Solving the Math Problem Think-Pair-Share

Professional Leadership

Algebra Connections Teacher Education in Clear

Instruction and Responsive Assessment of Student

Learning of Algebra Patterns and Problem Solving

This project is a research grant funded by the

Institute of Education Sciences, U.S. Department

of Education, Award Number R305M040127 DePaul

University Center for Urban Education http//teach

er.depaul.edu/AlgebraConnections.html

The Problem Situation

- NCLBtestachievementnewmatholdmathalgebrafractionde

cimallimited - TimeunlimitedpressureNCLBthinkingvalueaddedatandab

ovedatadrivenNCTMteacherleadershiptimeNCLB - timeNCLBachievementproblems

Limited Teacher Preparation

- Mathematics
- Formative Evaluation

Limited Teacher Preparation Limited

Student Development

Source National Assessment of Educational

Progress, 2005 National Assessment Results

http//nces.ed.gov/nationsreportcard/nrc/reading_

math_2005/s0027.asp?tab_idtab2subtab_idTab_1pr

intverchart

A Comprehensive ResponseTeacher Development

- Three graduate courses in mathematics
- One course in Formative Evaluation
- Teaching coaching and collaboration
- Scaffolds for student problem-solving

ADDING VALUE THROUGH TEACHER DEVELOPMENT math

strategies scaffolds for problem solving

formative evaluation teacher collaboration

School Progress

Co-Presenters

- WHAT Barbara Radner, Ph.D., Director DePaul

University Center for Urban Education - HOW James Lynn, Project Manager New Leaders

for New Schools - Molly Reed, Teacher Leader Gray Elementary

School - NEXT STEPS Mirna M. Diaz Ortiz,

Principal Nobel School

- math thinking
- relationships proportion shape and size

estimation - which operation to use sequence strategies

value

Program Neutral

A Scaffold for Learners and Teachers

- Researchers have shown that self-explanations

during learning or problem solving are positively

correlated with learning and problem-solving

measures (p. 197). Neuman and Schwarz suggest

three broad categories of self-explanation

clarification, inference, and justification.

Clarification entails explaining the problem

space. Justification entails giving reasons

that a particular solution step was taken.

Inference entails generating new knowledge

having the general form of Ifthen. - Neuman, Y Leibowitz, L., Schwarz, B. (2000).

Patterns of verbal mediation during problem

solving A sequential analysis of explanation.

The Journal of Experimental Educational, 68(3),

197-213.

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Teacher CompetenceTeacher Commitment

?Student Progress

Fifth Grade ITBS Math Gains 2005 to 2006

ComparisonsTreatment and Limited Treatment

Groups (YEAR 2)

Eighth Grade ITBS Math Gains 2005 to 2006 (YEAR

2)Teacher Commitment 1 vs. 3 vs. 4

Sixth Grade ITBS Math Gains 2005 to 2006 (YEAR

2)Teacher Competence Gain 0 vs. 1 vs. 2

Ask the Learners How do you learn? What is

difficult? What helps you learn?

Summary of Student Attitudes and Beliefs, pre- to

post-treatment changes in frequency and character

count

Content Analysis Year 1

Summary of Motivation for Learning Math, pre- to

post-treatment changes in frequency and character

count

Content Analysis Year 1

Summary of Student Perception of Teacher

Techniques, pre- to post-treatment changes in

frequency

Content Analysis Year 1

Co-Presenters HOW James Lynn, Project

Manager New Leaders for New Schools

The Pond Border Problem

- A company specializes in installing fish ponds

for residential landscaping. A square pond, 10

feet on each side, is to be surrounded by a

ceramic-tile border. The border will be one tile

wide all around and each tile is 1 foot by 1 foot.

10 ft 10 ft 10 ft 10 ft

10 ft 10 ft 10 ft 10 ft

10 ft 10 ft 10 ft

10 ft 10 ft 10 ft

10 ft 10 ft 10 ft

- Your challenge is to figure out how many tiles

will be needed without counting the tiles

individually. - Write down as many ways as you can for doing

this, giving the specific arithmetic involved in

detail. - For each method that you find, draw a diagram

that indicates how the method works.

One Representation

This student had counted ten tiles along each edge of the pond and then added two tiles to each edge since the border extended one tile in each direction past the edge of the pond next the student multiplied by four since there were four edges finally the student subtracted 4 because four of the tiles had each been counted twice. This students arithmetic looked like that shown at the right.

Other Representations

Representing the Situations using Arithmetic

12 10 x 2 x 2 24 20 24 20 44 12 11 11 10 44 10 x 4 40 4 44 12 10 x 12 x 10 144 100 144 100 44 11 x 4 44

Generalizing the Pond Border Problem

- The company decides that it would like to have a

general formula for the case of the square pond

that gives the number of tiles needed as a

function of the size of the pond. - Using s for the length of a side of the square

pond, find a general formula for the number of

tiles needed.

s

- Find a formula for each of the different ways of

seeing the problem. - Make your formula match the arithmetic as closely

as possible.

One Generalization

Example For the case where the student had counted ten tiles along the edge, extended by one tile in each direction, and then subtracted four for the double counting, the matching formula would be

4 (s 2) 4

ARITHMETIC ? ALGEBRA

Building Bridges from Arithmetic to Algebra

12 10 x 2 x 2 24 20 24 20 44 12 11 11 10 44 10 x 4 40 4 44 12 10 x 12 x 10 144 100 144 100 44 11 x 4 44

Building Bridges from Arithmetic to Algebra

12 10 x 2 x 2 24 20 24 20 44 12 11 11 10 44 10 x 4 40 4 44 12 10 x 12 x 10 144 100 144 100 44 11 x 4 44

2(s2)2s

(s2)2(s1)s

4s4

(s2)2 - s2

4(s1)

The Pond Border Problem Extension 1

- Suppose the pond is not a square. For example,

what if the pond were 8 feet by 6 feet, as shown

in the diagram below?

Explore examples like this and then develop an

expression for the number of tiles needed for the

border of a pond that is m feet by n feet.

The Pond Border Problem Extension 2

- Consider the problem of creating a border 2 feet

wide. For example, for a pond 10 feet by 10

feet, the border would look like the diagram

below. How many tiles would be needed?

- In general, how many tiles would be need for a a

border like this for a pond that is s feet by s

feet? - And what about for a rectangular pond that is m

feet by n feet? - Generalize the problem further by considering a

border that is r feet wide all around.

Developing Algebraic Thinking

- Formal

Pre-formal

Informal

Algebraic Habits of Mind

Doing-undoing

Building rules to represent functions

Abstracting from computation

Driscoll, M. (1999). Fostering algebraic

thinking A guide for teachers grades 6-10.

Portsmouth, NH Heinemann.

Fundamental Components of Algebraic Thinking

Understanding Patterns, Relations, and Functions

Analyzing Change in Various Contexts

Exploring Linear Relationships

Using Algebraic Symbols

Burke, M., et al. (2001). Navigating through

algebra in grades 9-12. Reston, VA National

Council of Teachers of Mathematics.

Literacy in Mathematics Class

Powerful Practices in Mathematics Class

Modeling

Justifying

Generalizing

Carpenter, T. P., Lehrer, R. (1999). Teaching

and learning mathematics with understanding. In

E. Fennema T. A. Romberg (Eds.), Classrooms

that promote mathematical understanding (pp.

1932). Mahwah, NJ Erlbaum.

Toward a More Expansive View of Algebra

- Implications of our research findings include

the need to broaden teachers conceptions of what

it means for students to think algebraically so

that their focus shifts away from particular

representations (e.g., symbol use is inherently

algebraic) and towards the student thinking

behind these representations. Teachers who

understand these links will be better equipped to

facilitate student connections between

representations. - Asquith, P., Stephens, A., Grandau, L., Knuth, E.

Alibali, M.W. (2005). Investigating

middle-school teachers perceptions of algebraic

thinking. Paper presented at the American

Educational Research Association Annual Meeting,

Montreal, Canada.

Algebra Curriculum Foundational Concepts

Graphing in the x-y plane

Multi-step problem solving

ALGEBRA CONTENT

Understanding variables and patterns

Signed number operations

Exponents

Fractions, percents, and proportional reasoning

Algebra in Grades K 2

- Students begin their study of algebra in early

elementary grades by learning about the use of

pictures and symbols to represent variables. They

look at patterns and describe those patterns.

They begin to look for unknown numbers in

connection with addition and subtraction number

sentences. They model the relationships found in

real-world situations by writing number sentences

that describe those situations. At these grade

levels, the study of algebra is very much

integrated with the other content standards.

Children should be encouraged to play with

concrete materials, describing the patterns they

find in a variety of ways. - New Jersey Mathematics Curriculum Framework

Algebra in Grades 3 4

- Although the formality increases in grades 3 and

4, it is important not to lose the sense of play

and the connection to the real world that were

present in earlier grades. As much as possible,

real experiences should generate situations and

data which students attempt to generalize and

communicate using ordinary language. Students

should explain and justify their reasoning orally

to the class and in writing on assessments using

ordinary language. When introducing a more formal

method of communicating, such as the language of

algebra, it is helpful to revisit some of the

situations used in previous grades. - New Jersey Mathematics Curriculum Framework

Algebra in Grades 5 6

- It is important that students continue to have

informal algebraic experiences in grades 5 and 6.

Students have previously had the opportunity to

generalize patterns, work informally with open

sentences, and represent numerical situations

using pictures, symbols, and letters as

variables, expressions, equations, and

inequalities. At these grade levels, they will

continue to build on this foundation. - Algebraic topics at this level should be

integrated with the development of other

mathematical content to enable students to

recognize that algebra is not a separate branch

of mathematics. Students must understand that

algebra is an expansion of the arithmetic and

geometry they have already experienced and a tool

to help them describe situations and solve

problems. - New Jersey Mathematics Curriculum Framework

Algebra in Grades 7 8

- Students in grades 7 and 8 continue to explore

algebraic concepts in an informal way. By using

physical models, data, graphs, and other

mathematical representations, students learn to

generalize number patterns to model, represent,

or describe observed physical patterns,

regularities, and problems. These informal

explorations help students gain confidence in

their ability to abstract relationships from

contextual information and use a variety of

representations to describe those relationships.

Manipulatives such as algebra tiles provide

opportunities for students with different

learning styles to understand algebraic concepts

and manipulations. Graphing calculators and

computers enable students to see the behaviors of

functions and study concepts such as slope. - New Jersey Mathematics Curriculum Framework

Algebra in Grades 7 8

- Students need to continue to see algebra as a

tool which is useful in describing mathematics

and solving problems. The algebraic experiences

should develop from modeling situations where

students gather data to solve problems or explain

phenomena. It is important that all concepts are

presented within a context, preferably one

meaningful to students, rather than through

traditional symbolic exercises. Once a concept is

well-understood, the students can use traditional

problems to reinforce the algebraic manipulations

associated with the concept. - New Jersey Mathematics Curriculum Framework

HOW Molly Reed, Teacher Leader Gray

Elementary School

The Teacher Connection

At the classroom level Algebra

Projects Writing in Math Problem of the Week

(POW)

Algebra Projects

Algebra Projects

Writing in Math

Writing in Math

Problem of the Week

Problem of the Week

Problem of the Week

Problem of the Week

POW Gains1st Semester 2006-2007

Multiplying the Solution Across the

Grade-level Math Club Multi-age Grouping

Sharing ideas across the grade level

Sharing ideas across the grade level

Math Club

Multi-age Grouping

Multi-age Grouping

Multi-age Grouping

Multi-age Grouping

Multi-age Grouping

Making Math a Priority Math Night (grades

3-5) School-wide Inservices Curriculum

Backmapping

NEXT STEPS Mirna Diaz Ortiz, Principal

Nobel School

Next Steps for Your School

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