University of Massachusetts - Amherst

ScholarWorks@UMass Amherst Master's Capstone Projects

Center for International Education

1990

A Basic Math Curriculum: Issues and Design Eugenie Ballering University of Massachusetts Amherst

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A BASIC MATH CURRICULUM: ISSUES AND DESIGN

EUGENIE BALLERING MASTER'S PROJECT

CENTER FOR INTERNATIONAL EDUCATION UNIVERSITY OF MASSACHUSETTS, AMHERST MAY 14, 1990

TABLE OF CONTENTS

FOREWORD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

THEORETICAL FOUNDATION, SOCIAL CONTEXTAND ADULT LEARNING THEORY.

5

INTRODUCTION .

5

CLARIFICATION.

5

DEFINITION OF BASIC ~.ATH

6

THEORETICAL FOUNDATIONOF BASIC MATH INSTRUCTION

6

SOCIAL CONTEXT OF BASIC MATH

9

MATHANXIETY.

. . . .

9

GENDER AND MATHEMATICS

11

MULTICULTURALCONSIDERATIONS

14

CLASSROOMENVIRONMENT.

15

ADULT LEARNING THEORY

16

THE BASIC MATH CURRICULUM

19

INTRODUCTION . . . .

19

PERSONAL ASSUMPTIONS AND ITS IMPLICATIONS FOR INSTRUCTION.

20

THE NCTM CURRICULUMAND EVALUATIONSTANDARDS

21

NCTM's ASSUMPTIONS .

22

THE PURPOSE OF BASIC MATH.

22

CONTENT OF BASIC MATH. .

23

INSTRUCTIONAL APPROACHES

23

THE USE OF WHOLE LANGUAGETECHNIQUES IN MATH INSTRUCTION

24

ROLE OF THE INSTRUCTOR.

25

.

ASSESSMENT AND EVALUATION.

26

CLASSROOMMANAGEMENT. . .

27

CONCLUSION AND RECOMMENDATIONS .....................

29

ANNOTATEDBIBLIOGRAPHY.........................

30

FOREWORD This

project

personal this

involvement

paper

I decided

literacy,

Basic

combining

my knowledge

interviews

It

was very

share

everybody,

practices

after

on one particular because

(ABE) than

it

in nonformal

it

now gets.

education

to read,

hear

at

aspect deserves

the

and see

Basic

University

and basic

my students,

friends,

colleagues

thoughts

me in doing with

this

me.

3

the

project

USA.

In

much more attention enjoyed

research, Education

courses, instructor

of Massachusetts.

so many different

literacy

of

of adult

I have

with

my work as an Adult Program

many years

in and outside

in mathematical

and encouraged their

evolving

literacy,

Education

exciting

and practices

helped

to focus

and with

the Workplace

still

in literacy

Education

at

thank

and is

mathematical

in Adult

field

evolved

mathematics. and advisors,

theories I want who

and who were willing

to

to

INTRODUCTION In most ABE programs instruction. often

Teachers

only

trained

on different Besides tutors

reading

and tutors

content

areas

being

trained

often

don't

value

to do with

tutors

don't

to teach

like

math,

to teach

mathematics

classrooms

increasingly

curriculum,

math is Math is

and writing This

and

instructional

in a holistic reading

themselves

feel

will

convince should

educators have

first

part

outline

instructional evaluation,

that

teach on

in the ABE

in their

daily

these

lives

and

problems.

a Basic

or tutor.

and writing, base

the

Math In this

and real using

life

reading

key figures

Basic

math.

and classroom

the

role

teachers

Basic

second

of the

4

teacher,

goal

The workshop paper:

and inform

a Basic

adult

the

of

to

project

I describe

discussing

how mathe-

has

learning

considerations, part

is

and tutors

Math.

in this

foundation,

management.

for

The primary

which affect

Math curriculum,

workshop

This workshop

be discussed

multicultural

and in the

and

in changing

Math so that

doing

issues

an ABE teacher

Such a training

theoretical

gender,

approaches,

for

in the ABE classroom.

which will

such as the

a Basic

learners

manipulatives

each of them can teach

I describe

management; for

are

comfortable

math anxiety,

classroom

Math.

they

to demystify

two parts

Math curriculum theory,

because

and learned is

often

techniques. in Basic

such a workshop

Some of them a focus

I propose

by using

workshop

taught

in order

way, as a form of communication

training is

with

and

that

they

in solving

problem

to be a theoretical

and tutors

think

mathematics

math problems

with

which has

background.

by any ABE teacher

integrated

area

they

However,

is meant

matics

content

Also many ABE teachers

teach

differently

competencies.

do reading

project

teachers

the

taught

can be instructed

examples.

tutor

can be taught

focussing

as much as they

because

teaching

are

and

in the ABE classroom this

and writing,

and in college,

encounter

support

which

school

Education

teachers

memorization.

to overcome

Curriculum

at

they

of

mathematics,

you have to have a mathematical and if

Basic

workplace

as a separate

and writing.

and rote

In order

skill-and

reading

computation

which

life

mathematics

the way they were taught

ask for

the main areas

in reading

in how to teach

even have math anxiety,

often

like

They see mathematics

nothing

are

who work in Adult

in how to instruct

not

writing.

and writing

and a general

content, assessment

the and

in

THEORETICALFOUNDATION, SOCIAL CONTEXTAND ADULTLEARNING THEORY INTRODUCTION Each aspect basic

of the general

math curriculum,

theoretical context issues

is affected

foundation and adult

impact

them accordingly, institution's

depending

on their

vision

and belief

mentioned

in this

such

must

so thats/he

own, the

the as the

in ABE, social

An ABE instructor

curriculum,

also

by issues

approaches

theory. on the

and therefore

and informed

of instructional

learning

and their

ABE curriculum,

identify

can act

learner's,

the

upon

society's

and

system.

CLARIFICATION The terms the

social

Adult

context,

Basic

divided

Education.

into

critical.

three

role

learning

in which

skills,

is

as adults

classes. --

or never

into

--

education

defined

as a guide

content,

teacher

gender,

interpretive,

in particular

environment. the

The adult

processes

of adult

Adult

"late

training

school

the

drop-outs

programs

developers";

the basic

Basic

who have a reading

high

skills

necessary

(Schmitt, purpose,

and evaluation

of Basic

management.

5

Education level

to support

the

back

forgotten success

Curriculum

instructional Math,

from K

literacy

who may have

1990). the

understand-

who "drop

and basic

people

a basic

considerations,

interest,

learners

and the

affect

multicultural

or values.

and

math can be

new knowledge,

describing

and classroom

of basic

the

describes

or job retraining

assessment

Math Curriculum,

foundation that

foundation,

men and women acquire

job skills

learned

Basic

positivist,

which

Most of them are the

theoretical

and the classroom

theory

as those

They are

further the

adult

defined

up to 8th grade. in"

the

the

theory,

factors

math anxiety,

attitudes,

are

the

context

of the math teacher theory

learners

paradigms:

are

learning

ing,

learning

are

The theoretical

The social

math curriculum the

adult

project

in

is

approaches, role

of the

DEFINITION OF BASIC MATH Some people Literacy".

call

Basic

Numeracy is

means of communication

public

Numeracy for public

deeper

schools

beauty

are

the

"invisible

infrequently

hidden

Numeracy

and Professional is

includes as well

Professional (Steen,

1990).

as being

information

areas In our

taught

the most,

Numeracy in which,

to analyze the numbers presented the knowledge with which to make data you need to analyze those those statistics (Frankenstein,

of Mathematics having

from

Numeracy,

Numeracy are

on Civic

of Teachers

literate

and use quantitative

be called

communicati-

and Cultural the focus

of mathematics

description,

Numeracy,

Council

mathematically

Math is a

Civic

math helps you develop the confidence by others about issues of importance, decisions about the kinds of numerical issues, the skills to research and use 1989, p 261). The National

"Mathematical

to language.

for

insights

it

Math or Numeracy in different

Practical

and in some colleges

call

Basic

Numeracy,

Leisure,

is

tool

Math is can also

We can divide

such as Practical

others

to math as literacy

of our age because view.

"Numeracy",

and a powerful

on, and representation. culture"

Math,

states

that

an appreciation able

of the value

and inclined

(NCTM, 1989).

being and

to appreciate

In this

paper

Basic

Math

isdefinedastheMathrelevanttotheABElearner. THEORETICALFOUNDATIONOF BASIC MATHINSTRUCTION A Basic literacy

math curriculum institution's

is

shaped

ideology

by a teacher's

and society's

belief

theoretical

system,

the

frameworks;

A theoretical framework can change the depth and types of questions one considers when thinking about one's practice. Theory can strengthen our energy in the struggle for humanization by focussing our attention on the interrelationships between our concrete daily teaching practice and the broader ideological and structural context (Frankenstein, 1983, p.324). Fay's

and Giroux's

ideologies

analysis

can be used

of different

to look at different

math.

Fay and Giroux

are using

Giroux

views

as a "dynamic

ideology

which meanings

and ideas

of knowledge, artifacts." social

cultural

are

political theoretical

theory.

different

produced,

to show that

every

social

refers

both

and ideologies.

Giroux

6

to the way in in forms

and cultural

theoretical

consider

in

same paradigm.

and embodied

science

and

approaches

the

practices,

the different

Fay and Giroux

frameworks

that

mediated, social

frameworks

instructional names for

concept

experiences,

And Fay describes sciences

theoretical

frameworks

theory three

calls

in

has an implied

different them the

instrumen-

tal, the

the

interactive

positivist,

interpretive

The first Fay's

category,

positivist

on, efficiency, and external class

and the reproductive

This

and technical

control.

race,

and seeing

(Frankenstein, social

the uses

1988).

Math.

and if

pass

the Graduate

need

for

Basic

learners' goals

Basic

offered,

their

neutrality.

which

and rote

the nature

and

concentrate memorization.

of math as a way of the world

are

relationship

of

math pedagogies

methods

used

between

are

often

Some instructors

rely

for

measurement

Basic

of the

is used

Math and

category,

interpretive

Giroux's

ons of knowledge,

viewing

approach

as opposed

central

meaning, problem.

This

pedagogies

which

instruction

at

on process

over

focus

the

product

to understand

point

of view of the

positivist

and problem

social

involved

interpretive

or In this

regarded

as the

mathematics individualizing

solving

primarily

in the framework

with

approach

a stress one

from the

social critique

and view math as a culturally 7

and

goals.

is

In this

world

use

gender

ideology

"humanistic"

1988).

directly

and the

view of objectivity

the

but also

construction.

math anxiety,

own pace,

which

on the human dimensi-

of content,

underlies

in Fays's

methods

their

reach

ideology.

formation,

centers

the

fit

of interaction

as a social

(Frankenstein,

actors

class

Such a

learners

and drills

to reach

to mastery

and explain

ideology

learners

on alleviating

to help

to

Instructio-

achieving

instruction

framework,

ideology

based.

instrumental

computation

knowledge

students'

tries

Interaction

and Giroux's

category

theoretical

job.

which partially

considering

the

is

to

term goals

a better

successfully

educators

offered

long

competency

on remedial

approaches to help

is

practices

heavily

seldom

(GED) to get

towards

memorization,

nonformal

The second

geared

is

of learners'

Diploma

framework

on rote

which

is because

instructional

theoretical

sometimes

it

The roles

using

concentrate

Math instruction

Equivalency

goals.

positivist

Fay's

for

equivalents

Math in the curriculum

approaches

race

subjectivity

of math in understanding grade

of predicti-

seen as objective

of the

traditional

or

integration.

learners,

their

stripped

into

knowledge,

is

proficiency,

is a direct

In many ABE programs

nal

Knowledge

them

model.

on principles

instructional

School

There

focus

underlies

any insight

Fay calls

theoretical

and celebrated

cognition-mechanical omits

while

of instrumental

are

and gender

in remedial

perspective

of Basic

Facts

perspective

expression

thinking

category

to the knower.

on lower-order This

Giroux's

framework

ideological

has its

and the critical

theoretical

formations,

ideology

process. the organized

system

of skills

with

shifts

the

of attention

focus

critiques

universal

developed

in this

math's

field

interaction

idealogy

is

development

it

notions

among socioeconomic exaggerated larger

1988).

social

(Giroux,

that

short-

and moral

and power differences and gender

groups

collapse

into

an

towards

and cultural

category

those

disintegration•

of math anxiety.

The

of these

programs

for

category,

of the

optimal

framework

are

geared

educators

results.

is

towards

elimina-

to be learner

Such goals

are well

suited

and ideology.

the critical

to social

with based

approaches

role

learners,

demand competency

elimination

and the

because

Math goals,

and the

theoretical

committed

this

term Basic

and to strive

The third

Though

as a blindness

learning,

math anxiety,

to this

ethnic,

economic

perspective

dissonance

"power and freedom

promote

would fit

and long

instructional centered

process.

have

p.338).

individualized ting

and methods

conflict

as well

It and

theoretical

cognitive

and racial,

of human will

Many ABE programs their

learning

with

settings.

to subjectivity

in the

of political

That is,

forces

1982,

concerned

classes

notion

changes

of math and the

omits

in specific

Many new approaches

including

the nature

learned

from objectivity

standards.

concerning

(Frankenstein,

values

ideology

and critical

framework,

are

change.

This ideological perspective, which owes much to Freire's pedagogy of the oppressed, extends the complex role schools play as institutions that mediate and sustain the logic of the state and the imperatives of capital, to include the important concept of human agency where, in a dialectical process, people both participate in their own oppression and struggle to resist (Frankenstein, 1988, p6).

As Giroux "the

insight

also

the

the

says, that

1982,

tools

to produce,

they

need

literacy

a moment of domination reinvent,

to break

them from transforming

Math involves

to deepen

issues.

The ability

instead

of critical

is but

and create

through

to myths

an oppressive

social

p354).

to Basic

perception

notion only

oppressed

questions

Math approaches.

not

in some ABE program

approach

and economic

meaning

the

prevent

Math instruction

people's

of Freire's

contains

for

that

(Giroux,

statistical

core

and material

and structures

critical

the

culture

possibility

ideological

reality•

at

of those In these

of meaning

one's

could

this

the ability

is

also

data

important

math is

constructing

math. 8

of political,

this

ABE programs

category

because

a

to ask basic

appreciation

to present

issues

fit

used

social

to change in critical

Basic

to construct

The purpose

of Basic

Math

is

seen as to gain

political These

popular,

and social

structures

instructional

educational those

approaches

help

control

are

very

to develop

of their

new social

economic,

but

they

go beyond

The critical

relations

1983).

to the nonformal

ideology

intent.

the

(Frankenstein,

similar

from interactive

in terms

over

of our society

approaches

approaches

should

democratic

in the

Math approach

struggle

for

humanization.

SOCIAL CONTEXTOF BASIC MATH Many social affect

context

Basic

factors

affect

Math instruction

considerations,

and the

interrelated

ABE.

like

math anxiety,

classroom

and influence

each

However,

some in particular

gender,

environment.

All

multiculural

these

factors

are

other.

MATHANXIETY Math anxiety leading

to the

experience, noticed but

is

avoiding

that

rather

anxiety

and this

affects

important

for is

with

better

Math anxiety formal

Basic

is

send to special

put

than

with

are

each

teaching

experiences. math anxiety

environment the

content

other

the

not

math,

at

It

its

is

causes

teacher

and

and tutor

with that

it

addressed

well

real

life

be true

in these 9

safe,

and beliefs

used

in the

about

anxiety

math anxiety

learners

teaching

based the

respected

Having

experiences

try

lie

in many of these

that

or are

and tutors

feel

to be conceptually

up with might

teachers

non-threatening

are

in

in many of the

and teachers

other

label

math courses

feelings

among students

this

math classes

learners

of these

where

supposed

However it are

for

a more

who have

workshops

where

approaches

match

help

Nevertheless

and math anxiety

Math is

instructors

that

I have

have math anxiety,

in remedial

workshops.

in combination

and workshops. that

are

The power and success

rather

will

often

a learning

keep a journal

it

system

to create

anxiety

and to look

and learners

math courses

Program,

Math to ABE learners.

label

remedial

shared

because

In my own

Math instructors.

math anxiety

and in control.

Basic

often

a negative

school

Education I think

to math anxiety problem

1978).

have a special

themselves

teach

in math and

(Tobias,

in general.

teachers

the

of failing

fronts

don't

learning

to pay attention

in becoming

devices

often

that

fear

in a Workplace

the way they

ways of dealing

are

of math on all

ABE students

factor

process

of enormous

as a ABE instructor

important

the

a syndrome

classes

which means

and people's deeper

causes

workshops;

own of

Students are initially relieved that their feelings about mathematics is so common that educators "have a name for them. But in fact this label focuses the problem, and therefore the solutions, on individual failure rather than on the broader societal context which plays such a significant role in producing personal "math anxiety." (Apple, 1979, p67). 11

Also

labels

development relations

in educational

of critical which

hidden ce,

used

consciousness

they

(Freire,

describe,

math curricula

such as the

use of rote

so that

assume

have

which

ments

(Hilton,

1980).

Math anxiety

about

learning

created

and sustained

believe

in these

without

examining

holding

that

the

about

the

the

situations

and complexity

which

because

and are

how math is

is

(1983)

tiny

often

caused

learning

and other

*

Math can only

*

Smart

people

be learned

during

do math fast

specific

and in their

math myths

can result

1990). (1990)

People

to harm the

person

in false

Some math myths are:

have a mathematical mind or you haven't. to talent and to genetic factors.

attributed

compart-

and society.

in math myths

and Frank

and

by misconceptions

potentiality

(Frank,

unmotivated

water-tight

by schooling

done

math avoidan-

in math education,

into

and its

belief

causes

memory dependance,

about

evidence

by Frankenstein

* Learners questions the right

features

math can be divided

misconceptions

belief

* You either is

causality

authoritarianism

tests

mentioned

work against

by mystifying

calculation,

and applications,

impressions

sometimes

1983).

Traditional exercises

settings

times

Math

in one's

ability

life,

heads.

think that they are stupid if they make mistakes or ask and these learners also think that they don't how to ask questions.

*

There is only one correct each math problem.

*

Cheating

is

*

Men are

better

* Math requires

answer

bad and looking in math than

and one correct

up the women.

a good memory.

10

answer

in the

way of solving

key is

also

bad.

These math myths because

they

society.

are

are

not

created

easy

to get

rid

and sustained

Frankenstein

says

of and are

very

by the majority

pervasive

of schooling

and

that:

Understanding the deeper causes of math "anxiety" involves an examination of how the structures and hegemonic ideologies of our society result in different groups being more affected than others by this "anxiety.• It also involves recognition that, to some extent. people participate in their own mathematical disempowerment (Frankenstein, 1983. p. 328). She states

that

math anxiety confidence

once learners

humanistic

and teachers

pegagogies

and feeling

will

in control

know the deeper

help

the

learner

causes

of

in getting

own learning.

of his/her

GENDERAND MATHEMATICS A person's

sex has always

individual's achieve

achievement

their

students. female race

full

students their

are

to demonstrate

science.

Biology

it

p.6).

factors,

school

related

behavior

Often at

any innate

largest

school

for

application

1988,

p.31).

Another

found

in the

an object 1982,

p.6).

performance comparison

area

or pattern The older in spatial with

order

related

curricula,

"This

Research math and

majority

of women Langbert

has

and

by societal

attitudes,

consistent

achievement

tasks"

which

imagination the

is

application,

boys.

11

(Reyes

is

and Stanic,

in math ability the ability

(Skolnick, bigger

to become noticeaachievement

sex difference

skills,

get

starts

differential

and problem-solving

ability,

in math

processes.

level.

girls

up to

to learn

caused

boys and girls

in the

achieve

work (Skolnick,

teacher

male and a student's

of things.

why such a large

male

many black

don't

inability

between

of spatial

like

students,

are most likely

and classroom

factors

status

with

between

These differences

female

explain

mathematics

high

female

by the natural

Differences

the difference the

from other

Like

away form math and science

Day, 1982,

ble

difference

the

don't

students

understand

status.

cannot

female

in comparison

isolate

caused

to an

in mathematics

in mathematics.

not

influential

Often

of low socio-economic

potential

performance

steer

economic

and students full

failed

to fully

we cannot

and socio

is

in mathematics.

potential

In order students

been and still

to manipulate

Langbert

the difference and problem

has been

and Day, in math

solving

in

Societal of

girls

to

and

(SES)

attitudes in

their

about

family,

the

implicit

groups.

messages

influences

in

several

results

article

in

human of

the

in

from

the

roles

can are

(Apple,1979).

parental,

teacher,

of

different

of

the

mass

of

prevail of

and

do change,

powerful

and

and

are

media

and

particular those

existing

persistent

Beckwitch

societal

Status

influences

by members

Jon

messages

their

societal

pattern

held

beings

Journal

of

confidence

Socio-economic

institutions,

history

lessened

appropriateness

influences

moment

and

Examples

result

societal

the send

race,

the

religious

societal

particular

and

for

attitudes

sex,

in math.

that

other

Although

any

level

community,

and

These

aptitude

a high

occupational

be responsible

of different

their

the

also

ability.

students

at

the

at

math

regarding

achieving

can

(1983)

attitudes

gave in

his

Education:

Girls were more likely than boys to attribute their success in math to hard work rather than ability. Conversely, girls would more often than boys, invoke lack of math ability to explain their failures in math. This was true at every level of achievement. They also concluded from earlier work that girls as compared to boys are less confident about their math ability and consistently underestimate their ability to solve mathematical problems (Beckwitch, 1983, p260). There

is

a large

indicates

that

Beckwitch

mentioned

the

body

teacher

discouragement

of

literature

attitudes several

of

on teacher affect

examples

girls

in

student of

the

expectations

that

achievement.

Jon

encouragement

of

boys

and

math;

In 33 second grade classes, Leinhardt, Seewald, and Engel (1979) found that teachers made more academic contacts and spent more "cognitive time" with girls in reading and with boys in math. Becker (1981) showed that in high school geometry classes, teachers favored boys over girls in a number of ways including "cognitive levels of questions,• "praise and criticism," "encouragement," and "individual help" (Beckwitch, 1983, p 262) . .John Ernest (eleven than

men and

ability

that boys

certain than

student's

four

women) towards

curricula of

the

courses,

40% of

that are

prospective

likely

math

may vary

to

to

students

in

topics

and

the

transmit

students.

depending

school

teachers

something

A teacher's on her/his

school.

activities

A teacher are

more

less attitude

beliefs

about

might appropriate

girls.

In addition, achievement

found

attitude

school

general

think for

the

also

sixty

a positive

towards the

(1976)

the related

attitude

math

curricula

behavior. and

affect Confidence

achievement

related 12

a student's in

learning

behavior,

attitude

and

math,

which

affects

math

is

a

achievement. sex

Sex difference

difference

been

in

identified

math

as

differences

influenced

by parental,

related

behavior

I think have

the

students, about teacher this

teachers

and

towards

the

major

Basic

Math

schools

and

to

new ideas

experiences

at

women a change can

isn't for

math,

equality

for and

because

and

it,

feel

that

and

tutors

men. should

The teachers have

by focussing valuing

such

a way that

raising professions

gender

can

be geared

group role

in which

work

and

awareness math

is

regarding required

13

expect

the

one

are

should

a

a

they

also

can

therefore

in

such

be fun

had

open

and

give

a way that that

it

show an enthusiasm the

towards and

work typical

(Skolnick,

1982).

as

math

arranging

independent

have

values

of

curriculum.

confidence

materials,

can

so many negative

reflect

Math

on building

manipulative

which

sensitive

Many ABE

they

should

a Basic

approaches

had

of male

be very they

same

experience

an attitude

ABE

learners;

in math

hand

it

a male

math.

curriculum

to

The instructional

risks,

Math

math,

enjoy

to

because

the

have

appreciate

of men and women in

approach

they

anxiety

preference

have

other

Female

have

male

experience

On the

don't

math.

that his

towards

less

be

my ABE class

often

with

needs

or

again

towards

tutor

school

more

The Basic

in

teaching/learning and

understanding

Achievement

however,

math

has

girls.

with

attitude

attitude

is

attitudes

to

the

against.

school.

worked

does

some formal

rebelled

and

mathematical

which

understand

only

about

students'

had

and

always

learners

on their

for

noticed

The teacher

curriculum

in

they

with

attitudes.

boys

in

might

societal

with

of math

variables

ABE classroom,

I have

something

who have

the

associated

usefulness

usefulness

positive in

general.

learners.

impact

learners

students

female

This

I have or

usually

important

and

students

worked

say

most

different

negative

in

I have might

often

male

learning

the

teacher

female

like

of

is

Perceived

in math.

is

a particular

confidence

achievement.

one

sex-related

in

a problem

reducing social are male

fear

of

taking

arrangements

in

validated, and

Langbert

solving

by

female and

Day,

MULTICULTURAL CONSIDERATIONS Many people

believe

Some people

say

mathematics

we can

shaped

the

activity

workers.

Later,

two

took

only

for

which in

only

had

to

However obvious, done and

Educators

in

a topic

also

during

in

became change

schools the

division

different

for

order

age

two decades

in

history

favor

of

the

cultures,

people

lies

the

predominance

because

right

have

1985). has the

become mathematics

groups,

gender,

of mathematics

is

borderline

cultural

two advocate

acces

now for

division

and

they the

to

we have

on the

Math

for

and

not

aristocracy,

(D'Ambrosia,

field

of mathematics

combining

lay

socio-economic This

the

economical

mathematics

the

era,

reasons.

of

a human

of

complex

social

math

manual

industrial

and

another

Western

brackets.

the

as

for

effectively

society

Ethnomathematics

of

to work

in

in Western

other

the

separate

increasingly

children

of have

a convergence the

for

social

recognized

between

from

its

history factors

as

Europe,

also

the

activity

during

the

at

started

with

for

resulted

last

pace

but

area

as well

in

culture-free.

political

a practical

dealing

manuals,

and

we look

Math

in

a topic

ethnomathematics. discipline

increased

also

if

and

Ages

as

our

the

Middle

but

This in

the

keep

by people from

as

It

and

and

to

education.

everybody

elite

be prepared

aristocracy. to

true

universal

social-economic

the

this

know math

is

of mathematics.

instruction

a new order,

needed

not

of necessity

seen

to

is

that

for

and

and

not

see

field

reasons

mathematics

this

during

place

machinery was

that

whole

scholarly

that

race

called

between

the

anthropology.

a broader

interpretati-

on of mathematics. Now we include as mathematics, apart from the Platonic ciphering and arithmetic, mensuration and relations of planetary orbits, the capabilities of classifying, ordering, inferring and modelling. This is a very broad range of human activities which, throughout history, have been expropriated by the scholarly establishment, formalized and codified and incorporated into what we call academic mathematics. But which remain alive in culturally identified groups and constitute routines in their practices. (D'Ambrosia, For the Learning of Mathematics 5, l, 1985, p 44-48) The researchers cal

who study

procedures

practiced Secondly differences the

period

structures

which

around

the

favor

ethnomathematics computation

and of

various

look

cultures'

and

rote

languages.

at

the

methodological

mathematics,

comma in mathematical other

first

at

the

memorization,

pedagogiconunonly

world.

ethonomatematicians in

look

For

notation example, 14

such or in

as

and cultural

conceptual use

classification the

African

language

of

Setswana, are,

things

as

in

and

the

People

bad

tion.

say

these

brackets, classes

look

from

in is

rote

classes

in

and

fail

to

will

the

of

the

language

math

inappropriate

what

of

because

knowing learning

that

Because

math

occurred

or not

at

other

the

are

for

have

been

and

labor

status

(SES),

Many people

which

hidden

cultures,

economic

so on.

contributions

math with

difficulties

rather

1985).

difficulties

which

do,

they

appear. physical

of

instruc

difficulties

people's

of

cognitive

1985).

social and

traditional

difficulties

researchers

by people

in many such

curriculum

they

(Berry,

conditions,

(Berry,

Thirdly

by what

languages

these

classroom

an unsuitable

math

that

However

structure

testing

US, many of

will

often

classified

Euro-European

memorization outside

are

from

distorted

groups, gender

these

recognized

contribution

different

age

race,

professional

and

groups

have

such

because

as

to

made valuable of

their

background. Basic

Math

instruction

can

incorporate

some of

these

findings

by:

* Multiculturalize the Basic Math curriculum by appealing to a diversity of cultural backgrounds and learning styles, rather than dismissing the students as incapable of learning math (Zavlavsky, 1989). * Incorporate methodological and conceptual differences in the curriculum as much as possible. Schmitt (1985) has written a handout for math instructors on the different ways learners do or write mathematical computation. Also letting learners become aware of their own and other people's ethno math. Learners in the USA can have a different mode of cognition and different method for performing operations. The instructors should incorporate or acknowledge this as much as possible.

*

Acknowledge hidden and distorted part student to unknown parts of math history by mathematicians the mathematics field economic groups, gender and class.

of history by exposing such as the contribution form other cultures, socio

to

CLASSROOM ENVIRONMENT Many ABE classes learners, week, tutors

ranging two hours

receive

great

challenge

often

many come

twice

are

classes

ABE teachers

are

15 hours to

of

only

a week

teach able

and

have

to

tutor to work 15

in once with

Math the

advanced

meet

only

teach

a week. and most

a week

are

twice

two different

a

reading Tutors

faced

the

often

with

appropriate

while

GED

by volunteer

assisted

4 hours

Basic

a learner

often

in how to

2 or

teach

and

often

training

tutor

in how to

how to

intermediate

10 learners;

willing

training

tutors

6 to

3 to

are

of

beginning,

time.

had

and

don't

from each

who have

and writing,

Also

constitute

the

way, learners

tutors.

The

result not

of this only

because

different the

is

teaching

of learner

levels

quality

that

Many learners

don't

them to come into

that

the

learner

feels

and that The classroom

messages

about

Attentive

independently? experiment construct

and respond

an environment

that

(NCTM, Professional

and it

is

very

and

safe

in the

curriculum

a hidden

and doing able

to justify

a big

learning

appreciated

math:

and

with

neatness?

speed?

a solution?

working

to make conjectures,

to solving

mathematical kinds

is

important

forms

approaches

Standards,

It

to learn

these

tutors,

is

students

to others'

of the

way of knowing

being

fosters

experiences

again.

in learning

alternative

because

and emotionally

environment

listening?

a challenge,

of volunteer

school

learner's

counts

quite

tutors.

a classroom

If we want with

but also

the availability

physically the

valued.

accuracy?

size,

have previous

for

what

Math becomes

of volunteer

step

environment

group

of learners,

of training

Basic

problems,

arguments,

of activities

and then

is

creating

essential

1989).

ADULT LEARNING THEORY Another

issue

theory.

Adults

those

considers

affecting are

defined

adults:

citizen,

soldier,

perceive

themselves

basic

by the

the

and the

math instruction

roles

like.

social

roles

of worker, Adults

to be essentially

is

are

adult

typically

spouse, also

learning

parent,

defined

responsible

assigned

for

to

responsible

as persons

her

or his

who

own

life. In adult

learning

theory

there

are

four

asswnptions

about

adult

learners: As individuals mature: 1) their self-concept moves from one of being a dependent personality toward being a self-directed human being: 2) they accwnulate a growing reservoir of experience that becomes an increasingly rich resource for learning; 3) their readiness to learn becomes oriented increasingly to the developmental tasks of their social roles: and 4) their time perspective changes from one of postponed application of knowledge to immediacy of application, and accordingly, their orientation toward learning shifts form one of subject-centeredness to one of performancecenteredness (Knowles, 1981, p44 45). Adult

learners

have

lives

have authentic

lives

avid

language

and eager

planners, Crowley,

teachers, e.a.,

within users,

outside the

the

classroom

classroom; learns,

self-disciplinarians

only

resource

and must then

1988).

16

can students

persons,

and friends

therefore be

thinkers,

(Gilles,

Bixby,

Adults

have

much time. learning ques

lots

The role because

can be used

adults,

of experience,

who are

experience

adults

are

in the

adult

rich

group

learner

but

Adults, tasks,

also

too,

have

readiness

products

implies

for

practice

that

centeredness their

creating

developmental

homogeneous

effective.

while

learning

others

in separate

Many problems were

within role

dependent

the

directing

and the

input,

of thinking skills

about

in

(1)

and free the

to adult

take

to develop

ease

learning

the

need

carpentry,

learners

an inner

learners

conflict of the

to be selfinto

our

completely

in getting

program

focus

into

and to develop

(Knowles,

if

model

need

to build

don't

learner,

on

a new way

some new

1981).

to be addressed

in order

to have

adult

place:

Adults want.

educators individuals experiences

to the

must

climate

to feel

The teacher

say and be sensitive

of the

be in

can be very

about

adult

psychological

have

tial

task

intellectual

learning.

Adult

learning

of learning,

induces

learning

expressions. concerns

this

and psychological

they

and

to know more about

treating

which

of the

need

to say what

students

is

learning

experiences The physical

deep

experiences

self-directed

conducive

start

There

role

of adult

in learning

preconditioned

adults'

grouping

be recommended.

for

gradually

the

Some key conditions learning

would

1981).

but

want

more interested

this

some preparatory

learners'

learners

1981).

of application

to developmental

when teachers

between

(Knowles,

designs

are

adult

personalities,

adults

of learner

if

timing

the

and the

For some kinds

according

groups

arise

the

are

(Knowles,

The immediacy

adult

developmental

Adults

of learning

be on

lives.

resulting

timing

that

should of the

day-to-day

roles.

own

critical

experiences

of social

tasks.

groups

For example,

gardening

they

demands

methods,

moments.

enable

their

The emphasis

of growth,

be done carefully.

techni-

will

into

have

in adult

Experiential

case

to their

the

role

to tap

on the

and teachable

and don't

room and it

etc.

drawing

evolution

performance with

exercises

of the

should

class

using

own phases

oriented

done.

learning,

them apply

their

of learners

step

for

only

to learn,

largely

have

discussion,

not

helping

they

goal

an important

education

simulation

application,

very

plays

what

resources

processes,

practical

This

of experience

by having

incident

are

has

learners'

to

really verbal

attuned

and institutions

they

will

concerns. 17

to be

comfortable,

be primarily

that

has

be articulated

respected listen

what

and facial to the

serve with

existen-

and be able these

(2) The learner's adult

learning

organizing the adult

self-diagnosis

which

the

curriculum

must always

while

in a learning

start

resource.

process

the

teacher

the

knowledge

learner

is not

the

medium evoking

the

(Freire,

p.

1971,

privately and talk

teacher about

teacher-of

the

about they

p 67).

students

and the

The teacher

has

equal

time

perspective

4) Conducting learners making

to all. learning

and teacher. a person

learn,

will

experiences

about

which

of it

the

students

in the process

with

is

teacher

object

of thinking,

"Through

As they dialogue,

cease

to connect,

the

to exist

student-teachers" doesn't

"a

may store

dialogue.

as an impartial

1986,

object

one

and students"

thinking

the

It means to try

a mutual

Rather,

teacher

students-of

but acting

A teacher

the

merge.

to trust

(Belenky,

says,

roles

be

and content is

the

have

of needs.

should guide

in a public

teacher-student

of view points,

student's

thinking

the adults

function

so that

to

in these

diagnosis

teacher.

teacher

through

between

learner

of both

must engage

their

a variety air

of the

publicly

together,

and a new term emerges: 1971,

are

the

As Freire

of the it

self

in

relevant

that

dialogue

this

reflection

and students what

and talk

teacher.

are

experiences

as a procedural

performing

Instead

which

learner's

property

critical

and talking

both

for

importance

support

and concerns

one in which

private

).

the

serving

and the

units

A continuous

help is

Responsibility

between

think

will

of crucial

learning

ideas

environment.

3) The planning with

the

is

instructor's

thematic of the

with

and learner

involved,

around

The des

units

it,

can be done through

learner.

instructor

of needs

(Freire

mean just

tolerating

refugee,

assuring

to enter

into

each

p.227). are

cannot

but can only help

also

mutual

really

teach

another

responsibility in the

person

sense

of of

learn.

Educators can help learners develop their own authentic voices if they emphasize connection over separation, understanding and acceptance over assessment, and collaboration over debate; if they accord respect to and allow time for the knowledge that emerges from firsthand experience, if instead of imposing their own expectations and arbitrary requirements, they encourage students to evolve their own pattern of work used on the problems they are pursuing. These are the lessons we've learned in listening to the learners' voices (Belenky, 1986, p.229).

18

5) Evaluation which

of learning

the

teacher

themselves

about

goals.

In this

program

itself

should

devotes the

energy

progress

process,

the

undertaking,

as must be all

Evaluation

an on-going because

it

notion

learning

discovery

learning,

do for

a sense

other

this

of release

educational

has

facilitated is

a mutual

learning

learning

experien-

needs,

or as

as more constructive

of learning,

reinforcing

Fortunately,

responsibility

facets

educational

of the

adult

process.

can take

evidence

their

of how it

of the

a new cycle

they

for

get

So, evaluation

see

a continuing

that

as they

experience

is

adults

as re-diagnosing

often

them into

in

toward

in terms phases

be seen

of self-evaluation,

and weaknesses

students.

other

Learners

the

making

strengths

of the

can also

launches

that

make the

learning

process.

are

be assessed

or inhibited

ce.

to helping

they

the

must also

be a process

of their

and exhilaration

once adults

for

lives,

the

their

they

(Knowles,

own

often

1980,

p.46).

for

a Basic

THE BASIC MATHCURRICULUM INTRODUCTION In this

part

of the

Math Curriculum. after

the

evolve.

This

instructor

Basic

first

part

class

is

class;

of the

each

the

curriculum

I will

Basic

Math Curriculum discussion instructional evaluation,

approaches, and classroom

the

has

at different

the

a typical typical

I'll

cover

of the

teacher,

19

Math

of literacy tutors

Basic

regarding

then

to The of

Math Basic

issues

the

standards.

assessment

Math.

mentioned

describe

of Basic

of

in the

acquisition

the

on the

content

version

goals.

to the

the

management.

type

different

and evaluation

change

Basic

volunteer

assumptions

paper.

create,

mentioned

describing

on my opinion

role

factors

only

curriculum

more definite

better

Before

project

I will

the

with

and writing

based

because

must

levels,

learner

NCTMcurriculum

of curriculum

learners

regarding

my personal

are

of this using

on all

skills.

describe

to be general

The final

and each

workplace

part

with

Math teacher all

outline

can a more definite

My assumption

Many of my assumptions first

learners

together

from reading

and

a general

is meant

depends

learners,

can range

give

curriculum.

6 to 10 learners,

some life-

in the

the

and tutors

of my paper.

work with goals

Curriculum

Math Curriculum

that

I will

meets

Teachers

and be in charge the

paper

Basic In the

Math, and

the

PERSONALASSUMPTIONSAND ITS IMPLICATIONS FOR INSTRUCTION Teachers'personal for

assumptions

instructional

assumptions

*

affects

practice

one considers

math.

Theory

by focussing

Basic

Math have

implications

Math Curriculum.

and can change

when thinking

can strengthen our attention

concrete

daily

(Marilyn

Frankenstein,

Curricula

one needs

check

and the

Basic

My personal

are:

Theory

tions

practices

regarding

it

instructional

in the

and the

theoretical

in harmony with

for

framework

or implementing

Basic

Math

and ideologies

own belief

one's

our

ideological

frameworks

in

hun1anization

between

broader

of ques-

practices

struggle

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and types

interrelationships

practice 1983).

is

depth

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our energy on the

teaching

whether

about

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statement

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or

vision.

* The goal of education struggle

for

humanization

of injustice this

with

learners

structure

need

deal

*

and hegemonic

learners

with

it

curriculum

* Adult school

curriculum

with learners

to learners'

required needs

discussing

for

underlying

causes

the

possible. because

of

Math instructors of math anxiety

own math anxiety

often

The Basic this

ans and giving

an enormous

classroom

about

what need

reservoir

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and day-to-day and teachers.

learners

have

and be able

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environment

counts

in learning

to share

their

hidden

Basic

encouragement

for

20

goals.

they

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way they

have

to be sensitiviway.

a certain which

reflect

experiences, as a group

to address examples female

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labelled

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that

incorporating

negated

affecting

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and day-to-day

need

and positive

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always

made to believe Math instructor

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and learning

who have

cultures,

of negative

experiences Basic

experiences

in length, extra

of the

messages

to be made to find

histories,

Women are

at math.

Basic

full

Sharing

others

of society.

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experiences.

ABE tutors

*

than

is

learners.

have

perceive

learnrers'

by math anxiety

aspect

with Basic

their

cultures,

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is necessary. dialogue

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and emotional

histories,

ty is

learners

as the world

make a honest

ideology

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mathematics.

and life

sensitive

change

in the

effectively.

a hidden

and doing

the

will

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forms

will

relations

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1971).

more affected

with

new social

relationships,

and others are

to explore

so that

to develop

(Freire,

and unequal

* Some groups the

is

this

are issue

of female

learners.

not

good

by mathematici-

* Instructors school.

key figures

Instructors

who totally have

are

can be strong

control

the

to be a role

learning

model

* Basic math deserves reading teaching

using Basic

for

influences

their

and learned

without

being

Basic

in

"superiors"

Math instructors

learners. attention

An integrated

reading

taught

environment.

as equal

and writing.

necessary,

in how math is

in the ABE curriculum

approach

and writing

with

reading

instructional

as

and writing

techniques

is

for

Math.

THE NCTMCURRICULUM AND EVALUATION STANDARDS The curriculum NCTM, which based

came out

teaching

standards

and evaluation in 1989,

approach

are

also

classroom.

These

dominated

curriculum

advocate

to math.

applicable standards

oriented

approach

involvement

as problem

solver,

technology,

and the

is

changing

citizens als

change,

to live

of all

races

doors

This

sexes,

must cover

responsible

represents adulthood

to a computationpromote

learning,

applications, was that

rate,

using

a more active

utilization

of

those

and further

to increase.

competencies

society.

will

our technological

need

All

competence

education

(NCTM, 1989).

21

needed

by individuincluding

in essential that

to remain

competence

as

As the demands

students,

competencies

the mathematical

world

and our responsibilities

continue

essential

in that

to employment

mathematics for

and both

in the ABE

of number sense.

affairs

so do the

and evaluation

response

life

of the

conceptually

Math Curriculum

of rote

the NCTMstandards

productively

mathematics. the

real

at an ever-increasing

of society

mathematics

The new standards

instead

development

in international

curriculum

a Basic

schools.

school

a more holistic,

were a reform

in the

for

for

These

for

conceptually

The rationale

standards

are

areas

necessary

open.

students

those of for

Essential will

need

NCTM's ASSUMPTIONS NCTM's standards

are

based

on four

assumptions:

1. The goal of teaching math is to help all students develop mathematical power which means that the learners will have the ability to explore, conjecture, and reason logically, as well as the ability to use a variety of mathematical methods effectively to solve non routine problems and the self-confidence and disposition to do so (NCTM, 1989, pS). 2. What students learn is fundamentally connected with how they learn it. Students' opportunities to learn mathematics are a function of the setting and the kinds of tasks and discourse in which they participate (NCTM, 1989, p.10).

3. All p.10).

students

can learn

to think

4. Teaching is a complex recipes or prescriptions

mathematically

practice and, hence, (NCTM, 1989, pll).

(NCTM, 1989, not

reducible

to

THE PURPOSE OF BASIC MATH The NCTM standards (1)

that

their

they

they

(4)

learn

is

they

depends

goals

of the

taking

tics

to do math that

itself

to four

workers, electorate.

is

lifelong

issues

learner,

the

understanding

stance

the nation's

societal

goals

learning, educating is

students.

sequence mathematics competence (1) tical

Problem

areas solving,

Reasoning,

Alertness

to the

areas

don't

interrelated;

areas.

(4) Applying reasonableness

present

providing

and is

providing

(Schmitt,

an

1990). competence

imply an instructional the

twelve

essential

in each area of essential

mathematical

ideas,

of results, 22

its

components

mathematics

literate

on mathematical

competence

(2) Communicating

pledged

for is

citizens

In fact,

The twelve

NCTM

and an informed

children,

areas

and

of the mathema-

mathematically

to learn,

critical

paper

in mind.

work force

as informed

that

Math

and has

of opportunity,

among topics.

are

in other

Standards:

opportunity

Basic

One of this

work force

of the

The critical

or a priority

future

and (5)

each

improvement

parents

twelve

all

the

in

problem

can be kept

today's

The NCTMcame up with

students:

become confident

in Part

to equip

issues

all

mathematically,

mentioned

equity

giving

they

Although

in its

the

for

become mathematical

regarding

of today's

for

that

NCTMstandards

for

learning,

they

goals

to communicate

ABE is helping

challenges,

(2)

mathematically. on the

lifelong

general

math;

(3) that

learn

a proactive

preparation

five

to value

to reason

curriculum the

learn

ability

solvers

articulate

to everyday

requires math are (3) Mathema-

situations,

(6) Estimation,

(7)

(5)

Appropriate ment,

computational

(10) Geometry,

skills,

(11)

(8) Algebraic

Statistics,

(12)

thinking,

(9) Measure-

Probability

CONTENTOF BASIC MATH The ABE learner mathematical based.

needs

on the

problem-posing

and the tutor learner's

food labels,

types

of data.

especially children

and gender

articles

related health

area

The math activities

Resources like

information,

numbers tax-forms,

charts,

or many

math materials,

their

children.

oriented

Because

math activities

which can also

should

also

on

for

containing

can be school

come and more will,

theme to focus

materials

want to work with

around

or competency

which

term goals.

many conceptually

have already learners.

skill

labels,

content

when the adults

the new NCTMstandards adult

life

medicine

Another

and long

evolves

posing

to decide

can be newspaper

concepts,

which

problem

have

short

materials

and mathematical bills,

Math Curriculum

themes which can be either

The learner

depending

a Basic

for

be used

consider

of

for

multicultural

issues.

INSTRUCTIONALAPPROACHES In order

to teach

them critically

through

techniques. materials,

Other

instructional

nwnbers, out

their

individually applying

the

intent.

learners can also in both

content

language

integration,

themes,

cultural

Basic

using

The instructional

include

who created actively

in exploring,

approach

a mathematical

23

Math can

and socio-economic of certain

them and to figure

involving

conjecturing,

and

the

one can look at the meaning

to discuss

teaching

the use of manipulative

can be taught

the political,

I favor

whole

upon the

and principles

and in groups, mathematics

are

Once decided

For example It

using

problem-solving,

by exploring

asking

and writing

approaches.

of math.

from the themes

strategies

work,

concepts,

become critical dimensions

reading

and computers.

skills,

different

Math evolving

instructional

cooperative

calculators facts,

Basic

students analyzing,

and a real-world

context.

and

THE USE OF WHOLELANGUAGETECHNIQUESIN MATHINSTRUCTION The instructor skills,

can use language

concepts,

two types

and principles.

of language:

language

to teach

we talk

spoken

and listen

the mathematical

In order and written

and for

to teach

math I distinguish

language.

the written

facts,

For the

language

spoken

we write

and

read. While often

the

the

working

learners

spoken

language

in small

explaining language

investigate

groups

is used. they

Reading

the use of written

resources;

tion,

applications,

definitions,

with

summaries,

The reading

is

primarily

for

about

Math to answer

questions

that

investigations.

The learners

as skimming,

scanning,

interpreting

tables

Writing charts

with

are

labels,

predictions, reflect,

by writing

summaries,

notes,

to record

successes,

and the development,

(3)

learning

and symbols.

It

requesting

their

strategies

such

and

also

good recording

gives

the

and give

(Richards,

24

learning

is

to

processes

and celebrations an insight

thats/he

values

(2) questioning,

1990).

of

of mathematical

and strategies.

feedback,

personal

can be to identification

learners

by showing

that

descriptions,

of writing

their

sharing

skills

or writing

reports,

evaluations,

writing the

by developing

Types of writing

use and understanding

recording

more information

board.

and demonstrate needs,

Writing

a newsletter

The purpose

use of resources,

can encourage

(1) praising

conduct

or each other,

lists,

of larners'

and future

instructor

and

information

Math Curriculum.

definition,

explanations.

The outcome

their

facts

of contents,

as publishing

translations,

instructions,

to clarify,

own thinking,

reading

teacher

on a bulletin

of present language

through explana-

specific as they

and tables

to the

as well

arguments,

and outcomes.

arise

in the Basic

or discoveries

can be used writing,

frequently

explanations,

announcements

role

and diagrams.

can occur

can be encouraged

indexes

The written

rules,

gathering

need specialized

using

other

instructions,

reports,

most

are

a significant

can read

questions. Basic

to each

have to do.

can play

learners

Math themes,

when learners

and listen

the math they

be used.

Basic

Especially

talk

and understanding can also

the different

in their The it

and by

and by

ROLE OF THE INSTRUCTOR For a Basic

Math instructor

from several

domains:

how students

learn

society.

the

Knowledge part

knowledge

particular,

about

adult

concept

in a given

from these pursue

for

is

domains

how to represent

balance

multiple

variety

of factors

governed

goals

by rules

NCTMhas defined Classroom

will

cannot

role

of the

these

environment

* Set goals and select achieve these goals.

*

Stimulate instructor

Instructors

don't

and how to

having

to

depend

abstract

on a

nor

p.11).

instructor

in the

to support

teaching

or create

mathematical

and manage classroom discourse are clearer about what·is being

don't

teachers

then

previous

idea,

decisions

in the

tasks so that learned.

* Analyze learner learning, the mathematical ment in order to make ongoing instructional

they

knowledge

Basic

Math

as to:

* Create a classroom of mathematics.

Instead,

a mathematical

themselves

be determined

of thumb (NCTM, 1989, the

find

be

to a learner's

mathematical

and considerations;

that

can only

using

how to respond

a particular

Instrucotrs

must also

must weave together

to decide

and

For instance

instance,

must make about

of

school

specific.

for

Instructors

learners,

however,

context

in general,

on knowledge

of classroom,

Instructors,

context.

the new topic.

context

teaching

draws of diverse

is general.

an instructor

different

question,

of the

learners

of the decision

mathematics

of mathematics,

mathematics,

Such knowledge

consider

teaching

organize perceptions.

have

opinions

and should

be able

teach must

learners

listen

and present However, which won't to suggest

to think;

to the

influence themes

25

students

learners

the

learners

and the

think

to discover

their

which

learners'

instruction,

learners

already.

themes

are not neutral

to the

important.

to help

to challenge

instructors

learning

tasks, and the environdecisions. (NCTM, 1989).

learners

as problems

and the

they

objects they judge

as

who will

Instructors totally

can

be strong

control

opinion,

he

the

says

influences

learning

without

being

"superiors•

Freire

environment.

also

who

has

this

that:

The opposite of manipulation is not an illusory neutrality, neither is it an illusory spontaneity. The opposite of being directive is not being non-directive that is likewise an illusion. The opposite both of manipulation and spontaneity is critical and democratic participation by the learners in the act of knowing of which they are the subjects (Freire, 1981, p.28).

ASSESSMENT AND EVALUATION Assessment Basic

Math

learners and

and

Curriculum

to

to

how they

evaluation

find

Basic of

evaluation

but

then

evaluation

of

makes the

what

the

Math the

adult

done

doing

still

and

of

important

ABE learner

and

and

learner also

instructor's

by themselves,

much of

evaluation

the

together,

and

of an A

because

learner

very

especially

the

the

However

responsiblity

be encouraged

use

uncomfortable in

experience.

most can

of

it

For

is

grasp

feel

evaluation

should

type

evaluation

would

learning

same

the

judgement.

learners

many ABE learners

and

ABE learners

instructor the

know

self-evaluation.

on the

a

the

to

of

the

for

both

want

evaluation

and

regarding

Math

they

instructor,

of

and

assessment

instructor-learner

knowledge

self

of

effective

because

aspects

instructor

what

types

learner,

dependent

leave

the

know,

instructor

very

new Basic to

important

a traditional

completely

their

most

both

by the

not

one,

the

different

by the

is

evaluation

beginning

learners

by the

I think.that

the

the

a collaborative

judgement

materials.

enables

are

program

be a successful

their

it

possible:

learners

ABE learner

some of

material

evaluation

the

likely

out

There

grasp

traditional

are

because

learn. of

learners'

evaluation

it

is

with

the

as much as

possible. Assessment necessary

and changes

NCTM developed used

for

* Assessing tics. Having

can

school

Basic

standards what

assessment

has

to

be an ongoing

be incorporated

fourteen

teaching

evaluation

*

evaluation

Math

students

be an

evaluation in

mentioned

during

the

each

ABE setting.

know and what

they

integral

of

part

of mathematical

26

so

tutoring

standards

by NCTM (1989)

* Focussing on a broad range holistic view of mathematics

process

session.

which These

that

can

also

fourteen

be school

are: think

about

teaching

and

tasks

and

mathema-

learning taking

a

* Developing problem number of mathematical

situations ideas

that

* Using multiple assessment demonstration formats.

*

Using

calculators,

require

techniques,

computers,

the

including

and

Using tors of

standardized achievement program outcomes.

NCTM proposes

*

Assessing

*

Having sole

the

to

put

what

less students

assessment purpose of

Using skills.

exercises

*

only

Using

* Excluding assessment

*

written

*

Using program

the

from

program

and

evaluation

teams

problems

only

have

sources

portfolios,

assessment information

one

correct

skills

of

many

answers

organized

requiring

and

on the

achievement

educators

various

of

computers,

standardized outcomes.

Many Math

and

on

indica-

on tests

for

by a content-

only

one

or

two

tests

calculators, process

Evaluating

only

oral,

know

counting grades

number

or word

as

a

on:

do not

be simply assigning

* Focusing on a large behavior matrix

*

emphasis

in

collecting

tests

of

written,

manipulatives

* Evaluating the program by systematically outcomes, curriculum, and instruction. *

applications

extended

basis

tests

the

(e.g.,

opinion

projects); of

of the

that

observation,

consisting

manipulative

from

test

scores

only

indicator

evaluation

various

must

journal

evaluations parties

of

data

interviews, and

the

must

involved

come

writings,

be conducted (Portia

Elliot,

1990).

CLASSROOM MANAGEMENT The the

Basic

Math

environment

ABE learners.

variety

of

ample

quantities

connecting

has

to

ABE classrooms

physical

materials of

such

links;

base

geometric

model.

rules.

fraction;

pieces;

and

need and

materials ten,

to

grid,

and

be

supplies. as

and colored and 27

with

Classrooms

pattern rods;

dot

emotionally

equipped

counters;

attribute,

spinners. graph,

physically

paper

interlocking blocks; geoboards; Simple

support a wide

should

have cubes;

tiles; balances; household

by

object,

such

cartons,

also

Emotionally have

to

feel

as

buttons;

dried

can

be used.

the

environment

safe

learners,

tutors,

community

has

to

and

respected

and

teachers

support

each

beans,

also the need other

shells,

need

to

way they to and

learning.

28

egg

be

and

supportive, are.

established. help

cartons,

each

for

milk

learners

A community The members other

in

their

of of

the

CONCLUSIONAND RECOMMENDATIONS I started

this

Curriculum

for

outline

for

tutors

for

my Basic

it

should

help

the

Math Curriculum

depending

on the

learner

ce that

social

and time

is

ABE materials

believe

that

Basic

universities focus

Basic

Math courses

the

which Math for

with

upon.

are

be very

should

ABE learners,

tutors,

of a more holistic, to the

conceptually ABE learner.

a one holistic

field

Maths/he

Math learning a great

deal

Basic

So let's of Basic

29

teaches

of life is

adult

to develop I from

NCTM and Universities

Math in ABE.

become available teachers, with

Also,

to ABE

administrators the are

other already

Math instruction, work together Math for

experien-

learners.

attention if

The

goal-oriented,

techniques

Math instructors Basic

deserves.

occurs.

learner

to collaborate

based

and

an instructor

useful

and publications

Some Basic

time

deserves

towards

field.

Basic

to the

ABE learners would

the

presented

relevant

as

The new NCTM

by ABE teachers

The adult

therefore

must work together

mathematics

come up with

classroom

research

and tutors.

more relevant

Basic

I have

some of their

makers

the

be built

and NCTM. It

could

policy

each

differ

context

an issue.

special

teachers

will

useable

math form an important

used

get

Math.

Math

a general

to be directly

school

ABE learner

comes to the

can and must

for

a Basic

to give

in Basic

If my curriculum

The Basic

adult

this

training

standards

affecting

attempted

I intend

teacher

Math.

issues

and I also

and evaluation

for

the

a curriculum.

base

curriculum

describing

ABE learners,

such

a theoretical

bais

paper

and

people

in favor which

in order

everybody.

in

to

is

ANNOTATEDBIBLIOGRAPHY

Beckwith, J. and Durkin, Girls, Boys and Math". 32-5, 1981.

J.

Berry, John W. "Learning Cultural Issues•. FLM Publishing Association,

Mathematics

Cocking

Rodney

(1981).

in

Montreal,

R. and Mestre Jose Hillsdale

..c...c.;;.._c::;..;;;..;;c..:::..:;=.;:.:=_;_;;=--:c;;.;;.;:c:=:.:..=:::..::..~•

" Science

and the Attack on Women: 6-9, September. October:

a Second

Quebec,

Language: Some Cross, 5, 2 (June 1985). Canada.

P. New Jersey,

"Curriculum and Evaluation tics". The National Council of Teachers Virginia, 1989.

1988.

Standards For of Mathematics,

D'ambrosio, Ubiratan. "Ethnomatematics and Pedagogy of Mathematics". For the Learning (February 1985). FLM Publishing Associaton,

School

MathemaInc.

its Place in the History and 5, 1 of Mathematics, Montreal, Quebec, Canada.

Elliot, Portia. "The Journal of Creative Behavior, Vol. 20, #3, Quarter, 1986". Right (or left) Brain Cognition, Wrong Metaphor Creative Behavior: It is Prefrontal Lobe Volition that Makes the (Human/Humane) Difference in the Release of Creative Potential. Ernest, Monthly,

J., et al. (1976). 83, 595-614.

"Mathematics

and

sex".

Fasheh, Munir. "Math, Culture and Authority". For the Learning of Mathematics, 3, 2, (November Association, Montreal, Quebec, Canada. Frankenstein, Committee,

Marilyn (1990). Umass Boston".

"Memorandum

American

1982).

written

Frankenstein, Marilyn (1988). Learning, U-Mass Boston.

Overcoming

Frankenstein, Application 165, #4, p.

"Critical Mathematics Epistemology". Journal

Marilyn (1983). of Paulo Freire's 315 - 339.

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to

Math Anxiety

Third for

Mathematical

FLM Publishing

The Futures

by Learning

About

Education: An of Educaton, Vol.

Frankenstein, Mathematics, Association

Marilyn (1989). "Multicultural A Different Third R - Radical Books, London, United Kingdom.

Considerations". Relearning Maths", VolumE! 1, Free

Frankenstein, Marilyn (1989). "Relearning Mathematics, A Different Third R - Radical Maths". Volume 1, Free Books, London, United Kingdom.

Association

Frankenstein, Marilyn and Powell, Arhur. "Empowering-Traditional College Students: The Dialectics of Society and Math Education",

Paper

Freire, Paulo (1987). "Literacy, Reading the Word and the World". Critical Studies in Education Series, Bergin and Garvey Publishers, Inc. Ma. Freire, Critical

Paulo (1985). "The Politics of Education". Studies in Education Series, Bergin and Garvey

Publishers,

Inc.

MA.

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