Provide ELLs with Access to Cognitively Demanding Mathematics Tasks

AMATYC 40th Annual Conference—Mathematics: Music to My EarsNovember 15, 2014

By Susie W. Håkansson, Ph.D.President, TODOS: Mathematics for ALL

Description

Challenges English language learners face

Conceptual understanding of proportional reasoning

Mathematics discourse community for English language learners

TODOS: Mathematics for ALL

Questions

Frontloading Language

Self-efficacy

Conceptual understanding

Discourse

Metacognition

Zone of proximal development

Equity and excellence

Challenges Facing Students

Access to the language of mathematics

Access to the mathematics content

Expectations

Self-efficacy

Equity and excellence

Why Is English So Hard?

The soldier decided to desert his dessert in the desert.

Upon seeing the tear in the painting, I shed a tear.

After a number of injections, my jaw got number.

A minute is a minute part of a day.

Why Is English So Hard?

There is no egg in eggplant and no ham in hamburger.

How can a slim chance and a fat change be the same, while a wise man and a wise guy are opposites?

Did you say thirty or thirteen?

Did you say two hundred or two hundredths?

Did you say fifty or sixty?

“Even”

Social registerThe floor is even (smooth/liso)

The picture is even with the window (leveled/nivelado)

Sleep provides even rhythm in our breathing (regular/uniforme)

The dog has an even temperament (calm/calmado)

If we share equally, we will be even (balance/igual)

I looked sick and felt even worse (comparative/aún)

So simple, even a child could do it (comparative/incluso)

Social registerGot even

To be even

Even out

Break even

Not even

Even-Steven

“Even”

Mathematics registerNumber: even numbers (e.g., 2, 4, 36, 58)

Number: even amounts (e.g., even amounts of flour and sugar)

Measurement: exact amount (an even pound)

Function: even function (e.g., y = 5x2 – 3; y = cos x)

“Even”

Math Word Meaning 1 (Math) Meaning 2

Solution The answer Two or more substances mixed together

Exercise Math problems to solve Physical movement to stay fit

Product The answer when you multiply two or more numbers

Something made by humans or machines

Expression Math statement with numbers and/or variables

To show emotion

Dual Meaning Words

Mathematics Language vs. English

Multiplying . . .English: repeated addition—largerMathematics: larger, smaller, or neither

Dividing . . .English: cut into piecesMathematics: same as multiplication (dividing by a non-zero number is multiplying by its reciprocal)

“Ameobas multiply by dividing”

Teachers learn to amplify and enrich—rather

than simplify—the language of the classroom,

giving students more opportunities to learn

the concepts involved.

Aída Walqui, Teacher Quality Initiative

Mathematics Content

Let’s go to mathematics content. We must always begin any lesson with the mathematics content we want our students to understand. We then ask the following questions:

What do we want our students to know and be able to do?

What mathematics language do we expect students to know before the lesson?

What mathematics language do we plan to use?

What instructional materials would support the students in learning the mathematics content?

How do we plan to assess what students learned?

Frontloading LanguageProportional ReasoningDirect proportion

Inverse proportion

Invariant

Covariation

Multiplicative thinking

Reasoning

Reasoning suggests that we use common sense, good judgment, and a thoughtful approach to problem solving, rather than plucking numbers from word problems or any problem and blindly applying rules and operations.

Proportional reasoning will refer

To the ability to scale up and/or down in appropriate situations and

To supply justifications for assertions made about relationships in situations involving simple direct proportions and inverse proportions.

Proportional Reasoning

Proportional Reasoning

In colloquial terms, proportional reasoning is

reasoning up and/or down in situations in

which there exists an invariant (constant)

relationship between two quantities that are

linked and varying together.

Proportional Reasoning

Many students who have not developed their proportional

reasoning ability have been able to compensate by using rules

in algebra, geometry, and trigonometry courses, but, in the

end, the rules are a poor substitute for understanding. We

need to get away from having students start with a/b = c/d. We

need students to understand proportional reasoning, and we

need to develop proportional reasoning in students by

knowing how to approach proportional reasoning without

using a/b=c/d.

Proportional Relationships

Proportional relationships involve some of the

simplest forms of covariation, where two

quantities are linked to each other in such a

way that when one changes, the other one

also changes in a precise way with the first

quantity. Let’s look at a couple of problems.

Laundry Detergent

If a box of detergent contains 80 cups of powder and your washing machine recommends 1¼ cups per load, approximately how many loads can you do with one box?

Laundry Detergent

1¼ cups 1 load

5 cups 4 loads

40 cups 32 loads

80 cups 64 loads

The ratio of cups to loads is 1¼:1 or 5:4

In a direct proportion, the direction of change in the related quantities is the same. We say that “y is directly proportional to x” or that “y varies as x.”

Men Working

It takes 6 men 4 days to complete a job. How long will it take 8 men to do the same job, assuming that they all work at the same pace?

Think: 6 men work 4 days

1 man doing the work of 6 men all by himself takes 24 days

8 men, dividing up the work that 1 man did, take 3 days

Men Working

In inverse proportion, the direction of change in the related quantities is not the same. We say that “y is inversely proportional to x” or that “y varies inversely as x.”

# men # days # man days

6 4 24

1 24 24

8 3 24

4 6 24

2 12 24

Multiplicative Thinking

Differentiate between additive and multiplicative situations and apply whichever transformation is appropriate.

Process of addition is associated with situations that entail adding, joining, subtracting, separating, and removing.

Process of multiplication is associated with situations that involve such processes as shrinking, enlarging, scaling, duplicating, exponentiating and fair sharing.

Couple of ProblemsIf 6 chocolates cost $.93, how much do 22 cost?

Guillermo can mow Ms. Rivas’ lawn in 45 minutes. Guillermo’s little brother Roberto takes twice as long to do the same lawn. How long will it take them if they each have a mower and they work together?

ChocolatesIf 6 chocolates cost $.93, how much do 22 cost?

6 chocolates $ .93

2 chocolates $ .31

20 chocolates $ 3.10

22 chocolates $ 3.41

The ratio of chocolates to cost is 1:15½ or 2:31

ChocolatesIf 6 chocolates cost $.93, how much do 22 cost?

6 chocolates $ .93

24 chocolates $ 3.72

2 chocolates $ .31

22 chocolates $ 3.41

The ratio of chocolates to cost is 1:15½ or 2:31

Lawn MowingGuillermo can mow Ms. Rivas’ lawn in 45 minutes. Guillermo’s little brother Roberto takes twice as long to do the same lawn. How long will it take them if they each have a mower and they work together?

Time elapsed

Amount of Lawn Guillermo mowed

Amount of Lawn Roberto mowed

Total Amount of Lawn by G & R

45 min 1 whole ½ 1 ½

15 min 1/3 1/6 3/6 = 1/2

30 min 2/3 1/3 1 whole

Lawn Mowing

Time elapsed

Amount of Lawn Guillermo mowed

Amount of Lawn Roberto mowed

Total Amount of Lawn by G & R

45 min 1 whole ½ 1 ½

90 min 2 whole 1 whole 3 whole

30 min 2/3 1/3 1 whole

Guillermo Guillermo Roberto

Instructional Strategy

What did I do or could have done to support the language and content?

Engaged you in a high cognitive demand task.Walked around and posed questions.Allowed for discourse.Allowed you to explore the why—the metacognition.

Discourse: Focus on an Effective Mathematics Community for ELLs

Four Areas of the Learning EnvironmentBeyond Setting High ExpectationsTaking Time to Listen, Observe, and LearnUnderstanding Individualistic vs. Collectivistic Value SystemsAffirming ELLs’ Cultures in the Mathematics Classroom

Beyond Setting High Expectations

All instructors set high expectations of their students.

What practices are aligned with high expectations?

What practices are not aligned with high expectations?

How do you or might you align your high expectations with your practices?

Taking Time to Listen, Observe and Learn

Listen with both a “mathematical ear” and a “personal ear.”

What spaces do you create for listening to, observing, and learning about students’ personal and mathematical experiences?

In the face of the many daily demands placed on instructors, how do you consistently take note of what students are struggling with personally and mathematically?

Understanding Individualistic vs. Collectivistic Values

There are two value systems that may distinguish different ways of being, knowing, and interacting: individualist and collectivistic.

“Every culture has both individualist and collectivistic values.” (Rothstein-Fisch and Trumbull)

Traditionally, schools in the U.S. tended to be more aligned with an individualistic value system, which represents mainstream U.S. culture.

Understanding Individualistic vs. Collectivistic Values

When ELLs are trying to make meaning of mathematics, they are more successful if they operate in a mathematics discourse community. They also need opportunities to work independently.

Instructors need to provide students with opportunities to participate both individually and in a group.

How do you become aware of the type of value system that is emphasized in the students’ home life? How does that translate to your interactions with students?

Observing ELLs’ Cultures in the Mathematics Classroom

It is critical for instructors to consider the culture of their students to find out ways in which they have engaged in past mathematics classrooms. Instructors should affirm students’ cultures and ways of doing mathematics.

Affirm students in the mathematics classroom by:

Validating Celebrating

Valuing Nurturing

Acknowledging Showing Interest

Observing ELLs’ Cultures in the Mathematics Classroom

What actions do you take to affirm students’ identity?

What do you do to build relationships with ELLs in the mathematics classroom?

Underwater Dreams

Underwater Dreams, written and directed by Mary Mazzio, and narrated by Michael Peña, is an epic story of how the sons of undocumented Mexican immigrants learned how to build an underwater robot from Home Depot parts. And defeat engineering powerhouse MIT in the process.

Here is the trailer that is on the web: http://www.underwaterdreamsfilm.com/.

Equity and Excellence

We want to provide access to ALL students (equity). We also want ALL students to make sense of rigorous, high quality, and cognitively demanding mathematics (excellence). We want them to approach the zone of proximal development, not the zone of minimal effort. We want equity AND excellence.

Equity without excellence is meaningless. Excellence without equity is unjust. We must always ask ourselves, what can we do to incorporate both?

Key References

“Teaching Fractions and Ratios for Understanding” by Susan J. Lamon

“Beyond Good Teaching: Advancing Mathematics Education for ELLs” Edited by

Sylvia Celedón-Pattichis and Nora G. Ramirez

TODOS Mission

The mission of TODOS is to advocate for equity and high quality mathematics

education for all students—in particular, Latina/o students.

TODOS Goals1. To advance educators' knowledge and ability that

lead to implementing an equitable, rigorous, and coherent mathematics program that incorporates the role language and culture play in teaching and learning mathematics.

2. To develop and support educational leaders who continue to carry out the mission of TODOS.

3. To generate and disseminate knowledge about equitable and high quality mathematics education.

TODOS Goals

4. To inform the public and influence educational policies in ways that enable students to become mathematically proficient in order to enhance college and career readiness.

5. To inform families about educational policies and learning strategies that will enable their children to become mathematically proficient.

TODOS Membership

JOIN TODOS for only $25 for a one-year membership, $70 for three years!

TODOS Website:

http://www.todos-math.org

Contact Information

Susie W. Håkansson, Ph.D.

shakans@g.ucla.edu

See me for a business card!

Join TODOS!

S140

Questions?