The date is July 24, 1964 and the time is
This is subject 13, tape 7, track B,
The subject's name is IMVHHMHMHNHHI
two 6 f clock.
3he is 20 ye&?s old and is
a senior at Carnegie ciajoring in phychology.
Instructions:
This fxm experiment is not designed
We are interested in hew people sclve problems.
to test your problem-solving ability but simply to discover wh%t methods you would
use to attack an algebra word problem.
you attack these problems.
single unknown
We are interested in how and not how well
All the problems involve one algebraic equation with a
You should not attempt solutions using more than one unknown.
A
problem is completed when you are able to dictate to me an algebraic equation which
is equivalent to the verbal statement.
In other words, you are asked to produce an
equation which if the actual arithmetic were carried out would yield the correct
numerical answer.
In order to follow your thoughts, we ask you to think aloud,
explaining each step as thoroughly as you can.
It is a well-known physchological
fact that such verbalization is not a handicap to problem solving.
presented on five by nine cards.
Problems will be
You should begin by reading the problems aloud.
Try to mention even details of the thought sequence which you consider trivial since
they may be useful in understanding your thoughts.
To help you remember your place
during the solution, I'll write down on the blackboard parts of the equation, formula,
etc, that you feel useful in solving the problem.
to write.
At the end of your solution you should dictate the completed equation to
me and I 1 11 write it on the board.
will be answered.
problem.
You nust specify exactly what I am
Once you have begun solving a problem, no questions
You must tell me when you feel that you are finished with the
Do not look to me for help, encouragement or additional information.
will not be told whether or not the solution is correct.
following
four
instructions particularly.
You
Please keep in cdnd the
Keep talking reading the problem aloud,
do not ask questions of the experimenter, specify clearly what you want written on the
board and signify when you are through with each problem.
Is that clear?
-2-
S:
Now whats, what can I ask to be written on the board?
E:
anything
S:
You said no questions after I start solving the problem.
E:
that's right.
Is that contradictory?
I'm sorry if that is.
You can write
anything on the board, formulae, parts of equations..
S:
I mean... oh., that's the way I solve it... I see
E:
It's just to make up for the fact that you don't have a pad and pencil
S:
Oh
E:
All right.
And I'd like to know what you write on the pad and pencil.
So if
you just tell me and specify what you want written down on the board, I'll write
it down for you.
I won't answer questions and I won't clarify as you read the
problems, incidentally.
Okay?
Now just to get you uh experienced with this
particular situation, I have three practice problems and then there will be five
problems.
So if you have... solve each one of these and if you have questions,
ask then, on these first three.
Practice problem No. 1.
S:
I read it out loud?
E:
You read it cut loud.
S:
If a certain number is miltiplied by six and the product increased by forty four
the result is sixty eight.
sixty eight*
Find the number.
Well, six X plus forty four equals
That what you do?
E:
Six X
S:
Plus forty four equals sixty eight.
E:
Ck^y
S:
Six X, oh, I keep... oh, that's all you want?
E:
That's all I want.
Just the equation*
0)
I'd prefer if you did try to work them
out numerically.
S:
Oh
E:
That's the final step.
And if you can tell me a little more about what you're
doing iay while you're solving .. this would be of help too.
No. 2
Practice problem
-3S:
If three more than a certain number is divided by five, the result is the same
as twice the number diminished by twelve.
What is the number? X plus three
over five equals two X minus twelve.
E:
S:
Well I don f t know what you expect me to say
E:
Well, I don't know either, possibly you'll say something more in later problems.
Practice problem No. 3»
S:
The difference between two numbers is twelve and seven times the smaller number
exceeds the greater by thirty.
Find the number.
Well you f d say, I dcn f t ...
I don't know hew else to... X ironus Y equals twelve.
Oh you just want one
equation.
E:
You're not allowed., you're not allowed to use two equations.
S:
Uh, Uh.
E:
All of these can be solved with one unknown.
S:
Well, well seven X minus Y equals thirty.
E:
That involves two unknowns.
S:
Oh.... uh... (pause)
E:
Please try to keep talking.
S:
I don't even know exactly what I'm thinking. Well, if you'd say seven, seven Y
Is that rigiit?
plus thirty equals Y but that's that's not... I guess that's the equation with
two unknowns but just using the same letter,
E:
I beg your pardon?
S:
I'm not allowed to write.
E:
Yes, so we can _____ your answer.
S:
Oka/.
(pause) I can't...
You have to write.
Write uh, X minus Y equals twelve.
Can I work with?
I can't work with
two..
E:
I'd rather .. wish you'd try ... if you can do it any other way, do it that
way but I prefer that you do it with one unknown.
and solved rathsr easily in one unknown.
I assure you it can be solved
-4-
S:
Okay, X plus twelve uh, equals seven X plus thirty.
Wait, seven., yah, wait, uh
equals seven X,. X plus twelve equals seven X minus thirty.
E:
I'm going to wait until you tell me you f re finished.
I fm finished.
All right.
My waiting
signifies nothing otherethan the fact dbkst you haven't told me you're done.
S:
Okay.
E:
And., to get everything straight, please try to keep talking, read the problem
at the beginning, solve them with just one unknown if you possibly can.
That
was not too difficult a solution with one unknown and don f t ask me any questions,
because I 1 11 try to reiBain mute.
S:
Problem No. 1
A nan has seven times as irany quarters as he has dimes.
exceeds the value of the quarters by two fifty.
(pause)
The value of the dimes
How many has he cf each coin?
Uh, how many .... Ten, ten X plus two fifty
(pause)
maybe..
E:
What are you doing now?
S:
Well now is./.I wasn f t paying any attention to the first sentence
TT tjrffl o
so., wait..
seven times as inany quarters., okay ten X equals two fifty plus seventy X.
E:
Do they...
S:
No wait, ten X equals two fifty plus seventy X? Am I allowed to figure it out
________ equation ______m
Et
No
S:
No
uh.. okay... (pause)
ten cents is the value of the dime.
fSilxkke X
is the number and that number is the value of the quarters plus two fifty.
So
the value of the quarters is is seven times X right., times twenty five so that
.... if it's a hundred and seventy five X instead c,f seventy., no couldn't be
yeh.. write that down
E:
What is ________
S:
One seventy five
(nauable) ten X,
Ten X is the value cf the dimes and that f s
equal to two fifty plus the value of the quarters which is .. yeh that's my
answer.
E:
You want to read that off ti me.
-5-
S:
Ten X equals two fifty plus one seventy five X
E:
Problem No. 2
Ss
Mr, Stewart decides to invest four thousand dollars, some at three percent and
the rest at four percent.
incomes?
How much did he invest at each wate to produce equal
Uh.. okay., point, point 03 X equals point 04 times four thousand
minus X, I guess,
point 03 X.4
E:
Problem No. 3
S:
A stream flows at the rate of two miles per hour.
rate of eight miles per hour in still water.
A launch can go down at the
How far down the stream does the
launch go and return if the upstream trip takes half as much time as the downstream
trip?
Uh.. (pause)
if the upstream trip takes half as imich time .. then ...
(mumble) ycu f d have eight ndles per... ten miles per hour...I guess you'd have
ten miles an hour going downstream, six going up? Maybe I'd better (mumbli)
okay, if you had, let's see, ten, let's see, ten... ten X equals no two times
ten X so .. now, let's see...a launch can go eight miles plus the two miles, it'd
be ten ... ten .. miles an hcur... X miles, ten miles an hour times X miles would
be ..
EJ
Would you speak a little louder please?
S:
oh., wait., upstream you'd have two times X miles equals now ixJt&E two times,
two times X miles wamtazeqmai travel/IS ten miles an hour ..is equal to the
same X ... two times X.,. twenty X... well, I'll say two, twenty X is equal to
now., the X is the mi... the distance., you have twenty times time times ten
miles an hour times X, X the distance, is equal to _____ six miles an hour
times, that wouldn't work.
Twenty times..
E:
fifould you please try to talk a little bit ______
S:
I don't know what I'm doing, that's the problem
E:
Ss
Well, if you would... I think you would go upstream at ten miles an hour and I
don't about the downstream.
Would that distance &&&
six miles?
Well you
can't answer that, can you? Well, we'll say that., uh.. X amount of time
-6-
it goes a certain distance., at six miles an hour and half the amount of time
it goes the same distance at ten miles an hour so it should be . . wait will you
write something?
Ez
Uh hum
3:
Say, Say X no but I can't, I keep using two unknowns.. I say X times six yeh times
uh, wait, X distance times six HJLles an hour plus two equals equals X times
uh, tiroes ten? ______ well, write that down, let me see that.. X is the
distance at six miles an hour ... that couldn't be right., that., all right,
the distance is what you f re trying to find., so .. say he f d go ten miles an
hour... this way I'd say six .. wait., you multiply the miles per hour times
get
the time you just /miles. (prarafr That's what I want (pause)
Es
Can you tell me what you're working on right now?
S:
I don't know.
I f m tryi... the mi... I'm tryi... if you nultiply the miles per
hour you're going times JSEKX the miles you __________ you get miles
and
$K& that's the distance... that what I wa... that's what I'm trying to find
On the upstream you're going ten miles an hour times X miles., now wait*, ten
miles an hour times, okay, wait., to get it, should say six miles an hour times
X hours wait, six miles an hour times no wait, ten miles an hour times X hours
is the number of miles and that's equal to I still get, I keep getting the same
thing. £xnxi ten X equals twelve X and that couldn't be right* Six, ten cdles
an hour times qh no, wait, I don't want that., wait ten miles an hour times
ten miles an hour times, times X hours is equal to the miles you go and that f s
equal to six miles an hour times two X hours
Can I come back to it? ...
Chhh .. (sigh) I don't even think... I don't even know if I have the miles per
hcur right cause I don't know how that would work... ____________^
Ten miles an hour times X hours is equal to ten miles an hour... so many miles..
six idbcsji miles an hour times times two X hours . . ______m is ten miles an
hour times X hours equals six miles an hour times two X hours.
E:
Problem No. 4
That's all I can..
-7S:
A board was sawed into two pieces*
One piece was two thirds as long as the
whcle board and was exceeded in length by the second piece by four
feet. How
long was the board before it was cut? Uh.. (pause) uh, two thirds
well if the
second piece was four feet longer than .. second piece was four feet
longer than
two thirds of the board .. doesn't make sense., it doesn't make sense.
, it says
one piece is two thirds as long as the whole beard and the other piece
is longer
than that. Makes more than one. One piece is twc.. (pause) well
the only
think
just
way I can think/of is if there wasn f t/sawed in half. If it was ..
if you took
uh, a piece like a rectaggle out of it and left one., and left the
length of it
there.
E:
S:
In that case, one third X equals four or .. well, that's the only
way
I can think of to dc that, one third X equals... That's assuming youtak
e out well
Well, please explain how you did it.
Okay, If you just saw it in two pieces I don't see how you can solve
the problem.
If you just, you know, saw right through it. But if you left., if
you like you
took a... uh, if you cut it, started to cut it, half the long way
but you didn't
cut all the way to the end so one piece., it wouldn't be two straig
ht pieces,
one would be shaped like an L and that would have the whole length
of the board
and the other piece would be two thirds of the length. That's ray
answer,
E:
S:
Okay, problem No. 5
ninety
wit
A car radiator contains exactly one liter of a jfo alcohoperc
l-water mixture. So
what quantity of water will change the liter to an eighty percent
alcohol
mixture? Uh.. let r see., we have., wait., ninety percent of a liter.
, wait
see., point 9, point 9 liters of a one liter and you want point eight
liters
over X, Is that, is that exactly... point nine, point nine equals
point eight
over X. Yeh.
E:
Okay. I'd like to uh drop back and get you to tell me how you did
some of these
problems. ___________ listen, hear too clearly. Let's try No.
1. Would
you explain to me what you did?
S:
Uh.. well, a man has seven times as many quarters as he has dimes
and the values.,
the value of the dimes exceeds the value of the quarters by two fifty.
So, if you
have, you have ten cents times X number of dimes is equal to uh seventirnes
that X number which is jsaoerx the number of quarters times twenty five cents,
which is the value of each quarter, plus £wo fifty,/y^knoS^, if the dimes
exceeded ihe value of the quarters..
E:
How &&&
what was that equation &&&
you gave it to me.
has changed since JOSH the last time
Want to give it to me again?
S:
Did I?
E:
Uh hum
S:
Uh, well ten .;ents is the value of the dime plus X amount of dimes, 1 mean X
number of diiues .
E:
Well tell me what to write
S:
10 X is equal to two fifty plus seven X times twenty five, or twenty five times
seven that's..
E;
Oh, I see you multiplied ____
S:
Okay?
E:
Fine, uh hum.
S:
(laugh)
E:
_______ and let us do No. 4
S:
about the piece of wood?
S:
the piece of wood.
S:
Is that supposed to be?
E:
Uh.
two is solved and then three..
did them both.
You didn't like that one very well.
Are you supposed to be able to solve that?
Tell me.
S:
E:
________ diagram .. have to do with this first problem.
that diagram?
S:
A quarter's worth two and a half times as a dime.
E:
Okay now does that Jch change anything about the problem?
S:
Uh.
E:
Well that's what I asked you.
Should it?
Because... well I have the...
Should it?
How do you interpret
3:
I don't think so cause I have X as the number, the number of coins and there's
seven times as iaany quarters, ten times the value, you f ve got the ten over
twenty five ___.
S:
Okay, let's look at No. 3 again ______there's seems to be no clue as to
what you were dcing.
S:
Cculd you try to tell me what you were up to?
I don't... I didn't know how to figure it out...well look... well a boat goes
eight miles an hour in still water plus the two miles of the flow of the
stream, okay he goes ten miles upstream.
downstream.
I don't know how to figure out the
I subtracted the two, I don't knew if that's right.
E:
Okay
S:
And I couldn't... I couldn't get any right answer cause I fug* figured it goes
.. upstream he's going ten miles an hour times wait... upstream takes half as
much time so we'll say downstream it's six miles an hour times X, X hours and
that should give me the ajoxumfc miles
E:
Uh hum
S:
And so... and that should equal ten irdles any hour times two X cause it takes..
you know, that's to balance it out.
Takes half as much time.
No wait, well
I was thinking the mi....
E:
What is... what's troubling you?
S:
I don't know... (laugh) I don't know..what I'm doing.
lou look puzzled.
If the downstream goes
six miles an hour times, tines X miles, it should be... the answer.
But the X
doesn't give yv.u even the miles, so it wouldn't be the distance would it?
mean it would be the time and not the distance.
I
So that's not right anyway.
E:
Well don't worry about that particular point, if you find the time, you can find
S:
the distance.
/okav
Ch ,,/so six miles an hour times X times should equal ten miles an hour times
two X.
It tcok half as much so tc get the equation to balance ____
E:
That isn't going to be right..
S:
_______ half as much time with ten miles an hour so it took half as much
time. So to make it equal to the same distance, you'd have to double the time,right?
-10-
T.jVn o 4*
E:
So it's six A equals/ twc times what was it?
S:
Oh waitj I had it the opposite it equals.
E:
les
S:
two times ten X.. six miles an hour times, six miles an hour times X hours
is twice as much as ten miles an hour times X hours
E:
S:
_________ .
E:
Twice as much time?
S:
Yes
takes twice as much time
'i at six miles per hour takes twice as much time?
S:
Uh hum
2:
Or it takes half as much time between the twc of them to i&ake it equal?
S:
Six miles an hour.
The trip takes... the trip at six miles an hour takes,
takes twice as much tine as the other cne so you'd.. wouldn f t you double the
other side? To ir-ake it even.
E:
Uh hum
S:
That's what I'm trying to do except I did it backwards., before
E:
S:
You did it backwards before?
I guess cause
las.. JBU& I have the six, I have the twc with the six..
E:
Okay I have.. I'd like to show you a couple more diagrams____
S:
Okay
£:
You obviously know that
S:
feh
E:
Okay.
That's diagram No. 1, that's diagram No. 2 Okay, you Know that I assume
too.
Yeh, that's pretty obvious.
Okay.
Here's diagram No. 3
S:
Yeh
E:
You notice anything about that diagram?
S:
The boat goes with the ____
Es
Okay.
This is diagram No. 4.
In particular.
-II3:
Wait, this is downstream.
The upstream ta... humm. . No I assume that ... oh
wait
E:
I wish you'd tell pe what you f re thinking in that..,
3:
No I was thinking that... oh I'm sorry.,, well the upstream,., if the upstream
trip took half as much time I thought that was when it was with the current.
E:
Say that again please.
S:
trip only
Because the upstream trip/took/half as much time I thought that that was the
trip with the current
£:
So going downstream usually means going down the stream.
S:
I don f t know if that meant..
E:
Kksd& Is that the assumption you were working on the first time you solved the
problem or when you Just solved it now?
S:
Both times.
E:
S:
3oth t~mes, but yet you put the two on a different side of the equation this time.
just
Well. It was my assumption I ./ done it work the first time.
E:
You done it wrong the first time so you were making the same assumption ___?
S:
Uh hum
E:
The other way around ______ .
current?
S:
Okay.
And you assume down,, upstream meant with the
Now how do you make this _______ ?
Well, now... well downstream I guess you f d still be going JSLEJoaxi ten irdles an
hour.,
turn., you don't have to count the turning point do you? you don't have
to count the time turning around?
is:
Oh no... no no. ____________
S:
Wait . * . downstream it, ' s going. . .
E:
S:
I don't..
E:
__________ okay let's uh _____ Problem No. 4, you didn't like very well,
did^ you?
S:
No
-12-
E:
S:
Well, if a board is cut like that
Is that supposed tc be /straight across
E:
Yeh, something like this.
3:
You know I don't understand how_____this can be four feet longer if this is
two thirds the length of the board.
E:
It is difficult to see isn't it?
S:
Yeh.
E:
Well, they, they can all be solved.
S:
Oh okay well uh no..
E:
Well, let's go on up ____
S:
Okay
E:
Is it supposed tc be., tobe able to be solved?
It's possible tc set up an equation
^__________ point I'm interested in... Now the last problem, uh let's see,
would you go to the blackboard please.
And draw me a picture of the situation
as you visualize it.
S:
frhat was i the problem I forgot.
E:
Humm it was No. 5.
B:
Draw a picture of a car radiator?
E:
Well do whatever you want.
The one about the car radiator..
S:
E:
S:
How did you think about the problem?
Did you have some kind of a picture in nind?
You have a radiator with a liter of mixture 90 percent of it is alcohol so so
the point 9 in one liter so you want point eight (mumble)
E:
Yeh, what if
3:
E:
The amount of water you added?
S:
No it would be the one
You have point eight over X
you'd have to add water
-13E:
Well look if you say that X is the liter of water plus the amount cf water you
added, then ... no that's not right..
S:
E:
Well., want to s4t down again and I 1 11 give you some more diagrams, okay.
S:
E:
________________ if ycu f d only talk more we wculdn f t have this problem.
Okay.
Does this help you any?
You already knew that.
S:
Yeh
E:
Ckay.
S:
That's what I thought I was trying to do.
E:
Well, any new insights?
S:
Wait., let me think., ninety percent in one liter, you want Bighty percent..
How about this one?
well if you put, put one plus X that wouldn't work?
E:
I'm afraid it wouldn't.
Let's....
in rny equation?
(laugh)