The competition between numbers and structure: why expressions with identical algebraic
structures trigger different interpretations - Statistical Data Included
Liora Linchevski
The common perception of school algebra as generalized arithmetic implies that algebraic structural rules are
perceived as rules that draw their legitimization and meaning from rules that are valid in the world of numbers
(Buxton, 1984; Davis, 1985; Smith, 1997). This widespread view has generated the search for a model that
describes the relations between students' understanding of the number system and of the algebraic one (Collis,
1971; Lee and Wheeler, 1989; Linchevski and Herscovics, 1996b). In the context of the school curriculum, it
has motivated a teaching approach that may be described as teaching arithmetic for "algebraic purposes"
(Davis, 1985; Arcavi, 1994; English and Sharry, 1996; Milton, 1999). The proponents of this approach maintain
that difficulties students experience with algebra originate in the lack of a suitable arithmetic foundation and
claim that the failure of students to appreciate the rules of working with "letters" is largely due to their failure
to understand the rules of workin g with numbers. They assume that understanding of the structural rules in
arithmetic guarantees understanding of the corresponding parts in algebra (Kuchemann, 1981; Booth, 1984;
Kieran, 1989; Lee and Wheeler, 1989; MacGregor, 1996; Milton, 1999).
This widespread view of school algebra has also motivated the ongoing research on students' "non-algebraic"
behavior (structural "bugs") in numerical contexts. However, although in some cases there are grounds to
suggest that certain "non-algebraic" behavior of students may be attributed to problems with arithmetic, by the
same token it has also been suggested that the presence of numbers does not always make the task "easier"
(Lins, 1990; Kieran and Sfard, 1999).
Non-algebraic behavior in a numerical context
The following describes one of many actual instances where students seem to be disregarding structural
aspects of numerical expressions when solving numerical problems. These seventh-grade students (1) had
been investigating in class, for quite some time, equivalent numerical expressions. They had been introduced to
the conventions related to the order of operations, to the role of brackets and to the possibilities of "removing"
and "inserting" brackets with and without changing the numerical expressions, such as the commutative,
associative and distributive laws, and had manipulated numerical expressions using their newly acquired
knowledge.
In one of the questions on a test, they were asked to evaluate the expression: 53-3x5+15. Rather than the
expected answer of 53, Ron arrived at 23. His work showed the following calculations: 53-3x5+15=5315+15=53-30=23. We assume that, in the first step, Ron followed the correct order of operations and
multiplied 3 by 5. However, after inserting the result (15) back into the expression, he "detached" (Herscovics
and Linchevski, 1994; Linchevski and Herscovics, 1996a; Linchevski and Livneh, 1999; Kirshner, 1989) the first
15 from the subtraction sign, put mental brackets around the two 15s and got 23 as an answer. Yet, in another
exercise with the same algebraic structure: 46-8x3+7, Ron followed the correct order of operations and got the
expected answer: 29. His written work showed: 46-8x3+7=46-24+7=22+7=29. Does Ron know the order of
operations or not?
Other students in Ron's class also arrived at different interpretations of expressions with identical algebraic
structures. For example, Rachel's answer for 160/(5x2), was l6; she multiplied 5 by 2 and then divided 160 by
10. Is she over-generalizing the order of operations, thinking that multiplication takes precedence over
division? Yet, in another item on her exam paper, 48/8x4, Rachel did not start to evaluate the expression by
first multiplying 8 by 4 and then dividing 48 by 32 (getting 3/2 as an answer), as she had done in the previous
case. This time she correctly followed the order of operations; she divided 48 by 8 and then multiplied the
obtained answer (6) by 4, and arrived at the correct answer (24).
What about Sarie who wrote: 53-3x5+15=1000, or Noam who found that 35-10-10+35 equals -20? They both
decomposed the expressions at unexpected points. Sarie viewed the expression as (53-3)x(5+15), and Noam
as (35-10)-(10+35). It certainly looks as if the students' procedures are accidental, random and inconsistent
and that they are completely blind to the mathematical structure.
It seems that, even after being taught structural aspects of numerical expressions, a considerable amount of
students did not perceive numerical expressions with the same algebraic structure as algebraically the same.
They were unable to "ignore" the specific numbers given in a numerical expression and grasp them as "any"
number. Thus, according to the arithmetic-algebra link, they are not prepared to think and act algebraically. Is
there any reason why the structure of the expressions often does not become transparent to the students
(Booth, 1984), why expressions with the same algebraic structure trigger in students different interpretations?
We hypothesize that how students interpret a numerical expression, as well as the alternative structures they
spontaneously associate with it, are partly dependent upon the unique numerical combination at hand. We
conjecture that certain number combinations often shift the focus of attention from the structure to the
numerical properties of the given terms in such a way that a wrong numerical value is assigned to the
expression. Certain numerical combinations encourage a correct interpretation, while others encourage a wrong
one, and this is probably one of the reasons why Rachel interpreted the expressions 160/5x2 and 48/8x4 in two
different ways. This assumption also raises questions about cases in which the interpretation given by a student
was correct. The properties of the specific numbers at hand -- and not the structural properties of the
expressions -- may very well have induced the correct interpretation.
Biasing number combinations
We contend that certain numerical combinations "seduce" naive solvers into working in a given order of
operations. Consider, for example, the following two expressions:
(1)217-17+ 69
(2)267-30+ 30
We assume that the numbers in Expression 1 lead to sequential computation from left to right (the 17 almost
"begs" to be subtracted from 217) much more easily than the numbers in Expression 2. The latter encourages
operating on the 30s before dealing with the 267. While the first expression will probably not evoke any
alternative scheme to the sequential one, whether correct or wrong, the second expression may evoke the
correct alternative scheme of -30+30 "giving zero" (that is, subtracting 30 and then adding 30 is the same as
doing nothing). It may also evoke the incorrect alternative of performing 267-60, that is, detaching the 30+30
from the subtraction operation (Herscovics and Linchevski, 1994; Linchevski and Herscovics, 1994; Linchevski
and Herscovics, 1996a; Linchevski and Livneh, 1999). Expressions like 530-10+10+10 should greatly increase
the probability of evoking this sort of incorrect scheme.
Similarly, we assume that an expression like:
(3) 12 x 5/2 x 17
would evoke an entirely different scheme, in some people, than the expression:
(4) 150 x 2 / 2 x 150
although both have the same algebraic structure. It is quite plausible that the correct solution - computation
from left to right - will be chosen most often to solve Expression 3, while there will be a tendency to choose
alternative schemes, which are not necessarily correct, in the case of Expressin 4 (e.g., (150 x 2) / (2 x 150)).
A similar contrast should obtain with the following two expressions:
(5) 100 / 2x25 / 5
(6) 35 / 5x2+7
Whereas, in the first pair (Expressions 3 and 4), the different schemes will sometimes lead to a correct solution
and sometimes to an incorrect one, in the second pair (Expressions 5 and 6), the ultimate result will not
necessarily reveal the use of alternative schemes. It is likely that a student who solved 100/2x25 / 5 by
inserting "mental brackets" - calculating (100+2) x (25+5), which gave 50x5=250 - was not doing this for the
right structural reasons, out of a flexible view of the algebraic structure a/bxc/d = (a+b)xc/d = (a/b)xc/d =
[(axc)/b] / d = (axc/b)x(1/d)=(axc)/(bxd) = (a/b)x(c/d). It is more likely that the particular numbers that
appeared in this expression led the student to perform the calculation in this specific order.
In fact, we should not be completely surprised by this phenomenon. The role of biasing number combinations in
the context of multiplication and division has been investigated by several researchers. For instance, Bell and
colleagues (Bell, Swan and Taylor, 1981) have shown that, when children are presented with problems with the
same structure, they use different operations to solve the problem, depending on the specific numerical data
given. Fischbein and colleagues (Fischbein, Deri, Nello and Marine, 1985) emphasize that this phenomenon
occurs even after learners have had a solid formal algorithmic training. Linchevski and Livneh (1999) have
found that different number combinations in expressions with the same algebraic structure are at least partly
responsible for the alternative structures the students spontaneously associate with these expressions.
The same difficulty with number combinations can be inferred from research on students' thinking processes in
generalization activities, such as number pattern generalization or visual pictorial generalization (e.g.,
MacGregor and Stacey, 1993; Orton and Orton, 1994; Taplin, 1995; Lee, 1996; Garcia-Cruz and Martinon,
1997). The data presented in these studies shows that students use different generalization processes for
different items. Our own analysis (Sasman, Olivier and Linchevski, 1999; Linchevski, Olivier, Sasman, and
Liebenberg, 1998) reveals that different items with identical functional relations often trigger different
generalization processes. Moreover, the generalization strategy that students suggest is partly dependent upon
the unique numerical combination at hand, rather than upon functional relations.
For example, the most widespread strategy suggested by students, regardless of the correct functional relation,
is the proportional multiplication strategy, i.e., if [n.sub.2] is k*[n.sub.1] then f([n.sub.2]) is k*f([n.sub.1]).
Thus, in a question like: Complete the missing values f(1)=3,f(2)=5,f(3)= 7,f(4)=9,f(5)= ?,f(20)=?,f(100)=?, a
considerable number of students, after finding correctly that f(5)=11, would calculate f(20) mistakenly as
4x11=44 (Linchevski, Olivier, Sasman, and Liebenberg, 1998).
While Vergnaud (1983) claims that this is an over-generalization of the many direct proportional relationships
that students are intuitively aware of from an early age, we argue that this occurs exclusively in the context of
specific number combinations - which we call "seductive numbers". The use of seductive numbers like
[n.sub.2]=20 while [n.sub.1]=5 stimulates the error. The students are trapped by the multiplicative
relationship between 20 and 5. It is quite possible that other numerical combinations like [n.sub.1]=5. while
[n.sub.2] is 17, 27 or 83, would trigger different, correct or incorrect, strategies. Indeed, a study by Sasman,
Olivier and Linchevski (1999) showed that the introduction of "non-seductive numbers" discouraged the
proportional multiplication strategy and gave rise to other strategies, many of which were inappropriate. These
examples suggest that the choice of a number combination can determine, to a certain extent, the strategy either correct or incorrect - that a student uses.
It is important to note that the above reported results were detected among different students, coming from
different age groups - from elementary to high-school - and from completely different educational systems.
Thus, they are probably pointing at some cognitive obstacles and not just a result of some particular teaching
strategy.
Studies conducted in other areas, such as logic and language, support our observations. In many instances,
sentences with the same logical structure were found to convey different meanings when they had different
linguistic content (Linchevski and Nesher, 1978). Even after subjects had studied the truth tables of the
sentences and practiced analyzing sentences and propositions according to the criteria of their logical structure,
the verbal context still had a considerable biasing effect on their perception of the logical structure of the
sentence and the truth value they assigned it. Walkerdine and Corran (1979) argue that cognitive development
might be viewed as shift of attention from the metaphoric axis - the situation, content and context-to the
metonymic one - structure and syntax. Leonard (1994) uses Walkerdine and Corran's metaphoric/metonymic
axes linguistic perspective as a theoretical frame in her analysis of students' difficulties in understanding
mathematical sentences (Walkerdine, 1982; Walkerd ine and Corran, 1979). She demonstrated how specific
words in a sentence can be replaced on the "metaphoric axis" without changing the logical structure of the
sentence, while changes along the "metonymic axis" lead to a change in the structure of the sentences even if
its key words remain the same. In the second case, it is obvious that the change generally alters the meaning
of the sentence, but Leonard notes that, even in the first case, when the structure of the sentence remains the
same, substituting wards in this axis changes the way students interpret the sentences.
We assume that numbers create a "numerical" context exactly as words create a "verbal" one, and thus have a
considerable biasing effect on students' perception of the algebraic structure, one that is comparable to the
biasing effect words have on the perception of sentences. Along these lines, we conjecture that the properties
of the specific numbers in an expression will determine, to a certain extent, the structural interpretation that
students assign to it. We therefore decided to investigate the relations between number combinations and
structure interpretations in the context of numerical expressions.
Rationale For the Design of the Tasks
To investigate the above assumptions, we designed some student tasks. All the tasks were numerical
expressions constructed according to the following considerations: We chose algebraic structures that the
students were familiar with, for each of which we prepared three different numerical versions. In one version,
the numerical combination encouraged operating according to the correct algebraic structure ("goes with"). In
the second version, the numbers encouraged operating in a way that was opposed to the correct algebraic
structure ("goes against"). In the third version, the numerical combination was "neutral"--that is, in our view, it
did not encourage the solver to operate in either of these ways.
Consider, for example, the following algebraic structure: a-bxc
(a) The following numerical combination "goes with" the structure: 21-5x2
It is "easier" to first multiply 5 by 2 and then to subtract the 10 from 21.
(b) The following combination "goes against" the structure: 127-27x15
It is "easier" to subtract 27 from 127 than to multiply 27 by 15.
(c) A "neutral" numerical combination could take the following form: 20-5x3
as the two possibilities, multiplying first (5x3) or subtracting first (20-5), are equally appealing.
A different sort of "neutral" combination is illustrated by: 13l/17x6
where both possibilities are equally unappealing.
Method
Stage 1
This study is one in a series of studies focusing on pre-algebraic thinking that were carried out in Canada and
Israel as part of a long-term co-operation in this research area (Herscovics and Linchevski, 1994; Linchevski
and Herscovics, 1994; Linchevski and Herscovics, 1996a; Linchevski and Livneh, 1999). In the first stage of the
study, we interviewed all sixth graders (mean age=1 1.5) in two classes, that were compatible in terms of the
curriculum studied, of the public school system, one in Israel (N=31) and one in Canada (N=28). All the
students had learned the order of operations in class prior to the interview and had plenty of opportunities to
drill and discuss equivalent numerical expressions.
Each student was interviewed individually on two consecutive days. The length of each interview was 30 to 45
minutes. An observer was present at each interview to take notes, and also participated later in the analysis of
the students' responses. Taking into consideration Kirshner's comments regarding the possible influence of
visual cues on students' syntactic decisions (Kirshner, 1989), all items were typed and printed with equal
spaces between operations and terms to avoid, as much as possible, any misleading visual cues.
The expressions were ordered in a random way rather than in blocks of expressions with the same algebraic
structure. The student was presented with each expression separately and told: Here is an exercise. Can you
show me an "efficient" way to find the answer to this exercise? (It was clear to the students that by "efficient"
we meant easy, not demanding a lot of paper work. There were no time limitations.) A simple calculator(2) was
available to be used as a number-facts table when needed.
During the interviews we documented, as accurately as possible, the students' procedures. We did not want to
influence their procedures by overly interacting with them. However, after a student had completed the set of
activities, we initiated a less structured discussion in order to gain more insight into his/her considerations.
Since the differences between the students' results in the two classes were non-significant, we combined the
data of both classes.
Stage 2
In the second stage, we administered a written questionnaire (with the same items as in the interview) to 78
seventh graders (mean age = 12.5) in two classes that were compatible in terms of curriculum studied. All
students had learned the order of operations and the relevant structural rules in class beforehand.
The questionnaire was prepared in two versions: open-ended and multiple-choice. In both versions, the order of
the items was the same as in the interviews. In the open-ended version of the questionnaire, the students
(N=38) were asked to calculate the expressions and to write in their answers.
In the multiple-choice version, the students (N=40) were asked to select one of three possible answers: the
correct interpretation, the "biased" interpretation and a third interpretation based, when possible, on wrong
answers students gave during the interviews. In both versions, they were asked to record all steps in as
detailed a manner as possible.
Results
Table I displays success rates for each mode of data collecting (interview, open-ended questionnaire, multiplechoice questionnaire) and for each numerical version given to the algebraic structures ("goes with," "goes
against," "neutral"). To facilitate analysis, we put the data for each set of three numerical versions together.
The actual expressions and the different structural interpretations the students gave in the interview to each of
the expressions are presented in Table 2.
As Table 1 clearly shows, the numerical combination of an expression influences the way students interpret its
structure. In all items, the success rate changes as the number combination changes (with the exception of
Item 1 in the written questionnaires). Moreover, when the numerical combination "went with" the structure, the
number of students who interpreted the structure correctly was usually higher than the success rate when the
numerical combination "went against," For example, while 24+3x5 (Item 3A in Table 2) was interpreted
correctly according to the algebraic structure by 91% of the students that were interviewed, only 62% of the
students did so in 240/%15x2 (Item 3B in Table 2). In the first case, they calculated from left to right while, in
the second, many of them first multiplied and then divided the first number by product. In the former case, the
"seductive" numbers "pushed" them in the correct direction, while, in the latter, the numbers "distracted" them
from it.
Further discussions with the students
This same tendency was found in all three modes of data collecting. However, we hesitate to compare the rates
obtained in the interviews to those obtained from the questionnaires, since the populations were different. In
respect to the comparison between the two questionnaires, we assumed that the alternative answers presented
in the multiple-choice version would trigger alternative structures and induce structures that were not the first
ones to be evoked by the expression. We did not know, however, what direction would take precedence, from
the mistaken interpretation to the correct one or the other way around. The lower success rates in the multiplechoice questionnaire suggest that the students' structural knowledge is not stable, and being presented with a
wrong alternative distracts them from the correct answer more frequently than the other way around.
As mentioned above, immediately after a student was interviewed on the items appearing in Table 1, a less
structured discussion was initiated. In these talks, we presented the student with numerical expressions that
were very likely to induce a high rate of misinterpretation of the algebraic structure. During the discussions,
many alternative interpretations - either correct or wrong - of the algebraic structure were recorded. The
findings support our assumption: The numerical combination of an expression often governed student
interpretations.
Dror, for example, was presented with the exercise: 136-36+29. According to our categorization, the numbers
in this expression "go with" the structure, since they encourage the correct order of operations. Dror, indeed,
evaluated the expression correctly by going from left to right and gave 129 as an answer. Later on, during the
discussion, he was asked to evaluate the expression: 154-20+20.
Dr: First I will do 20+ 20, it 40, 154-40 equals 114.
Dr: It doesn't make any sense, we better go here [pointing at 136-36+29], from left to right.
I: I have a question. In a previous item, 136-36+29, you did something that was a bit different. You solved it
from left to right; 136-36 equals 100, 100+29 equals 129. In this case [pointing at 154-20+20] you first added
the 20s. Would it be OK also here [pointing at 136-36+29] to first add 36+29 and then to subtract the sum
from 136?
I: But you yourself, in the other example [pointing at 154-20+20], you first added 20+20.
Dr. Here it makes sense because here it's easy I mean here it doesn't make any difference if I first add 20 and
20 and only then subtract it [the sum] from 154.
I: I don't follow, so why in the first one it does?
Dr: I don't know, you have to think before you do, maybe I could do it like that, but it's easier the other way.
It is obvious that Dror violates the syntactic rules and changes strategies according to the specific numbers in
the given expressions. He knows that he is supposed to go from left to right, but he also knows that sometimes
"it makes more sense" to do things differently. When and why is not so clear. The two schemata he has in mind
compete with each other. From the following conversation, it is obvious that Dror is not even aware that the
two strategies would produce different numerical answers.
I: Do you think that in both ways you would get the same result?
Dr: [completely surprised] What do you mean?
I: I mean, if we calculate by first adding the 20s and then subtracting 40 from 154 [pointing at the expression]
or we calculate by going from left to right, 154 subtracting 20 and then adding 20, will we get the same result
in both ways?
Dr: It does not matter, the result is the same. ... My way is easier.
I: Let's try. [Dror calculates in both ways, and realizes that he gets two different answers, He checks again.]
Dr: It doesn't make sense. I have to think. Maybe you have to go from left to right...
Inbal was asked to evaluate the three following expressions (which were not presented to her consecutively):
(1) 90%5x2; (2) 36%3x2; (3) 27%9x2. In the first two expressions, Inbal multiplied before dividing, getting 9
as an answer for the first expression and 6 for the second. In the third expression, however, she first divided
27 by 9 and then multiplied the obtained 3 by 2, getting 6 as an answer. When asked to explain her moves,
Inbal said that in the third expression she would also have first multiplied, but she "didn't want to get 1 and
something as an answer" (she estimated that 27 divided by IS would not produce a whole number), and she
"saw that here it's better to do it the other way " In her decisions of how to interpret an expression, Inbal used
her "number sense." The less attractive interpretation is abandoned regardless of structural considerations.
Ron was given the following exercises (not presented consecutively): (1) 540-20+20; (2) 90-20+30; (3) 13636+29. He solved exercises 2 and 3 sequentially, according to the correct order of operations, but in item 1, he
first added the 20s and then subtracted the sum from 540. Upon being challenged, Ron explained: "Addition
and subtraction are the same, so I see what is more convenient." It seems that Ron is aware of the order of
operations but over-generalizes the rules, interpreting the phrase "multiplication and division before addition
and subtraction" as "it does not matter what comes first, addition or subtraction, so we are allowed to do what
is more convenient." In Ron's case, there is an interplay between numerical considerations and structural rules.
He gives priority to numerical considerations, but he is aware of structural rules.
Dan was given the same exercises as Ron. In the first two exercises he violated the order of operations by
adding before subtracting (540-40,90-50). He was then told that some other student did it differently; the
other student calculated the exercise by "going from left to right." Dan did not seem to experience any conflict.
Da: That is also OK, but my way is easier
I: Do you think that you and the other student would get the same answer?
Da: Yes, ... I don't know [Dan was then asked to calculate in both ways. He got two different numerical
answers.] He [the other student] did it his way and I did it my way Both ways are OK.
Da: Take the case of Tern, who was asked:
I: Is it OK to get two different results to the same exercise?
I: Is it correct or incorrect that 100/5%5=100+1?
Te: [looking at the numbers very carefully] Yes, it correct.
I: Will you please explain to me why it is correct?
Te: Because it is the same numbers: S and S. If it were 4 and 5, it would not be correct.
(The interviewer is not sure if Terin means that the order of calculation will still be correct, but the result of 4
divided by 5 won't be 1, or, in the second case, it won't be correct to first divide 4 by 5 and only then to divide
100 by 4/5.)
I: You mean that since here we have 5 and 5, it's correct, but if we had 4 and 5, it won't be correct? Can you
please explain why?
Te: Well, if it's 5 and 5, it gives I and it makes sense. 4 divided by 5, it's... it's simple the other way around,
100 divided by 4, it's simple, and then you just divide by 5.
There were several instances in which students inserted mental or actual brackets where they previously had
not existed, leading them to misinterpret the problem. For instance, Ortal solved 24+3x2 by inserting brackets
aroun 3x2. When asked to explain, she said: "I put the brackets here so it will be easier to solve, there is no
rule that prohibits it. " Roni put brackets around 7+3 in the expression 25-7+3, justifying it by: "There is no
multiplication and division here, so there is nothing I have to give priority to, I do what is easier," Asaf, who
solved 28-8%4+3x2 by putting actual brackets around 28-8, said: "I put brackets, since I see that it is 28-8,
and you first do brackets, I solved it according to the rule." Being distracted by the 28-8, Asaf put brackets
around 28-8, reconstructing the expression as (28-8)/4+3x2, and then declared that "you first do brackets."
Discussion
The results support our hypothesis that the specific number combination in each expression encourages the use
of certain sequences of operations, either correct or incorrect, and discourages others, either correct or wrong.
The particular number combination in the expression competes with the algebraic structure. Our results indicate
that the specific number combination, often shifts the focus of attention from the structure to the numerical
properties of the given terms in such a way that the meaning of the expression is changed and a wrong
numerical value is assigned to it. It seems that certain number combinations, especially when combined with
some specific mathematical operations, trigger deeply ingrained cognitive schemata which probably compete
with the new structural perspective.
The interaction between the structure and the specific number combination seems to explain, at least partially,
what Greeno (1982) sees as random and inconsistent mistakes. Greeno claims that the difficulty confronting
beginning algebra students is that they partition algebraic expressions into component parts in a way that
seems aimless, mistaken and arbitrary. Our analysis suggests that this difficulty may stem at least partially
from the competition between the structure and the biasing number combinations, which frequently encourage
decomposition of the expression at the wrong points.
The findings of our study suggest that numbers create a "numerical" context exactly as words create a "verbal"
one. Numbers are loaded entities that have a considerable biasing effect on students' perception of the
algebraic structure, comparable to the biasing effect words have on the perception of sentences. The numerical
scheme and the structural scheme coexist, and the student is often unable to prioritize them or to integrate
them as separate, yet interrelated schemata that sometimes might and sometimes might not be activated
simultaneously. It seems that the traditional approach to the learning of the formal structure of expressions and
propositions does not address the contextual influences.
Part of the problem may be related to the development of number sense, which is an integral part of the early
stages of teaching arithmetic. One of the components of number sense is mental calculation, which involves the
invention of non-standard methods of calculation based on the properties of the specific numbers at hand (e.g.,
Markovitz and Sowder, 1994). Number sense entails using computational procedures flexibly in accordance with
the given numbers. Thus, in developing number sense, attention is focused on the particular numbers involved
in the calculation. This means that the bulk of the teacher's and students' attention in activities intended to
develop number sense is devoted to the specific numbers used in the expression. This factor is the primary
justification for replacing the standard algorithmic procedure with an alternative one. Furthermore, the goal of
developing number sense often pushes the teacher to carefully choose the specific numbers and operations to
be used in expressions presented to the students, so that non-standard solution procedures will lead to more
elegant and efficient calculations. In other words, the development of number sense largely involves taking
advantage of the properties of the numbers in the specific example.
Number sense is developed in those educational stages in which considerations of algebraic structure are still
only a secondary aspect of the learning process. The examples presented to students axe intended to provoke a
search for relations between the numerical terms and encourage taking advantage of these relations. Often,
these examples are designed, intentionally or not, to avoid any conflict between the immediate, spontaneous
alternative procedure, governed by the type of numbers involved, and the algebraic structure. Thus, in
evaluating an expression like 31+4+6, a solution process in which the 4 and 6 are grouped and evaluated
before the 31 is added, as in 31+(4+6), will definitely be encouraged. Since children at this stage are not
usually confronted with conflicting examples -where the structure of the expression restricts certain "tempting"
grouping - they might over generalize the permitted flexibility. We have to be aware of Fischbein and
colleagues' (Fischbein et al., 1985; Fischbein and Baltsa n, 1999) observations that children generalize the way
they were initially taught in school before they develop a critical attitude, and that some mental behaviors tend
to act beyond any formal control because these behaviors shape the facts at hand in a meaningful way.
Fischbein and Baltsan (1999) challenge a widespread approach to the construction of mathematical schemata
among children, whereby students are presented at earlier stages with activities and experiences that result in
the construction of schemata that will have, at later stages, to be replaced by alternative ones. These
researchers point out that this "replacement" process often does not lead to the extinction of the old scheme
and its replacement by the new one, but rather that this old scheme re-emerges and the new one does not
establish itself. They claim that many times an initial scheme becomes a tacit model. The strength of a tacit
model is in its comprehensiveness; it acts as a structure, as a whole. In such a case, they argue, the init ial
scheme molds the students' reasoning despite repeated instruction of alternative schemata later on.
We find Fischbein and Baltsan's analysis relevant not only instances in which the old scheme is expected to
become extinct and replaced by a new once but also to instances where the new and the old are expected to
coexist. In the current study we found that numerical aspects of mathematical expressions often do not yield to
structural aspects learned at later stages. The numerical perspectives of arithmetic that are the focus at early
stages function later on as a tacit model and probably interfere with the establishment of "structure sense".
The biasing number combination is, of course, not the only explanatory factor. Many wrong interpretations of
the algebraic structure were observed even when the numbers and the structure supported each other rather
than competed. For example, when calculating 43-5x2, 39% of the students mistakenly calculated the
expression sequentially (subtracted 5 from 43 and multiplied the difference by 2) although the numbers at hand
do not encourage this incorrect interpretation. Nonetheless, the numerical combination of an expression is one
factor that may explain students' judgments and misinterpretations (on other factors and obstacles associated
with students' difficulties in tackling the algebraic structure, see, e.g., Matz, 1980; Greeno, 1982; Booth, 1988;
Kieran, 1988, 1992; Kirshner, 1989; Lins, 1990). This new factor offers a different angle for looking at
students' interpretations of algebraic structures, and may suggest some alternative explanations for previously
documented algebraic mistakes.
Introducing school algebra as generalized arithmetic provides the opportunity to situate the structural rules of
algebra in a meaningful context, since the rules may be justified on semantic grounds--in our case the
numerical value of the expression--rather than syntactic ones--in our case formal algebraic structure (BloedyVinner, 1998). The students can make sense of the rules by experimenting with them in the "laboratory of
numbers," and thus establish their meanings and validation via procedural views. However, the influence of the
numerical contexts on students' perceptions should be taken into consideration when this route to algebra is
chosen.
Moreover, it is important to note that, while research on students' structural "bugs" in algebra is quite advanced
and research on students' structural "bugs" in arithmetic exists, systematic research on the cognitive
isomorphism between the two is limited. The link between arithmetic and algebra, although widely accepted,
has not yet been thoroughly examined. The question of whether systematic work on "structure sense" within
the world of numbers will lead students to a better understanding of algebra is still to be answered.
Table 1
Student Interpretations (percentage of correct answers) per Number
Combination and Method of Data Collection
Item Structure
1)
2)
3)
4)
5)
6)
7)
8)
9)
a-bxc
a-b+c
a/bxc
axb/cxd
a/b/c
a-b+c+d
a-b/c+dxe
a-bxc+d
a-b-c
Item Structure
1)
2)
3)
4)
5)
6)
7)
8)
a-bxc
a-b+c
a/bxc
axb/cxd
a/b/c
a-b+c+d
a-b/c+dxe
a-bxc+d
Mode of Data Collection + Number Combination
Interview
Open Ended
goes
goes
neutral
goes
goes
with
against
with
against
61
87
91
62
83
73
69
73
91
33
69
62
47
62
56
44
49
66
52
78
67
47
71
77
49
51
87
92
81
84
49
100
90
44
65
90
Mode of Data Collection + Number
Combination
Open Ended
Multiple Choice
neutral
goes
goes
neutral
with
against
92
84
73
39
91
94
32
53
87
75
80
60
82
78
38
67
87
60
58
49
82
77
29
56
89
71
69
58
84
84
18
56
92
71
62
40
86
79
29
41
75
9) a-b-c
Table 2
89
86
72
81
Items Presented in Interview and Student Interpretations
Item
A
"goes with"
B
"goes against"
C
"neutral"
Item 1: a - b x c
Multiplication first
Subtraction first
43 - 5 x 2
61%
39%
47 - 7 x 5
33%
67%
47 - 3 x 5
52%
48%
Item 2: a - b + c
Subtraction first (a-b)+c
Addition first a-(b+c)
27 - 7 + 5
87%
13%
27 - 7 + 3
69%
31%
28 - 5 + 3
78%
22%
Item 3: a + b x c
Division first (a+b)xc
Multiplication first a+(bxc)
24 + 3 x 5
91%
8%
240 + 15 x 2
62%
38%
24 + 3 x 2
67%
33%
Item 4: a x b + c x d
19x16+15x17
Left to right
(axb)+(cxd)
Other
8x5+2x17
150x2+2x50
62%
38%
0%
47%
49%
4%
Item 5: a + b + c
Left to right
a+(b+c)
Other
75+25+3
83%
11%
6%
75+9+3
62%
38%
0%
Item 6: a - b + c + d
Left to right
a-(b+c+d)
Other
168-20+10+30
73%
20%
7%
130-10+10+10
56%
38%
6%
Item 7: a - b + c d x e
a-(b+c)+(dxe)
(a-b)+c+(dxe)
Other
147-16+4+3x5
69%
21%
10%
37-5+2+4x3
44%
25%
31%
Item 8: a - b x c + d
a-(bxc)+d
(a-b)xc+d
Item 9: a - b -c
286
(a-b)-c
a-(b-c)
56-2x3+20
73%
26
8725-725-386
91
9
57-7x4+20
49%
51
9420-575-575
66
34
47%
53%
0%
64+8+4
71%
29%
0%
450-25+15
77%
23%
0%
28-8+4+3x2
49%
31%
20%
54-2x8+20
51%
49
676-54787
13
(1.) The examples are taken from exam papers of seventh graders in two public schools in Israel. Neither these
students nor their schools were included in our research population.
(2.) By simple calculator we mean a calculator that does not follow conventions related to the order of
operations.
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The study was partly funded by the S. Amitzur Unit for Research in Mathematics Education, at the Hebrew
University of Jerusalem. The authors would like to thank Ms. Patricia Lytle for collecting the Canadian data, Ms.
Rachel Bohadana for her contribution to Stage 2 of the study, and Ms. Helene Hogri for her editorial assistance.
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