P o s i t i v e EFFects oF Computer Programming
On Students' Understanding oF V a r i a b l e s and Equations
John C l e m e n t
Jack Lochhead
Elliot
Soloway **
* * Department oF
Computer and InFormation Science
Unversity oF Massachusetts
Amherst, Mass., 01002
C o g n i t i v e Development ProJect
Department oF
P h y s i c s and AstT'onomy
U n i v e r s i t y oF Massachusetts
A m h e r s t , Mas~., 0 1 0 0 2
geometry by way oF computer programming.
Underlying t h e i r view i s a r e c o g n i t i o n oF
the
importance oF " d o i n g , " oF a c t i v i t y ,
and oF p r o c e d u r e .
Educators
From Dewey
to
Piaget
have emphasized t h a t i n o r d e r
to understand a concept students need to
take an a c t i v e r o l e .
ABSTRACT
There i s a common i n t u i t i o n among those
in
computer science t h a t prograa~ing
h e l p s to
develop
good
problem
solving
skills.
Our
work
has
attempted
to
isolate
the
specific
Factors
in
programming
which
e n h a n c e mathematical
problem s o l v i n g a b i l i t y .
We have Found
that
a s u r p r i s i n g number oF c o l l e g e
students have d i F F i c u l t y w i t h very simp]e
algebra
word
problems.
However,
s i g n i f i c a n t l y more students are a b l e
to
solve these w o r d problems c o r r e c t l y in
t h e c o n t e x t oF w r i t i n g
computer programs,
than in the c o n t e x t oF simply w r i t i n g an
a l g e b r a i c equation.
We obtained s i m i l a r
results
in
comparing
the
reading
oF
algebraic
equations
within
computer
programs and the reading oF a l g e b r a i c
equations
by
themselves.
Computer
programming
a p p a r e n t l y p u t s an emphasis
precisely
on
the
active,
procedural
semantics o F equations t h a t many stude;jts
lack.
I.
This pedagogical i n t u i t i o n needs
to
be i n v e s t i g a t e d e m p i r i c a l l y so t h a t i t
can be a r t i c u l a t e d more
precisely.
A
step
in
t h a t d i r e c t i o n has b e e n made
r e c e n t l y by Howe, O / S h e a , and
PIane
[1979]
in
a s e r i e s oF experiments based
on t h e F o l l o w i n g p a r a d i g m :
a course
in
mathematics i s taught i n the standard way
without
incorporating
computP~
programming, and simultaneously, the same
course
is
taught
with
computer
programming.
Students" masterq oF the
subJect matter is
then
compared
across
the
two
groups.
In
such
experiments
t h e r e seems t o be a c o n s i s t e n t
eFFect
in
Favor
oF
incorporating
computer
programming
The
above
work
might
be
characterized
as
experiments
on t h e
"macro" level;
in
contrast,
the
work
r e p o r t e d h e r e has F o c u s s e d on t h e " m i c r o "
level.
That is,
we h a v e attempted t o
develop
topls
which
w o u l d e n a b l e us t o
isolate
specific,
critical
Factors
Introduction
i
There i s a common i n t u i t i o n among
those in computer science education t h a t
computer
programming e n c o u r a g e s
the
development
oF
good
problem
solving
skills.
Papert [1971] and the
LDCO
p r o j e c t were e a r l y proponents oF t h i s
view;
t h e y d e v e l o p e d a method
to
teoch
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©1980 ACM 0-89791-028-I/80/1000/0467
I
This research was supported i n
p a r t bg
NSF
grant
SED78-22043 in
the J o i n t
National
Institute
oF
Education - N a t i o n a l Science Foundation
Program
oF
Research
on
Cognitive
Processes
and t h e S t r u c t u r e
oF K n o w l e d g e
i n Science and Mathematics.
$00.75
467
T h i s r e s e a r c h was a l s o s u p p o r t e d i n
pawt
by
a
grant
From t h e U.S. Army R e s e a r c h
I n s t i t u t e For the B e h a v i o r a l and S o c i a l
Sciences,
ARI
grant
No.
DAHC-19-77-g-O012.
Any o p i n i o n s ,
findings,
conclusions
or
recommendations
expressed in this report
are those oF the authors,
and do not
n e c e s s a r i l y r e f l e c t the views oF the U. 5.
government.
c o n t r i b u t i n g t o the above r e s u l t s .
Thus,
r a t h e r than s t u d y i n g an e n t i r e course, we
have Focussed
on s i n g l e
problems.
We
shall
present
results
(section
]I)
concerning
the
surprisingly
puo~
performance oF c o l l e g e s t u d e n t s on two
o s t e n s i b l y s i m p l e a l g e b r a word problems.
These results
suggest several
hypotheses.
One i s t h a t t h e e r r o r s r e s u l t e d F r o m the
students' Failure to
give
a procedu]-~l
i n t e r p r e t a t i o n t o the a l g e b r a i c e q u a t i o n .
A second
set
oF e x p e r i m e n t a l r e s u l t s
(section
IV)
p r o v i d e s s i g n i f i c a n t , new
support
For
this
hypothesis.
Namely,
students
do
signFicantly
better
on
c e r t a i n a l g e b r a word problems w h e n they
occur i n a programming c o n t e x t , than when
t h e same problems occur i n a t r a d i t i o n a l ,
algebraic
(non-programming) c o n t e x t .
We
go
on
to
suggest
several
aspects
oF
programming
which
could account
Fo+" t h e
way i n which t h i s a c t i v i t y F o s t e r s a more
active
interpretation
oF
a l g e b r a by
students.
We c o n c l u d e w i t h a d e s c r i p t i o n
oF
Future
research
which
we h o p e w i l l
Further
explicate
the
benefits
oF
programming.
The e r r o r s made on problems 1 and
were l a r g e l y oF one k i n d ;
i n both cases
68% oF t h e e r r o r s were " r e v e r s a l s " :
6~ =
P i n s t e a d oF S = 6P and 4C = 5S i n s t e a d
oF 5C = 4S.
The c o n s i s t e n c y oF these
error
p a t t e r n s argues a g a i n s t the i d e a
t h a t such e r r o r s were caused
s i m p l y by
carelessness.
This
idea
is
also
discounted
by t h e F a c t t h a t r o u g h l y
half
the
s u b j e c t s were g i v e n t h e F o l l o w i n g
h i n t w i t h both problems.
"Be c a r e f u l :
some students put
a number i n t h e wrong p l a c e i n
the e q u a t i o n . "
This h i n t
did
not
have a s i g n i f i c a n t
eFFect~
it
was a s s o c i a t e d w i t h
an
increase in
the
p e r c e n t a g e oF c o r r e c t
s o l u t i o n s by o n l y 3% and 5% r e s p e c t i v e l y .
~II.
Interpretation
E.x~eriments
~
Alaeb~a
How i s i t p o s s i b l e For s t u d e n t s w i t h
such
weaknesses t o
s u r v i v e high school
and c o l l e g e s c i e n c e courses?
It
appeat's
that
these
students
have
developed
special
purpose t r a n s l a t i o n
algorithms
which
work
For
many t e x t b o o k
problem~,
b u t w h i c h do n o t
involve
anything
that
could
reasonably
be
called
a semantic
u n d e r s t a n d i n g oF
algebra.
M a n y wul-d
problems are constructed
so t h a t
they can
be
solved
through
a
trivial
word-to-symbol
matching
algorithm.
Others,
such
as
physics
problems,
~re
given
in
a highly
restricted context,
where
there
are
only
two
or
th~.ee
pretaught equations to
choose betwee11.
T h i s c h o i c e can be made e i t h e r by p i c k i n g
the
one e q u a t i o n which c o n t a i n s a l l oF
the
given v a r i a b l e s
or
through
units
analysis.
While these t e c h n i q u e s may he
p a r t i a l l y successful
in
many classroom
situations,
they
are
t o o p r i m i t i v e and
u n r e l i a b l e t o be t r u s t e d i n any
but
the
most r o u t i n e a p p l i c a t i o n s .
II~
E x p e r i m e n t s w i t h Word p r o b l e m s i n
Traditional Aloebraic Settioq
In
a
previous
study,
Clement,
Lochhead,
and Monk [ 1 9 7 9 ] uncovered two
o s t e n s i b l y s i m p l e problems w i t h
which
s t u d e n t s had g r e a t d i F F i c u l t y .
I n Table
1 we
list
the
two
problems
and
the
performance
results
gathered
From
a d m i n i s t e r i n g them t o
a group
oF 150
Freshman
students
at
a major s t a t e
university.
F u l l y 37% missed t h e
First
p r o b l e m w h i l e 73X missed the
second!
Even more d i s t u r b i n g i s t h e Fact t h a t a l l
the
students
in
this
sample were
e n g i n e e r i n g majors.
DiFFiculty
with
algebraic
manipulation
did
not
seem
responsible
For
these
results;
almost
all
students
answered correctly
problems
which tested
For
this
skill
[Clement,
Lochhead,
Soloway 1979].
N o r do we F e e l
t h a t the students" d i F F i c u l t y
could
be
explained
by
saying
that
the
problem
contained
"tricky
wording."
Evidence
a g a i n s t t h i s v i e w stems From t h e r e s u l t s
obtained
on p r o b l e m s
such as 3 in Table 1
(also
given to engineering
majors).
Here
students are given a p i c t u r e
description
oF
the
problem
and
asked
to write
an
a l g e b r a i c equation;
this
"non-language"
problem
was
missed
by 68% oF t h e
students.
Finally,
we
note
that
colleagues
at
two
other
colleges
and
u n i v e r s i t i e s have t e s t e d
similar
groups
and
obtained
comparable
results
[Kaput
1 9 7 9 a , Monk 1 9 7 9 ] .
In order to
pursue
the
source
oF
these
errors,
we
conducted
audio
and
video-taped interviews with
20 s t u d e n t s
who
were asked to think
out loud as they
worked these and other related
problems.
On t h e " S t u d e n t s
and ProFessors
" problem
we were a b l e t o i d e n t i f y
two s t r a t e g i e s
which
led to the r e v e r s a l e r r o r .
In the
F i r s t , the student
s i m p l y assumed t h a t
the
order
or contiguity
oF k e y w o r d s i n
the E n g l i s h
language p r o b l e m s t a t e m e n t
mapped d i r e c t l y i n t o t h e o r d e r oF symbols
appearing in the a l g e b r a i c equation.
For
example,
one s t u d e n t w r o t e 6S = P and
explained:
"Well,
the
problem s t a t e s
it
right
oFF:
"6 t i m e s s t u d e n t s . '
So i t w i l l be s i x
times S is
equal to professors."
468
about the
s e m a n t i c s oF t h e
algebraic
equation.
F o r example, one s u b J e c t w r o t e
'6S = 1P" and e x p l a i n e d :
In t h i s t y p e
oF
strategy,
the
student
appears t o
be
using
t h e s y n t a x oF t h e
E n g l i s h p r o b l e m s t a t e m e n t - - - and n o t
~n
u n d e r s t a n d i n g oF t h e p r o b l e m i t s e l f
.....
on which
to
base h i s / h e r
translation
process.
Weaknesses
in
this
type
oF
direct
translation
strategy
have
previously
been a n a l y z e d bg P a i g e and
Simon[19&b].
"There's
six
times
as
many
students,
which
means i t ' s
six
students
to
one
professor
and
t h i s ( p o i n t s t o bS) i s s i x t i m e s
as many s t u d e n t s
as
there
are
professors
(points
to 1P). "
On
the
other
hand,
in
a
second
incorrect
strategy,
students
acted as iF
they
did
utilize
an
accurate
representation
oF
the
meaning oF
the
problem.
However,
reversal
errors
appeared
to
arise
because oF c o n f u s i o n
When asked
to
draw
a
picture
to
illustrate
h i s e q u a t i o n , the s t u d e n t dre~
From r i g h t t o l e f t one c i r c l e w i t h a
"P"
in
it,
an
equal
s i g n , and s i x c i r c l e s
with
"S's"
in them.
SubJects
such as the
II
Problem
W r i t e an e q u a t i o n u s i n g t h e
variables
S and P t o
r e p r e s e n t the
Following statement:
"There are
~ix
times
as m a n y s t u d e n t s as
p r o f e s s o r s a t t h i s U n i v e r s i t y . " Use S For t h e number oF s t u d e n t s
and
P For t h e number oF p r o f e s s o r s .
Problem
Sample S i z e
% Correct
% Incorrect
150
63
37
2:
W r i t e an e q u a t i o n u s i n g t h e
variables
C and S t o
r e p r e s e n t the
Following statement:
"At
M i n d y ' s r e s t a u r a n t , For e v e r y Four p e o p l e
who order
cheesecake,
there
are Five people who ordered
strudel."
Let
C r e p r e s e n t t h e number o# cheesecakes and S r e p r e s e n t t h e number oF
strudels.
Sample S i z e
% Correct
% Incorrect
150
27
73
~ T o b l e ~ 3_.L
S p i e s F l g o v e r t h e Norun A i r p l a n e M a n u F a c t u r e r s and
a e r i a l p h o t o g r a p h oF t h e new p l p n e s i n t h e yard.
return
with
an
They a r e F a i r l y c e r t a i n t h a t t h e y have p h o t o g r a p h e d a r e p r e s e n t a t i v e
sample oF one w e e k ' s p r o d u c t i o n . W r i t e an e q u a t i o n u s i n g t h e l e t t e r s
R and B t h a t d e s c r i b e s t h e r e l a t i o n s h i p between t h e
number oF r e d
a i r p l a n e s and t h e number oF b l u e p l a n e s produced. The e q u a t i o n s h o u l d
a l l o w you t o c a l c u l a t e t h e number oF b l u e p l a n e s produced i n a month
i F you know t h e number oF r e d p l a n e s produced i n a month.
Sample S i z e
% Correct
% Incorrect
34
32
68
Table I
469
I n t h e above a n a l g s i s , a c o m p a r i s o n
was m a d e between two ways oF v i e w i n g
equations.
W h i l e t h e d i s t i n c t i o n may be
subtle,
it
is nonetheless critical.
We
stress
this
issue,
since,
a~
mathematically literate
adults,
it
is
diFFicult
to
imagine n o t
viewing
an
equation
as
specifying
o p e r a t i o n s on
variable
quantities.
Nonetheless,
oulinterview
data
suggest
that
this
viewpoint is
abstract
and
elusive
Peer
many s t u d e n t s .
above seem t o
use an a c c u r a t e model oF
the p r a c t i c a l s i t u a t i o n , but
they
still
Fail to symbolize that understanding with
the correct
equation.
A p p a r e n t l y such
subjects
interpret
the
reversed
equation,
'6S = P~,
as
stating that a large
group
oF s t u d e n t s
are
associated
with
a
small
g r o u p oF
professors.
To t h e s e s t u d e n t s t h e l e t t e r
"P"
stands
For "a professor"
rather
than
" t h e n u m b e r oF p r o f e s s o r s "
and the
ee..q__ua].
~iqn
exoresses
a
comoarison
or
a s s o c i a t i o n r a t h e r than
an e q u i v a l e n c e . .
The
Fact
that
the
"S"
side
oF
the
e q u a t i o n has a " b " on i t
indicates
that
it
i s l a r g e r t h a n t h e "P" s i d e which has
no m o d i f i e r .
Thus, t h e r e
appear t o
be
more S ' s
than
t h e r e are P's.
Thus t h e
student attempts to write
the
algebraic
equation
"6S = P~
as
a
"Figurative"
s t a t e m e n t , d e s c r i b i n g a p a s s i v e p_~.~.t~e
in
which
r e l a t i v e s i z e s oF t h e e n t i t i e s
are represented.
IV.
Computer Proqrams vs.
A~qeb~ajc
Equations:
Experimental Results
On t h e
basis
oF
the
Foregoing
analgsis,
we d e v e l o p e d t h e
Follow~ng
hypothesis:
i F s t u d e n t s were
placed
in
an e n v i r o n m e n t which c o u l d i n d u c e them t o
t a k e a more a c t i v e ,
p r o c e d u r a l v i e w oF
equations,
then
t h e e r r o r r a t e on t h ~ s e
problems
should
go
down.
One
clear
c a n d i d a t e For such an e n v i r o n m e n t i s t h a t
oF
computer
programming.
That
i~,
a
computer
program
is
a
definite
p r e s c r i p t i o n For a c t i o n ;
it is a set
oF
commands
which
produces
some result.
Below, we p r e s e n t e m p i r i c a l t e s t s oF t h i s
hypothesis;
i n t h e n e x t s e c t i o n we s h a l l
p r e s e n t o u r a n a l y s i s oF t h e s e r e s u l t s . .
This
contrasts
to
the
correct
equation
"S = 6 P ' ,
which
needs t o
be
v i e w e d as e x p r e s s i n g an a ~ t i v e
operatJ~+~
b e i n g p e r f o r m e d on one number ( t h e number
oF p r o f e s s o r s ) i n o r d e r t o o b t a i n a n o t h e r
number
(the
number oF s t u d e n t s ) .
]he
correct
equation,
S = 6P,
does
not
d e s c r i b e s i z e s oF t h e groups i n a l i t e r a l
o r d i r e c t manner.
Rather,
it
describes
an e q u i v a l e n c e r e l a t i o n t h a t would o c c u r
iF
one
were
to
make
the
group
oF
professors six
times
larger.
In o t h e r
words,
the
e q u a t i o n S = 6P
is
not
a
direct
description
oF
the
actual
situtation,
but r a t h e r , i t r e p r e s e n t s the
h y p o t h e t i c a l s t a t e oF a F F a i r s which ~ o u l d
result after performing the operation
oF
multiplying
the
current
number
oF
p r o f e s s o r s by
6.
While
some students
Find
the
correct equation through trial
and
error
bg
writing
the
reversed
equation
First
and
then
plugging
~n
numbers as
a
checks
our
analysis
oF
protocols
From
successful
solutions
indicates
that
the
key
to
Fullq
understanding
the
correct
translation
lies in viewing
the
number
six
as an
o p e r a t o r which
t r a n s f o r m s t h e number oF
p r o f e s s o r s i n t o t h e number oF s t u d e n t s .
One s u b j e c t who c o r r e c t l y w r o t e S = 6P
said:
Exoeriment
In
this
experiment,
our
subjects
were
17 p r o f e s s i o n a l e n g i n e e r s , w i t h 10
t o 30 y e a r s e x p e r i e n c e , who were t a k i n g a
one week
intensive
c o u r s e on t h e BASIC
programming l a n g u a g e .
At
the
beginning
oF t h e
First
day oF t h e c o u r s e , beFoT-e
any i n s t r u c t i o n had begun on BASIC,
they
were asked t o s o l v e p r o b l e m 1 i n T a b l e ~.
We were s u p r i s e d
to
Find
that
47% oF
these
practicing
e n g i n e e r s missed t h i s
problem!
On
the
second
day
oF
the
course,
after
the
s t u d e n t s had w r i t t e n
and
run
programs
using
assignment
statments,
conditional
s t a t e m e n t s , and
For-next
loops,
and
without
ang
discussion
oF t h e
answers t o t h e a b o v e
questions, the
s t u d e n t s were asked
to
s o l v e p r o b l e m 2 i n T a b l e 2.
All subjects
answered t h i s
question correctly
using
the
s t a t e m e n t LET B = ( 1 1 - H ) / 6 ( o r sonle
v a r i a n t ) i n t h e i r program.
Note t h a t t h e
Form oF t h i s s t a t e m e n t i s e q u i v a l e n t t o
t h a t oF t h e c o r r e c t answer t o
the
Fi~'st
equation.
Although
this
result
could
c o n c e i v a b l y have been due t o a
"practice
eFFect"
From h a v i n g
done t h e p r e v i o u s
p r o b l e m , we s t r o n g l y Suspect t h a t such an
eFFect alon~, c o u l d n o t be r e s p o n s i b l e Pot
so large
a jump in performance.
" I F you ~ a n t
to
even o u t
the
number oF s t u d e n t s t o t h e number
oF
professors,
you'd
have
to
have
six
times
as
mang
professors."
The e q u a t i o n i s
thus
interpreted
in
a
p r o c e d u r a l manner as
an i n s t r u c t i o n t o
a~t.
470
Experiment
In
this
e x p e r i m e n t , our
subjects
were p r i m a r i l g Freshmen and sophomores ~n
a course on machine and assembly language
programming.
This
time,
however, h a l f
the c l a s s was g i v e n problem 1 i n Table 3,
while
the
o t h e r h a l e was s i m u l t a n e o u s l y
g i v e n problem 2 i n
T a b l e 3.
The o n l y
difference
in
the q u e s t i o n s i s t h a t t h e
l a t t e r asks ~or a computer p~og~am wh~]e
the
Former
asks
For
an a l g e b r a i c
equation.
As
i n d i c a t e d in
Table
3,
significantly
more s t u d e n t s c o u l d ~oIve
problem 1 than
could
s o l v e p r o b l e m ~.
Probability
oP these r e s u l t s
on t h e
assumption t h a t e r r o r s
on each
problem
were e q u a l l y l i k e l y i s p { .05.
Experiment
The above 2 experiments
explored
thP
writinq
oF
computer
programs
o~
equations.
However,
in
the
study
mentioned e a r l i e r , Clement, Lochhead and
Monk
[1979]
observed
that
~e~di~tg~
e q u a t i o n s a l s o gave s t u d e n t s a g r e a t de~l
oF t r o u b l e .
That
is,
m a n y students
~ailed
t o w r i t e a c o r r e c t e x p l a n a t i o n oF
Problem
the
relationship
expressed
by
the
equation.
Following
the
hypothesis
o u t l i n e d above, we wanted t o compare the
results
oF
students
reading
~nd
explaining
an
equation,
which
~as
embedded
in
a
computer
program,
with
students reading
and
explaining
an
equation,
which
stood alone.
The two
q u e s t i o n s i n Table 4 were g i v e n as p a r t
oF an
11 q u e s t i o n t e s t
t o 87, m o s t l y
Freshman,
engineering
students.
The
diFFerence
between
the
groups
u,h i c h
answered one c o r r e c t l y
but
the
other
incorrectly
is
quite
interesting.
Namely,
the
group
o~
students
~ho
answered t h e
computer problem c o r r e c t l y
( p r o b l e m 2, Table 4 ) ,
but
the
equ~titm
problem i n c o r r e c t l y (p~oblem 1, Ta b l e 4)
was more than 3 t i m e s as
l a r g e as t h e
g~oup
who
answered the equation
problem
correctly,
but
missed
the
computer
problem.
T h i s d i f f e r e n c e i s s i g n i P J c a r ,t
a t the .005 l e v e l .
Here a g a i n ,
we see
that
the
programming
environment
F a c i l i t a t e d the s t u d e n t s " unde~standiT,g.
1:
g i v e n the F o l l o w i n g s t a t e m e n t :
"At t h e l a s t F o o t b a l l game, Fo~ every 4 p e o p l e who bought sandwiches,
t h e r e were 5 who bought h a m b u r g e r s . "
W r i t e an e q u a t i o n which r e p r e s e n t s the above s t a t e m e n t . Use S #or the
number o~ p e o p l e who bought sandwiches, and H For number oF p e o p l e
who b o u g h t h a m b u r g e r s
Sample S i z e
% Co~rect
X Incorrect
17
53
47
Problem
g i v e n the ~ o l l o w i n g s t a t e m e n t :
"At the last
hard liquour,
company cocktail
party,
~ol, e v e r y
6
people
t h e m e wewe 11 p e o p l e who d ~ n k
b e e r . '°
who
drank
W r i t e a computer program i n BASIC which w i l l
output
the
number oF
beer d r i n k e r s when s u p p l i e d ( v i a user i n p u t a t the t e r m i n a l ) w i t h the
number oF hard l i q u o u r d r i n k e r s . Use H Po~ t h e number oP p e o p l e who
drank hard l i q u o u r , and B ~o~ the numbel- o~ p e o p l e who drank beer.
Sample S i z e
% Correct
% Incorrect
17
100
0
Table 2
471
P ~ g b ! e m 1_:
g i v e n the F o l l o w i n g s t a t e m e n t :
" A t t h e l a s t company c o c k t a i l p a r t y , Fo~+ evewy 6 p e o p l e
hard l i q u o u ~ , t h e r e were 11 p e o p l e who d¢~nk beew."
who
drank
W r i t e a computer p~og~am i n BASIC which w i l l
output
the
number
oF
beer d~inke~s when s u p p l i e d ( v i a user i n p u t a t t h e t e r m i n a l ) w i t h the
number oF hard l i q u o u ~ d r i n k e r s . Use H Fu~" t h e number oF p e o p l e who
drank hard l i q u o u ~ , and B For the number, oF p e o p l e who d~ank beer.
Sample S i z e
% Co~ect
% Incorrect
52
69
31
P~Qblem 2_/.
g i v e n the F o l l o w i n g s t a t e m e n t :
" A t t h e l a s t company c o c k t a i l p a t t y , Foe every
b people
hard l i q u o u ~ , t h e r e were 11 p e o p l e who drank b e e r . "
who
drank
W r i t e an e q u a t i o n which ~ep~esents t h e above s t a t e m e n t . Use H Fow the
number oF p e o p l e who drank
hard liquou~+ and B Fo~ t h e numbe~ oF
p e o p l e who d~ank beer.
Sample S i z e
% Co~ect
% Incorwect
51
45
55
P r o b a b i l i t y oF these ~ e s u l t s on the a s s u m p t i o n t h a t
problem were e q u a l l y l i k e l y i s p < . 0 5
e~o~s
on
each
Table 3
II I
I
II I
III
~.~oblem ~_L
W~ite a sentence i n E n g l i s h t h a t g i v e s the same i n f o r m a t i o n
Following equation:
as
the
A = 7S
A i s t h e numbe~ oF assemblews i n a FBctc,~y.
S i s t h e number oF s o l d e r e r s i n a Fa¢+to+.y.
Problem
2_/.
P~og~am K a y a k
Input I
K= I -2
Print
K
End
For t h e above computer p~og~am d e s c r i b e i n E n g l i s h
the
mathematical
r e l a t i o n s h i p which e x i s t s between I , this number oF I g l o o s , and K, the
n u m b e r oF K a y a k s .
CQmDarison ~ . P r o b l e m ' ! and Problem
a.
b.
Numbe~ oF p e o p l e who g o t 1 compeer, b u t 2 inco~wect
Numbe~ oF p e o p l e who g o t ~ , o r . f e e t , b u t 1 i n c o r r e c t
5
18
P r o b a b i l i t y oF these ~ e s u l t s on t h e a s s u m p t i o n t h a t case a and b were
e q u a l l y l i k e l y i s < .005
Table 4
472
V~_ Whq ! Proqramminq C o n t e x t Decreases
R e v e r s a l Ewrors;
Some HUDotheses
4.
programming
a
o_.f.
~eb,u,q~
and
programming
education.
5.
The p r a c t i c e o._p. decomoosinQ
problem i n t o e x p l i c i t steps._ A
number oF s t u d e n t s solved
the
computer
program
problem
by
writing
down a two s t e p s e q u e n c e
oF o p e r a t i o n s ,
h y p o t h e s e s , any
number
oF w h i c h
could
explain
why
students
could
solve
the
in
practice
proa~ams.
While
s t u d e n t s may
n o t be encouraged t o " r u n
their
equations"
in
typic~l
mathematics
courses,
this
concept oF a c t u a l number t e s t i n g
is
an
integral
part
of
The ~ange oF e x p e r i m e n t s we have
carried
out
has
provided
us w i t h
c o m p e l l i n g evidence as t o
the
positive
c o n t r i b u t i o n oF a programming e n v i r o n m e n t
to
certain
t y p e s oF problem s o l v i n g .
These r e s u l t s a r e even more s t r i k i n g when
one r e a l i z e s t h a t , a p r i o r i ,
one would
think
that
writing
a computer program
would be more d i F F i c u l t than w r i t i n g
an
equation.
We
have
F o r m e d several
problems
better
environment:
T,he
programming
X = B/6
11-X
One i n t e r p r e t a t i o n
For
this
phemenon m i g h t be t h a t s t u d e n t s
"saw" p a r t i a l r e s u l t s "produced"
on the way t o t h e s o l u t i o n .
B =
1.
Unambiauous
semantics
proqramm,,~ q
~F
lanq.ua~e.
constructions.
While
va~'ious
m a t h e m a t i c a l symbols ( e . g . , the
e q u a l s - s i g n ) are o f t e n open t o a
variety
oF
interpretations
in
mathematics (see [ K a p u t 1979b]),
programming
languages
VI.
require
be
symbol,
t h a t o n l y one i n t e r p r e t a t i o n
associated
with
each
The e m p i r i c a l evidence
described
herein
i s i n g e n e r a l agreement w i t h t h a t
o b t a i n e d by
the
°'macro" study
kited
earlier.
However,
our
r e s e a r c h method
has a l l o w e d us t o
develop
specific
hypotheses
concerning
Factors
in
a
programing language which
contribute to
improved p r o b l e m s o l v i n g .
C u r r e n t l y , ~e
are c o n t i n u i n g t o v i d e o - t a p e s t u d e n t s as
T h i s Fact i s u s u a l l y
emphasized
in
programming
language
instruction.
For
example,
the
meaning oF ' = ' i n ' I = I + I ' i s
explicitly
d e f i n e d as an a c t
oF
replacement,
i . e . , the v a l u e oF
the r i g h t s i d e oF the
equation
becomes the
new v a l u e oF the
v a r i a b l e on the l e ~ t .
A l so , t h e
interpretation
oF v a r i a b l e s ~s
clear,
i.e.,
they
stand
For
numbers which a r e a c t e d on by
operators.
2.
they
3.
Viewin9
a_n
prooramminq
students".
~'equatiqn"
lanquaqe
ir_~
a_s
a!~
active
input/output+
transformation.
That
is,
the
r i g h t hand s i d e oF the
equation
(the
input)
is
o p e r a t e d on t o
produce a v a l u e
For the
left
hand s i d e
(the
problems,
and
hope
to
In
summary, r e s u l t s
From written
tests
and c l i n i c a l i n t e r v i e w s have shown
that
m a n y science
oriented
college
s t u d e n t s have s e r i o u s d i F F i c u l t y w i t h t h e
semantics oF a l g e b r a i c
notation --- a
diFFiculty
in l e a r n i n g to view equations
as
active
operations
on
variable
quantities
rather
than
as
statements
which d e s c r i b e a s t a t i c
scene.
Perhaps
t h i s i s n o t so s u r p i s i n g , c o n s i d e r i n g t h e
s t r o n g emphasis i n secondary s c h o o l s on
e q u a t i o n m a n i p u l a t i o n i n word problems.
Symbol
manipulation
rules
can
theoretically
be l e a r n e d i n
school as
"legal"
p a t t e r n s oF l e t t e r
movements
without
any
semantic
underpinning.
Computer programming, however,
puts
a
n a t u r a l emphasis p r e c i s e l y on the a c t i v e ,
p r o c e d u r a l semantics oF e q u a t i o n s t h a t so
The Fact
that
one must w r i t e
"6-S ~ r a t h e r
than
s i m p l y "6S ~
m i g h t serve
to
prompt
one t o
view t h a t expression o p e r a t i v e l y
as meaning " s i x t i m e s the number
oF s t u d e n t s " r a t h e r than F a l l i n g
i n t o the
error
oF v i e w i n g i t
as " s i x
solve
establish exactly
which
aspects
oF
programming
are
most
important
to
overcoming the
reversal
error.
Our
p r e l i m i n a r y c l i n i c a l r e s u l t s i n t h i s area
p o i n t t o F a c t o r s 1,2 and 3 above as
the
most i m p o r t a n t , but F u r t h e r c l i n i c a l d a t a
is required to confirm this observation.
E.xplicitoess
required
.~L
the
s q n t a x .~. p r o a r a m m i n q ! a n q u ~ q e ~ .....
descriptively
Conc.l.udinq Remarks
output).
473
many students a p p a r e n t l y lack.
]hu~,
w h i l e ouw c u r r e n t ~ e s u l t s must be viewed
as p r e l i m i n a r y , they d i ~ e c t l g suggest
that
it
would
be
beneficial
to
incorporate
computer programming i n t o
high
school
algebra
courses,
and,
we
suspect,
i n t o other mathematics courses
as w e l l .
ReFerences
Clement, d., Lochhead, d., and Monk, g.
(1979)
" T r a n s l a t i o n D i F F i c u l t i e s in
Learning
Mathematics,"
Technical
Report,
Cognitive
Developmer+t
ProJect, Department oF Phgsics and
Ast~onomg,
Universitg
oF
Massachusetts, Amherst.
Clement, d., Lochhead, d. and Soloway, E.
(1979)
" T ~ a n s l a t i n g BetweenSymbol
Systems: I s o l a t i n g Common D i F F i c u l t y
i n Solving Algebra Wo~d Problems,"
COINS Technical
Report
79-19,
oF
Computer
~nd
Department
InFormation Science,
U n i v e r s i t y oF
Massachusetts, Amherst.
Howe,
J . A . M . , O ' S h e a , T.
and
Plane,
d.
(1979) "Teaching Mathematics Through
Logo P r o g r a m m i n g , "
DAI
Research
Paper
115, D e p a r t m e n t oF A r t i F i c i a l
Intelligence,
University
oF
Edinburgh.
Kaput,
d.
(1979a) Personal communication.
Kaput,
d.
(1979b)
"Mathematics
arid
Learning:
Roots
oF E p i s t e m o l o g i c a l
Status,"
Coqnitive
P~ocess
Instruction
(d.
Clement and d.
Lochhead, Eds.), F r a n k l i n I n s t i t u t e
Press, Philadelphia.
Monk, O.S. (1979) Personal communication.
Paige, d. and Simon, H. (1966) " C o g n i t i v e
Processes
in Solving Algebra Word
Problems," Problem S~)vinq R e s e a r ~ L
Method
and
Theory
(B.
Kleinmutz,
Ed.), dohn Wiley and Sons, New York.
Papert, S. (1971) "Teaching C h i l d r e n t o
be Mathematicians versus Teaching
a b o u t M a t h e m a t i c s , " MIT A I Lab
Memo
249, C a m b r i d g e .
474