DINA TIROSH, RUHAMA EVEN and NAOMI ROBINSON

SIMPLIFYING ALGEBRAIC EXPRESSIONS: TEACHERAWARENESS AND TEACHING APPROACHES

ABSTRACT. This study investigates four seventh-grade teachers’ awareness of students’tendency to conjoin or ‘finish’ open expressions. It also investigates teachers’ ways ofcoping with this tendency. Three types of data were collected: 1) lesson plans, 2) lessonobservations, and 3) post-lesson interviews. The analysis showed that the two experiencedteachers were aware of this tendency and some of its possible sources, while the novices wereunaware of either. Teaching approaches related to this tendency also differed considerably.In conclusion we analyze these teaching methods in light of the existing literature anddiscuss possible short- and long-term implications of the use of each approach.

All teachers encounter situations in which they need to decide how theypresent the subject matter to their students. They may do this either ontheir own initiative or in response to a student’s comment. In making suchdecisions, teachers’ knowledge of the subject matter obviously plays amajor part. Also important is their acquaintance with, and understandingof, students’ ways of thinking, as well as various alternative representationsof specific topics.

Shulman (1986) distinguishes between two kinds of subject-matterunderstanding that teachers need to have –knowing that(something isso) andknowing why(it is so). We have suggested (Even and Tirosh,1995) that these terms are also useful when dealing with teachers’ know-ledge about students’ ways of thinking.Knowing thatin this context refersto research-based and experience-based knowledge about students’com-mon conceptions and ways of thinking about the subject matter.Knowingwhy refers to general knowledge about possible sources of these concep-tions, and also to the teachers’ understanding of the sources of a student’sresponse in a specific case.

This paper is an initial attempt to deal with the issue of teachers’ know-ledge of students’ ways of thinking in the context of algebra. It stemsfrom a larger research project that examines differences between experi-enced and novice teachers with respect to teaching the topic of equivalentalgebraic expressions. In the course of the study students’ experiencesmainly centered on the issue of simplifying algebraic expressions. Lessonobservations revealed that students tended to conjoin or ‘finish’ algebraicexpressions. For example, students tended to write the expression 2x+ 3as 5x or 5.

Educational Studies in Mathematics35: 51–64, 1998.c 1998Kluwer Academic Publishers. Printed in Belgium.

Gr.: 201007286, PIPS Nr. 139740 HUMNKAP

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This tendency is documented in the mathematics education literature,which also describes some possible sources and methods for dealing with it.However, the literature does not investigate teachers’ familiarity with thistendency and its possible sources. Also missing in the literature are relevantdescriptions of real classroom teaching and student/teacher interactions.

In this paper we explore four teachers’ awareness (knowing that) of stu-dents’ tendency to conjoin or ‘finish’ open expressions. We also consider,when teachers are aware of this, to what they attribute it (knowing why).Furthermore, we describe these teachers’ relevant actual classroom teach-ing methods and their on-the-spot responses to students’ related behaviorin class. Finally, we analyze the observed teaching methods in light ofthe existing relevant literature. We shall first briefly survey several sourcesdescribed in the literature for students’ tendency to conjoin or ‘finish’ openexpressions.

1. SOURCES OFSTUDENTS’ TENDENCY TOCONJOIN OR‘FINISH’ OPENEXPRESSIONS

The literature suggests several different sources to students’ tendency toconjoin open expressions. One of them has to do with conventions innatural language. For instance, Tall and Thomas (1991) mention that dueto similar meanings of ‘and’ and ‘plus’ in natural language it is common forstudents to consider ‘ab’ to mean the same as ‘a+b’. Stacey and MacGregor(1994) remark that students may erroneously draw on previous learningfrom other areas that do not differentiate between conjoining and adding,e.g., in chemistry adding oxygen to carbon produces CO2.

Another frequent explanation for this tendency is that students facecognitive difficulty in ‘accepting lack of closure’. They conceive openexpressions as ‘incomplete’ and tend to ‘finish’ them (Booth, 1988; Collis,1975; Davis, 1975). The latter explanation still leaves room for the ques-tion, Why do students perceive open expressions in this way? A typicaljustification is that the ‘behavior’ of algebraic expressions is expected tobe similar to that of arithmetic expressions. Similarity to arithmetic canbe reflected either in expecting a final single-termed answer (e.g., Booth,1988; Tall and Thomas, 1991), or by interpreting symbols such as ‘+’ onlyin terms of actions to be performed, thus leading to conjoining the terms(e.g., Davis, 1975). (Similarly, at higher grade levels students feel reluctantto accept 3+ 2i as a ‘complete’ number, Tirosh and Almog, 1989.)

Another, somewhat broader explanation for the same behavior is relatedto the dual nature of mathematical notations:processand object (e.g.,Davis, 1975; Sfard, 1991; Tall and Thomas, 1991). In algebra, the symbol

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5x+ 8 stands both for the process ‘add five times x and eight’ and for anobject. Often, students grasp 5x+ 8 only as a process to be performed and‘add’ 5x+8, in what seems to them a reasonable way, getting expressions,such as 13x, or 13. (Note that even though 13x is an open phrase andapparently includes a multiplication operation, many students accept theterm as ‘complete’ and do not feel the need to perform a process.)

2. METHODS

2.1. Participants

Four seventh-grade teachers participated in this study. Two were noviceswith one or two years of experience, and the other two had more than 15years of experience each. The experienced teachers were known amongstudents and colleagues as excellent teachers. All teachers were teachingalgebra in regular seventh grade classes in different schools, with a low-middle class socioeconomic population. All teachers agreed to participatein the study on the basis of their familiarity with one of the researchers.The teachers were not told what the aims of the study were.

The four subjects were teaching from the same textbook (Robinson,1992) which mainly emphasizes the process facet of algebraic expressions(called ‘open phrases’). An open phrase is described as follows: ‘5�a+ 3is an open phrase in which ‘a’is a variable. If we substitute�2 in the openphrase we get 5�(�2)+3 = �7’ (p. 99). Later, when the issue of equivalentalgebraic expressions is introduced, the following definition is presented:‘Two open phrases are equivalent if for each number substituted, the samenumber is obtained. Only numbers that belong to both substitution setsmay be substituted’ (p. 119).

The teachers used a traditional approach to the teaching of algebra – onethat is common in Israeli schools – with an emphasis on formal language,procedures and algorithms. Their lessons were teacher-centered with noemphasis on students’ investigations.

2.2. Data Collection and Analysis

As part of the larger project three types of data were collected in the secondhalf of the school year. They included: 1) lesson observation – the threeinitial lessons that each of the four teachers taught on equivalent algebraicexpressions (open phrases) were observed by one of the researchers; fieldnotes were taken during observations and were supplemented by audio-taperecordings; 2) lesson plan – each teacher was asked to submit a lesson planbefore each lesson, either in writing or on audio-tape; and 3) post-lesson

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interviews were conducted after each lesson. The teachers were asked toreflect on their lesson plan, expectations from students, lesson objectivesand general feelings about the lesson. Teachers were also asked to explainspecific situations that occurred during the lessons. All interviews wereaudio-taped. In addition to the data from the largest project, specificallyfor this study, each teacher was asked the following questions: What arethe main difficulties that students encounter when they learn to simplifyalgebraic expressions? What are the sources of these difficulties?

The taped lesson plans, observed lessons and interviews were tran-scribed. For the purpose of this study, the transcripts were analyzed withreference to the students’ tendency to conjoin or ‘finish’ open expressions.We looked only for manifestations of this topic, and for all such mani-festations (e.g., in-class student behavior and teacher responses, relevantcomments in the lesson plans, reflections during the post-lesson inter-views). Several dimensions were central in our analysis, such as teacherawareness of student conceptions related to the tendency to conjoin or‘finish’ open expressions, teacher explanations of the sources of this beha-vior, teacher reactions to relevant student comments, and choice of relatedteaching methods.

3. TEACHERAWARENESS ANDTEACHING APPROACHES

In this section we report the data for each teacher and their analysis. Foreach teacher, we first present whether he/she explicitly mentioned thatstudents tend to conjoin or ‘finish’ open expressions. If so, we presentthe teacher’s notions of possible sources of this behavior. Then, a typicalreaction (or reactions) of each teacher to situations in which studentsconjoined or ‘finished’ algebraic expressions, or in which they chose toraise this issue, is described. The teachers’ names were altered.

3.1. Benny

Benny is a novice teacher in his first year of teaching. His behavior suggeststhat Benny is unaware of students’ tendency to conjoin or ‘finish’ openexpressions. He does not mention this issue in an interview, when askedto describe students’ difficulties related to learning algebraic expressions,nor does he address it in his lesson plans.

In respect to simplifying algebraic expressions, Benny plans to providestudents with a rule of ‘adding numbers separately and adding lettersseparately’. (Notice that his phrasing of the rule is problematic becauseaccording to it 5m+ 2 can be equal to 7+ m or 7m or another variation.)

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During the lesson he states the rule and keeps repeating it. When anincorrect response is given, he often states that this is wrong, and repeatsthe rule. The following fragment shows what happened in his class whenhe tried to apply his plan.

Benny writes the expression 3m+2+2m on the board and asks: ‘Whatdoes this equal?’ confirming immediately with the rule: ‘Add the numbersseparately and add the letters separately.’ Then he suggests coloring the‘numbers’: 3m+ 2 + 2m, and writes 5m+ 2. A student asks: ‘And whatnow?’ Another student suggests: ‘7m’. The teacher (rather surprised bythis answer) says: ‘No! 5m+ 2 doesnot equal 7m.’ And he repeats therule again: ‘The rule is: add the numbers separately and add the lettersseparately.’ Then he gives the students another example and colors the(free) numbers: 4a+ 5 � 2a+ 7. The teacher emphasizes the rule bydictating it to the students and asking them to repeat it out loud. The rest ofthe lesson is devoted to work on similar exercises. The students continueto experience difficulties. For example, towards the end of the lesson, astudent asks: ‘Why doesn’t 10+ 2b equal 12b?’ The teacher does notrespond to the question.

In his reflection on the lesson Benny expresses his dissatisfaction andfrustration with the way he has taught this material. He explains that hesenses there is a problem but he does not understand its sources. He addsthat giving a rule without explanation was problematic and that he does notreally understand what his students’ difficulties were. During his interviewshe mentions several times that he will have to reflect on his teaching andon students’ responses in order to make decisions about future teaching.

3.2. Drora

Drora is a novice teacher in her second year of teaching. From the way sheplans and teaches her lessons and from her interviews it seems that she isaware of various difficulties related to working with algebraic expressions,such as understanding that 1x= x, x � 5 = 5x,�1x = �x. She attributesthese difficulties to, for instance, students not understanding the meaning ofcoefficients of variables, difficulties associated with the order of operations,and obstacles related to the use of the distributive property. Yet she doesnot mention that students tend to conjoin open expressions.

Her way of teaching how to simplify algebraic expressions consists ofusing the ‘like terms’ and the ‘fruit salad’ teaching approaches. Duringthe lessons, when her students give an incorrect response, Drora oftenresponds that this is wrong and provides the correct answer. The followingobservations illustrate this.

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In one of the lessons the class discusses the simplification of the expres-sion 7x� 5 � x + 12. Drora asks: ‘What are the like terms that we canadd?’ One student answers: ‘7x and 5.’ The teacher replies by stating:‘7x and –x, isn’t it? 5 does not have an x nor does 12.’ She immediatelywrites the expression 7x� x � 5+ 12 on the board, and continues withthe idea of like terms, asking again: ‘What are the like terms? What can Iuse here?’ She immediately answers her own question: ‘I have 7x and -x.’When a student asks why we add 7x and –x, i.e., why was 5 left out, Droraexplains: ‘Because I add only numbers that have an x, don’t I? I add onlylike terms, terms that have an x, and 5 doesn’t have one.’ (Notice that Drorauses inaccurate language which may do for the current problem but not forcases such as 7x and 5x2, for example, which are ‘numbers that have an x’and, therefore, may be interpreted by students as like terms according toher explanations.)

On another occasion, a student, while working at the board, writes:5t � 3t+ t + 2 = 11t. The teacher uses the distributive property to get3t from 5t� 3t+ t. Then she asks: ‘Can we add 3t and 2? Can we adda numeral to an algebraic expression?’ She immediately answers her ownquestion: ‘No! 3t+2 is the final expression.’ She continues, using the ‘fruitsalad’ approach:

What is 3t? 3 times t. Is it a number? No! It’s an algebraic expression. And what is 2? 2 is aconstant number. We can add 3t only to the same thing. If I have 3 pears and 2 apples, canone say that I have 5 pears? We are left with 3 pears and 2 apples. 3t is an expression and2 is a number. So we cannot add them.

During the interviews, Drora mentioned that she realized that somestudents in her class experienced difficulties. But she was unable to specifyany. Then she claimed that everything she does helps only the good studentsas the weak ones will not understand, no matter how hard she tries.

3.3. Gilah

Gilah is an experienced teacher. When asked during an interview to mentionvarious difficulties related to the learning of algebra, she specifies, amongother things, students’ tendency to ‘simplify’ expressions such as 3x+4 to 7x. She further explains: ‘Students tend to make it as simple aspossible. They tend to ‘finish’ it [the expression].’ In her opinion, thisis the main obstacle in teaching how to simplify algebraic expressions.Therefore, she planned a comprehensive activity, devoted to getting thestudents acquainted with the notions of like and unlike terms to be taughtbefore the lessons on simplification of algebraic expressions, and she spenttime and effort on teaching and directing the students towards the use ofthis one specific method. In an interview she claims:

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I think that differentiating between like and unlike terms should precede the issue ofsimplifying algebraic expressions. There is a need to work extensively on the topic of likeand unlike terms.

Her introductory activity consisted of two main parts. In the first one,‘Identifying like terms’, students are told that ‘like terms are terms thathave an identical combination of variables’ and they receive a variety ofexamples of like and unlike terms (e.g., 2x2 and 4x2, 3ab and 6ab, 5a and6a2, 2bc and 3ac, 3ab and –2ba). Then they practice and discuss identifyinglike and unlike terms with a great variety of examples. In the second partof this activity ‘Collecting like terms’, students are told that ‘in order tosimplify algebraic expressions, one can collect like terms.’ The studentsthen receive a variety of examples that illustrate how to collect like terms,starting with 4a+ 2a = 6a and gradually reaching more complicatedexpressions such as 2xy+ 4x+ 1:5y+ 6xy+ y = 8xy+ 2:5y+ 4x. Theexamples are accompanied by written descriptions which highlight thelike terms and the result of their collection. After discussing the examples,the students practice simplifying algebraic expressions by collecting liketerms.

As the class progresses with its algebra work, Gilah and her studentskeep referring to the notions of like and unlike terms. They use them todetermine if and how a given algebraic expression can be simplified. Thestudents seem comfortable with this notion and way of work and rarelymake mistakes of the type: 5+ 3x = 8x. When they do, a very quickcomment suffices to clarify the situation. For example, during a lessonthe class arrives at the expression 4x+ 6. One student (A) suggests thatthis equals 10x. Another student (B) objects: ‘4x and 6 are unlike termsbecause 4x is a number with an x and 6 is only a number.’ The teacherpushes further: ‘So what?’ Student A replies confidently: ‘The answerstays 4x+ 6.’

3.4. Batia

Batia is an experienced teacher. Her lesson planning, her teaching, as wellas her interviews all make it obvious that she is aware of students’ tendencyto ‘finish’ open expressions. For example, in her written lesson plan shewrites: ‘I expect difficulties in problematic cases [such as] 2x+ 3 = 5x’.Also, in an interview, when asked about difficulties that students commonlyencounter when studying algebraic expressions, she mentions, among otherthings, the tendency to add 2a+3 and get 5a. When the interviewer probedher for the reasons for this difficulty, she stated: ‘They need to get ananswer, it does not seem finished to them.’

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In her lessons on equivalent algebraic expressions, Batia includes vari-ous activities and she uses several strategies, namely, substitution, orderof operations and going backwards. She bases her approach on creatingconflicts in her students’ thinking about the issues they raise. Batia offersthe students opportunities to express their ways of thinking about the issueeven when they are wrong and she treats them with respect. The followingexamples illustrate her approach.

The teacher writes 3+ 4x on the board and asks the class what canbe done with this expression. One student says: ‘It should remain as itis’. Although this is a correct answer, probably the one that most teacherswould expect at this point, the teacher continues to probe for other reactionsthat students may come up with. Another student says: ‘No, that’s not true,3+ 4x equals(3+ 4)x.’ The teacher then suggests one way to check thisby substituting x= 5, i.e., 3+4 �5 :6= (3+4) �5. She continues to presentsimilar expressions for students to work with.

In another lesson, the teacher uses the idea of the order of operations.

Teacher: What is 3+ 4x?

Student: 7x.

Teacher: How about 7?

Student: Maybe?!

Teacher: Well, let’s see again. 3+ 4x. What is the operation between 4 and x.

Student: Multiplication.

Teacher: So, first we have to determine what 4� x could be. Can we know that?

Student: No!

Teacher: So, can I first add the numbers?

Student: No! OK, I got it.

(Note that this explanation is problematic as according to it 3x+ 4x, forinstance, may not seem to allow addition either.)

In still another lesson, the class discusses the expression x2 + 3x. Onestudent asks if this could be 4x2. The teacher repeats the question: ‘Couldit be 4x2?’ in a neutral tone, not implying the correct answer. Anotherstudent correctly replies: ‘No.’ The teacher then asks a ‘going backwards’question: ‘In order to get 4x2, what do we need there?’ One student answers:‘It should have been 3x2 and not 3x.’ The first student further insists: ‘Whycan’t I add them? I still don’t understand.’ The teacher repeats the question.The second student continues to explain: ‘We don’t know what x2 and 3xare. Therefore, one cannot add them. We should first multiply.’ The firststudent seems satisfied with the explanation.

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4. DISCUSSION

Our study shows differences between teachers in terms of awareness ofstudents’ tendency to conjoin or ‘finish’ open expressions, and in the waysthey dealt with this. In the following we discuss each of the methods theteachers used in light of the mathematics education literature.

4.1. Collecting Like Terms

Three of the four teachers used some version of the ‘collecting like terms’approach, which is commonly used when teaching simplifying algebraicexpressions. The two novice teachers, Benny and Drora, started to usethis method as they introduce the topic of simplifying algebraic expres-sions. In their classes students seemed unwilling to accept expressionsincluding a ‘+’ sign (such as 3+ 2a) as final products. Gilah, one of theexperienced teachers, on the other hand, devoted an extensive period oftime to practicing ‘collecting like terms’ before dealing with simplifyingalgebraic expressions. In her class students seemed to have mastered thisskill. Gilah’s teaching approach consisted of what Davis (1989) refers toas a course in which the student is asked to perform some fragmentary,individual, small rituals. These skills are presented to students as ‘ritualsto be practiced until they can be executed in the proper, orthodox fashion’(p. 117). Davis claims that in such an approach the student sees no purposeor goal in the activity. ‘Consequently, the student sees no reason why theritual is performed in one way and not another.’ Davis mocks the theoryunderlying such courses which assumes that ‘if the students spend enoughtime practicing dull, meaningless, incomprehensible little rituals, sooneror later something WONDERFUL will happen’ (p. 118).

Considering Gilah’s success in helping students master the rather prob-lematic topic of simplifying algebraic expressions and Davis’ criticism ofthe long-term implications of such a method, faces us with a dilemma. Onthe one hand, one may want to fulfill what can be considered the students’immediate needs. On the other hand, one cannot ignore the students’ gen-eral conception of mathematics as it develops throughout the school years.How the teacher confronts this dilemma largely depends on the importanceshe attributes to mastering skills in mathematics.

4.2. ‘Fruit Salad’ Approach

This commonly practiced approach was used in our study only by Drora.The rationale for this approach rests on the assumption that an expression,such as, 2a+ 5b could be regarded as 2 apples and 5 bananas, and be seen

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by students as an entity in its own right. (Obviously, there can be manyvariants of the ‘fruit salad’ approach, such as, a ‘furniture’ approach, etc.)

Pimm (1987) criticizes the ‘fruit salad’ approach on the grounds that‘it leads to confusion betweena being apples anda being ‘the numberof apples” (p. 132). He claims that there should be a distinction betweenthe objects themselves and the number of them. ‘The algebraic expres-sion is not an analog of 5 apples, nor is 5 apples a possible interpretationof 5a: : : the letters themselves are standing for numbers’ (p. 132). Booth(1988) adds that this method may not be useful as ‘not only does it encour-age an erroneous view of the meaning of letters, but it can also be usedby students to justify their [wrong] simplification’ (p. 26). She provides anexample from her research where the same number of students simplified2a+ 5b to 7ab basing this on ‘2 apples plus 5 bananas is 7 apples-and-bananas’, as those who used this illustration to justify why 2a+ 5b couldnot be simplified further. Also, this approach may lead to the belief thatone cannot multiply algebraic expressions such as 2a and 5b because onecannot multiply apples and bananas.

It is a conunon approach in mathematics teaching to refer to concreteobjects when facing difficulties. However, there are cases, such as the ‘fruitsalad’ illustration, when this may cause more harm than benefit.

4.3. Going Backwards

This method, used by Batia, bears some similarity to Chalouh and Herscov-ics’ (1988) process-object approach in algebra. These researchers describea teaching experiment with six students which builds on students’ tend-ency to ‘close’ algebraic expressions. In an attempt to help students viewalgebraic expressions not only as processes but also as objects, they sug-gest ‘completing’ them whenever possible. For instance, students werepresented with the problem of finding the area of a rectangle whose widthis 3 units, and whose length is given in two parts: one part is unknown (c)and the other is 2 units. The students were asked to complete the statement‘Area= —’. The right-hand side expresses the process while the left-handside closes it by giving it a name. The researchers report that the subjects‘constructed meaning for algebraic expressions’ (p. 43). However, in thisexperiment each student was individually instructed and followed for ashort period of time. Thus it is impossible to draw conclusions from thisexperiment to a classroom situation.

4.4. Order of Operations

Batia used this approach which we have not found in the literature. Severalresearchers (e.g., Crowley, Thomas and Tall, 1994; Davis, et al., 1978;

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Sfard, 1995) have shown that children have difficulty accepting a symbolsuch as ‘x+ 3’ because they see it as representing a process and not amental object – a process they cannot carry out because they do not knowwhat x is. Batia’s approach might unintentionally reinforce this tendency.This approach focuses on one facet of algebraic expressions, the processfacet, and does not encourage a conception of an expression as an object.Consequently, Batia’s students may have been helped to view 3+ 4x notas a possible final answer, but rather as a partial answer which they cannot‘finish’.

4.5. Substitution

This approach, commonly mentioned in textbooks, was used by Batia. Itis based, again, on the process facet as the students substitute numbersin two expressions and check if the resulting numbers are the same. Likethe ‘order of operations’ approach, this one does not encourage a viewof algebraic expressions as objects and as final answers. However, it doeshelp in seeing that the expressions are not equivalent.

5. CONCLUSION

As we have seen in this study, not all four teachers were aware of students’tendency to conjoin or ‘finish’ open expressions. The two who were (‘knewthat’) explained this (‘knowing why’) by referring to students’ eagernessto ‘finish’ expressions. Yet none of them went into deeper levels of explan-ation, such as attributing this need to ‘finish’ open expressions to previousexperiences in arithmetic, or to the tension between the process and objectfacets of mathematical concepts.

How can teachers be helped to become acquainted with the existenceand sources of this tendency as well as with the impact of various waysto dealing with it in the classroom? The mathematics education literatureseems to be an obvious source of information about students’ ways of think-ing. The CGI project (Fennema, et al., 1996) and the MCS-cases Project(Markovits and Even, 1994) show evidence of success in making teachersknowledgeable about research findings regarding specific student concep-tions and developing new ways of teaching that take this knowledge intoaccount. However, preservice and inservice teacher education programsattempting to raise teacher awareness of student conceptions should makesome decisions regarding which student conceptions to concentrate on.

A spontaneous solution may be to choose the most salient ones. However,students’ conceptions related to specific mathematical topics may differ

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according to which curricula they study. The extent to which specific diffi-culties are embedded in a specific approach to learning and teaching is notclear. For instance, it is possible that the tendency to conjoin open expres-sions will be found only in classes which use the traditional approach toteaching algebra, like those observed in our study. In algebra today, thereare several innovative programs, including the function-based approachwhich is built on a top-level view of algebra and formulates the subjectof algebra parsimoniously, basing everything on the concept of function(Schwartz and Yerushalmy, 1992). Other examples are the English OpenUniversity approach with an emphasis on expressing generalities, promot-ing the dual perception of process and object of algebraic expressions(Mason, 1985), and the Dutch Freudenthal Institute approach which buildson realistic contexts or situations (van Reeuwijk, 1995). These innovativecurricula attempt to provide meanings to the various activities studentsare experiencing when learning algebra. Providing students with a broadercontext in which not completing the expression makes sense, offer someadvantage, and not simply remain another formal exercise may contributeto students’ accepting ‘lack of closure’.

Several recent studies support the hypothesis that students’conceptionsare related to the specific instruction. For example, Sutherland (1991)conducted an experiment in which students operated variables in a Logoenvironment. She found that the majority of the participants accepted lackof closure in algebraic expressions. Similarly, Tall and Thomas (1991)worked in a computer-based environment, using the Dynamic AlgebraModule. They report that this approach was successful in overcoming the‘lack of closure obstacle.’ Payne and Squibb’s (1990) paper on algebra mal-rules and cognitive accounts of errors also gives evidence that students’misconceptions and difficulties are linked and specific to the curriculumand instructional approach used in their studies. Therefore, rather than tofamiliarize teachers with an ‘inventory’ of student misconceptions, a mainobjective may be to raise their general sensitivity to students’ ways ofmaking sense of the subject matter and the instruction.

While one can use the mathematics education literature in order to raiseteachers’ sensitivity to students’ ways of thinking about open expressionsand their tendency to conjoin or ‘finish’ them, this literature does notoffer enough information and discussion regarding the impact of variousapproaches and teaching methods related to this conception. As our studyindicates, teachers use various methods when dealing with their students’difficulties. In spite of the existence of a variety of teaching methods, twoof the four teachers used only one method in the observed lessons. It isimportant for teachers to be acquainted with various teaching methods and

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to be aware of their pros and cons in different contexts with different teach-ing aims, and with different students. Teachers should consider possibleshort- and long-term implications of the use of each method on students’knowledge and understanding of specific facts, procedures, concepts, andideas as well as on their knowledge about the nature of mathematics. Here,as in other domains, the literature does not look to the rich source ofmethods that teachers create and use to build a vivid inventory of possiblemethods. Such an inventory should be supplemented with rich descrip-tions of classes in which these various methods were applied, and criticalanalysis of their pros and cons. This inventory and these descriptions mayalso be used to raise teacher awareness of the need to employ a variety ofmethods, depending on teaching aims, student understanding, and otherrelevant considerations.

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