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
Stable URL: http://www.jstor.org/stable/3027122
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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
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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
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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
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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