Issues in Addressing The Needs of

English Language Learners in Context of Mathematics

Yolanda De La CruzArizona State University

Presented at Claremont Graduate University

November 17, 2007

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The Big PictureWhat are you are striving to accomplish? Academic gains in mathematics English language proficiency Enrich Ells mathematics academic language in

the native language Close the Achievement Gap Increase the percentage of Ells that obtain a

High School deploma

Nearly 40,000 seniors from the Class of 2006 did not pass the exit exam.

High School Exit Exam

.Numbers of Teachers in the California Workforce

250,527272,459

305,855 306,548 307,864309,773306,940301,361283,975 292,012

0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

1996-97 1997-98 1998-99 1999-2000 2000-01 2001-02 2002-03 2003-04 2004-05 2005-06

Public School Enrollment, 1990 to 2015

-

500,000

1,000,000

1,500,000

2,000,000

2,500,000

3,000,000

3,500,000

1990-911992-931994-951996-971998-992000-012002-032004-052006-072008-092010-112012-132014-15

Number of K-12 students

K-5 enrollment 6-8 enrollment 9-12 enrollment

.

Number of Underprepared Teachers by Credential Type

17,83920,399

28,139

37,309

41,73942,42740,587

0

10,000

20,000

30,000

40,000

50,000

1999-2000 2000-01 2001-02 2002-03 2003-04 2004-05 2005-06Number of teachers without full credentials

More than one underprepared credential type or missing credential informationUniversity or district intern credential Emergency permit, pre-intern certificate, or waiver

.

.Distribution of Interns, by School-Level Percentage of Minority Students, 2005-06

Intern teachers are maldistributed– 75% of interns are assigned to high minority schools.

Only 25% of interns are assigned to low minority schools.

Persistent Inequities

7%

18%

31%

44%

Lowest minority quartile Second minority quartile

Third minority quartile Highest minority quartile

Students in the lowest performing schools are the most likely to get novice and underprepared teachers.

Underprepared First- and Second-Year Mathematics and Science Teachers,

2005-06

23

29

40

29

35

0

5

10

15

20

25

30

35

40

45

All teachers Middle schoolmathematics

High schoolmathematics

Middle schoolscience

High schoolscience

Percent of novice teachers without full credentials

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What Do These Graphs Mean?

Persistent gap in academic achievement between Caucasian students and those from culturally and linguistically diverse groups:

Many teachers are underprepared to make content comprehensible for ELs.

Few teachers trained to teach initial literacy or content-area literacy to secondary ELs.

ELs are tested in mathematics and reading under No Child Left Behind; and in 2007-08, tests in science have been added to the battery of assessments they must take.

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Sections that Promote Mathematics Excellence For English Learners

1. Learning Atmosphere & Physical Environment

2. Instructional Practices

3. Mathematics Content & Curriculum

4. Language Practices

5. Family & Community Involvement

6. Assessment of Student Learning

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Overview Mathematics is taught as a problem-solving

approach focuses on the use of oral; and written language to communicate problem-solving procedures and mathematics reasoning. Student learn computation skills in context of meaningful problem-solving applications.

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Areas that Require Work Scaffolds for entry, engagement, and

extension in mathematics and academic language

Enrichment of curriculum Native language mathematics instructional

materials Teacher fluency with mathematics, math

pedagogy, and discourse in English and language (for bilingual teachers)

Understanding student work

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ELLs and Mathematics Mathemtics is not langauge-neutral Language issues:

Medication among native language, English, and academic mathematical English

Specialized vocabulary and syntax Everyday vocabulary used differently

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Understanding the Complexity of ELLs’ Culture, Language, and Knowledge

We need to build on background knowledge

Identify cognates(Spanish-English) in mathematics

Language production patterns Develop language within mathematics Vocabulary building (enrich) Clarify the academic language

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What is Presently Being Taught Math in Primary grades consist of numbers, place

value, addition, subtraction, and multiplication with whole numbers.

Intent of instruction is to make the operations conceptually familiar, automatic, and accurate.

Math in Intermediate grades consists of multiplication, division, decimals, fractions, ratios, proportions, and percents. Some measurement and geometry.

Graphing, probability, and statistics and use of calculators are introduced.

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What is Presently Being Taught …….continued

The math curriculum develops students’ ability of understand concepts through a variety of problem-solving experiences with manipulative materials or pictorial models prior to working with abstractions.

A foundation of basic facts and skills as in addition and subtraction, through understanding a concrete model of the mathematical operation.

Computational skill is developed through practicing the procedure under a variety of circumstances.

Problem solving strategies are developed by applying factual information to realistic problems in a variety of different settings.

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Actual Class Instruction Middle school teachers spent 70% to

75% teaching skills in addition, subtraction, multiplication, and division.

Of the time is spent on skills development, half was spent developing conceptual teaching and the other half was spent on teaching problem-solving using story problems.

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1. Learning Atmosphere & Physical Environment

A caring classroom atmosphere of mutual respect and support is facilitated by the teacher who: Knows each child as an individual, Embraces languages, customs, and cultures of ELL

students, Provides culturally rich learning materials, Encourages self-expression and provides positive

recognition, Builds student confidence and esteem, Fosters an emotionally safe environment that allows

students to feel secure and to take risks.

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1. Learning Atmosphere & Physical Environment cont…..

The classroom is visually rich to support student learning Incorporates displays of student produced work,

whenever possible, Is colorful and thought stimulating, Contains pertinent, real-world information and

applications, Reinforces math-specific vocabulary and concepts, Provides color-coded learning supports when

appropriate. Room arrangement facilitates student interaction and group

work.

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2. Instructional Practices Instructional practices foster cooperation

and collaboration. Concepts are presented accurately,

logically, and in engaging ways. Multiple representations incorporate

mathematics learning levels: concrete, semi-concrete, and abstract.

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2. Instructional Practices cont.

The teacher employs student-centered instructional practices. Approaches content from a concept-oriented

constructivist method, Surrounds students with different modalities, Connects new concepts to prior learning or prior

knowledge, Encourages students to refine and reflect about their own

work and verbalize concept understanding “in their own words”,

Chooses homework to optimize individual content development,

Provides extra help and resources on an individual basis.

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2. Instructional Practices cont.

Students are frequently partnered with peer learners to enhance learning opportunities. To develop math content, To aid English language development, To insure sustained active participation in the

class, To welcome new students into an established

learning community.

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2. Instructional Practices cont.

Instructional activities are varied and support diverse learning styles and multiple intelligences, including for instance: Frequent use of models and/or manipulatives, Music as a motivator and anchor, Mind maps, poster-walks, and word walls Key vocabulary and cognates presented in different

forms, Vivid adjectives, Graphic organizers.

Graphic Organizers Description-test describes or defines information

Organizers-webs, features charts, comparison chartsEnumeration-text lists information about several related items, (e.g., events, characters, objects) and provides supporting evidence or details

Organizers-tree diagrams, branch diagrams, webs, outlines, comparison charts Comparison-contrast-text comments on similarities and differences among facts, people, events, and uses comparative adjectives and transitional markers (e.g, “on the one hand…on the other,” “both…only one”)

Organizers-Venn diagrams, comparisonsChronological or sequential-text organized in a time sequence and uses temporal markers, such as dates, prepositional phrases of time, sequence words (e.g., first, next, then)

Organizers-timelines, story summariesCause-effect-text describes cause-effect reactions, how one thing occurs as the result of another and uses causative words (e.g., so, as a result, therefore)

Organizers-flow charts, sequence chains, and cyclesProblem-solution-text presents a problem, and one or more solutions, word choice relates to options, alternatives, consequences, and results

Organizers-decision-making diagrams, semantic maps

Four Square forMathematical Story Problems

Decide on the OperationWhich do you choose and why?

ComputeAdd, subtract, multiply or divide.

Check your work.Does your answer make sense?Check it by doing the opposite operation.

Write to explain what you did and why.

Survey the question.Key words:___________________________Numbers:____________________________What are you looking for:________________

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What’s Different in Mathematics for English Learners?

Discourse structure may be vary different from students’ previous English experience.

Grammatical forms and structures in textbooks becomes increasingly complex.

All four academic language skills are required. The language of mathematics is highly

specific, misunderstandings are remarkably persistent.

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Importance of Addressing The Needs of English Learners

Most ELLs need 4-7 years to learn English before they reach average academic performance levels.

As ELLs, they are by definition not proficient.

But they are tested before they are proficient in English.

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Specialized Vocabulary English students may become confused with unique

terms such as addend and quotient, and terms with specialized meanings, such as altogether, round, table.

English learners have few cues to the meaning of words or phrases apart from the limited and often abstract context that is provided in the words and symbols in the problem statement.

There are also combinations that have special meanings, such as square root, multiplication table, and least common denominator.

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Specific words in mathematics single the

use of certain mathematics operations.

Addition is suggested by the following words:

add and plus

sum total combine

Subtractions is suggested by the following words:

minus less less than

difference more than

decreased by

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Differences in Math Symbols

U.S System

4,2321,258,125

4-1/2 = 4.5

Spanish-Speaking Countries

4.232

1.258.125

4,5

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Contextualize to Subject Matter Scaffolds Activity

Task: Think-Pair-Share

(bridging, schema building)

Write the following sentence stem for all to see. Ask students to take two minutes to jot down a few notes in response:

When I see or hear the word “geometry,”

I think…. Ask students to share their responses with a partner.

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3. Mathematics Content & Curriculum Glossary of mathematical terms is always

available for reference. English-Spanish Dictionary or Math Glossaryhttp://www.mathnotes.com/aw_span_gloss.htmlhttp://math2.org/math/spanish/eng-spa.htm

Content is aligned to appropriate grade-level, mathematics TEKS and professional standards.

Content is based on diagnosed student needs.

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3. Mathematics Content & Curriculum cont….

Content is systematically designed to incorporate sound learning principles. To incorporate increased complexity, To present a cohesive big-picture through chunking, To connect concepts through bridging and scaffolding, To emphasize multidisciplinary understandings, To reflect on inherent patterns by comparing and

contrasting concepts.

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3. Mathematics Content & Curriculum cont….

Curriculum is challenging, relevant,

age-appropriate, and well-paced To include contextually-based problems, To incorporate student realities, To involve interactive problem solving.

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4. Language Practices Language support is offered without supplanting

English instruction. Support is aligned with student’s diagnosed

language needs. Language used is appropriate to age and grade

level and presented in a socially meaningful context.

Mathematics-specific vocabulary is explicitly and implicitly taught and reinforced through repetition.

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4. Language Practices cont…. Teachers are knowledgeable about the second

language acquisition theories and best practices. Ideally, dual language instructional support should be

offered. When dual language teachers are not available,

sheltered instruction should be utilized to provide strong language support by addressing content

through ESL.

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5. Family & Community Involvement

Schools connect to student’s family-life by embedding contextual experiences and skills in teaching and curriculum.

Projects are relevant and promote family interaction.

Opportunities are available for English-speaking higher grade-level students to mentor ELL lower grade-level students either in an in-school or after-school program, as appropriate.

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5. Family & Community Involvement cont….

Teacher engages in frequent communication with families About activities and events in which parents can

participate, About student progress.

Teacher utilizes services provided by a community liaison and is knowledgeable about community resources.

Parents are informed about the benefits of using their most cognitively advanced language at home.

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6. Assessment of Student Learning Classroom assessment is designed to foster

student success. Assessment methods allow students frequent

opportunities to demonstrate mastery in a variety of ways.

Various assessment techniques are used to measure student understandings.

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6. Assessment of Student Learning cont…. Grades are oriented to promote and

emphasize valid step-by-step logical reasoning processes.

Assessment data and results shape instructional planning.

Flexible time allotments are given to demonstrate concept mastery.

How to Teach Problem-Solving Steps Understand the question-teach students to understand

the problem through elaboration and imagery. Find the needed information-Help students use

selective attention to find needed information. Make a plan-have students identify the operation and

what the problem calls for, then choose a plan (write a number sentence, identify parts of the problem, work with a peer, make a table, make a list).

Solve a problem-Students write out the steps of the problem and solve it, using cooperation to review the steps they have taken.

Check the answer-Students use a variety of approached o verify that their answer makes sense.

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Geometry Activity California State Board of Education Geometry

Content Standards Grades Eight Through Twelve http://www.cde.ca.gov/be/st/ss/mthgeometry.asp

This activity will help students learn to identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes.

Vocabulary Flip Net

Edge

Turn

Face

Slide

Finding a Suitable DefinitionSlide 1

1. These are pentominoes.

(a) (b) (c)

2. These are not pentominoes.

3. Which of these are pentominoes?

(d)(e) (f)

(g) (h) (i)

4. How can you define a pentomino?

(j)

Finding a Suitable DefinitionActivity 1

1. Show Slide 1 and have students find a suitable definition for a pentomino.

2. Write their definition on the board or on chart paper.The formal definition means little to English learners; however, students can develop their own definitions using Slide 1 through group discussions and consensus. This helps in giving ownership of the definition to the students while disclosing only two of the twelve pentominoes.

A pentomino is a two-dimentional shape made from five congruent squares such that each square has at least one of its sides in common with another square.

A Suitable Definition

They should come up with something that includes the following information:

Finding the Twelve Pentominoes Activity 2

1. Have students work in small groups to find the twelve different pentominoes.

2. The new shapes should be made with five square tiles to test their definition of a pentomino.

3. Each whole new shape should be cut out of squared paper.

4. Each group should check to make sure they do not have duplicate shapes by flipping, turning or rotating similar shapes.

5. Each group will need a set of twelve pentominoes for the activities that follow.

The Twelve PentominoesF L I

V

T

X

NP

U Y ZW

Fold into an Open Box

Activity 3

1. Have students guess which of their pentominoes form a network for a cube without a top.

2. Fold the pentominoes to check their conjectures.

F, L, N, T, W, X, Y, and Z form such an open box.Transferring between the second

and third dimensions is important in many occupations such as architecture, drafting, and design.

This activity facilitates the student’s understanding of the process.

The Pentominoes that form an Open Box

Exploring SymmetryActivity 4

1. Have students place a Mira on all or some of the pentominoes so that reflection is the same as the behind the Mira. This can also be done by folding the shapes.

2. Ask student if all pentominoes have a line of symmetry?

3. Have them identify those that have a line of symmetry and those that do not.

4. Have the identify those that have multiple lines of symmetry.

Exploring TessellationsActivity 5

1. A tessellations is created when a shape is repeated over and over again covering a plane without overlaps.

2. Ask students if they can find the pentominoes that tessellate.

3. Have them identify those that tessellate and those that do not.

What can you do with these 12 puzzle pieces? You would be amazed!(Remember, each piece contains 5 squares, so there are 60 cubes total.)Using all 12 pieces, you can make: 1. A 6 x 10 rectangle 2. A 5 x 12 rectangle 3. A 4 x 15 rectangle 4. A 3 x 20 rectangle 5. An 8 x 8 square with 4 pieces missing in the middle 6. An 8 x 8 square with 4 pieces missing in the corners 7. An 8 x 8 square with 4 pieces missing almost anywhere 8. A 3 x 4 x 5 cube 9. A 2 x 5 x 6 cube 10. A 2 x 3 x 10 cube 11. A 2D replica of each piece, only three times larger 12. A 5 x 13 rectangle with the shape of 1 pentomino piece missing in the

middle 13. Shapes with jagged edges 14. Tessellations using a pentomino 15. Hundreds of other shapes!

http://www.xs4all.nl/~gp/PolyominoSolver/Polyomino.html

What’s Difficult in Math forEnglish Learners?

Language Dependence in Mathematics-Specialized terms and terms used in unique ways; syntactic features of word problems.

Non-Linguistic Difficulties-Students must process complex concepts and different used of otherwise familiar terms.

Cultural differences-Cultural differences in the use of decimals and comas, in fractions, or in the strategies students use to solve word problems.

Instructional implications-Special procedures are needed that enable students to test hypotheses about language use and other potential problem areas. Cooperative learning activities are necessary to give students practice in verbalizing new concepts.

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What Can We do To Help Our English Learners in Mathematics

Glossary of mathematical terms is always available for reference Content is aligned to appropriate grade-level professional

standards Content is based on diagnosed student needs Mathematics-specific vocabulary is explicitly and implicitly taught

and reinforced through repetition. Content is systematically designed to incorporate sound learning

principles. To incorporate increased complexity, To present a cohesive big-picture through chunking, To connect concepts through bridging and scaffolding, To reflect on inherent patterns by comparing and contrasting

concepts.

Websites•http://www.math.clemson.edu/~simms/java/pentominoes/

•http://www.coolmagnetman.com/pent.htm

•http://www.johnrausch.com/PuzzlingWorld/

•http://www.mathleague.com/help/geometry/geometry.htm

•http://www.math.com/homeworkhelp/Geometry.html

•http://www.coolmath4kids.com/geometrystuff.html

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Websites California High School Math Standardshttp://www.cde.ca.gov/re/pn/fd/documents/math-stnd.pdf Further Informationhttp://www.tsusmell.org AAA Math-Activites to review basic skillshttp://aaamath.com/index.html Algebra Story and Word Problemshttp://www.hawaii.edu/suremath/intro_algebra.html

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Websites cont…. A+Math-Interactive flashcardshttp://www.aplusmath.com/ Arithmetic-Interactive math lessonshttp://hometown.aol.com/iongoal/mathlessons.html Ask Dr, Math-math questions & answershttp://mathforum.org/dr.math/ Ask ERIC-math lessons in specific topicshttp://www.askeric.org/cgi-bin/lessons.cgi/Mathematics

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Websites cont…. Basket Math Interactive-in Spanish & Englishhttp://www.scienceacademy.com CanTeach-Math activities and lessonshttp://www.canteach.ca/elementary/math.html Ministerio Espanol de Educacion y Cienciahttp://descartes.cnice.mecd.es/ EdHelper.com-database of lessonshttp://www.edhelper.com/

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Websites cont….

A to Z teacher Stuff-Lesson Planshttp://www.atozteachstuff.com/Lesson

Plans/Mathatics/index.shtml Matemática Interactiva-Spanish math lessonshttp://www.eduteka.org.MI/master/interactive/ PBS teacher Source-Math lessonshttp://www.pbs.org.teachersource/math.html The Educator’s Reference desk-math lessonshttp://www.eduref.org/cgi-bin/lessons.cgi/Mathematics

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Websites cont…. Lesson plans and resources

http://www.cloudnet.com/~edrbsass/edmath.htm Math Power Points

http://www.pppst.com/index.html

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Questions?

I’m still unclear about….