Focusing on Informal Strategies When Linking Arithmetic to Early Algebra
Author(s): Barbara A. Van Amerom
Reviewed work(s):
Source: Educational Studies in Mathematics, Vol. 54, No. 1, Realistic Mathematics Education
Research: Leen Streefland's Work Continues (2003), pp. 63-75
Published by: Springer
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BARBARA A. VAN AMEROM
FOCUSING ON INFORMAL STRATEGIES WHEN LINKING
ARITHMETIC TO EARLY ALGEBRA
ABSTRACT. In early algebrastudentsoften struggle with equationsolving. Modeled on
Streefland'sstudies of students' own productionsa prototypepre-algebralearningstrand
was designed which takes students'informal(arithmetical)strategiesas a startingpointfor
solving equations.In orderto make availablethe skills and tools needed for manipulating
equations,the studentsare stimulatedand guided to develop suitable algebraiclanguage,
notationsand reasoning.One of the results of the study is thatreasoningand symbolizing
appearto develop as independentcapabilities.For instance,studentsin grades6 and 7 can
solve equations at both a formal and an informallevel, but formal symbolizing has been
found to be a majorobstacle.
KEY WORDS: arithmetic,early algebra, equations, history of mathematics,reasoning,
symbolizing
INTRODUCTION
Many secondary school students experience great difficulty in learning
how to solve equations.Arithmeticand elementaryalgebraappearto be
a world apart. Comparedto arithmetic,algebraic skill requires another
approachto problem solving. Several researchprojectshave reportedon
these learning difficulties related to algebraic equation solving (Kieran,
1989, 1992; Filloy and Rojano, 1989; Sfard, 1991, 1995; Herscovics and
Linchevski, 1994; Linchevski and Herscovics, 1996; Bednarz and Janvier, 1996). Key difficulties mentioned in these reportsare constructing
equationsfrom word problems,as well as interpreting,rewritingand simplifying algebraicexpressions.
In 1995 a project'Reinventionof algebra'was startedat the Freudcnthal
Instituteto investigatewhich didacticalmeans enable studentsto make a
smooth transitionfrom arithmeticto early algebra.The project resulted
in a PhD Thesis (Van Amerom, 2002) that describes two possible ways
of approachingthe difficultiesstudentshave with this transition.The first
way, which was strongly influenced by Streefland'sinterest in students'
own productions,is to startfrom the students'informal strategiesand to
build more formal methods out of these. How this was researchedand
which resultscame out is reportedin this paper.
#
EducationalStudies in Mathematics 54: 63-75, 2003.
? 2003 KluwerAcademicPublishers. Printed in the Netherlands.
64
BARBARA
A. VANAMEROM
The second approach,which functionedratherin the backgroundof the
project,is to use inputfrom the historyof mathematics.This aspect of the
projectfollows recentresearchon the advantagesandpossibilitiesof using
and implementinghistory of mathematicsin the classroom (Fauvel and
Van Maanen,2000). Streefland(1996) also emphasizedthe use of history
as a sourceof inspirationfor instructionaldevelopment.Accordingto him,
looking backat the originsmakesone betterpreparedto teachin the future.
ALGEBRA AND ARITHMETIC
Similaritiesand differences
There is not one generally accepted point of view on what algebrais or
how it shouldbe learned,because it has so many applications.But for the
sake of practicality,it is useful to distinguishfour basic perspectives:(1)
algebraas generalizedarithmetic,(2) algebraas a problem-solvingtool, (3)
algebraas the studyof relationships,and (4) algebraas the studyof structures. Each of these operatesin a differentmedium, where, for example,
lettershave a specific meaningand role (Usiskin, 1988). In this study we
restrict ourselves to linear relationships,formulas and equation solving.
The proposed learning activities correspondwith the second and third
perspectives of algebra as mentioned, and they depend on the dialectic
relationshipbetween algebraand arithmetic.
A closer look at the similaritiesand differencesbetween algebraand
arithmeticcan help us understandsome of the problemsthatstudentshave
with the early learningof algebra.Arithmeticdeals with straightforward
calculationswith known numbers,while algebrarequiresreasoningabout
unknown or variable quantitiesand recognizing the difference between
specific and general situations.There are differencesregardingthe interpretationof letters, symbols, expressionsand the concept of equality.For
instance, in arithmeticletters are usually abbreviationsor units, whereas
algebraicletters are stand-insfor variableor unknownnumbers.Several
researchers(Booth, 1988; Kieran, 1989, 1992; Sfard, 1991) have studied
problemsrelatedto the recognitionof mathematicalstructuresin algebraic
expressions. Kieran speaks of two conceptions of mathematicalexpressions: procedural(concernedto operationson numbers,workingtowards
an outcome) and structural(concernedwith operationson mathematical
objects).In the 1960s it was alreadyclearthatdiscrepanciesbetweenarithmetic andalgebracan cause greatdifficultiesin earlyalgebralearning.The
difficulty of algebraiclanguage is often underestimatedand certainlynot
self-explanatory.Freudenthalexplainsthat
FROMARITHMETICTO EARLYALGEBRA
65
[ilts syntaxconsistsof a largenumberof rulesbasedon principleswhich,partially,contradictthoseof everydaylanguageandof the languageof arithmetic,
and which are even mutuallycontradictory.
(Freudenthal,1962, p. 35)
He then says:
The most strikingdivergenceof algebrafrom arithmeticin linguistic habits is a
semanticalone with far-reachingsyntacticimplications.In arithmetic3 + 4 means
a problem. It has to be interpretedas a command:add 4 to 3. In algebra 3 + 4
means a number,viz. 7. This is a switch that proves essential as letters occur in
the formulae.a + b cannoteasily be interpretedas a problem.
(ibid.)
The two interpretations(arithmeticaland algebraic)of the sum 3 + 4 in
the citationabove correspondwith the termsproceduraland structural(or
operationaland structural,Sfard, 1991, 1995).
But despite their contrastingnatures,algebraand arithmeticalso have
definiteinterfaces.For example,algebrarelies heavily on arithmeticaloperationsand arithmeticalexpressionsare sometimes treatedalgebraically.
Arithmeticalactivities like solving additionproblemswith a missing addend, and the backtrackingof operationspreparethe studentsfor studying
linearrelations.Furthermore,the historicaldevelopmentof algebrashows
that word problems have always been a part of mathematicsthat brings
togetheralgebraicand arithmeticalreasoning.
Cognitiveobstacles of learningalgebra
An enormnous
increasein researchduringthe last decade has producedan
abundanceof new conjectureson the difficult transitionfrom arithmetic
to algebra.For instance, with regardto equationsolving there is claimed
to be a discrepancyknown as cognitive gap (Herscovics and Linchevski,
1994) or didactic cut (Filloy and Rojano, 1989). These researcherspoint
out a breakin the developmentof operatingon the unknownin an equation.
Operatingon an unknownrequiresa new notionof equality.In the transfer
from a word problem(arithmetic)to an equation(algebraic),the meaning
of the equal sign changes from announcinga resultto statingequivalence.
And when the unknownappearson both sides of the equalitysign instead
of one side, the equationcan no longerbe solved arithmetically(by inverting the operationsone by one). Kieran(1989) reportsthatMatz and Davis
have done researchon students'interpretationof the expressionx + 3. Students see this as a process (adding3) ratherthana finalresultthatstandsby
itself. They have called this difficultythe 'process-productdilemma'.Sfard
(1995) comparesdiscontinuitiesin studentconceptionsof algebrawith the
historicaldevelopmentof algebra.She writesthatabbreviated(or so-called
syncopated)notations in algebraare linked to an operationalconception
of algebra,whereas symbolic algebracorrespondswith a structuralcon-
66
BARBARA A. VAN AMEROM
ception of algebra.' Da Rocha Falcao (1995) suggests that the disruption
between arithmeticand algebrais containedin the approachto problem
solving. Arithmeticalproblemscan be solved directly,possibly with intermediateanswersif necessary.Algebraicproblems,on the otherhand,need
to be translatedandwrittenin formalrepresentationsfirst,afterwhich they
can be solved. Mason formulatesthe difference between arithmeticand
algebraas follows:
Arithmeticproceedsdirectlyfrom the known to the unknownusing known computations;algebraproceedsindirectlyfrom the unknown,via the known,to equations and inequalities which can then be solved using established techniques.
(Mason, 1996, p. 23)
THE LEARNING STRAND
In this studywe set out to findout whethera bottom-up-approach
(starting
frominformalmethodsthatstudentsalreadyuse) towardsalgebracan minimize the discrepancybetweenarithmeticand algebra.In orderto do so we
designed a learningstrandfor the primaryschool grades5 and 6, and the
secondarygrade 7. The general topic of the primaryschool lesson series
is recognizingand describingrelationsbetween quantitiesusing different
representations:tables, sums, rhetoricdescriptionsand (word) equations.
No priorknowledgewas requiredapartfromthe basic operationsandratio
tables. The mathematicalcontentof the studentmaterialsincludes:
-
-
comparingquantitiesand equalizingthem in differentways;
discussingconflicts of notationsand differentmeaningsof symbols;
practicingwith operations(+, -, x, . ) and invertingoperations;
problemsolving by reasoningwith multipleconditions;
reasoningaboutexpressions(substitution,inversion,andglobal interpretation).
In the secondaryschool units these topics were repeatedin a compressed
way and then extendedwith solving systems of equations.
In the learningstrandthe studentsare challengedto solve pre-algebraic
problemsmanyof which were inspiredby history.Historically,wordproblems form an obvious link between arithmeticand algebra.Although algebra has made it much simplerto solve word problemsin general, it is
remarkablehow well specific cases of such mathematicalproblemswere
dealt with before the inventionof algebra,using arithmeticalprocedures.
Some types of problems are even more easily solved without algebra.
The history of algebrahas inspiredus to design the learningmaterialas
FROMARITHMETICTO EARLYALGEBRA
67
follows. Word or story-problemsoffer ample opportunityfor mathematizing activities. Babylonian,Egyptian,Chinese and early Westernalgebra
was primarilyconcerned with the solving of problems situatedin every
day life, althoughthese civilizations also showed interestin mathematical
riddles and recreationalproblems.Barter,fair exchange,money, mathematical riddles and recreationalpuzzles were rich contexts for developing
handy solution methods and notationsystems and these contexts are also
appealingand meaningfulfor students.
Anotherpossible access is based on notationuse, for instanceby comparing the historical progress in symbolizationand schematizationwith
that of modernstudents.Rhetoricalalgebra(writtenin words) finds itself
in between arithmeticand symbolic algebra,so to speak:an algebraicway
of thinking about unknowns combined with an arithmetic(procedural)
conceptionof numbersand operations.
The naturalpreferenceand aptitudefor solving word problemsarithmeticallyformsthe basis for the firsthalf of the learningstrand,wherestudents' own informalstrategiesshouldbe adequatelyincluded.The transfer
to a more formal algebraicapproachis instigatedby the guided development of algebraicnotation,especially the change from rhetoricalto syncopated notation, as well as a more algebraicway of thinking.We were
interestedin determiningwhetherthe evolutionof intuitivenotationsused
by the learner shows similarities with the historical developmentof algebraic notation.The bartercontext in particularappearsto be a natural,
suitable setting to develop (pre-)algebraicnotations and tools such as a
good understandingof the basic operationsand their inverses, an open
mind to what letters and symbols mean in different situations, and the
abilityto reasonabout(un)knownquantities.
The following Chinese barterproblemfrom the 'Nine Chapterson the
MathematicalArt' inspiredStreefland(1995, 1996) and the authorto use
the contextof barteras a naturalandhistorically-foundedstartingpoint for
the teachingof linearequations:
By selling 2 buffaloesand 5 wethersand buying13 pigs, 1000 qian remains.One
can buy 9 wethers by selling 3 buffaloesand 3 pigs. By selling 6 wethers and 8
pigs one can buy 5 buffaloesand is short of 600 qian. How muchdo a buffalo,a
wetherand a pig cost?
In modernnotationwe can write the following system:
2b + 5w 13p + 1000
3b + 3p =9w
6w + 8p + 600 = Sb
(1)
(2)
(3)
68
A. VANAMEROM
BARBARA
where the unknownsb, w andp standfor the price of a buffalo, a wether
and a pig respectively.From the perspectiveof mathematicalphenomenology, Streeflandand Van Amerom (1996) posed a numberof questions
regardingthe origin, meaningand purposeof (systems of) equations.The
example given above is interestingespecially when looking at the second
equation,whereno numberof 'qian'is present.In this 'barter'equationthe
unknownsb, w andp can also representthe numberof animalsthemselves,
instead of their money value. The introductionof an isolated numberin
the equations (1) and (3) thereforechanges not only the medium of the
equation (from number of animals to money) but also the meaning of
the unknowns(fromobject-relatedto quality-of-object-related).
Streefland
(1995) found in his teaching experimentin which students had to find
out how packages can be filled with candies of differentprices, that the
meaning of literal symbols is an importantconstituentof the progressive
formalizationof the pupils. Accordingto Streefland,the studentsneed to
be awareof the changes of meaning that lettersundergo.In this way, the
children'slevel of mathematicalthinkingevolves. Streeflandmentionsthe
example of asking for the meaning of the equality sign, which indicates
'costs so much', as well as the usual 'is equal to'. These considerations
indicatesteps in the conceptualdevelopmentof variables.
We also aimed to investigatehow notationand mathematicalabstraction are related. The categorizationrhetorical- syncopated- symbolic
is the result of the currentconception of how algebradeveloped, and for
this reasonit is often mistakenfor a gradationof mathematicalabstraction
(Radford,1997). When the developmentof algebrais seen from a socioculturalperspectiveinstead, syncopatedalgebrawas not an intermediate
stage of maturationbut it was merely a technical matter.As Radfordexplains, the limitationsof writingand lack of book printingquite naturally
led to abbreviationsandcontractionsof words.Studentsmay revealsimilar
needs for efficiency when they develop theirown notations(fromcontextboundnotationto an independent,generalmathematicallanguage),butthis
need not coincide with the conceptualdevelopmentof letteruse.
CLASSROOM EXAMPLES
The field testingof the units of the learningstrandwas done in two rounds.
In the firstroundthe materialswere triedout in two primaryclasses each
consistingof 30 grade5-6 students(10-12 year olds). In the second round
the revised materialswere proposedto three grade 6 primaryclasses and
two grade7 secondaryclasses (12-13 yearolds). Duringthe trialdatawere
collected in severalways: observinglessons, gatheringstudentsnotes, ad-
FROMARITHMETICTO EARLYALGEBRA
69
ministratingpost-tests, and having questionnairescompleted by students
and teachers.
In this paperwe will focus on the issues of symbolizingand reasoning
by taking one case from the first roundsituatedin grade 5-6 and another
case from the second roundin grade7.
Case 1: Symbolizingand the meaningof letters
Evoking shortenednotationsforms one of the spearpoints in the learning
strand.The introductionof shortenednotations in grade 5-6 is done in
the context of a game of cards. In this activity the students work with
cards displaying the scores of each round. There are four types of representations:a descriptionof what happensin each round, the scores of
all the playersin points, a descriptionof a relationbetween the scores (for
example, 'Petrahas 5 points more than Jacqueline')and a word formula
expressinga certainrelation('points Petra= points Jacqueline+ 5'). The
childrenare asked to find the right cardsfor every round,and then design
their own cardsfor two more roundsof playing cards. Sometimesthereis
more than one card to describe the scores, when the scores are related
in two different ways (for example, 'twice as much' and '5 more'), in
which case a switch of perspectiveis needed. The following classroom
vignette shows how a pupil carriedout the task of shorteningnotationstoo
rigorously(VanAmerom,2002, p. 132).
Classroomvignette 1
Robert:
Points Petraplus Anton is Jacqueline.
Teacher:
Somethingis not rightthere .. .. Well, maybe the calculationis.
Tim:
Points Petraplus points Anton.
Teacher:
Indeed!You have shortenedtoo much. It is all aboutthe points
of these people. Youcan'tjust addpeople to the points thatthey
get, that'simpossible!
(Laughingin the class.)
Teacher:
It is actually aboutthe numberof points: the numberof points
that Petra has plus the numberof points that Anton has is the
numberto points thatJacquelinehas.
Renske:
That'swhat I had!
Teacher:
Yes, we talkedaboutthatfor a little bit yesterday.It is not wrong
what you say, Robert,but it is not clear when it is too short.It
is importantthatyou say it clearly.
In this way the pupils and the teacherascertainedtogetherthat when you
use abbreviations,it must be clear to everyone what you mean. At the
same time one can wonderif it is necessaryto be so precise at this stage,
70
BARBARA A. VAN AMEROM
because the pupils realize exactly what the letters are about. Thereforea
compromiseshouldbe found between precise and unambiguousnotations
on one hand, and intuitive(probablyinconsistent)productionsof children
on the other.
Sometimes the teacher let a good opportunitypass by, as shown in
classroomvignette 2. In one of the lessons childrensuggested what could
be the meaning of the expression 'pA = 3 x pJ'. Our decision to use this
kind of symbolism is based on otherpupils' free productionsin a preliminary try-out.The letter combinationmaintainsthe link with the context:
the letterp standsfor 'numberof pointsbelongingto' andthe capitalletter
stands for the person in question,in this case Annelies and Jeroen.In the
expression, such a letter combinationbehaves like a variablefor which
numberscan be substituted.When the score of one of the playersis given,
the expressionbecomes an equationthatcan be solved. The teacherasked
the childrenfor an examplethatwill illustratethatthe relationbetweenthe
variablespA andpJ is '3 times as much'.
Classroomvignette 2
Teacher:
If we think of points, what would be possible? You have to
write it down in a handy way,just like in PocketMoney,which
numbersare possible.
Yvette:
3 and 9.
Teacher:
Who has 3 and who has 9?
Yvette:
Annelies has 9.
Teacher:
How would you write it down? Why don't you show us on the
blackboard.
Yvette writes: A-9p
j-3p
Sanne:
I would write an equal sign, not a line.
The teachermissed a good opportunityto discuss three samples of inconsistent symbolizing:Yvette's choice to write a capitalletterA and then a
small letterj, her use of the letterp as a unit even thoughit is alreadypart
of the variable,and Sanne's suggestionat the end. Apparentlyit was not a
problemto the childrenthatlettersmean differentthings at the same time.
As long as the pupils and the teacher are all conscious of this fact, the
developmentandrefinementof notationsis a naturalprocess. On the other
hand, it is not our intentionto cause unnecessaryconfusion regardingthe
meaningof letters.It was decided that if childrenhave a naturaltendency
to use the letterp as a unit, it shouldnot be includedin the expressionsand
formulas.
The lesson materialswere adjustedand tested again in 1999. The dual
characterof the learning strand- to develop reasoning and symbolizing
71
FROMARITHMETIC
TOEARLYALGEBRA
OX
cx-& .~
iAL V-OW*.,,,
S,,
6
K
I
+
LAxR1,5C)
.
~.. s.
8
..Ut TCki I.
)_.,
.7
Figure 1. Symbolic notationsand informalreasoning.
skills in the study of relations - was maintainedbut placed in a more
problem-orientedsetting and with a more explicit historical component
at primaryand secondarylevel.
Case 2: Symbolizing versus reasoning
Anotherimportantissue to be studiedwas the students'reasoningin connection with their symbolizing. In the following we will discuss two examplesof studentworkfromthe secondroundof the classroomexperiment
to demonstratethat (pre-)algebraicsymbolizingtends to be more difficult
for studentsthanreasoning.
Encouragedby the ideas and results of the classroom experimenton
candy by Streefland(1995), a grade 7 unit on equation solving was designed based on mathematizingof fancy fair attractionsinto equations.
One of the tasks in the writtentest was:
Sacha wants to make two bouquetsusing roses and phloxes. Theflorist replies:
"Uhm... IO roses and 5 phloxesfor 15,75, and 5 roses and IOphloxesfor 14,25;
that will be 30 guilders altogetherplease."
Whatis the price of one rose? And one phlox? Show your calculations.
One of the outcomes of the experimentis that algebraicequationsolving
need not necessarilydevelop synchronouslywith algebraicsymbolization.
BARBARA
A. VANAMEROM
72
kIodbIaadje
L 66%
wk
1211
.tXh
+
x z;z
zh Xy2<
zk
Figure2. Informalnotationsand algebraicreasoning.
For instance, we have observed student work where a correct symbolic
system of equationswas followed by an incorrector lower orderstrategy,
or where the studentproceededwith the solutionprocess rhetorically.The
student whose work is shown in Figure 1 (Van Amerom, 2002, p. 230)
mathematizesthe problemby constructinga system of equations,andthen
applies an informal,pre-algebraicexchange strategywhich is developed
in the unit. Below the equationsshe writes: "Weget 5 roses more and 5
phloxes less, the differenceis 1.50. We get 1 rose more and 1 phlox less,
the differenceis 0.30."The calculationsshow thatshe continuesthe pattern
to get 15 roses for the price of 17.25 guilders,and then she determinesthe
priceof 1 rose and 1 phlox. The level of symbolizingmay appearto be high
at firstdue to the presenceof symbolic equations,but the studentdoes not
operateon the equations.The equationsmay have helpedher structurethe
problembutthey arenot a partof the solutionprocess.And even thoughthe
unknownnumbersof flowers are an integralpartof her reasoning,the lettersrepresentingthemarenot neededin the calculations.Thereis a parallel
here with the historicaldevelopmentof symbolizing the solution. In the
rhetoricaland syncopatedstages of algebrathe unknownwas mentioned
only at the startand at the end of the problem;the calculationswere done
using only the coefficients.
Alternativelythe solution in Figure 2 (Van Amerom, 2002, p. 227)
illustrateshow the level of reasoningcan be higherthanthe level of symbolizing. This studentsolves the system of equations
2 x h + 2 x k = 66
3 x h + 4 x k = 114
FROMARITHMETIC
TOEARLYALGEBRA
73
by doublingthe firstequationandthen subtractingthe second fromit. First
he deals with the right hand sides of the equations (66 x 2 and 132 114). In between the two horizontallines we observe how he multiplies
the termswith the unknowns.Then he writes:"Butthe task says 3h so 18
is 1 h." Finally he substitutesthe value 18 to solve for k. A remarkable
contrastpresents itself. This studentsuccessfully applies a formal algebraic strategyof eliminatingone unknownby operatingon the equations,
while his symbolizingis still at a very informallevel. Again the unknown
is only partiallyincluded in the solution process; it appearsonly where
necessary.In otherwords,both examplesof equationsolving illustratethat
competenceof reasoningand symbolizingare separateissues.
EPILOGUE
Difficulties in the learningof algebracan be partiallyascribedto ontological differencesbetween arithmeticand algebra.Ourstudyuses informal,
pre-algebraicmethodsof reasoningand symbolizing as a way to facilitate
the transitionfrom an arithmeticalto an algebraicmode of problemsolving. We have shown some examples in which informalnotationsdeviate
from conventionalalgebrasyntax, such as the inconsequentuse of letters
and the pseudo-absenceof the unknownin solving systems of equations.
These side effects bring new considerationsfor teaching: how can we
bridge the gap between students'intuitive,meaningfulnotationsand the
more formallevel of conventionalsymbolism?The observationthat symbolizing and reasoningcompetenciesare not necessarilydevelopedat the
same pace - neitherin ancientnor in modem times - also has pedagogical
implications.It appearsthat equationsolving does not dependon a structuralperceptionof equations,nor does it rely on correctmanipulationsof
the equation.
In retrospectwe can say that knowledge of the historicaldevelopment
of algebrahas led to a sharperanalysisof studentworkandthe discoveryof
certainparallelsbetweencontemporaryandhistoricalmethodsof symbolizing. Streefland'snotice to look back at the origins in orderto anticipate
has turnedout to be a valuablepiece of his legacy.
ACKNOWLEDGEMENTS
The author acknowledges helpful comments from Jan van Maanen and
Marja van den Heuvel-Panhuizenand two anonymous reviewers on an
earlierdraftof this paper.
74
BARBARA A. VAN AMEROM
NOTE
1. According to Struik (1990) this classification of algebra problems into rhetorical,
syncopated and symbolic algebra appearedfirstly in Nesselmann, Die Algebra der
Griechen,Berlijn 1842.
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Aidadreef12,
3561 GE Utrecht,TheNetherlands
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