Teaching Math to Students

Who Barely Speak English –

Yes We Can! by

Elmano Costa, Ed. D.

ecosta@csustan.edu

209-667-3638

California State University, Stanislaus

1 University Circle

Turlock, CA 95382

Comprehensible Input

Activate Prior Knowledge (or build it if it does not exist)

Use Concrete Objects/Manipulatives

Do Demonstrations/Provide Visual Clues

Teach Vocabulary

Simplify and Paraphrase Instructional Language (but not content/expectations)

Use Cooperative Learning

Use Graphic Organizers

The Language of Math

What is:

A table? In everyday life? In math?

A row? In a classroom? In math?

An expression? In everyday life? In math?

Homophones: Sum, some?

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MATH IS NOT A UNIVERSAL LANGUAGE - IT IS A UNIQUE LANGUAGE.

THE LANGUAGE OF INSTRUCTION CAN BE A BARRIER TO LEARNING THE CONTENT

Stages of English Development

Early Beginning Point to …

Find the …

Put the ___ next to the ___

Give the to ___

Who has the ___?

Do you have a ___?

Who wants the ____?

Beginning Yes/no questions (Is Jimmy the tallest?)

Either/or questions (Is this larger or

smaller?)

One word responses to questions (What

is three times two?)

General questions which encourage lists

(What are all the shapes you see?)

Two word responses

Intermediate Why?

How?

Tell me about … Talk about …

What do you think about …?

Describe …

How would you changes this part of the

answer?

Describe/compare …

How are these different/similar?

Early Advanced What would you recommend/suggest

we do next?

How do you think this problem will be

solved?

What would happen if …?

Which do you think …?

Krashen’s Four Quadrants

Cognitively Undemanding

Context Imbedded Context Reduced

Cognitively Demanding

Cooperative Strategies for EL

Students

WAYS TO GROUP PURPOSE

By primary language To learn new content -

able to use primary

language

Mixed EO and EL of

various languages

To practice English

when content and

vocabulary are known

Sample List of Vocabulary for

Functions Unit Conceptual Specific Process

Function

Pattern

Linearity

Nonlinear function

Rate

Slope

Change

Growth

Input

Output

Nth Term

General Term

Independent Variable

Variable

Expression

Coefficient

Constant

Graph

Perimeter

Area

Axis

Describe

Analyze

Compare

Contrast

Extend

Investigate

Explore

Interpret

Predict

Justify

Explain

Proposition 227 Five Year Report

Our findings suggest that it is not the language of instruction but rather the quality of instruction that matters most (to the achievement of English learners). (emphasis added)

» Robert Linquanti (2006) - co-author

Effects of Proposition 227 on the Education of English Learners, American Institute for Research and WestEd, 2006. www.WestEd.org/Prop227

I hear and I forget

I see and I learn

I do and I understand

- Chinese Proverb

Go teach;

and if you must, use words.

- Francis of Assisi

Gráfico

Tampa Circumfêrencia Diámetro Resultado

da Divisão

1

2

3

4

5

Metaphor for Teaching

Teaching is like walking a child from a

lighted area into a dark cave. You must

move slowly to allow the eyes time to

adjust. You must help the child feel his/her

way, making sure of his/her footing before

proceeding onwards. And you must

encourage the child to overcome his/her

fear of the unknown.

The average teacher explains.

The superior teacher demonstrates.

But the truly great teacher inspires.

- Thoreau

According to Glasser and others

WE LEARN:

10% of what we only read

20% of what we hear

30% of what we see

50% or what we both see and hear

70% of what is discussed with others

80% of what we experience personally

90% of what we teach to others

NCTM’s Principals and Standards

for School Mathematics Content Standards

1. Number and Operations

2. Algebra

3. Geometry

4. Measurement

5. Data Analysis and Probability

Process Standards

6. Problem Solving

7. Reasoning and Proof

8. Communication

9. Connections

10. Representation

California Mathematics

Framework

BALANCED MATHEMATICAL

PROGRAM

• Procedural and Computational Skills

• Conceptual Understanding

• Problem Solving

Principles of Math Instruction

Skills without conceptual understanding

are meaningless

Conceptual understanding without skills

is inefficient

Skills and conceptual understanding

without problem solving is useless.

CREDE Five Standards Teachers and Students Working Together

– Use instructional group activities in which students and teacher work together to create a product or idea

Developing Language and Literacy Skills across all Curriculum

– Apply literacy strategies and develop language competence in all subject areas

Connecting Lessons to Students' Lives

– Contextualize teaching and curriculum in students' existing experiences in home, community, and school

Engaging Students with Challenging Lessons

– Maintain challenging standards for student performance; design activities to advance understanding to more complex levels

Emphasizing Dialogue over Lectures

– Instruct through teacher-student dialogue, especially academic, goal-directed, small-group conversations (known as instructional conversations), rather than lecture

Characteristics of Effective

Instruction for EL Students Hands-on, meaningful learning tasks drawn from the

core curriculum

Prior Knowledge - frequent, systematic use

Plentiful, systematic student-student interaction and cooperation

Structured opportunities for students to use their primary language as a means of facilitating their learning and promoting positive self-esteem

Integration of language arts (listening, speaking, reading, writing)

Adapted from University of California, Equals, Lawrence Hall of Science

Characteristics of Effective

Instruction for EL Students (Cont) Rich variety of authentic and meaningful language

experiences

Frequent use of contextualization rather than simplification to facilitate student comprehension

Frequent and systematic opportunities for students to negotiate meaning

Variety of multimedia, multisensory learning experiences

Integration of multicultural perspectives throughout the core curriculum

Focus on authentic assessment (more than traditional paper-pencil assessment)

Adapted from University of California, Equals, Lawrence Hall of Science

Academic Challenge in High

Poverty Classrooms Observed 140 classrooms in mathematics, reading, writing

Compared skills oriented and meaning-oriented classrooms

Students exposed to instruction emphasizing meaning are likely to demonstrate a greater grasp of advanced skills at the end of the school year (6-7 NCEs better on test of mathematical understanding

Amount of time spent on instruction, attention to discrete basic skills, the teachers general proficiency at managing instruction and background characteristics of the teacher did not alter the findings on type of instruction and achievement

Meaning-centered instruction did not impede mastery of basic skills, and may eve facilitate it (children in classrooms focusing on meaning performed 6.1 NCEs better on measures of computational ability than students being taught skills only)

Results held for very low achieving students as well as the higher achieving students in high poverty schools.

From Knapp, M. S., Shields, P. M. & Turnbull, B. J. (1995).

Lesson Planning for EL Students Into (Planning Phase)

Through (Teaching Phase)

Beyond (Extending Phase)

Find out what

students already know

Plan experiences to

build the link - prior

knowledge to lesson

objective

Identify key

vocabulary

Identify ways to

make the lesson

comprehensible

Incorporate ELD

objective (content

based ELD)

C 1992 C Williams, F. Sanchez, R.

Walqui

Adapted by E. Costa (2002)

Connection to prior

learning (review)

Sequenced lesson delivery

Build on incremental steps

Comprehensible input -

manipulatives,

demonstration, etc.

Scaffolding

Students reconstruct the

information

Students interact with the

language (vocabulary)

Students reinforce their

learning by working in native

language cooperative

groups

Debriefing of

learning

Teacher

facilitates in getting

students to think

about their

learning

Self-evaluation

by the students

Teacher

assesses student

learning to use in

planning new

instruction

(feedback loop)

Swings in Mathematical

Instructional Practices

1900-1935 Focus on basics

1935-1938 Meaningful mathematics

1958-1970 Discovery learning

1970-1980s Back to basics

1990s Meaningful mathematics

and discovery learning

Current Back to basics

Stages of English Development

and Mathematical Questioning

Early Beginning Point to É. Find the É Put the ___ next to the ___ Give the ____ to ___. Who has the ____? Do you have a ____ Who wants the ___? Who has the ___?

Beginning Yes/no questions (Is Jimmy the tallest?) Either/or questions (Is this larger or smaller?) One word responses to questions (What is three times two?) General questions which encourage lists (What are all the shapes you see?) Two word responses

Intermediate Why? How? Tell me aboutÉ Talk aboutÉ What do you think aboutÉ ? Describe É How would you change this part of the answer? Describe/compareÉ. How are these different/similar?

Early Advanced What would you recommend/suggest we do to next?? How do you think this problem will be solved? What would happen if É? Which do you thinkÉ .?

Lesson Planning for EL Students

Academic Objective: To be able to _________________________________

Into (Planning Phase)

Through (Teaching Phase)

Beyond (Extending Phase)

Teacher builds on prior knowledge

Find out what students already know

Connect prior knowledge to the lesson

Plan experiences to build the link

Identify key vocabulary

Identify ways to make lesson comprehensible

Incorporate ELD objective (content based ELD)

Copyright 1992 C. Williams, F. Sanchez, A. Walqui Adapted for math by E. Costa (2002)

Sequenced lesson delivery

Connect to prior learning

Build on incremental steps

Attention to comprehensible input Ğ manipulatives, demonstration, etc.

Scaffolding Students

reconstruct the information

Students interact with the language (vocabulary)

Through peers student reinforce instruction by conversing in their native language

Students create, plan, research, and project

Teacher facilitates in getting students to think about their learning

Self evaluation by students

Debriefing of learning

Teacher assesses student learning to use in planning new activities/lessons (feedback loop)

Bloom’s Taxonomy of

Cognitive Thinking

EVALUATION Uses given criteria to judge; appraises alternatives

SYNTHESIS Puts ideas together in a new way; asks: ÒWhy not do something in another

way?Ó

ANALYSIS Identifies overall patterns; recognizes relationships

APPLICATION Distinguishes between where can and cannot be used; tells where

information has been or could be used; applies information to specific situations

COMPREHENSION Predicts effects and consequences; interprets facts; explains in own words

KNOWLEDGE Can repeat and remember (facts, knowledge, definitions, rules, trends)

Bloom’s Taxonomy of

Cognitive Thinking EVALUATION

Uses given criteria to judge; appraises alternatives

SYNTHESIS Puts ideas together in a new way; asks. “Why not do something in another way?”

ANALYSIS Identifies overall patterns; recognizes relationships

APPLICATION Distinguishes between where can and cannot be used; tells where information has been or could

be used; applies information to specific situations

COMPREHENSION Predicts effects and consequences; interprets facts; explains in own words

KNOWLEDGE Can repeat and remember (facts, knowledge, definitions, rules, trends)