The Arithmetic Connection
Author(s): Lesley Lee and David Wheeler
Reviewed work(s):
Source: Educational Studies in Mathematics, Vol. 20, No. 1 (Feb., 1989), pp. 41-54
Published by: Springer
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LESLEY LEE AND DAVID WHEELER
THE ARITHMETIC CONNECTION
ABSTRACT. From test and interview data obtained during an investigation into Grade
10 students' conceptions of algebraic generalisation and justification, we have extracted
evidence of the extent to which these students have coordinated the "worlds"of arithmetic
and algebra, and can move freely between them. The data show more dissociation than
we expected, even among students who were successfulat standardalgebraictasks. Conceiving algebra as "generalisedarithmetic" may obscure the many genuine obstacles that the
learner has to overcome in moving from fluent performancein arithmeticto fluent performance in algebra while achieving and maintaininga smooth coordination of both modes of
action.
Historically,algebragrew out of arithmeticand
it ought so to grow afreshfor each individual.
(Mathematical Association, The Teaching of
Algebra in Schools, p. 5)
Our investigations into the algebraic thinking of high school students
show that the connection between the algebra and arithmetic in their
minds is not always as direct and transparentas the quoted precept might
suggest. Introducing algebra to beginning students as "generalisedarithmetic" may be a sensible strategy but there are distinct pedagogical
difficulties to be faced if it is adopted. Our account of some of the
misconceptionsthat can occur at the arithmetic/algebrainterfacemay give
some pointers to the characterof these pedagogicaldifficulties.
For this article we have drawn on the results of an open-ended test
administeredto 350 Grade 10 students (ages 15-16), on interviewswith 25
of these students, as well as some interviewsfrom a pilot project undertaken two years ago. Our main study was designed to examine students'
conceptions of generalisationand justification. Preliminaryfindings have
been reported in Lee and Wheeler (1986), and a fuller report is available
in Lee and Wheeler (1987). Here, in this article, we focus on what our
data tell us about the coordination of arithmetic and algebra in the
students' minds as this is indicated by the extent to which they seem able
to move into and out of each of these worlds at will. In general we find
serious lapses in their coordination of these two worlds though of course
not all students exhibit or express the same degree of dissociation between
them.
EducationalStudies in Mathemalics20 (1989) 41-54.
? 1989 by KluwerAcademicPublishers.
42
LESLEY LEE AND DAVID WHEELER
MOVING FROM ALGEBRA INTO ARITHMETIC
One of the four types of questions we used in testing and interviewing
students confronted the student with algebraic statements such as:
2x +l
2x+1+7
1
1
1
1
8' 6n 3n 3n'
and (a2+b2)3=a6+b6.
We asked in each case "Is the statementdefinitelytrue?possibly true?never
true?Say how you know." There are two types of response to the request
to "say how you know". For the first statement, about equal numbers of
students solved the equation to get x = 0 or quoted a rule concerning the
cancellingof the 2x's. In the next, studentseither took the left hand side of
the statementand manipulatedit, correctlyor incorrectly,to see if it could
be made to look like the right hand side, or they quoted an algebraicrule
or law (often of their own invention)which they felt covered the situation.
Half of the students given the third statement justified its "truth" by
producingan exponent law and a third of these recognisedthere would be
"mixed terms". Of the 268 students given one or other of these problems
only 10 made any attempt to "check with numbers"and only one of these
was responding to the third question involving exponents where the
production of a single counter-example would have been much more
efficientthan the cubing of a2 + b2.
The rule-boundapproach of these students can be illustratedby looking
at responses to this third statement. In defence of the truth of the
exponential statement we were offered some very elaborate explanations
such as this written one given by a student from an acceleratedgroup:
"This statement is definitely true. There are several laws in dealing with
exponents. And the one that applies here is you multiply the number
(outside the bracket) with those exponents inside the bracket. You don't
add themlike you normallydo. If you had an examplelike a2 + a3, you
add them so you get a' but the bracket tells us to multiply."
or this oral defence of the truth of the statement:
"Well the numbers and letter or letters that the number is attached to is
timesed by the number outside the bracket. Since there is a plus sign it
takes each number separately. Let's say the equation read as follows
(a6+ b6)3 then the answer would read a'8 + b'8 but if the equation read
(a4b4)4then the answer would read a 16b16, without the plus the numbers
are sort of pulled together.The reason I believe this is true is becauseif you
took (a2 + b2)3 and added each one like this (a2 +b2)(a2 +b2)(a2+b 2)
THE ARITHMETIC CONNECTION
43
then by adding each a-equation in each bracket your answer would read
a2 + a2 + a which equals a6 and the same goes for b. It all depends on the
sign and the number outside the brackets."
Other students were content to say "It's a rule" or to write a2x3 + b2x3.
Rules were also produced to justify a' + b5 and a8 + b8.
And yet studentsgave some hints that they see these "rules"as somewhat
arbitrary. One student who had given the rule which would make
(a2 + b2)3= a8 + b8 said: "On the otherhandI couldforgetgrade 8 mathand
it couldbe that whenyou have a numberinside the bracketyou multiplyit by
the outer number. . ." whichwouldgive a6 + b6. Otherssaid, "It's definitely
true,from whatwe havelearnedin school but it isjust a theorylike everything
in math"; "The statement is definitelytrue if I were to base my answeron
today'smathematics."One studentindicatedthat if the expansionweredone
by findingthe product of (a2 + b2) (a2 + b2) (a2 + b2) therewould be mixed
terms, but if it were done by multiplyingexponents, a2x 3+ b2X 3, then the
statement as given would be true.
The one student who did substitute numbers also succeeded in demonstrating the truth of the statement. Using a = I and b = 2 he wrote
(12+22)3(
+4)3= 1+64=65.
In spite of the use of numerals the
student did not move into arithmeticsince the 1 and the 4 were not added
beforecubing. One might say that he too remainedin the world of algebra.
In fact all the students who worked this question did so from a strictly
algebraicperspectivein spite of the definiteadvantagesof looking at it from
an arithmeticperspectivewherea single numericexamplecould immediately
settle the matter.
In the interviews,when studentsproducedan erroneous"rule"conclusion
we asked them to try a few numbers.One might say we pushed them into
the world of arithmetic. The substitution request was often made quite
specific in that simple values were suggested for the variables. Where
students had declared algebraic equality, substitution with numbers produced inequality.This did not always generate the expecteddisequilibrium
in the students. Some students when questioned about this indicated that
they did not reallyexpect the same result in arithmetic.A studentdoing the
(a2 + b2)3 problem in our pilot project explained that numbersand letters
behaveddifferentlyand so could be expectedto give differentresults.In most
of the interviews,however,we told studentsthat both their resultscould not
be correct and in essence forced them to choose between their algebraand
their arithmetic. Most in this case voted for the result given by their
arithmeticand suspectedthat in the algebrathey had used the "wrongrule".
44
LESLEY LEE AND DAVID WHEELER
In the case of the statement
2x+ 1
2x+1+7
1
8'
we asked all three interviewstudentswho cancelled the 2x's to check their
response for a certain value of x. This led to general confusion. One said
"It's not the same whenyou substitutea number",but confronted with the
possibilitiesof crossing out or not, he tended to favor crossing out. On the
other hand he did waver, considering,"This is a whole equationso I can't
just take part of it and cancel out." He realized that he did work with the
numberexample and the algebra in differentways. When asked "Whichis
the right way to test that one?", he replied,"That's whereI'm always stuck
in math." Another student who expected her substitution x = 1 to work
out, immediately questioned her algebraic cancelling when it didn't. She
said, "I guess I shouldn'thave takenthe 2x and 2x outfor some reason",but
didn't really know why. She admitted that in arithmeticyou can't cancel
the 2's (she was putting x = 1) "becauseof the rest of the stuff aroundit. It
gets in the way". In her second attempt at the problem she cancelled the
2x + 1 and referringto her previous attempt, said "Iforgot to cancel the I
in thefirst try". When a second substitutionattempt did not work out she
eventuallyrememberedthat "I can only cancel withmultiplying".The third
student also suspected that cancelling the 2x's was the problem after two
numerical examples didn't work out. She too could not say why and
decided to abandon the problem.
Several points are worth underlining here. Firstly, not one of these
students(like their test counterparts)had establisheda "substitution"reflex
to check their work. Secondly, when a check was imposed on them it did
not seem to serve as a correctivetool. Rather it placed them in a dilemma
and seemed to force them toward a choice between their algebraic and
arithmeticbehaviors. Two of the three intervieweesseemed to trust their
arithmetic behavior slightly more but could not use it to resolve the
dilemma or to throw any light on their algebra. The third seemed to be
used to living with arithmetic/algebracontradictions but leaned more
toward algebraic behavior.
We see here another illustrationof how algebra and arithmeticare two
dissociated worlds for these students. They do not spontaneouslythink of
going from algebrainto arithmeticand when they are pushed to do so their
algebrais not instructedby their arithmeticas one would suppose it ought
to be if they perceivedalgebra as generalisedarithmetic.
THE ARITHMETIC CONNECTION
45
A problem of a type similar to the above three was worded slightly
differently. Students were given an erroneous algebraic development of
5/(2 - x) + 5/(2 + x) = 4 which led to the conclusion that 20 = 4 and asked
to "explain this result". Only a quarterof the students we tested indicated
either explicitly or implicitly that 20 cannot be equal to 4:
"As far as I know in mathematics 20 cannot be equal to 4. Maybe
5/(2 - x) + 5(2 + x) # 4."
"The answer is fine. It works. But it is not logical. 20 cannot = 4. Plain
numberscan't equal other plain numbers!"
A third of the students went along entirelywith the erroneous algebraic
developmentand its conclusion, 20=4:
"Well by following all the proper proceduresto get to the result of four
should explain everythingfor instance 5 x 4 = 20 which is true. But if you
want step by step explanation how you got to this answer well then,
step.
1) find lowest common denominatorwhich is (2- x)(2 + x)
2) then multiply them together with 5.
3) You'll end up with 5(2 + x + 2- x)
4) which then equals 5(4) = 4
Result 20 = 4."
"Then 5(4) 5 x 4 is equal to 20 which shows your value for 4. In this case
4 is used as a variable not using its original meaning."
And an eighth of the studentsfelt that one cannot stop at 20 = 4 but should
either simplify this:
"The answer will be 4-20 or 20- 4"
"Dividing by 4, 5 = 1"
"0 = -16"
"Yes it could be 20 = 4 or 1/5 dependingon the sort of answerasked for."
or introduce an x into the answer.
"Results should be x = 4." "x = 4 or x = 20"
As in the previous problems,students gave a justificationby rule for the
algebraicdevelopment. That these "rules"could lead to a result which is
nonsense in arithmeticdid not appear to be a problem for the majorityof
these students. For many the main problem with "the answer" was that
LESLEY LEE AND DAVID WHEELER
46
they did not get a value of x. It was the unexpectedalgebraicresult that led
many students to question the algebraic development, although very few
were able to identify the error. Once again students behaved as though
algebra were a closed system untroubledby arithmetic.
The general acceptance of 20=4 was not the only manifestation of
something having gone wrong arithmetically.On looking at the arithmetical work done in this series of four problems, one suspects that some
regressionhad taken place. One does not expect even grade seven students
to exhibit some of the arithmeticalbehaviourof these grade 10 students.As
we have remarked before, some students did not even seem to see the
possibility of effecting simple sums and differencesof whole numbers:for
example,(I + 4)3 waswrittenas I + 64, (4 + 9)3 as 43 + 93, andthe (8 - 1)
in the expression 2 x (8 - 1) was not read as 7. Some work with fractions
was also strange. A student who was putting n = 1/4 in the expression
1/6n - 1/3n = 113nwrote the following:
61
6-)
141114
=
3( ')
Zil)
*-
=3
6--
4
Examplesalso occurred of the "classical"error of subtractingfractions by
subtractingtheir numeratorsand denominatorsseparately.One student we
interviewedwas asked to do 1/6 - 1/3. Subtractingnumeratorsand denominators she got 0/3. Similarlyshe reduced 1/4 - 1/2 to 0/2. Then looking at
her algebraic work (where 0 had been replaced by 1) she changed the
numerators in both cases to I because "0 somethings still has to be
something".Yet when given 1/3 - 1/6 she proceeded to find a common
denominatorand correctlycarriedout the subtraction.Asked why she used
a different method here, she said "3 minus 6, 1 can't do it". Zero is a
number that takes on a mysterious realm of meanings for these algebra
students. The above student felt that when 0 occurredin a numerator it
ought to be replaced by 1. Many students did not feel 0 is a full-fledged
candidate for a solution to an equation. Some interpreteda 0 solution as
the null set or as an exception: "No becausefor 0 it wouldbe true because
Omakes everything0", "0 wouldbe the only exception", "x belongs to the
null set except whenx is 0". One student used 0/4 in a long substitution
attempt without ever treating it as 0 and another said that an x cannot be
equal to 0. Not only did these students move with great difficulty from
algebra into arithmeticin these problems but their arithmeticappearedto
have been disturbed by their algebra.
THE ARITHMETIC CONNECTION
47
MOVING FROM ARITHMETIC INTO ALGEBRA
If we look at another series of problems which requiredstudents to move
in the opposite direction, from arithmetic into algebra, we see that very
similar problems occurred. We refer to three problems, one involving
consecutive numbers,
The sum of two consecutivenumbersis always an odd number.Theproduct
of two consecutive numbers is always an even number. Are these two
statementstrue?If they are can you show why?
and two others of the "pick-a-number"type,
A girl multipliesa numberby 5 and then adds 12. She then subtractsher
starting numberand dividesthe result by 4. She notices that the answershe
gets is 3 more than the numbershe startedwith. She says, "I thinkthat would
happen,whatevernumberI started with."Is she right?Explaincarefullywhy
your answeris right.
Choose any numberbetween 1 and 10. Add it to 10 and write down the
answer. Take thefirst numberawayfrom 10 and writedown the answer.Add
your two answers.Whatresultsdo you get? Willthe resultbe the samefor all
starting numbers?Explain why your answeris right. (See Note.)
These problems did not explicitly request the students "to use algebra"
although students were aware that this was an algebra test. Three-quarters
of the 352 students who were given one or two of these problems on their
test did not use any algebraat all. In the consecutivenumbersquestion, for
example, 78 of 118 students avoided any algebraicwork, most giving one
or two examples as a demonstrationwith about half of these giving some
even/odd arguments.Typical examples of this were:
"Well: 2 + 3 = 5v/
3 + 4 =71
4+5=9v/
5 + 6 = 1I1v/
Also: 2 x 3=61
3 x 4= 12v
4 x 5=20v/
5 x 6 = 30v/"
"Let I + 2 be the two consecutive numbers.- this adds up to 3 - odd.
2 x I - let's use the same numbers.
=2 this is an even number.
Yes these two statementsare true
Using the same numbers 1 and 2.
You can see added = 3 odd
multiplied= 2 even."
48
LESLEY LEE AND DAVID WHEELER
When this problemwas worded "Show, using algebra,that the sum of two
consecutivenumbersis always an odd number"27% still avoided algebra,
giving examples again. And when we examine the work of the 25% and
73% respectivelywho did in fact write some algebraicsymbols (generallyx
and x + 1) on their papers, we see that very few of these students actually
carriedout the complete algebraicdemonstration.For example,
"x + (x + 1) =
2+(2+ 1) =
2 + 3 = 5,/odd
3+(3+ 1) =
3 + 4 = 7,/odd
x(x + 1) =
2(2+1) =4+2=6/even
3(3 + 1) = 12,/even"
In fact only 10% of the students successfullyused algebra to demonstrate
the odd-ness of the sum of consecutive numbers.
Results were similar on the other two questions. Only 44 of the 116
studentsgiven the "girl question"used some algebraand half of these used
their algebraic formulation only to create examples. When the same
question was asked in the form "Using algebra, show that she is right", a
third of the students still wrote no algebraicstatements.Very few students
actually carriedout a complete algebraicdemonstrationbut relied instead
on a few examples. The "choose a number" question elicited the lowest
algebraic response rate with only 9 of 118 students even attempting an
algebraicexplanation.
The series of problems asked the students to establish some fact about
numbers. We expected of course that students would move from the
arithmetic number situation into the algebraic in order to establish the
arithmeticgeneralisation.We found that the studentstended to stay in the
arithmeticmode when the problem involved numbers. In fact their reluctance to leave the arithmetic mode here was almost as strong as their
reluctanceto abandon the algebraicmode in the first series of questions.
Students do not appear to see algebra as generalisedarithmetic.Indeed
the question arises whether or not they believe that arithmetic can be
generalised.There were indications in both the interviewsand tests that
they do not. For instance many of the students who did use some algebra
in this series of questions, when asked whether their algebraic work
demonstratedthe truth of the proposition, responded negatively or indicated that examples would be preferable.One student, when confronted
with a completealgebraicproof said that if she were explainingthe problem
to a friend she would "solve it" with algebra and then "prove it by
examples".Another studentwho had been pushed and proddedthroughan
THE ARITHMETIC CONNECTION
49
algebraicdemonstrationsaid "It works . .. 'CauseI tried it witha number".
The algebraic work was often considered on a par with a single numeric
example or, in a few cases, rather less reliable.
Even when students believedgeneralisationwas possible they did not see
algebra as a tool for establishing such a generalised statement about
numbers. Comments such as:
"This statementcould possibly be true because with numbersthere is great
variations in the way that you go about solving any math question.
Numbers have differentpossibilities."
"Numbers do go on and on and we can't check them all."
"I can't go on tryingevery single numberto find out ... numberscan go on
to a certain extent."
"You would never think or realize that you can have statementsthat are
always true no matter what numbersyou take."
certainlymake one wonder whether these students believe that generalisation is possible.
VARIATIONS IN THE EXTENT OF ARITHMETIC/ALGEBRA
DISSOCIATION
There were of course considerableindividual differencesin the degree of
algebra/arithmeticdissociationexhibitedby the students;many of these are
evident in the examplesgiven above. Those studentswho were quite willing
to accept and who even expected differentanswers in their algebraic and
arithmeticsolutions to a problemcould be said to have the highest degree
of algebra/arithmeticdissociation. An example from our pilot study is
typical of this. (See Note.)
5
c
2
Asked for the area of the rectangularshape with sides 5 and c + 2, the
student replied "lOc".
S: I just took length times width and I took 2 times 5 is 10 and I just put
IOcwhatever c is for the variable c.
50
LESLEY LEE AND DAVID WHEELER
I: Okay, um, let's suppose,just for the sake of argument,that c is equal to
4. Okay? Now, what would your answer give if c is 4?
S: I'd put 30.
I: Good. Uh, how did you get 30?
S: Well 4 and 2 is 6 and 6 times 5 is 30.
I: Okay. Now does that agree with what you would give if you gave that
answerwith c equal to 4? Forget the figure.What would that (points to
1Oc)be if c was equal to 4?
S: It would be 40.
I: Um, so one way you get 30 and the other way you get 40.
S: Yeah.
I: Do you want to change your ... if you want to have another go at an
answer then you can.
S: No.
Students such as the one above felt no need to justify or correct incoherences that arose when they were asked to check their algebrawith numbers.
Nor did they see any link between their arithmetic work in the second
question series and the algebraicsolution that was either imposed or done
for them. One student for example after having been dragged through an
arduous algebraic demonstration was asked whether it meant that the
generalisation was now true. Totally ignoring the algebra she replied
"Probably,if you pick a numberout of anywhereand they do workout well,
there's a major chance that all the other numbersare going to work out as
well."9
Other students seemed to expect some degree of algebra/arithmetic
coherence.In the first series of problemsthey appearedslightly uncomfortable or confused when algebraand arithmeticgave two differentresultsand
wondered whether one or the other was wrong. For instance, all three
students we interviewed who had cancelled the 2x's in the problem
involving
2x + I
2x+1+7
1
8
immediately realised something was wrong when they were asked to
substitute a number for x: "Oops", "I-guess I shouldn'thave taken the 2x
and 2x out for some reason", "It's not the same when you substitute a
number... That's where I'm always stuck in math". Similarly, in the
problem which concluded with 20 =4, 24 of the 89 students given this
question indicatedeither explicitly or implicitlythat 20 # 4 and 13 of these
embarked on a check of the algebra in an attempt to find the error.
THE ARITHMETIC CONNECTION
51
A third group of studentsimmediatelyrejectedtheiralgebraicwork when
a numeric substitution did not work out. A couple even suggested that
algebracould be checked by tryingsome numbers.One student checkedthe
statement (a2 + b2)3= a6+ b6 by substituting a = 3 and b =4 and concluded "This statementis never true, also the left numberis always greater
than the right number.And here is proof." He then proceeded with an
algebraicdemonstration.A few students also used an algebraicdemonstration in the second seriesof problems:six studentsout of 118 did use algebra
in the problem "Choose any number between 1 and 10..." and felt that
the fact that (10 + x) + (10 - x) = 20 explained the constant result. This
last group exhibited the least degree of algebra/arithmeticdissociation.
CONCLUDING
DISCUSSION
The tendency of some students to justify algebraicstatementsby appeal to
a rule ratherthan to their experienceof the behaviourof numbersreminds
us of the 19th century debate in Britain about the nature of algebra. Was
algebra "universal arithmetic" and therefore governed by the known
behaviour of quantitative arithmetic or was it a "symbolic system" with
essentially arbitraryrules? (Pycior, 1984). The proper historical answer is
that algebrais not in any straightforwardsense either of these alternatives,
but just as most of the Britishmathematiciansin the second quarterof the
19th century seemed temporarilyunable to transcend the opposing positions in order to resolve them, so many of the studentswe examinedseemed
to be at an either/or stage of developmentin which their arithmetic and
algebraic behaviourswere not at all comfortably integrated.The question
we cannot yet answer, and which is obviously very important to try to
answer,is whetherany of these students- all, many, or some of them - will
in the natural course of events achieve the coordination they lacked at the
time we examinedthem. Our guess is that those who have already decided
that algebra just doesn't make any sense are unwittingly shielding themselves from the intellectualconflict that might push their understandinga
step further.
Davis et al. (1978, p. 127) point out that students doing algebra rarely
apply a "check with numbers"strategyeven when recommendedto adopt
it. Our analysis shows that the strategy presupposes a reasonably clear
understandingof the connection between the worlds of arithmetic and
algebra, and that many of the students we examined were not yet clear
about the relationship. And, indeed, there are some technical difficulties
that everyone has to resolve. First and foremost is the striking difference
52
LESLEY LEE AND DAVID WHEELER
between writing and manipulatingexpressions in algebra and writing and
manipulating expressions in arithmetic. In spite of the use of common
(operational) signs, what one actually does in the two cases is very
different, so different that one cannot be surprised if students do not
immediatelyspot the connection. An algebraicexpression may perhaps be
transformableinto equivalentforms, but its value cannot be computed.The
same expressionwith numericalvalues substitutedfor the lettersis immediately computableand "collapses",losing all its individualcharacter,into a
single numeral. The pedagogy of algebra (in so far as it exists) seems to
have nothing to offer to help studentsgrasp the arithmetic/algebraconnection that underlies all these differencesbetween two modes of symbolic
behaviour.
Indeed, some pedagogicaldevices produce a miasmic fog of their own. It
is not unknown in traditionalalgebra teaching to offer numericalinstances
- of combining arithmeticfractions, say - as clues to the processes to be
used in combiningalgebraicfractions.The device, of course, is intendedto
serve as a reminder,as a structuralmodel for the algebraicprocedure,not
as a validation of the procedure,but one must sympathise with students
who are confused about the purpose of exploiting the parallelism.Does it
not quite strongly suggest that an algebraic generalisationmay be established from numericalinstances?
The parallelismworks because, in a quite strong sense, the algebra of
combining fractions is already present in the arithmetic of combining
fractions. There is no question here of the algebraicform generalisingthe
arithmeticalprocedure:the arithmeticalprocedure is already fully ge1ieralised. Indeed, as with such items as the commutativeand distributivelaws,
say, the algebraic form is only the record of a generalisation which is
already known. A difficulty for students is to appreciate the purpose of
writing down such familiar information about numbers in such a formal
way.
The arithmetic-algebraconnection is decidedly not plain and simple.
Combine the above obscuritieswith the asymmetryembeddedin the use of
numericalsubstitutionas a means of validation - one numericalsubstitution may disprove an algebraic statement whereas no finite numberof
numericalsubstitutionscan prove it - and one has a situation which might
seem designed to confuse everyone but the angels.
The peculiararithmeticalbehaviourswe observedin some of the students
make us think of Filloy and Rojano's (1984) suggestion that the challenge
of dealing with the syntax of a new and unfamiliar (algebraic) language
tends to destabilisestudents'semanticcontrol of the familiar(arithmetical)
THE ARITHMETIC CONNECTION
53
language. We did not pinpoint this issue in our research so we are not
really in a position to confirm or deny this intriguinghypothesis. It does,
however, seem plausible to us that the shift from arithmeticallanguage to
a formal algebraiclanguage that needs to be coordinatedin complex ways
with the first language could well cause temporary lapses in attention to
meaning that would give rise to aberrant arithmeticalbehaviours. All of
our subjectswould certainlyhave identified20 = 4 as an invalid statement
if it had appearedon its own or in an arithmeticalcontext, but setting it
at the tail end of a sequence of algebraic statements disturbed the students' normal interpretations.Those students who did try to dissolve their
residualunease looked without exception for syntactical"solutions"- e.g.
by treating the statement as an equation and dividing both sides, moving
one of the numbers to the other side of the equals sign, etc.
It surprises us that these aberrant behaviours are still present in students who have been learningalgebra for at least two years. Lapses which
might be anticipated in the work of students just embarking on algebra
(as with those studied by Filloy and Rojano) now seem to us more
serious because they are still occurring after students have attained more
surface fluency in the standard algebraic routines.
The picture our data yields shows the track leading from arithmeticto
algebra to be litteredwith procedural,linguistic,conceptual and epistemological obstacles. It is tempting to describe high school algebra as it is
unveiled in our research as a disaster area - an impression that would
probably be confirmed by reading reports of the CSMS and SESM
researches (Hart, 1981; Booth, 1984) or the writings of Stella Baruk
(1977, 1985), say. But the point we would stress in coming to the end of
this brief exploration is that the obstacles are real and not all trivial.
The students we worked with, in most cases, were neither lazy nor
foolish; their teachers, in most cases, were neither lazy nor foolish either.
At the end of the research we have an enhanced appreciation of
the difficultiesthat have to be overcome in bridging the worlds of arithmetic and algebra, and perhaps a greater dismay at the inability of
traditional pedagogy to give teachers or students much help in overcoming them.
NOTE
The "add and take" problem comes from Alan Bell's (1978) doctoral study, 'The learningof
general mathematicalstrategies',and the rectanglequestion from materials produced by the
CSMS MathematicsTeam. Sources for the other problemscannot be acknowledgedas we are
no longer sure where we first encounteredthem.
54
LESLEY LEE AND DAVID WHEELER
REFERENCES
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Baruk, S.: 1985, L'dge du capitaine:de l'erreuren mathematiques,editions du Seuil, Paris.
Booth, L. R.: 1984, Algebra: Children's Strategies and Errors, NFER-Nelson, Windsor,
Berkshire.
Davis, R. B., E. Jokusch, and C. McKnight: 1978, 'Cognitive processes in learning algebra',
Journalof Children'sMathematicalBehavior2(1), 10-320.
Filloy, E. and T. Rojano: 1984, 'La aparici6n del lenguaje arithmetico-algebraico',L'Educazione MatematicaV, 278-306.
Hart, K: 1981, Children'sUnderstandingof Mathematics,John Murray, London.
Lee, L. and D. Wheeler: 1986, 'High school students' conception of justification in algebra',
Proceedingsof the Eighth AnnualMeeting of the N. AmericanChapterof the International
Groupfor the Psychology of MathematicsEducation,East Lansing, Michigan.
Lee, L. and D. Wheeler: 1987, 'Algebraicthinking in high school students:their conceptions
of generalisationand justification', Research report, Concordia University, Montreal.
MathematicalAssociation: 1945, The Teachingof Algebra in Schools, London, G. Bell and
Sons. (The report was first written in 1929.)
Pycior, H. M.: 1984, 'Internalism,externalism and beyond: 19th century British algebra',
Historia Mathematica11, 424-441.
MathematicsDepartment
ConcordiaUniversity
Montreal, Quebec
CanadaH4B IR6