history of mathematics || the history of mathematics in the classroom

The History of Mathematics in the ClassroomAuthor(s): Paul ErnestSource: Mathematics in School, Vol. 27, No. 4, History of Mathematics (Sep., 1998), pp. 25-31Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211871 .

Accessed: 07/04/2014 11:24

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 141.212.109.170 on Mon, 7 Apr 2014 11:24:42 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=mathas

http://www.jstor.org/stable/30211871?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


THE HISTORY OF MATHEMATICS

in the Classroom by Paul Ernest

There are many reasons why using the history of mathematics in the classroom is a good idea, and this issue of Mathematics in School indicates many of them, both in theory and in practice. An historical approach can help to improve perceptions of mathematics and attitudes to it, by making it inter- esting, alive and part of human history and culture. It can show by example the importance of problem solving and processes in mathematics. It can also show up certain con- ceptual sticking points and obstacles in the development of knowledge, problems that can recur in learning.

In his survey, John Fauvel (1991) suggests that the use of the history of mathematics in teaching has the following outcomes.

" Helps to increase motivation for learning * Makes mathematics less frightening * Pupils derive comfort from knowing that they are not the

only ones with problems * Gives mathematics a human face * Changes pupils' perceptions of mathematics

In addition to improving attitudes, a benefit is that the myth of mathematics as a perfectly finished body of knowledge is challenged. This has been one of my main research interests as my recent book shows (Ernest, 1997). Working with the history of mathematics in the classroom can also show that mathematics is multicultural in origins, and not just a product of Europe as many seem to believe. It also shows the inter-disciplinary nature of mathematics-how it is relevant to every aspect of human life, from religion, politics, government and warfare to art, music, architecture and the wildest dreams of the human imagination.

Problems and problem solving are central to both the history of mathematics and the learning of mathematics. In both areas they stimulate knowledge growth. Historical problems, such as the K6nigsberg Bridge Problem, which stimu- lated Euler to create Topology (Wolff, 1963) can introduce students to network theory in today's classroom. Mathema- ticians in history struggled to create mathematical processes and strategies which are still valuable in learning and doing mathematics. Descartes (1628) offered a wonderful set of 'rules for the direction of the mind' which are as valuable today as when he first used them. Polya's (1945) heuristics have long been in use in mathematical classrooms. In the following worksheets a number of historical problems are posed as classroom tasks.

The parallel between the historical evolution of mathematics and an individual's learning of mathematics has been noted since the time of Darwin. As Freudenthal (1981) said 'The young learner recapitulates the learning process of

Mathematics in School, September 1998

mankind, although in a modified way.' Thus, the history of mathematics can suggest a fruitful order of development and can point out likely sticking points. Thus, historically the concept of zero took a long time to develop, and this is still the source of many learners' problems. Likewise, expanding the system of natural numbers to include the integers and rationals was historically a source of difficulty. Children learn that the multiplication of natural numbers (greater than 1) is a procedure that produces larger numbers. But this ceases to be true for rational numbers or integers. Multiplication no longer always 'makes bigger'. Many examples like this are indicated in the history of mathematics providing teachers with a tool for anticipating psychological problems in the learning of mathematics. For example, taking a larger natural number from a smaller (negative answer) was problematic in history for a long time; as was writing out algebraic equations symbolically and also defining functions in abstract ways without using an explicit expression. Each of these examples is a sticking point in learning mathematics, just as it was in history.

As a mathematics teacher educator I get my student teachers to design classroom worksheets drawing on the history of mathematics as a way to open their eyes to its value. The following are a few examples of what they have pro- duced. Of course some of the ideas are borrowed from other publications, but none the worse for that. None of the worksheets are flawless, but they do contain some good ideas. Credit is due to Elizabeth Ford, Richard Perring, Nicola Pinson, Nicola Squires, Kirsty Welch, and others for produc- ing them. *

References Descartes, R. (1628) 'Rules for the Direction of the Mind.' In Philosophical

Works, Vol. 1, 1995 Dover Press, New York. Ernest, P. 1997 Social Constructivism as a Philosophy of Mathematics, State

University of New York Press, Albany, New York (ISBN 0791435873 h/b; 0791435881 p/b).

Fauvel, J. 1991 'Using History in Mathematics Education', For the Learning of Mathematics, 112, pp 3-6, 16.

Freudenthal, H. 1981 'Should A Mathematics Teacher Know Something About The History of Mathematics', For The Learning of Mathematics, 21.

Polya, G. 1945 How to Solve It, Princeton University Press, Princeton, New Jersey.

Wolff, P. (Ed) 1963 Breakthroughs in Mathematics, New American Library, New York.

Author Paul Ernest, Exeter University School of Education, Heavitree Road, Exeter, Devon EX1 1LQ. e-mail: [email protected]

25



26

The

following

workcards

are

for

secondary

school

pupils

working

in small

groups

discussing

the

problems

and

questions.

Charles

L Dodgson,

better

known

as Lewis

Carroll,

author

of Alice's

adventures

in

Wonderland

was

a well

known

mathematician

and

logician.

He posed

the

following

problem..... It is obvious

that

the better

of two

clocks

is the

one

which

shows

the

right

time

more

often.

Suppose

that

you

are given

two

clocks,

one

of which

loses

a minute

a day,

the

other

doesn't

work.

Which

clock

would

you

choose

and

whya

To

prove

that

two

unequal

numbers

are

equal.....

Let

a = b + c

Multiplying

by (a - b)

gives

a2 -

ab = ab + ac - b2 - be

Subtracting

ac from

both

sides

gives

a2 -

ab - ac = ab - b2 - be

Factorizing

gives

a (a - b -

c) = b (a - b - c)

Dividing

by a - b - c

gives

a = b

Thus

a, which

was

originally

greater

than

b, has

now

been

shown

to be equal

to it. Can

this

be correcta

Discuss

and

see

if you

can

find

the

fault

in the proof.

In the

German

town

of Konigsberg

ran

the

river

Pregel.

In the

river

were

two

islands,

connected

to the

mainland

and

each

other

by seven

bridges.

A frequent

topic

of conversation

in the

town

was

whether

or not

it was

possible

for a person

to walk

from

any

point

in the

town,

cross

each

bridge

once

only,

and

return

to their

starting

point.

The

mathematician

Euler

heard

of this

problem

and

found

its solution.

Discuss

the

problem

and

see

if you

can

also

find

the

solution

with

reasons.

The

bridges

looked

like

this

MATHEMATICS IN SCHOOL SEPTEMBER 1998

l ,Ma

In the

great

temple

at Benares,

beneath

the

dome

which

marks

the

centre

of the

world,

rests

a

brass

plate

in which

are

fixed

three

diamond

needles,

each

a cubit

high

and

as thick

as the

body

of a bee.

On

one

of these

needles,

at the

creation,

God

placed

sixty

four

discs

of pure

gold,

the

largest

disc

resting

on the

brass

plate,

and

the

others

getting

smaller

and

smaller

up

to the

top

one.

This

is the

Tower

of Bramah.

Day

and

night

unceasingly

the

priests

transfer

the

discs

from

one

diamond

needle

to another

according

to the

fixed

and

immutable

laws

of

Bramah,

which

require

that

the

priest

on duty

must

not

move

more

than

one

disc

at a time

and

that

he must

place

this

disc

on a needle

so that

there

is no smaller

disc

beneath

it. When

the

sixty

four

discs

shall

have

been

thus

transferred

from

the

needle

on which

at the

creation

God

placed

them

to one

of the

other

needles,

tower,

temple

and

Brahmins

alike

will

crumble

into

dust,

and

with

a thunder

clap

the

world

will

vanish.

If the

priests

move

one

disc

every

second,

how

long

will

it take

before

the

world

endsa

(Hint:

If there

were

2 discs

the

total

number

of moves

is 22

- 1, if there

are

3 discs

the

total

is

2 -

1, etc.....)

In the

5th century

BC,

Zeno

of Elea

posed

the

problem

of Achilles

and

the

tortoise...

Achilles

and

a tortoise

have

a race.

If Achilles

gives

the

tortoise

a head

start

then

he can

never

overtake

him

as Achilles

must

always

first

get

to the

point

from

which

the

tortoise

has

just

departed.

For

example,

if the

tortoise

has

a 100m

head

start

and

Achilles

moves

at 10m

per

second

while

the

tortoise

moves

at lm per

second

then

Achilles

travels

the

first

100m

in 10

seconds

while

the

tortoise

has

gone

10m.

Achilles

takes

one

second

to cover

this

distance,

while

the

tortoise

move

Im.

Achilles

covers

this

distance

in 1/10

of a second,

but

the

tortoise

is still

1/10m

ahead.

Does

Achilles

ever

actually

overtake

the

tortoisea




EGYPTIAN

FRACTIONS

The

Ancient

Egyptians

used

fractions

to divide

their

land.

Apart

from

one

or two

special

cases,

all

their

fractions

had

a numerator

of 1.

e.g.

1/2,

1/4,

1/5,

1/9 etc.

To

get

3/4

they

did

not

just

add

1/4

+ 1/4

+ 1/4

because

they

never

used

fractions

with

the

same

denominator,

so 3/4

= 1/2

+ 1/4

1) 2) 3) 4) 5) 6) 7)

How

could

they

make

3/8a

How

would

they

make

5/12a

Write

7/10

in the

Egyptian

way.

Make

5/6

into

an Egyptian

fraction.

See

how

many

other

fractions

you

can

make

in the

Egyptian

way.

Can

all other

fractions

be written

in the

Egyptian

waya

Are

there

any

general

patterns

or rules

you

can

spota

27

FRACTIONS

IN

ANCIENT

EGYPT

Instead

of using

today's

modern

numerical

symbols

eg 0, 1, 2, 3 etc,

the

Ancient

Egyptian

scribes

used

symbols

taken

from

the

Eye

of Horus

to record

measurements

of grain.

Eyebrow Left

part

of eye

Centre

of eye

Right

part

of eye

Down-swept

line

Central

decoration

= 1/8

= 1/2

= 1/4

= 1/16

= 1/32

= 1/64

Add

all the

fractions

in the

Eye

of Horus.

Subtract

this

from

1. What

fraction

is missinga

As only

unit

fractions

(fractions

with

numerator

1) were

represented

by parts

of the

Eye

of

Horus,

all other

fractions

had

to be written

as the

sum

of two

or more

unit

fractions.

eg

3/8

was

written

as 1/4

+ 1/8

Firstly

find

the

unit

fractions

represented

by the

following

parts

of the

Eye

of Horus

and

then

write

the

fractions

as they

would

be written

today.

1. 2. 3. And

write

the

following

fractions,

firstly

as unit

fractions

and

then

as the

ancient

Egyptians

would

have

done.

1.

3/16

2.

33/64

3.

10/32

4.

17/64



5=1+1+1+1+1,

5=2+1+1+1,

5=2+2+1,

5=3+1+1,

5=3+2,

5=4+1,

5 1 1 + + 1 5 2 1 + 1 5 + 2 + 1 1 5 3 2 5 5

4+

1, 5=5,

5=1+1+1+1+

1, 5=2+1+1+

1, 5=2+2+2+1,

5=3+1+

1,5=

1 + 1 + 1 + 1 + 5 = 2 + 1 + 1 1 5 2 + 2 + 1 5 3 1 1 5 3 2 5

RAMANUJAN'S

PARTITION

OF

NUMBERS

Ramanujan

was

a great

mathematician

born

in 1887

into

a poor

Indian

family.

But

his

mathematical

genius

was

recognised

at an early

age.

His

entire

higher

maths

education

came

from

just

two

borrowed

books.

He

liked

to work

out

his

problems

using

chalk

on a slate.

This

meant

he rubbed

out

all his workings,

leaving

only

the

solutions.

He was

encouraged

to send

some

of his mathematical

discoveries

to English

mathematicians

to be evaluated.

G H Hardy

wrote

back

inviting

him

to come

to

England. Ramanujan

was

particularly

good

at seeing

relationships

and

patterns

in number

sequences.

One

of the

amazing

number

theories

that

he found

was

a formula

giving

the

exact

number

of partitions

for any

positive

whole

number.

No

other

mathematician

thought

this

possible.

A partition

is a way

of expressing

a number

as the

sum

of other

numbers.

If you

look

at a small

number

like

5.

There

are

seven

different

partitions,

as shown.

5=1+1+1+1+1 5=2+1

+ 1 + 1

5=2+2+1 5=3+

1 + 1

5=3+2 5=4+1 5=5

1.

List

all the

possible

partition

for the

number

3.

2.

Discover

all the

partitions

for the

number

4. List

them.

Check

your

answers

with

somebody

else.

Look

at the

way

that

they

have

set out

their

answers.

Can

you

think

of a way

of writing

down

all the

answers,

without

missing

anya

3.

Use

this

way

of writing

down

the

answers

for the

number

6.

4.

Find

all 15 possibilities

for the

number

7.

The

number

of partitions

gets

bigger

very

quickly!

For

example,

the

number

100

has

190,569,292

partitions.

There

are

3,972,999,029,388

partitions

for the

number

200.

You

can

see

that

finding

a formula

to find

the

exact

number

of

partitions

was

quite

an amazing

feat!

And

you

will

understand

why

he was

often

called

"the

man

who

loved

numbers".

RANGOLI

PATTERN

1

Rangoli

patterns

are

symmetrical

patterns

used

by Hindus

at their

New

Year

(Divali)

celebrations.

A

Draw

a few

lines

in one

quarter

of

a grid

B

Reflect

along

mirror

lines

vertically

and

horizontally

C

Reflect

along

diagonal

mirror

lines

-

D

Make

your

own

Rangoli

pattern.

Find

out

all you

can

about

-

Rangoli

patterns.

C|

mathematics in school september 1998 28



PASCAL'S

TRIANGLE

AND

THE

BINOMIAL

EXPANSION

MATHEMATICS IN SCHOOL SEPTEMBER 1998 29

The

triangular

table

of numbers

that

is known

as Pascal's

triangle

was

named

after

Blaise

Pascal

(1623

- 1662)

subsequent

to the

composition

of his

famous

Trait6

du Triangle

arithm6tique,

published

in 1665.

However

this

arithmetic

triangle

appeared

in China

as early

as 1303

in Chu-

Shih-Chieh's

Precious

Mirror

of the

Four

Elements

on algebra

and

it is believed

that

Omar

Khayyam

(c.1100)

had

some

knowledge

of this

triangle

too.

None

the

less,

Pascal

proposed

this

triangle

in a new

form,

shown

below

and

investigated

its properties

more

deeply

than

his

predecessors. The

Chinese

used

Pascal's

triangle

as a means

of generating

the

binomial

coefficients,

this

is,

the

coefficients

occurring

in the

formulas:

(a+b)2

= a2

+2ab+b2

(a + b)3

= a3 +

3a2b

+3ab2

+ b3

(a +

b)4

= a4 + 4a3b

+ 6a2b2

+ 4ab3

+ b4

and

so on.

When

tabulated

it looks

as follows:

1 11 121 1331 14641 15101051

and

so on

Can

you

see

how

each

row

is generateda

The

numbers

in row

five

give

the

coefficients

for

the

expansion

of (a +

b)5

and

so on for

the

subsequent

rows.

Using

the triangle

we may

write

(a +

b)5

= a5 +

5a4b

+ 10a3b2

+ 10a2b3

+ 5ab4

+ b'5

Expand

the

following

binomial

expressions,

you

may

need

to expand

Pascal's

triangle

further.

1. 2. 3. 4. 5. 6. 7.

(1 +

2X)4

(a +

b)6

(1 + x)7

(2x

+ y)3

(1 +

3y)4

(1 +

y)3

(3 +

4y)2

29

ANCIENT

NUMBER

SYSTEMS

The

same

seven

numbers

are

shown

on this

page.

They

are

shown

in Roman

Numerals;

Egyptian

Script;

Mayan

(from

Central

American);

Babylonian/Sumerian

(now

where

Iraq

is in

the

Middle

East)

and

Hindu-Arabic

script

(the

one

we

normally

use).

Cut

out

the

numbers

and

arrange

them

together

with

the

same

numbers.

Which

is whicha

What

is 47 in each

scripta

What

about

other

numbersa

Are

any

systems

confusinga ^A

A

A

< < I

CCC

S999

58

0 .

--

QD

0 0

TV

<<

yY vvy vyY

LVIII

CIL <<

~~~

["~II

A

ill ill

I

0

27

149 9

AAA

^

I

300 75

A A 1

<<<

vyY 38

,,,, 000

A

A

A

^^^11I A

II

2

TV I I

LXXV

XXVIl

XXXVIll



EULER

AND

POLYHEDRA

Euler

(pronounced

oi-ler)

was

a Swiss

born

mathematician

and

physicist

who

lived

from

1707

to 1783

and

worked

in St Petersburg.

Amongst

other

topics

he was

interested

in the

study

of polyhedra.

In your

group

examine

the following

polyhedra

Tetrahedron,

cube,

rectangular

prism,

octahedron,

pyramid,

triangular

prism,

isosahedron,

dodecahedron,

hexagonal

prism,etc.

Complete

a table

showing

each

polyhedron

in terms

of the

number

of vertices,

edges

and

faces.

(see

below)

Polyhedron

Vertices

(v)

Faces

(f)

Edges

(e)

cube

8

6

12

Is there

a relationship

between

these

three

valuesa

If you

spot

one,

write

down

a formula

for it using

v, f and

e.


MATHEMATICS.

HISTORY

AND

MUSICa

99999999999999

Pythagoras

was

a Greek

philosopher

and

mathematician.

He was

born

around

580BC

in Samos

near

the

coast

of Asia

Minor.

Pythagoras

made

a discovery

of the

underlying

mathematics

behind

the

musical

scale.

He found

that

a marvellous

connection

existed

between

musical

harmony

and

the

whole

numbers

we count.

Pythagoras

found

that

if you

plucked

a string

and

sound

a note,

then

pluck

an equally

taut

string

twice

as long,

you

will

hear

a new

note,

just

one

harmonic

octane

below

the

first.

Starting

with

any

string

and

the

note

it sounds,

you

can

go

down

the

scale

by increasing

the

length

of the

string

according

to the

simple

fractions

expressible

as the

ratios

of whole

numbers.

Pythagoras

concluded

that

'all harmony,

all beauty,

all nature,

can

be expressed

by whole-number

relationships'.

When

a string

is kept

under

a constant

tension,

the

pitch

of the

note

given

(the

frequency)

depends

on the

length

of the

string:

as the

length

increases,

the

frequency

decreases.

It is known

that:

when

the

length

is multiplied

by 2, the

frequency

is divided

by 2;

when

the

length

is multiplied

by 3, the

frequency

is divided

by 3; and

so on.

In general,

when

the

length

is multiplied

by k, the

frequency

is divided

by k.

A guitar

string

of length

60cm

gave

a note

with

a frequency

of 200Hz.

Fill

in the

table

and

show

the

corresponding

length

and

frequency

for

the

same

type

of string

under

the

same

tension.

Length

I

Frequencyf

(in cm)

(in Hz)

60

200

30

400

15

20

5

500

120

300

3 Check

in the

table

that

1 x

f is always

the

same

u How

can

fbe

found

from

1a Write

a formula:

f=........

3 How

can

1 be found

from

fa Write

a formula:

1 = ........

L Plot

a graph

to show

the

relationship

between

I and

f

A string

of length

40cm

gives

out

a note

of frequency

260Hz.

1. What

frequency

would

be given

out

by a length

of 20cm

of the

same

string

under

the

same

tensiona

2.

What

is the

relationship

between

the

length

1 cm

of a string

of that

type

under

the

same

tension

and

the

frequency

f Hz of the

note

given

outa

3.

Find

fwhen

I is 50.

4.

Find

I when

f is 200.

30



CHUI

KUNG

SUAN

(9 halls

calculation)

mathematics in school september 1998

A magic

square

is a square

in which

numbers

are

arranged

so the

numbers

along

any

row,

column

or

main

diagonal

add

up to the

same

total.

Chinese

legend

says

the

first

magic

square,

the

"Lo

Shu"

was

brought

to Emperor

Yu

in around

2200BC

on the

back

of a sacred

turtle.

1.

Can

you

arrange

the

numbers

1,2,3,4,5,6,7,8

and

9 in this

square

so that

each

row,

column

and

main

diagonal

add

up to 15.

2.

How

many

ways

can

you

do thisA

Record

each

way

that

works.

3.

Can

you

spot

any

patternsA

ai~

1.

Use

square

counters

numbered

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15

and

16

to see

if you

can

arrange

these

numbers

so that

each

row,

column

and

main

diagonal

add

up to 34.

2.

How

many

ways

can

you

find

where

this

will

workA

There

are

a possible

880

ways!

Record

each

way

that

works.

C.

In 1275

Yung

Liu

published

a book

called

"Continuum

of Ancient

Mathematical

Methods

for

Elucidating

the

Strange

Properties

of Numbers".

He

gave

methods

of making

magic

squares

as you

have

been

doing.

One

example

is shown

below:

I #

5 1

9

13

2

6

10

14

3

7

11

15

4

8'1

121

16

1.

Arrange

the

numbers

1-16

in columns

beginning

in the

top

left

hand

corner.

2.

Exchange

the

corner

numbers

with

the

numbers

diagonally

opposite

' 1->16

4<->13

3.

Take

the

middle

square.

Exchange

the

corner

numbers

6<->11

7<->10.

6 10

7 11 I

16

5

9

4

2

11

7

14

3

10

6

15

13

8

12

1

You

will

end

up with

a magic

square.

41- 1.

Try

arranging

the

numbers

1-16

in a

different

way

eg in rows

instead

columns

4

3

2

1

8

7

6

5

12

11

10

9

16

15

14

13

2.

Can

you

now

rearrange

numbers

to make

a magic

square.

Record

what

you

did.

SYMMETRY

IN

ROMAN

MOSAICS:

These

are

all patterns

from

Roman

mosaics.

On

each

mosaic

draw

any

lines

of reflections,

you

can,

using

a hand

mirror

to help

you.

Do

any

of these

mosaics

contain

rotational

symmetryA

If they

do what

are

their

ordersA

A I

--/

-

",

! 4d

On

square

paper

design

your

own

Roman

mosaic

with

either

reflectional

or rotational

symmetry.

31



history of mathematics || the history of mathematics in the classroom

Documents