Silicon Valley Mathematics Initiative - · PDF fileSilicon Valley Mathematics Initiative. Principal As Instructional Leader Mathematics Network Meeting. Mathematics Instruction for

Silicon Valley Mathematics

Initiative

Principal As Instructional

LeaderMathematics

Network Meeting

Mathematics Instruction for

English Language Learners

What do English

Language Learners

need?

…learn vocabulary

…learn vocabulary

…learn the mathematics register

…learn vocabulary


…participate in discourse around mathematics

What doTeachers of

English Language Learners

need?

…teach vocabulary

…teach the mathematics register

…engage students in meaningful discussions around mathematics

My premise…

Planning lessons with dual goals of developing

mathematics understanding and academic language will

improve ELLs ability to become proficient in DOING mathematics.

TODAY

subtleties of language

mathematics register vs. social register

the function of language in reasoning about mathematics

Luis Harold Asturias Méndez

[email protected]

mailto:[email protected]

mailto:[email protected]

First, a bit more about me…

Norwood Street School, Los Angeles

Lawrence Hall of ScienceUniversity of California, Berkeley

Center for Mathematics

Excellence and Equity

"Excellence in mathematics education requires equity - high

expectations and strong support for all students."NCTM, Principles and Standards for School Mathematics (2000)

www.todos-math.org

TODOS: Mathematics for ALL

http://www.todos-math.org

http://www.todos-math.org

Literacy

Mathematics

Mathematics

Literacy

Mathematics

ELDLiteracy

Mathematics

ELD

MELD

Literacy

Two foundationaldefinitions

Language

MathematicsProficiency

What is Language?

A definition…

Language

is

a

System

for

communicating

concepts

ideas

feelings

and

thoughts…

through

sounds

gestures

signs

and

marks

A definition…

Language

is

a

System

for

communicating

concepts

ideas

feelings

and

thoughts…

through

sounds

gestures

signs

and

marks

Language is a system of finite arbitrary symbols combined according to rules of grammar for the purpose of communication. Individual languages use sounds, gestures and other symbols to represent objects, concepts, emotions, ideas, and thoughts.

en.wikipedia.org/wiki/Language

In mathematics, logic and computer science, a formal language is a set of finite-length words (i.e. character strings) drawn from some finite alphabet, and the scientific theory that deals with these entities is known as formal language theory. Note that we can talk about formal language in many contexts (scientific, legal, linguistic and so on), meaning a mode of expression more careful and accurate, or more mannered than everyday speech.

en.wikipedia.org/wiki/Language

What about…

Academic Language?

You write to ask me for my opinion of X, who has applied for a position in your department. I cannot recommend him too highly nor say enough good things about him. There is no other student of mine with whom I can adequately compare him. His thesis is the sort of work you don’t expect to see nowadays and in it he has clearly demonstrated his complete capabilities. The amount of material he knows will surprise you. You will indeed be fortunate if you can get him to work for you.

The

Language

of

each

discipline

used

to

communicate

about

that

discipline.

To communicate

what about that

discipline?

concepts

ideas

feelings

and

thoughts…

through

sounds

gestures

signs

and

marks

What is Mathematical Proficiency?

National Research Council. (2001) Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.)

Educating ELLs—The Vision

native-like levels of proficiency in English;

native-like levels of proficiency in English;

maintenance and promotion of students’ native language;

everyone in school is responsible for the education of ELLs;

everyone in school is responsible for the education of ELLs;

first-rate services and full access to those services;

knowledge of more than one language and culture is an advantage.

About Language…

Language is functional.


Language varies.


Language varies.

Language learning is cultural learning.

Language acquisition is a long-term process.

Language acquisition is a long-term process.

Language acquisition occurs through meaningful use and interaction.

Language processes develop interdependently.

Language processes develop interdependently.

Native language proficiency contributes to second language acquisition.

Social and academic language proficiencies are necessary for school success.

Why pay attention to langauge?

Travieso

SOLO

SOLO

PARTNER

SOLO

PARTNER

TABLE

SOLO

PARTNER

TABLE

WHOLE GROUP

How did it feel to have to solve a problem in language different from yours?

How did it feel to have to solve a problem in language different from yours?

Did you feel you had access to the problem? …to the language? …to the mathematics?

How do you design instruction with attention to English language development; in particular academic language development?

Isn’t that just good teaching?

English learners have a triple challenge…

For ELs, the ongoing triple challenge of handling “everyday” and [Academic] English, unfamiliar contexts/cultural norms, and content, all at the same time [particularly] during an on-demand setting can be quite daunting.

Carl Lager, UC Santa Barbara

…learn vocabulary

…learn vocabulary


…learn vocabulary



vocabulary

Sometimes a mob might grift all day without turning them over, but this is unlikely except in the case of a jug mob which takes a limited number of pokes. He

who has on his person more than one billfold is something of a hazard both to himself and to the

mob, for each billfold can count as a separate offense if he should be caught. Therefore, it is safer to have cash only. Class mobs usually count the money each

time they skin the pokes. One stall commonly is responsible for all of it, and an accounting in full is

made at the end of the day. When there is a woman with the mob, she usually carries the knock up

(Maurer, 1955, p.194)

Word Map Strategy

Dual Meaning Word Strategy

Cognate Strategy

Feature Analysis Strategy

Word Map Strategy

Dual Meaning Word Strategy

The scientist needs to stir the solution after adding mold and bark. Then test to see if there is a gas… and test to see if the material sticks.


Cognate Strategy

A veces los problemas de matemática incluyen números, sistemas de medidas u otros datos innecesarios para encontrar la solución. La información que no se necesita se llama información irrelevante. En algunos problemas hay que asumir cierta iformación para poder resolverlos.

ExponenteFórmulaTabla de frecuenciaNúmero racionalEcuación linealMultiplicarPropiedadNúmero negativo

ÁreaFraccionesCalcularApproximarSoluciónCírculoEcuaciónEstimarTriángulo

Feature Analysis Strategy

What does it mean to plan lessons with dual goals in mind?

In terms of language

Prior knowledge

Access content

Extend language

Prior knowledge

The procedure is actually quite simple. First you arrange items into different groups, of course one pile may be sufficient depending on how much there is to do. If you have to go somewhere else due to lack of facilities that is the next step; otherwise you are pretty well set. It is important not to overdo things. That is it is better to do too few things at once than too many. In the short run this may not seem important but complications can easily arise. A mistake can be expensive as well. At first the whole procedure will seem complicated. Soon, however, it will become another facet of life. It is difficult to forsee any end to the necessity for this task in the immediate future, but then one never can tell. After the procedure is completed one arranges the material into different groups once again. Then they can be put into their appropriate places. Eventually they will be used once more and the whole cycle will then have to be repeated. However, that is part of life. (Bransford & Johnson, 1972)

Students are not blank slates

Access content

Extend language

Which vocabulary words should be a focus of instruction in content areas that has English language learners in mind?

What does it mean to know a word?

When should vocabulary instruction happen for English language learners?

How do you teach content vocabulary to English language learners?

What shouldTeachers do?

Pick the right words


Present them in motivating ways (texts)


Present them in motivating ways (texts)

Provide learner-friendly definitions

Teach word-learning strategies (morphology, inferring from context)

Teach word-learning strategies (morphology, inferring from context)

Motivate ‘word awareness’

Teach all-purpose academic words:

Words for thinking: hypothesize, evidence, criterion

Words for thinking: hypothesize, evidence, criterion

Words for classifying: vehicle, utensil, process

Words for communication: emphasize, affirm, negotiate

Words for communication: emphasize, affirm, negotiate

Words for expressing relationships: dominate, correspond, locate

the mathematics

register

Academic Language

Is not anyone’s primary language.

Is not anyone’s primary language.

Does not work well in everyday situations.

It is an essential part of learning each subject.

It is an essential part of learning each subject.

To learn science includes learning the language of science.

Logic, analogy, and explanatory narrative dominate the text structures.

There are often pictures, demonstrations, diagrams, realia, experiments, and revisitation.

Many features of the academic language cross linguistic boundaries.

For example, mathematical equations and graphical displays are like sub-languages shared across languages.

The diagrams of how DNA works look the same in any language.

Everyday English (EE) vs.

Academic English (AE)

AE requires several proficiencies…

phonological

lexical

grammatical

discoursal

strategic

metalinguistic

Phonological

Phonological

Sounds:

Phonological

Sounds: tens vs. tenths

Phonological

Sounds: tens vs. tenths sixty vs. sixteen

Phonological

Sounds: tens vs. tenths sixty vs. sixteen sum vs. some

Phonological

Sounds: tens vs. tenths sixty vs. sixteen sum vs. some whole vs. hole

Phonological

Sounds: tens vs. tenths sixty vs. sixteen sum vs. some whole vs. hole off vs. of

Phonological

Phonological

Sounds:

Phonological

Sounds: T: How many halves do you have?

Lexical

Lexical

Three kinds of words:

Lexical

Three kinds of words: meaning in EE only

Lexical

Three kinds of words: meaning in EE only meaning in AE only

Lexical

Three kinds of words: meaning in EE only meaning in AE only meaning in EE and a different meaning in AE

Meaning in EE only

Meaning in EE only

cat

Meaning in EE only

cat taxi

Meaning in EE only

cat taxi because

Meaning in EE only

cat taxi because climb

Meaning in EE only

cat taxi because climb shelf

Meaning in EE only


dog

Meaning in EE only


dog lamp

Meaning in EE only


dog lamp etc.

Meaning in EE only


dog lamp etc.

The difficulty these words pose is the same whether we use them in EE or AE.

Meaning in AE only

Meaning in AE only

hypotenuse

Meaning in AE only

hypotenuse parallelogram

Meaning in AE only

hypotenuse parallelogram coefficient

Meaning in AE only

hypotenuse parallelogram coefficientquadratic

Meaning in AE only

hypotenuse parallelogram coefficientquadratichabitat

Meaning in AE only


erosion

Meaning in AE only


erosionphotosynthesis.

Meaning in AE only


erosionphotosynthesis.

Students encounter these words in school and must learn their meaning from the teacher, the

textbook, or someone else who can explain them.

Specialized meaning in AE








Is zero a “number”?


Is zero a “number”? “I own a number of algebra books.”


Is zero a “number”? “I own a number of algebra books.” “I have a number of friends.”


Is zero a “number”? “I own a number of algebra books.” “I have a number of friends.” “I have visited a number of places.”



What is a line?


What is a line? EE: any line segment


What is a line? EE: any line segment AE: an infinite line



Is a straight line a curve?


Is a straight line a curve? EE: not a curve


Is a straight line a curve? EE: not a curve AE: the simplest example of a curve



“Or”


“Or” Coffee or tea?


“Or” Coffee or tea? Are you coming or going?


“Or” Coffee or tea? Are you coming or going? Was that your husband or your boyfriend?


“Or” Coffee or tea? Are you coming or going? Was that your husband or your boyfriend?Do it now or later?



“Or”


“Or” EE: “or” is exclusive


“Or” EE: “or” is exclusive AE: (logic) by convention “or” is inclusive. Thus, “A or B” is true if A or B or both.



Multiplying:


Multiplying: EE: repeating addition, make things bigger


Multiplying: EE: repeating addition, make things bigger AE: bigger, smaller, or neither



Dividing:


Dividing: EE: cut into pieces, possibly equal pieces


Dividing: EE: cut into pieces, possibly equal pieces AE: the same as multiplication (dividing by a non-zero number is multiplying by its reciprocal).

“Amoebas multiply by dividing.”

Grammatical

Grammatical

Rules/syntax of English

Grammatical

Rules/syntax of English Context is crucial.

Grammatical

Rules/syntax of English Context is crucial. Some words and the mathematical concept they represent differs depending on the context.

Grammatical

Grammatical

Rules/syntax of English

Grammatical

Rules/syntax of EnglishFor example, base and square will mean different things if we are discussing space or number.

Grammatical

Grammatical

Rules/syntax of English:

Grammatical

Rules/syntax of English: prepositions

Grammatical


The temperature fell…

Grammatical


The temperature fell… to 10 degrees

Grammatical


The temperature fell… to 10 degrees by 10 degrees

Grammatical


The temperature fell… to 10 degrees by 10 degrees from 10 degrees

Discourse

Discoursal

Discoursal

Discoursal

Knowledge of complex discourse devices used in specific academic genres

Strategic

Strategic

Strategic

Knowledge of cognitive and metacognitive strategies for reading, writing speaking and listening

Metalinguistic

Metalinguistic

Metalinguistic

Conscious knowledge of ways to correct utterances in academic speech and writing

Surround the pool

Write an expression for the number of tiles needed to surround a pool that is s feet on an edge.

s

s

Student Responses

Student Responsess + s + s + s + 4


4(s + 1)


4(s + 1)

4(s + 4)


4(s + 1)

4(s + 4)

4s + 4


4(s + 1)

4(s + 4)

4s + 4

2s + 2(s + 2)


4(s + 1)

4(s + 4)

4s + 4

2s + 2(s + 2)

8 + 4(s-1)


s

s

S

S


s

s


s

s


s

s

S + 2


s

s

S - 1


s

s

very little mathematical talk,


vague use of words,


vague use of words,

incomplete explanations

Is communication

important in mathematics?

Is communication in mathematics important for

English learners?

What does it look like in the classroom?

Why is English so hard?

I could not disagree with

you less.

Free gifts with every purchase.

I personally feel better

advanced planningnever beforepast historytired clichessworn affidavitfinal ultimatum

“the whole piece”

Up to 40% discount and more.

The soldier decided

to desert his dessert

in the desert.

The buck does funny things when

the does are present.

Upon seeing the tear

in the painting I shed a tear.

I spent last evening

evening out a pile of dirt.

Even(a side trip)

Social Register

The floor is even (smooth/liso)

The floor is even (smooth/liso)

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

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

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

The dog has an even temperament (calm/calmado)

If we divide equally…we will be even (balance/igual)

If we divide equally…we will be even (balance/igual)

Looked sick and felt even worse (comparative/aún)

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

Got even

Got even

To be even

Got even

To be even

Even out

Got even

To be even

Even out

Break even

Mathematics Register

Number: Even numbers (e.g., 2, 4, 6, etc.)

Number: Even numbers (e.g., 2, 4, 6, etc.)

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

Measurement: Exact amount (e.g., an even pound)

After a number of injections

my jaw got number.

A minute is a minute part of a

day.

There is no egg in eggplant nor ham in hamburger.

Sweetmeats are candies

while sweetbreads, which aren't

sweet, are meat.

Guinea pigs are neither from

Guinea nor are they pigs.

If a vegetarian eats vegetables, what does a

humanitarian eat?

If teachers taught, why didn't

preachers praught?

How can a slim chance and a fat chance be the

same,

while a wise man and a wise guy are

opposites?

Avoid some pitfalls:

miscommunication

What is the difference between

11 and 6?

"Eleven has two numbers,

6 has only one." "6 is curly, 11 is

straight."

Compute 11 - 6

“Take 7 from 93 as many times as you can.”

“Take 7 from 93 as many times as you can.”

"I get 86 every time."

“You can take 7 from 93 thirteen

times”

Which is the best estimate for the weight of a garbage truck?

5 pounds

8 tons

14 ounces

1,400 ounces

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

Reflect on practice

How am I providing my students opportunities to use academic language?

How am I creating the need for students to use the language?

Am I teaching my students the language tools they need to use academic language?

How do I know what language tools they need?

First, ________

Then, ________

If, ________

Then, ________

Why should we do all these?

To help them communicate

about mathematics

What about it?

To

communicate

mathematics

concepts

ideas

feelings

and

thoughts…

through

sounds

words

phrases

sentences

essays

discussions

and

justifications

…learn vocabulary

…learn vocabulary


…learn vocabulary



To End…

English Rules of Thum (sic)

Don't use no double negatives.

Make each pronoun agree with their antecedents.

Join clauses good, like a conjunction should.

About them sentence fragments.

When dangling, watch your participles.

Verbs has got to agree with their subjects.

Just between you and i, case is important.

Don't write run-on sentences when they are hard to read.

Don't use commas, which aren't necessary.

Try to not ever split infinitives.

It is important to use your apostrophe's correctly.

Proofread your writing to see if you any words out.

Corect speling is esential.

A preposition is something you never end a sentence up with.

While a transcendent vocabulary is laudable, one must be eternally careful so that the calculated objective of communication does not become ensconsed in obscurity.

Fin

Documents

Silicon Valley Mathematics Initiative - · PDF fileSilicon Valley Mathematics Initiative. Principal As Instructional Leader Mathematics Network Meeting. Mathematics Instruction for