A Curriculum Suggestion for Teaching College Arithmetic-3027122-Stanley Schmidt

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A Curriculum Suggestion for Teaching College ArithmeticAuthor(s): Stanley SchmidtSource: The Two-Year College Mathematics Journal, Vol. 1, No. 1 (Spring, 1970), pp. 92-95Published by: Mathematical Association of America

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ClassroomNotes

Teaching xperience ftenproduces ome

technique r specialknowledgen mathe-

matics nstruction hichhasproved seful

in

the classroom. very

nstructoras one

or two such techniquesn his repertoire,

and it

is our hope thathe

will find his

section f theJournal suitable ehicle or

making hem vailable o

others.

A Curriculum

uggestion

for

TeachingCollege

Arithmetic

Stanley chmidt

CityCollege f

San Francisco

One of

the

greatest

ifficultieshat he

n-

structor

of

a junior

college

class

in

arithmeticaces

s

expressed y

the

ques-

tion: "How shall presenthe

content

f

the

ourse

o

the tudents?"

In

this

article wish

to

present n

approachto the teaching f these arith-

metic classes

which mightbe used

in

conjunctionwith

the

two traditional

methods

f

1) lecturing,

nd

2) assigning

problems

nd

helping

ndividualtudents.

developed

his

method,

alled

"The

Math

Tour,"

while

eaching

mathematicst

City

College

f

San Francisco

uring

he

spring

semesterf

1969.

In

its

original

orm he Math

Tour

was

used

during

he ast third

f the

semester

with classof fifteentudents nd using

no

student

ssistants.

Withthe

help

of

several f the studentsn the class,there

would

be

little

ifficulty

n

employing

his

teaching

method

with class of 50 to 70

members.

Basically,

The

Math Tour is a

pro-

gramed earning equence

whichties

the

student

not to

the text but to the

in-

structor.

ach student s

handed,

fter n

appropriate xplanation, sheet

of paper

like hefacsimileelow:

Math NAME_

San Francisco

The MathTour

Complete

nd

return.

et

ll of the

follow-

ing problems orrect nd you

advance

o

the next town on

your triparound

the

world.

f

youhave

n error his

paper

will

be returned

o

you.

You will

not

be

told

which

problem

s

wrong.

Hunt

and

find

yourerrors nd turn t inagainuntilyou

advance

o thenext

ity.

4496 699401

48

+

2861

-

248109

X 26

Those

studentswho hand

n

thefirst

sheet

with ll

the

problems

orrectly

one

receive second heet ntitled:Welcome

to

Oakland " containing two

simple

division roblems.

In

tsoriginal orm he Math

Tour on-

sisted f

52

steps.

Before

ach class

meet-

ing, wrote p enough teps n the our o

ensure

hat would tay head

of

the lass.

The

ob

of

writing p

the

tour an thus e

broken

p

into

manywork

essions ather

than

having

o be

completedntirely

n

ad-

vance.

During

he use

of The

Math

Tour,

often

verheardtudents

sking

ach

other

suchquestionss "Haveyou got o Boston

yet?"

or

stating

I'm

going

o

try

o

get

o

England oday."

feel

hat

here

s

perhaps

more

ncentive

n

this

system

f

learning

progress han

in the more

traditional

framework

f

finishing

o

manypages

n

a

given mount

f time.

Anyonewishing

o have

copy

of

The

MathTour

may

write o

me n

care

of

City

College

f

San

Francisco.

92



The

followingelectionsre from

n Mathematical

ircles y HowardW.

Eves,

published

yPrindle,

Weber

Schmidt,nc.,1969.

1060

The origin of

our word "sine."

The

meanings of

the

presentnames of

the

trigonometric unctions,

with the

exception of

sine, re

clear from heir

geometrical

nterpretations

hen the angle is

placed at

thecenterof a

circle of unitradius.

Thus, in

Figure 18, if the

radius of

the circle s one

unit, the

measures of tan 0 and

sec 0 are given

by the

lengthsof the tangent

egmentCD and

the

secant egment OD.

And, ofcourse,

cotangent

merelymeans

complement's angent,

and so

on. The

functions

angent,

cotangent, ecant, and cosecant

have been

known by

various

other names, these

presentones

appearing as late as

the

end ofthe

sixteenth entury.

The originof

the word

sine s curious.

Aryabhata called

it

ardha-jya

("half

chord") and also

yd-ardhj

"chord half"), and

thenabbreviated

the termby simplyusing yd ("chord"). From yd theArabs phoneti-

cally derived iba,

which, followingthe Arabian

practice

of

omitting

vowel

symbols, was writtenas

jb. Now

jiba,

aside

from

ts technical

significance,

s a

meaningless

word

in

Arabic. Later

writers,

oming

across

b as

an

abbreviation for the

meaningless

iba

decided to

sub-

D

C ~~~~~

o

A

FIGURE

18

stitute

aib

instead,

which

contains the same

letters

and

is a

good

Arabian

word

meaning

"cove" or

"bay."

Still later,

Gherardo of

Cremona

(ca.

1150),

when

he

made

his

translations

rom

the

Arabic,

replaced

the

Arabicjaib by

its

Latin

equivalent, sinus,

whence

came our

present

word sine.

93



1490

On the origin of > and < . During his stay of roughly a

year in America, Harriot took the opportunityto study the Indians

and to learn to speak their anguage. Upon returning o England, he

wrote a book entitledA Brief nd True Report f theNew FoundLand of

Virginia, f

the

Commodities,

nd

of

theNature

nd Manner

of

the Naturall

Inhabitants first edition 1588, second edition 1590). Captain John

White,who accompanied Harriot, made sketches f scenes and people

seen

by the two men. In the 1590 edition of Harriot's book appeared

engravingsmade by Thomas de Bry of some of the sketchesdrawn by

White. One of these

engravings hows

a

rear

view of

an

Indian chief

on whose leftshoulder blade appears the mark reproduced in Figure

22. If the small serif-likemarksare removed, and the resulting

ymbol

FIGURE

22

pulled apart in the horizontal direction, herewill appear

two

symbols

similar to those

that

Harriot

chose for

"is

greater

than"

and "is less

than." It is thus possible, as was pointed out by Charles L. Smithofthe

State

University f New York at Potsdam, that

a mark on

the

back of

an

Indian chief

suggested to Harriot

two mathematical

symbols

that

have now been

in

use formore than three centuries.

In

the absence of any statedor recordedmotivationon

the

part

of

Harriot, the above explanation

could well be the true one.

But there

is at least one feature of Harriot's early symbols perhaps militating

against

the

conjecture.

Harriot constructedhis

inequality signs

as

very

long, horizontally drawn-out symbols, and not at all like the short

stubby symbols ppearing on the Indian chief's back.

Of

course,

since Harriot

had

adopted

the

long

drawn-out

equality

sign

of

Robert

Recorde,

it could

be

that

his

long

drawn-out

nequality

signs

were so

designed merely

or

imilitude f

representation.However,

one

would

like

to think that Harriot had

a

more rational

motivation

forthe

origin

of

his

symbols

han an

adaptation

of

marks

appearing

on

the back of an Indian chief.

Such

a

rational and

easily

conceived

motivation

would

be

this:

In an

expression

like

2

=

2,

the

space

94



between the two

left nds ofthe bars of

the equalitysign s equal

to the

space

between

the two rightends of these

bars, and, also, the

number

on

the left of the

equality sign is equal

to the number on the

right.

Therefore, n designing a

symbol to

represent he qualitative

relation

between

4

and 2, say, since

the leftnumber4 is

greater than theright

number 2, why

not adopt a symbol

composed of two

convergingars, so

that the space between the two left ends of thesebars is greaterthan

that

between the two right

ends of the bars?

Because of

Harriot's

adoption of the

long equality sign, a

long inequality sign for

"is

greater than,"

composed of two

converging bars

like we have just

described,should, to circumvent

possible

misinterpretation,

ompletely

converge,yielding,over the

years,

4

>

2.

Whatever Harriot's

motivation

might

have been

for

the

origin

of

his

inequality

signs, the motivation

described immediately above

has

fine

pedagogical

value,

and

once

a

student hears this motivation

he

will neverconfoundthe two symbols > and <.

3520

Why

here s no Nobel Prize

in

mathematics. here is

a

Nobel

prize

n

several

of

the great fields

fstudy,

but none

in

mathema-

tics.

The

reason for this s

interesting.

At one time the

great

Swedish

mathematician

G.

M.

Mittag-Leffler

1846-1927)

was a man

of

con-

siderable wealth, and

in

accumulating

his fortune

he

antagonized

a

number

of people,

in

particular

AlfredNobel, who founded

the

five

great

prizes

for

annual

award forthe best work

n

Physics,Chemistry,

Physiology or Medicine, for Idealistic Literary Work, and for the

Cause

of Universal Peace. At

the

time the

prizes

were

set

up,

mathe-

matics

was

also under consideration.Nobel asked

his

advisers,

f

there

should

be

a

prize

in

mathematics,

n

their

opinion might

Mittag-

Leffler

ver win it?

Since

Mittag-Leffler

as

such

an

able and

famous

mathematician,they

had

to admit that such

would

indeed be

a

possi-

bility.

"Let

there

be

no Nobel

Prize in

Mathematics, then,"

Alfred

Nobel ordered.

Ze =0?' 9

95

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A Curriculum Suggestion for Teaching College Arithmetic-3027122-Stanley Schmidt