared first, second, etc.? Why? In what ways will the order of the solution paths help students make connections between the strategies and mathem atical ideas?

5.

How will you orchestrate the class discussion so that students:

o o make sense of the mathemati cal ideas being shared ? make conne ctions bet ween their solution strategy and those shared ?

Lesson Planning Template Adapted from Thinking Through A Lesson Protocol by Margaret Schwan Smith

1.

2.

Identify the mathematical goals of the lesson both short term and long term. Identify the all the ways in which the task can be solved.

a) Which of these methods do you think your students will use? b) What misconceptions might students have? How will you help students correct these misconceptions? c) What errors mig ht students make? How will you help students recognize and correct their errors?

3.

Launch: How will you introduce students to the activity so as not to reduce the demands of the task? What will you hear that lets you know students understand the task? 4.

Which solution paths do you want to have shared during the class discussion in order to accomplish the goals for the lesson?

a) Which will be shared first, second, etc.? Why? b) In what ways will the order of the solution pa ths help students make connections between the strategies and mathematical ideas?

5.

a)

How will you orchestrate the class discussion so that students:

make sense of the mathematical ideas being shared? b) make connections between their solution strateg y and those shared?

Implications for Practice

 Planning for instruction requires asking and answering questions focused on:  Understanding the variety of approaches students may use to solve a problem;  Deciding which solution paths you want shared and in what order; and  Planning how to facilitate the next step in the learning trajectory for students at differing levels.

What is the role of a teacher?

Working with student solutions:

Teachers must decide “what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge”.

NCTM, 2000, p.19

What is the role of a teacher?

Classroom Discourse

“Decisions about when to let students struggle to make sense of an idea or a problem without direct teacher input, when to ask leading questions, and when to tell students something directly

are crucial

to orchestrating productive mathematical discourse in the classroom.” NCTM, 1991, p.36

The Pool Problem

This is model of a square pool surrounded by a border of colored tiles. Each square tile in the drawing measures 1 foot by 1 foot. There are 10 colored tiles on each side of the pool.

Without counting every tile, find the number of orange tiles used to make the border.

What does MS Humphreys do to support the conversation and the learning?

What does MS Humphreys do to support the conversation and the learning?

 Why does Zack’s method of (4x8) + 4 make sense with the picture?

What does MS Humphreys do to support the conversation and the learning?

Turn and talk with a neighbor:

 What do you notice about the nature of the classroom discourse in this classroom?

 What does MS Humphreys do to support the conversation and the learning?

What does MS Humphreys do to support the conversation and the learning?

 Waits until many students have hands raised to call on student.

 T: Uhm, does anyone want to comment to her, see if you agree with her? Kayla, you call on someone.

 T: Is that the same as what you said?

 T: OK, Kay wanted to comment on Zach’s method.

Classroom Discussions:

Using Math Talk To Help Students Learn by Chapin, O’Connor, and Canavan Anderson 

Wait time



Teacher Revoicing

odd number?” - “So you are saying that is an 

Asking students to restate someone else’s reasoning

- “Can you repeat what he just said in your own words?” 

Asking students to apply their own reasoning to someone else’s reasoning

- “Do you agree or disagree?” 

Prompting students for further participation

”Would someone like to add on?” --

What is the role of a teacher?

Classroom Discourse

“Decisions about when to let students struggle to make sense of an ideas or a problem without direct teacher input, when to ask leading questions, and when to tell students something directly

are crucial

to orchestrating productive mathematical discourse in the classroom.” NCTM, 1991, p.36

“Are these differences that make a difference?”

“Can we find connections among the various instructional decisions needed to support a standards based learning environment?

Connections Among Instructional Dilemmas   How do you work with multiple student solution paths in the classroom?

How do you manage classroom conversations effectively? Other instructional dilemmas:  Identifying and working with the long/short term trajectory of mathematical goals  Making instructional moves based on mathematical goals   Assessing student understanding during instructional activities Anticipating and planning for possible student solutions and/or misconceptions  Judging when and how to allow students to grapple, while keeping them engaged in learning

Prior Research on Cognitive Demand

 Walter Doyle  Quasar Project  TIMSS Video Studies  Two relevant features of this research:

1. All tasks are not created equal!

What would a student potentially learn from each task?

Low Level Tasks Convert the fraction 3/8 to a decimal and a percent. High Level Tasks Shade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine each of the following: (a) the percent of area that is shaded, (b) the decimal part of the area that is shaded, and (c) the fractional part of the area that is shaded.

Research: Quasar findings around task selection

Memorization

The task requires the recall of previously learned information. No understanding is required.

Procedures With Connections Concept

The task provides a procedure for finding the solution but it connects the procedure to meaning.

Procedures Without Connections to Concept

The task requires an established procedure for finding the solution. There is no connection to meaning.

Doing Mathematics

There is no predictable pathway suggested by the task and it requires complex thinking.

2. Tasks are important, but teachers also matter!

What is the role of the teacher?

Tasks Impact Student Learning Tasks as they appear in curricular materials Student learning

What is the role of the teacher?

The Mathematics Tasks Framework Tasks as they appear in curricular materials Tasks as set up by teachers Tasks as enacted by teacher and students Student learning

Stein, Grover & Henningsen (1996) Smith & Stein (1998) Stein, Smith, Henningsen & Silver (2000)

Patterns of Set up, Enactment, and Student Learning Task Set Up A.

High Task Enactment High Student Learning High B.

Low Low Low C.

High Low, but… Moderate Stein & Lane, 1996

Teacher actions and reactions….

 influence the

nature and extent of student engagement

with challenging tasks, and  affect

students’ opportunities to learn

and through task engagement.

from

Instructional Decisions In A Problem Centered Learning Environment

Mathematics Maintaining Student Engagement Through Discourse

Maintaining Cognitive Demand Colors Multiple Aspects of Instruction

Mathematics Maintaining Student Engagement Through Discourse

Today’s Big Ideas

Student success in mathematics depends on more than excellent mathematics textbooks…  New ideas about preparing for an effective lesson  New ideas about facilitating effective student discourse  The Mathematics Tasks Framework Tasks as they appear in curricular materials Tasks as set up by teachers Tasks as enacted by teacher and students Student learning

Turn and talk with a neighbor:

 Share one thing you we discussed today that you can use to build your practice.

 Share one thing you would like to find out more about.

Find out more…..

 M. Stein, M. Smith, M. Henningsen, E. Silver

. Implementing Standards Based Mathematics Instruction: A Casebook for Professional Development

. Reston, VA: National Council of Teachers of Mathematics, 2000.

 M. Stein, and M. Smith. “Mathematical Tasks as a Framework for Reflection: From Research to Practice.”

Mathematics Teaching in the Middle School

3 (January 1998): 268-275.

Contact Information

Valerie L. Mills Oakland Schools [email protected]

What is the work of a teacher?

Formative Assessment

“To ensure deep, high-quality learning for all students, assessment and instruction must be integrated so that assessment becomes a routine part of the ongoing classroom activity rather than an interruption.” PSSM, 2000 “To maximize the instructional value of assessment, teachers need to move beyond a superficial "right or wrong" analysis of tasks to a focus on how students are thinking about the tasks.” PSSM, 2000