american printing house for the blind · american printing house for the blind ... a braille math...

American Printing House for the Blind

“Child in a Strange Country” Traveling Exhibit

Section 3: MathematicsRevised Graphic Proofs 03.08.13


“Child in a Strange Country” Traveling Exhibit Concept

Schultz

11-261 / Revisions 13-131

03-08-13

—

Mathematics Revised Graphic Proof Package

11-261 American Printing House for the Blind – Traveling Exhibit Graphic ElevationNot to Scale



Schultz

11-261 / Revisions 13-131

03-08-13

—


11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: MathematicsFinal Size: 35.4375” x 78”

As H

elen

Kel

ler’

s ed

ucat

ion

prog

ress

ed, s

he

mas

tere

d an

evo

lvin

g ar

ray

of c

ount

ing

and

calc

ulat

ing

tool

s de

velo

ped

for

blin

d le

arne

rs.

At th

e Pe

rkin

s In

stitu

tion,

she

dis

cove

red

“mod

ern”

mat

h ai

ds s

uch

as th

e Ta

ylor

sla

te

and

the

conc

ept o

f “m

enta

l ari

thm

etic

.”

To p

repa

re fo

r co

llege

, she

spe

nt m

ost o

f 18

97 s

tudy

ing

alge

bra,

geo

met

ry, a

nd s

cien

ce

at th

e Ca

mbr

idge

Sch

ool f

or Y

oung

Lad

ies.

Sh

e us

ed a

bra

illew

rite

r to

rec

ord

the

step

s

of h

er w

ork,

and

she

cre

ated

geo

met

ry

diag

ram

s us

ing

wir

es a

nd p

ins

on a

pad

ded

wor

k bo

ard.

Kelle

r’s

qual

ifyin

g ex

ams

for

adm

issi

on to

Ra

dclif

fe in

189

9 in

clud

ed g

eom

etry

and

al

gebr

a te

sts

in M

odifi

ed A

mer

ican

Bra

ille

— a

syst

em s

he h

ad le

arne

d on

ly th

e da

y be

fore

! Al

thou

gh h

er e

ffort

s w

ere

succ

essf

ul, s

he

ackn

owle

dged

that

ari

thm

etic

and

geo

met

ry

wer

e no

t her

favo

rite

sub

ject

s!

Mat

hem

atic

s

“Mis

s Su

lliva

n tr

ied

to t

each

me

to c

oun

t

by s

trin

ging

bea

ds in

gro

ups,

and

by

arra

ngin

g ki

nder

gart

en s

traw

s

I lea

rned

to

add

and

sub

trac

t.”

— H

elen

Kel

ler,

1903 Ph

oto

cour

tesy

Am

eric

an F

ound

atio

n fo

r th

e Bl

ind

MA

.01

– In

tro



Schultz

11-261 / Revisions 13-131

03-08-13

—



Illus

trat

ion

of S

aund

erso

n’s

Boar

d &

Key

dev

ice

Saun

ders

on’s

Pal

pabl

e Ar

ithm

etic

dev

ice

was

abo

ut o

ne s

quar

e fo

ot,

divi

ded

into

sm

alle

r squ

ares

. Eac

h sq

uare

was

div

ided

into

four

eq

ual p

arts

by

perp

endi

cula

r lin

es. P

ins

inse

rted

in th

e sm

all s

quar

es

repr

esen

ted

num

bers

. Stu

dent

s w

orke

d th

eir m

ath

prob

lem

s on

the

boar

d in

the

sam

e w

ay n

umer

als

are

arra

nged

on

pape

r.

“Pal

pabl

e Ar

ithm

etic

”N

icho

las

Saun

ders

on, a

mat

hem

atic

ian

who

lost

his

sig

ht a

s a

boy,

cr

eate

d th

e fi r

st k

now

n ca

lcul

atin

g de

vice

for

peop

le w

ho w

ere

blin

d. H

e ca

lled

his

met

hod

“Pal

pabl

e Ar

ithm

etic

.” P

alpa

ble

mea

ns

“tan

gibl

e” o

r “c

apab

le o

f bei

ng to

uche

d or

felt.

”

Alth

ough

he

neve

r at

tend

ed c

olle

ge h

imse

lf, S

aund

erso

n ta

ught

at

Cam

brid

ge U

nive

rsity

in E

ngla

nd, e

arni

ng th

e di

stin

guis

hed

posi

tion

of L

ucas

ian

Prof

esso

r of

Mat

hem

atic

s.

Befo

re m

athe

mat

ical

aid

s w

ere

inve

nted

, peo

ple

who

wer

e bl

ind

used

sm

all,

com

mon

obj

ects

– s

hells

, peb

bles

, or

stic

ks –

for

coun

ting

and

calc

ulat

ing.

Hum

an fi

nger

s, h

owev

er, h

ave

alw

ays

been

our

mos

t con

veni

ent c

ount

ing

tool

s, fo

r bl

ind

and

sigh

ted

lear

ners

alik

e! C

ount

ing

on te

n fi n

gers

led

to th

e de

velo

pmen

t of

the

deci

mal

sys

tem

(bas

ed o

n th

e nu

mbe

r 10

) and

the

coun

ting

tool

cal

led

the

abac

us.

Co

un

tin

g

Nic

hola

s Sa

unde

rson

(168

2-17

39)

MA_

A01

(Tex

as S

late

)

Texa

s Sl

ate

By

turn

ing

the

tact

ile ty

pes

in th

e sl

ate’

s oc

tago

nal

hole

s, u

p to

32

diffe

rent

mat

h sy

mbo

ls c

ould

be

repr

e-se

nted

.

MA

.02

– In

ter



Schultz

11-261 / Revisions 13-131

03-08-13

—



Late

r m

athe

mat

ics

devi

ces

calle

d sl

ates

us

ed p

egs

rath

er th

an p

ins.

Inve

ntor

s fo

cuse

d on

rep

rese

ntin

g nu

mbe

rs in

the

sim

ples

t way

w

ith th

e fe

wes

t peg

s.Ari

thm

etic

Sla

tes

The

Tayl

or

Slat

e

The

Reve

rend

Will

iam

Tay

lor

intr

oduc

ed h

is a

rith

met

ic b

oard

in

the

early

185

0s. W

ith th

is s

late

, stu

dent

s co

uld

mak

e 32

diff

eren

t sy

mbo

ls, i

nclu

ding

sig

ns, l

ette

rs, a

nd n

umbe

rs. T

o so

lve

a pr

oble

m,

user

s po

sitio

ned

type

in o

ctag

onal

cel

ls. T

he s

ymbo

l’s m

eani

ng w

as

dete

rmin

ed b

y th

e or

ient

atio

n of

the

type

. The

Tay

lor

Slat

e w

as th

e fi r

st a

rith

met

ic d

evic

e to

gai

n w

ides

prea

d ap

prov

al.

The

Texa

s Sl

ate

The

Texa

s Sl

ate

had

squa

re c

ells

and

use

d ty

pes

with

Ar

abic

num

eral

s si

mila

r to

thos

e us

ed in

pri

ntin

g. T

he

Texa

s sl

ate

fam

iliar

ized

stu

dent

s w

ho w

ere

blin

d w

ith

the

shap

es o

f Ara

bic

num

eral

s.

The

Pen

tag

on

al S

late

In 1

829,

Sco

tsm

an W

illia

m L

ang

of G

lasg

ow d

evel

oped

a s

late

w

ith a

pen

tago

nal p

eg a

nd h

ole

that

allo

wed

for

form

atio

n of

all

ten

fi gur

es. T

he A

mer

ican

Pri

ntin

g H

ouse

pro

duce

d an

upd

ated

ve

rsio

n of

the

Lang

des

ign

in 19

32.

Ari

thm

etic

Sla

te

Arith

met

ic S

late

, “Bo

oks

for t

he B

lind,

” Ca

ssel

l’s Il

lust

rate

d Fa

mily

Pap

er, 1

860

The

Tayl

or S

late

Key W

illia

m L

ang’

s pe

ntag

onal

sla

te

Gir

l wit

h Te

xas

Slat

e, N

ew Y

ork

Inst

itut

ion

for

the

Educ

atio

n of

the

Blin

d, 1

939

MA

.03



Schultz

11-261 / Revisions 13-131

03-08-13

—



Cu

bar

ith

m

The

cuba

rith

m, a

bra

ille

mat

h sl

ate

from

Par

is, F

ranc

e us

ed a

pla

stic

gri

d to

hol

d a

set o

f sm

all c

ubes

. Bra

ille

sym

bols

mar

ked

the

six

face

s of

eac

h cu

be, a

nd n

umer

als

wer

e fo

rmed

by

rota

ting

the

cube

s.

Ho

ff A

id

In 19

46, P

aul H

off,

a te

ache

r at

the

Min

neso

ta B

raill

e an

d Si

ght S

avin

g Sc

hool

, pat

ente

d an

upw

ard-

wri

ting

brai

lle

slat

e fo

r w

ritin

g m

ath

prob

lem

s. T

he H

off A

id p

ocke

t de

vice

wro

te b

raill

e fr

om le

ft to

rig

ht, l

ike

a br

aille

wri

ter,

rath

er th

an fr

om r

ight

to le

ft lik

e ot

her

mat

h sl

ates

.

Pate

nt il

lust

rati

on fo

r a

Brai

lle w

riti

ng d

evic

e de

sign

ed b

y Pa

ul W

. Hof

f

The

Cyp

her

ing

Bo

ard

As b

raill

e be

cam

e m

ore

wid

ely

used

, dev

ices

usi

ng b

raill

e co

de r

epla

ced

othe

r de

vice

s w

ith th

eir

own

uniq

ue s

ymbo

ls. I

n hi

s 18

67 b

ook,

Bra

ille

Syst

em fo

r the

Blin

d,

Hen

ry R

obyn

of t

he M

isso

uri S

choo

l for

the

Blin

d pi

ctur

ed a

n “A

ppar

atus

for

Wri

ting,

Cy

pher

ing,

and

Mus

ic.”

“Cy

pher

ing”

(cip

heri

ng) m

eans

cal

cula

ting

or fi

guri

ng.

Usi

ng a

Hof

f Aid

A T

aylo

r Sl

ate

By tu

rnin

g th

e ta

ctile

type

s in

the

slat

e’s

octa

gona

l hol

es,

up to

32

diffe

rent

mat

h sy

mbo

ls c

ould

be

repr

esen

ted.

A

cub

arit

hm b

raill

e sl

ate

A Ta

ylor

Sla

teA

cuba

rithm

bra

ille

slat

e

MA_

A04

(Cub

arith

m)

MA_

A02

(Tay

lor

Slat

e)

MA

.04

– Fl

ip/In

ter



Schultz

11-261 / Revisions 13-131

03-08-13

—



Men

tal A

rith

met

ic

Blin

d st

uden

ts w

ere

also

taug

ht to

so

lve

prob

lem

s m

enta

lly, o

r “i

n th

eir

head

s,”

befo

re th

ere

was

a s

ubst

itute

for

penc

il an

d pa

per

to r

ecor

d th

em. A

rith

met

ic b

ooks

w

ere

mad

e up

of w

ord

prob

lem

s.

Abra

ham

Nem

eth,

a p

rofe

ssor

at t

he U

nive

rsity

of

Det

roit

who

was

blin

d, c

reat

ed a

bra

ille

code

for u

se

in a

dvan

ced

mat

hem

atic

s. In

195

2, h

is in

vent

ion

was

acc

epte

d as

the

nat

iona

l sta

ndar

d br

aille

m

athe

mat

ics

code

. Alth

ough

Nem

eth

Code

use

s th

e sa

me

brai

lle s

ymbo

ls a

s lit

erar

y br

aille

, the

y no

w

expr

ess

mat

hem

atic

al a

nd s

cien

tifi c

mea

ning

s.

The

Nem

eth

Code

rep

rese

nts

Arab

ic n

umer

als

by

drop

ping

bra

ille

sym

bols

into

the

low

er tw

o-th

irds

of th

e br

aille

cel

l. Be

caus

e of

the

posi

tion

of th

e nu

mbe

rs, t

he b

raill

e nu

mbe

r si

gn is

not

alw

ays

need

ed to

dis

tingu

ish

num

bers

from

lette

rs. B

raill

e sy

mbo

ls r

epre

sent

mat

hem

atic

al o

pera

tions

suc

h as

plu

s, m

inus

, mul

tiply

, and

div

ide.

“…[W

]e g

ive

our

pupi

ls c

onsi

dera

ble

drill

in m

enta

l or

al w

ork.

The

cla

ss is

cal

led

to a

tten

tion

and

the

wor

k is

co

nduc

ted

in m

ilita

ry fa

shio

n. T

he p

upils

, if s

itti

ng, t

ake

erec

t pos

itio

ns a

nd w

hen

stan

ding

, hav

e bo

th fe

et o

n th

e fl o

or a

nd h

ands

by

thei

r si

des.

It is

not

icea

ble

that

whe

n a

girl

beg

ins

to fi

nger

her

cur

ls a

nd fi

x th

e bo

w o

n he

r ha

ir

that

her

min

d is

ther

e al

so. W

hat w

e ai

m fo

r is

per

fect

po

ise

wit

h th

e m

ind

who

lly c

ente

red

on th

e w

ork.”

—

Mis

s M

ary

B. S

choo

nmak

er, T

each

er

New

Yor

k In

stitu

te fo

r the

Blin

d, 19

20

Page

Rep

rodu

ctio

n fr

om T

he M

enta

l Ari

thm

etic

of W

arre

n Co

lbur

n

Page

Rep

rodu

ctio

n fro

m T

he M

enta

l Ari

thm

etic

of W

arre

n Co

lbur

n,

Amer

ican

Prin

ting

Hou

se fo

r the

Blin

d, 18

74

MA_

A05

(Men

tal A

rith

met

ic)

Nem

eth

key

and

prob

lem

usi

ng T

ACK

-TIL

ES®

Use

thes

e TA

CK-T

ILES

® to

dis

cove

r a m

ath

prob

lem

writ

ten

in th

e N

emet

h Br

aille

Cod

e. T

ACK-

TILE

S® a

re m

ovab

le ti

les

desi

gned

to h

elp

stud

ents

lear

n br

aille

.

TACK

TIL

ES® c

ourt

esy

of T

ACK

TILE

S® B

raill

e Sy

stem

s

MA_

A06

(Tac

k-til

es)

Abr

aham

Nem

eth,

Ph.

D. (

born

191

8)

The

Nem

eth

Bra

ille

Co

de

Nem

eth

key

and

prob

lem

usi

ng T

ack-

tiles

Use

thes

e TA

CK

-TIL

ES®

to d

isco

ver a

mat

h pr

oble

m

writ

ten

in th

e N

emet

h B

raill

e C

ode.

TA

CK

-TIL

ES®

are

mov

able

tile

s de

sign

ed to

hel

p st

uden

ts le

arn

brai

lle.

TAC

K T

ILES

® c

ourt

esy

of T

AC

K T

ILES

® B

raill

e Sy

s-te

ms

Page

Rep

rodu

ctio

n fr

om T

he M

enta

l Arit

hmet

ic o

f War

ren

Col

burn

Page

Rep

rodu

ctio

n fr

om T

he M

enta

l Arit

hmet

ic o

f War

ren

Col

burn

, A

mer

ican

Prin

ting

Hou

se fo

r the

Blin

d, 1

874

RW.0

5



Schultz

11-261 / Revisions 13-131

03-08-13

—



Pla

ne

Geo

met

ry

In 1

846,

John

Als

ton

desc

ribe

d m

akin

g a

tact

ile g

eom

etry

di

agra

m “

by fi

xing

a th

ick

thre

ad u

pon

the

pape

r by

m

eans

of g

lue,

with

spl

it pe

as a

t the

ang

les

to r

epre

sent

th

e le

tters

.” T

acks

and

str

ing

on a

boa

rd w

ere

also

use

d to

ill

ustr

ate

geom

etri

c fi g

ures

. Des

pite

occ

asio

nal p

in-p

rick

ed

fi nge

rs, p

ins

and

stri

ngbo

ards

are

stil

l use

d to

day

to

teac

h pl

ane

geom

etry

.

Soli

d G

eom

etry

Solid

geo

met

ry m

odel

s fo

r te

achi

ng h

ave

been

pro

duce

d si

nce

the

mid

-1800

s. G

eom

etri

c fi g

ures

em

boss

ed o

n pa

per

wer

e al

so u

sefu

l to

geom

etry

stu

dent

s.

The

Ab

acas

Des

pite

em

phas

is o

n “m

enta

l ari

thm

etic

” an

d m

athe

mat

ics

slat

es,

blin

d st

uden

ts h

ad n

o m

etho

d co

mpa

rabl

e to

sig

hted

stu

dent

s’

penc

ils a

nd p

aper

unt

il th

e in

trod

uctio

n of

the

Cran

mer

Aba

cus

in

the

1960

s. A

lthou

gh n

ot m

ade

prim

arily

for

use

by b

lind

stud

ents

, th

e ab

acus

pro

vide

d th

em w

ith a

han

ds-o

n w

ay to

cal

cula

te w

ith

spee

d an

d ac

cura

cy.

An a

ncie

nt to

ol, t

he a

bacu

s fi r

st a

ppea

red

arou

nd

800

year

s ag

o in

Chi

na. D

r. Te

rran

ce V

. Cra

nmer

, a

blin

d m

an, a

dapt

ed a

Japa

nese

aba

cus

so th

at

the

bead

s w

ould

sta

y in

pla

ce a

nd n

ot m

ove

acci

dent

ally

. The

Cra

nmer

Aba

cus

was

fi rs

t pr

oduc

ed b

y th

e Am

eric

an P

rint

ing

Hou

se fo

r th

e Bl

ind

in 19

63.

Geo

met

ric

Mod

el

This

woo

den

sphe

re is

cut

into

sec

tions

atta

ched

by

leat

her

hing

es. E

ach

sect

ion

can

be o

pene

d. T

his

mod

el, p

roba

bly

mad

e at

Cor

nell

Uni

vers

ity, w

as re

fere

nced

in th

e 18

72 a

nnua

l re

port

for t

he K

entu

cky

Scho

ol fo

r the

Blin

d.

Wor

king

wit

h an

aba

cus

Dr.

Ter

ranc

e Cr

anm

er

Geo

met

ry

MA_

A07

Tabl

etop

(Pin

& R

ubbe

r Ba

nd G

eom

etry

)M

A_A0

8 Ta

blet

op(M

itche

ll W

ire

Form

s)

MA_

A09

Tabl

etop

(Cra

nmer

Aba

cus)

MA_

A11

Tabl

etop

(Tal

king

Cal

cula

tor)

MA

.06

– Ta

ble/

Inte

r



Schultz

11-261 / Revisions 13-131

03-08-13

—


11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: Mathematics - Encased Object LabelFinal Size: 8” x 3”

Texas SlateThe Texas Slate had square-shaped cells and used types with Arabic numerals.



Schultz

11-261 / Revisions 13-131

03-08-13

—


11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: Mathematics - Tabletop Object LabelFinal Size: 6” x 3”

Cranmer AbacusUse the abacus to add and subtract.



Schultz

11-261 / Revisions 13-131

03-08-13

—



Mitchell Wire FormsMitchell Wire Forms with Matched Planes and Volumes, American Printing House for the Blind, 1951-1990



Schultz

11-261 / Revisions 13-131

03-08-13

—



Rubber Bands and PinsTry your hand at constructing a geometric fi gure. Take a rubber band from the bin and place it over the stationary pins on the board. What shape have you made?



Schultz

11-261 / Revisions 13-131

03-08-13

—


11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: Mathematics - Tabletop Instructional/Information PanelFinal Size: 7” x 14”

Beads are stationary to illustrate the correct positions. The columns of beads from right to left represent ones, tens, hundreds, and so on. The single bead above the bar represents 5 units, and the 4 beads below the bar represent single units. Beads are in use only when set against the center bar.

Now, follow the steps to do a calculation on the abacus. Your fi rst problem: 274+91 = 365.

Abacus Calculations: Step by Step

Step 1: Set 2 beads in the hundreds column for 200.

Step 2: Set 2 beads in the tens column plus the 5 bead from the tens column to make 270.

Step 3: Set 4 beads from the ones column to make 274.

Step 4: To add 91 to 274, fi rst add in the 90. Since there are not enough unused beads left in the tens column, you must add 1 bead from the hundreds column.

Step 5: Then, subtract the extra 10 to make a total addition of only 90 (by removing one bead from the tens column).

Step 6: Finally, add in the 1 by adding the 5 beads in the ones column and subtracting the 4 single units in that column. This leaves the correct answer of 365.



Schultz

11-261 / Revisions 13-131

03-08-13

—


11-261 American Printing House for the Blind – Traveling Exhibit Graphic Description: Section 3: Mathematics - Tabletop Information Panel and Object LabelFinal Size: 8” x 7”

Although the Cranmer abacus is still used, technology also plays a major role in helping blind or visually impaired students learn mathematics today. For example, talking calculators provide answers to calculations at the push of a button.

Talking Calculators

Orion TI-36X Talking Scientifi c CalculatorThe Orion TI-36X Talking Scientifi c Calculator is based on the popular TI-36 from Texas Instruments®.

american printing house for the blind · american printing house for the blind ... a braille math...

Documents