does cmp do it all? re-envisioning the work of mathematics teachers in a standards-based classroom....

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 DirectorsValerie Mills & Edward Silver

Project TeamAlison 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). Cases of mathematics instruction to enhance teaching Volume 2: Algebra as the study of patterns and functions. New York: Teachers College Press.

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

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.


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

Tree Circum. Height Spread

Giant Sequoia

83.2 feet 275 feet 107 feet

Coast Redwood


White Oak


Problem 1.3A. 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.

Working with multiple solution pathsMelinda’s Problem

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?


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

Lesson Planning TemplateAdapted from Thinking Through A Lesson Protocol by Margaret Schwan Smith

1. Iden t if y the mathemat ical goals of the le sson both short te rm and long term.

2. Iden t if y the all t he ways in which t he task can be solved. o Which of these methods do you think your students will use? o What misconceptions might students have? How will you help students correct these m isconceptions? o 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? Wh at will you hear that lets you know students understand the task?

4. Wh ich solution paths do you want to have shared during the class discussion in order to accomplish the goals for the lesson? o Which will be shared first, second, etc.? Why ? o In what ways will the order of the solution paths help students make connections between the strategies

and mathematical ideas?

5. How will you orchestrate the class discussion so that students: o make sense of the mathematical ideas being shared? o make connections between their solution strategy and those shared?

Lesson Planning TemplateAdapted from Thinking Through A Lesson Protocol by Margaret Schwan Smith

1. I dentify the mathemat ical goals of the lesson both short term and long te rm.

2. I dentify the all the ways in which the ta sk can be solved.

a) Whi ch of t hese meth ods do you th ink your st udent s will use?

b) What misconcept ions might st udent s have? How wil l you help st udent s cor r ect t hese misconcept ions?

c) What errors mig ht stu dent s make? How will you help st udent s r ecognize and corr ect th eir erro r s?

3. Launch: How will you intro duce st udents to the act ivity so as not to reduce the de mands of the task? What will you h ear that

let s you know students underst and the tas k?

4. Which solution pat hs do you want t o have shared during

the cl ass discussion in orde r t o accomplish t he goals for

the l esson? a) Which will be shared first, second, etc .? Why? b) In what ways will the orde r of the solution pa t hs help students make

connections bet ween t he st r ategies and mathe matical ideas?

5. How will you orchestr ate t he class discussion so that

students: a) make sense of th e mathe mat ical ideas being shared?

b) make connecti ons bet ween t heir solutio n strateg y and t hose 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.


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


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?


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


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?

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 Learnby Chapin, O’Connor, and Canavan Anderson

Wait time Teacher Revoicing -- “So you are saying that is an

odd number?” 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?”


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 DoyleQuasar ProjectTIMSS 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 High Leve l Tasks

Con ver t the fra ction 3/8 to a deci mal and a pe rcen t.

Shade 6 small squa res in a 4 x 10 rect ang le. Using the rec tan gle , explain how to deter mine each of the following: (a) the pe rcent of a rea that is shade d, (b) the decimal part of the area tha t is sha ded, and (c) the frac tional part of the a rea that is sha ded.

Research: Quasar findings around task selection

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

Procedures Without Connections to ConceptThe task requires an established procedure for finding the solution. There is no connection to meaning.

Procedures With Connections ConceptThe task provides a procedure for finding the solution but it connects the procedure to meaning.

Doing MathematicsThere 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 set up by teachers

Tasks as they appear in curricular materials

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

High

Low

Low, but…

Stein & Lane, 1996

Moderate

High

Low

A.

B.

C.

High

Low

High

Task Set Up Task Enactment Student Learning

Teacher actions and reactions….

influence the nature and extent of student engagement with challenging tasks, and

affect students’ opportunities to learn from and through task engagement.

Instructional Decisions In A Problem Centered Learning Environment

Assessing Student

UnderstandingUsing Multiple

Strategies

Mathematics

Anticipating Student

Thinking

Scaffolding Student

Thinking

Selecting Aspects

of Tasks To Highlight

Maintaining Student Engagement Through Discourse

Maintaining Cognitive DemandColors Multiple Aspects of Instruction

Assessing Student

UnderstandingUsing Multiple

Strategies

Mathematics

Anticipating Student

Thinking

Scaffolding Student

Thinking

Selecting Aspects

of Tasks To Highlight

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 set up by teachers

Tasks as they appear in curricular materials

Tasks as enacted by teacher and students

Student learning


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. MillsOakland 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

does cmp do it all? re-envisioning the work of mathematics teachers in a standards-based classroom....

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