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Consistencies and inconsistencies instudents' solutions to algebraic ‘single-value’ inequalitiesPessia Tsamir & Luciana Bazzinia Tel Aviv University , Israelb University of Turin , ItalyPublished online: 09 Aug 2006.
To cite this article: Pessia Tsamir & Luciana Bazzini (2004) Consistencies and inconsistencies instudents' solutions to algebraic ‘single-value’ inequalities, International Journal of MathematicalEducation in Science and Technology, 35:6, 793-812, DOI: 10.1080/00207390412331271357
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Consistencies and inconsistencies in students’ solutions toalgebraic ‘single-value’ inequalities
PESSIA TSAMIR and LUCIANA BAZZINI
Tel Aviv University, Israel, University of Turin, Italy
(Received 21 May 2003)
This paper describes students’ solutions to a commonly taught and notcommonly taught inequality. The findings showed students’ difficulties.Participants implicitly and explicitly exhibited two intuitive beliefs: inequalitiesmust result in inequalities and solving inequalities and equations are the sameprocess. Following the analysis of students’ written solutions, individualinterviews were conducted that gave a better insight into their reasoning andprovided some ideas for teaching. The concluding section of the paper offersrelevant educational implications.
1. IntroductionThere have beenmany calls for teachers to consider students’ mathematical ways
of thinking and their own errors, when designing and carrying out instruction [1–3].However, this is actually quite a complex and demanding task. One reason is thata prerequisite is familiarity with students’ reactions to a wide variety of relevanttasks and an understanding of possible reasons for these different reactions.
In this paper we offer an opportunity to zoom in on students’ reactions totwo very specific types of mathematical tasks, a ‘solve’ task and a reversed, ‘find-a-problem-for-the-given-solution’ task. We then analyse consistencies and incon-sistencies in students’ solutions, and suggest some ways to implement our findingsin instruction. The topic we chose for this purpose was algebraic inequalities.
Why this topic? Inequalities were identified as playing an important role inmathematics but not in the Israeli high-school mathematics curriculum. Whilebeing part of algebra, linear planning, the investigation of functions, calculus,and various other mathematical topics [4–6] in Israel, algebraic inequalities receiverelatively little attention and are commonly discussed only with students in highlyranked mathematics classes in upper grades of secondary schools. And even thesediscussions are usually limited, emphasizing the ‘how to solve it,’ algorithmicperspective of algebraic manipulations, providing students with rules for solvingthem, but neglecting issues such as ‘Why is this the solution?’ and ‘How can I besure that the solution I have reached is the correct solution?’
Another reason for choosing inequalities is that so far, research in mathematicseducation has paid only limited attention to students’ conceptions of inequalities[7–13]. A substantial number of the related articles dealt with teachers’ andresearchers’ suggestions for instructional approaches, not necessarily with researchsupport. They recommended, for instance, the sign-chart method [14], thenumber-line method [15, 16], and various versions of the graphic method [9, 17].The studies that dealt with students’ solutions, identified students’ difficulties in
International Journal of Mathematical Education in Science and TechnologyISSN 0020–739X print/ISSN 1464–5211 online # 2004 Taylor & Francis Ltd
http://www.tandf.co.uk/journalsDOI: 10.1080/00207390412331271357
int. j. math. educ. sci. technol., 2004vol. 35, no. 6, 793–812
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the use of logical connectives [16], and in the solution of inequalities with ‘R’ or ‘’’results [11]. They reported, for instance, on students’ tendency to regard trans-formable inequalities as being equivalent [10], and to multiply or divide both sidesof an inequality by a not necessarily positive expression without changing thedirection of the inequality sign [12, 13, 18].
Moreover, in professional working-group sessions conducted at mathematicseducation conferences, researchers reported, for various countries including Israel,students’ difficulties and teachers’ frustration when dealing with inequalities[19, 20]. Consequently, an Italian–Israeli collaborative study was designed inorder (a) to extend the existing body of knowledge regarding students’ ways ofthinking and their difficulties when solving various types of algebraic inequalities,and (b) to consider possible related instruction.
In this paper, we focus on Israeli students’ reactions to single-value solutionsto inequality tasks. The main related research questions were:
1. Do secondary-school students accept the expression {x|x¼ a} as the solu-tion of an inequality when (a) presented in a multiple-choice task? Or (b)presented in a ‘reversed’ task, asking whether a given set can be the truthsets (the solution) of an equation or of an inequality?
2. Are the students’ reactions to these two tasks usually consistent?
We conclude by suggesting ways to implement our findings in instruction.
2. Methodology
2.1. ToolsA 15-task questionnaire was administered. Students were presented with
standard and non-standard tasks that have similar underlying mathematicalideas. They were given six tasks presented in the manner to which they wereaccustomed in their classes, i.e., ‘solve’ tasks—designated ‘standard tasks’. Theywere also given nine tasks related to the same mathematical issues, which werepresented in a variety of non-customary ways and designated ‘non-standard tasks’.
We focus here on Task 1 (a non-standard task) and Task 9 (a standard task).
Task 1
Given the set S¼ {x �R: x¼ 3}. Consider the following statement:S can be the solution of both an equation and an inequality.Explain your answer.
Task 9
Indicate which of the following is the truth set (the solution) of 5x44 0(a) {x: x>0} (b) R (c) {x: x<�5} (d) {x: 0<x<1/5}(e) � (f )1 x¼ 0 (g) {x: x4 0}
Explain your answer.
1The students used in their classes either {x | x¼ 0} or x¼ 0 as the solutions toinequalities like x2 4 0 or equations like x2¼ 0, and we presented both options (either inwriting or on the blackboard when distributing the questionnaires) in order to make surethat they understood what type of solution we have in mind. The two expressions are usedin the paper as well.
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Task 1 demanded proving the existence of a case where a single-value, i.e. 3 is the
solution of an equation, and also proving the existence of a case where this single-
value is the solution of an inequality. This kind of reversed task, which asks the
students to examine the existence of a case where x¼ 3 is the solution to either an
equation or an inequality, and then, if possible, to provide expressions to match
a given solution, was not dealt with in the classes we investigated. However, since
x¼ 3 is in itself a type of trivial equation, and x� 3¼ 0 is quite an immediate
variation of it, we expected the ‘equation’ demand to be much easier than the
‘inequality’ demand. The needed inequality is an E(x)4 0 expression, i.e. ‘E(x)<0
OR E(x)¼ 0’, where the solution to E(x)<0 is the empty set and the solution to
E(x)¼ 0 is the single-value solution.
Task 9 was a standard ‘solve’ task, similar to other tasks presented in Israeli
classes. Consequently, we expected Task 9 and the first part of Task 1 to be
easy and assumed that most students would solve them correctly. On the other
hand, we expected the second part of Task 1, where the students had to examine
the existence of a case where x¼ 3 is the solution of an inequality, to be
problematic.
2.2. Participants
Participants in this study were 148 Israeli high-school students aged 16–17
years and enrolled in high-level mathematics classes. That is, we examined
students who intended to take final mathematics examinations in high school.
Success in these examinations is a precondition for acceptance to academic
institutions.
In their previous algebra studies, the participating students had studied the
topic of algebraic inequalities, including linear, quadratic, rational and absolute
value inequalities. The participating students were taught this topic in a traditional
way: they were presented with various methods for solving the different types of
inequalities—for example, parabolas or the number line to solve quadratic inequali-
ties, and ‘multiplying by the square of the denominator’ for the solutions of
rational inequalities.
After data analysis, 21 students were interviewed individually. These 21
students were selected on the basis of their written solutions to specific tasks.
In most cases, the interviews aimed to gain a better understanding of the students’
written solutions, which meant that students were usually shown their original
solutions and asked to explain in detail. Here we refer to four of the 21
interviewees, Abby, Betty, Carmel and David2, who were chosen on the basis of
their types of solutions to Tasks 1 and 9. That is to say, in these solutions to Tasks
1 and 9, we identified four types of reactions, namely, correct responses to both tasks,
incorrect responses to both tasks, a correct response to the first task but an incorrect one
to the second, and an incorrect response to the first task but a correct one to the second.
Following this categorization, the above mentioned four students were invited to
individually reexamine their responses—one student was chosen from each group.
2All names used in this paper are pseudonyms.
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2.3. Procedure
The mathematics teachers distributed the questionnaires during mathematics
lessons. Participants were given approximately one hour to complete the ques-
tionnaire. This usually sufficed. The researchers analysed, categorized and sum-
marized the different solutions.
As mentioned before, after analysis of their written solutions, and based on
their solutions to Tasks 1 and 9, four students were selected to be individually
interviewed about their solutions to these two tasks. These interviews had two
aims: understanding the students’ ideas, and trying to lead them in the ‘right
direction’. It was jointly decided by the teacher and the researcher to conduct
starting the interviews on a positive note, that is to say, when possible, to start the
interview by explicitly highlighting the students’ correct ideas. In this sense and in
the sense that any interview with a teacher or a researcher is a kind of intervention,
we relate to these interviews as ‘instructional interviews’.
At the beginning of their interview, each of the four students was given several
minutes to re-examine his/her solutions to Tasks 1 and 9, and directed to one of
his/her correct responses, when available. During the interview, the students were
offered paper, pen and a calculator for any necessary calculations. Each of the
four interviews lasted about 20 minutes, was audiotaped and then transcribed.
Some notes were taken by the researcher during the interview too.
3. Results
We first provide an analysis of students’ written solutions to Tasks 1 and 9
separately. Then, we discuss consistencies and inconsistencies in students’ written
solutions as well as in their oral interviews.
3.1. Students’ solutions to Task 1 and to Task 9
In this section we limit ourselves to the students’ solutions to each of the two
tasks separately, as expressed in their written solutions.
3.2. Students’ written solutions to Task 1
The findings indicate that students had no problem whatsoever in correctly
responding that x¼ 3 can be the solution of an equation. Most of them substan-
tiated their responses with an example, usually of a first-degree equation, such
as 2x� 6¼ 0. This, however, was not the case with the participants’ responses to
the question of whether {x | x¼ 3} can be the solution to an inequality.
When addressing the latter question, only about half (51%) of the students
correctly claimed that x¼ 3 can be the solution of an inequality, and of these
students, only a few (5%) were able to give valid explanations. These latter
explanations were usually the presentation of the following example of the
quadratic inequality (x� 3)24 0.
Students also tended to explain that the claim ‘x¼ 3 can be the solution of an
inequality’ was true, because x¼ 3 could be the solution of a system of inequalities.
Such justifications were often (16%) accompanied by an uncomplicated linear
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example such as:
2x� 64 0
x� 35 0
8<:
Another type of interesting justification given by a small number (3%) ofparticipants was that ‘the claim is true, because x¼ 3 can belong to the set of
solutions of an inequality’. This justification was accompanied by illustrationssuch as x� 1>0, and the further explanation that ‘x� 1>0 is an inequalitywhereas x¼ 3 is a solution. The solution of this inequality is {x: x>1}, and 3 is
one of the values that satisfies the condition. Therefore x¼3 belongs to the truthset of x� 1>0’. This issue is discussed in detail further on.
Still, usually, students (27%) provided no justification for their claim.
3.2 Students’ written solutions to task 9
In contrast to our initial expectations, and in spite of the fact that such ‘solve’tasks were common in the classes under study, only about 50% of the participants
who responded to this task correctly identified x¼ 0 as the solution of theinequality (see figure 1).
A substantial number of the participants claimed that the set of solutions
was empty (�� the empty set, or ‘there is no solution to the given inequality’).Typical explanations were ‘x4 is an expression with an even power and thus it can
never be negative’, where students ignored the ‘zero-option’. Another interestingphenomenon was the students’ tendency to answer that the set of solutions of
5x44 0 was x4 0, which was further explained by a number of them who claimed,for instance, that they ‘simply computed the fourth root of both sides of the
inequality’.As mentioned before, and as can be seen from figures 1 and 2, about half of
the participants claimed that ‘x¼ 3 can be the solution of an inequality,’ and abouthalf of the participants identified x¼ 0 as the solution of 5x44 0. That is, about
half of the participating students indicated the possibility of having x¼ a as thesolution of an inequality, either in Task 1 or in Task 9.
x = 0 x 0 Other0
10
20
30
40
50
60
70
80
90
100
Φ
%
≥
Figure 1. Frequencies of students’ solutions to Task 9 (in %).
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A question that naturally arose was: Were these the same students? That is to
say, did the students consistently express their understanding that x¼ a could be
the solution of an inequality in their reactions to both tasks by responding ‘true’ to
Task 1 and ‘x¼ 0’ to Task 9? Did they consistently reject a single-value solution to
inequalities in their reactions to both tasks by responding ‘false’ to Task 1 and ‘�’,
‘x4 0’ or any other solution to Task 9?
3.4. Consistencies and inconsistencies
We will relate to the consistency in students’ solutions to the two tasks, first
by analysing their written solutions and then by examining consistent and
inconsistent reactions which occurred during the individual interviews.
3.5. Students’ (in)consistent written solutions
Figure 2 shows that about half of the students (57%) provided consistent
responses to the two tasks: about a quarter of the students (29%) consistently
accepted single-value solutions for algebraic inequalities by pointing to a single-
value to the ‘solve’ task, and by responding that the claim that a single-value can
be the solution of an inequality is true. The other quarter consistently rejected, in
both tasks, a single-value option for algebraic inequalities.
More than 35% of the participants were inconsistent in their reactions to the
two tasks. Part of them correctly claimed that x¼ 3 could be the solution of an
inequality, but did not identify x¼ 0 as the solution of the inequality in Task 9.
More interesting were the inconsistent reactions of about 20% of the participants,
who claimed that x¼ 3 could not be the solution of an inequality, usually explaining
that ‘an inequality can only be the solution of an inequality’, even if within the
same questionnaire, they reached an x¼ 0 solution to the inequality presented
in Task 9.
True* Correct
False Incorrect
False Correct
True* Incorrect
0
10
20
30
40
50
60
70
80
90
100
Consistent
Inconsistent
%
Figure 2. Frequencies of consistent and inconsistent reactions to Tasks 1 and 9 (in %).
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More information about the consistency and inconsistencies in students’solutions to the two tasks can be found in the analysis of their responses duringthe oral interviews.
3.6. Students’ (in)consistent oral solutionsFirst, we will deal with the interviews of the two students who provided
consistent reactions, and then we will discuss the two students who reactedinconsistently.
3.6.1. Consistent reactionsAbby and Betty provided consistent solutions to both tasks. Abby correctly
claimed that x¼ 3 can be the solution of an inequality, and also marked x¼ 0 as thesolution of the inequality 5x44 0. She justified her correct response to Task 1 byvaguely stating that ‘there are examples to prove it’, and later on we found thatcorrect solution was based on a misinterpretation of the task. Betty, however,incorrectly stated that ‘x¼ 3 cannot be the solution of an inequality’, justifying itby saying that ‘only an inequality can be the solution of an inequality,’ and solving5x44 0 as x4 0.
Abby’s interview—consistently accepting a single-value solution
Abby was told that she had correctly marked x¼ 0 as the solution of theinequality 5x44 0. She was asked to elaborate on her solution, and she presentedthe following analysis:
Abby: I examined the expression 5x4 in terms of being positive ornegative. The important factor is the even power that causes theexpression to always be positive . . . [long pause] always except forx¼ 0. For x¼ 0 it’s zero. There are no values of x for which 5x4
is negative. That’s why the solution is x¼ 0.
Abby exhibited a good grasp of ‘how the even power influences the sign of theexpression 5x4’. Before the interviewer posed another question, Abby went on tore-examine her solution to Task 1, correctly stating again that x¼ 3 could be thesolution of an inequality. She was asked to illustrate to the researcher what kindof examples she meant when she wrote that ‘there are examples to prove it’.
Abby: Here I wrote that x¼ 3 can be the solution of an inequality. I foundthe inequality 5x� 10>0 where it works [pause]. There are othersuch inequalities but I gave an example [pause] like it’s enough.
Interviewer: Enough for what?Abby: Enough for proving that x¼ 3 can be the solution of an inequality.
This example shows that it’s possible to have an inequality wherex¼ 3 is the solution.
Abby exhibited a good grasp of the role of an example in validating an existencestatement [21]. However, the example she used indicated that she confused havinga value like 3 as ‘a solution,’ i.e., one of a possible number of solutions thatdetermine the truth set, and having a value as ‘the solution’, i.e., this number isthe entire truth set of the inequality. The interviewer wanted Abby to identifythe difference between the two situations.
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Interviewer: Is 3 the only solution or is 3 a solution, that is to say. . . ehh, one
of several or many solutions of the inequality 5x� 10>0?
Abby: A solution. [pause; looks at the task] It [the task] refers to the
solution. x¼ 3 has to be the solution. [pause] My example isn’t
suitable. I didn’t answer this question.
Interviewer: What do you mean?
Abby: The answer to the given question is no. x¼ 3 can be part of a range
that is the solution, but it . . . [pause] I mean a single number,
cannot be the entire solution of an inequality. The solution of an
inequality has to be a range.
Abby easily noted that she had not answered the posed question. But surpris-
ingly, after re-examining her response her reaction was a total withdrawal from
the original claim that x¼ 3 could be the solution of an inequality. Even though
she herself had previously discussed a case where an inequality resulted in x¼ 0,
she now explained her revised solution by saying, ‘the solution of an inequality
has to be a range’.
At this stage, Abby was expressing two incompatible ideas: the general view
that the solution of an inequality must be a range and the ‘x¼ 0 solution’, i.e.
a single-value solution to a specific inequality that she herself had solved correctly.
As mentioned before, her written solutions were in front of her. The interviewer
decided to direct her to her x¼ 0 solution to Task 9.
Interviewer: Can x¼ 0 be the solution of an inequality?
Abby: No. [pause] The case of x¼ 0 is similar to that of x¼ 3. It’s
basically the same question . . . [pause] Of course it can. I just
solved 5x44 0 [takes out a pen and paper and starts to scribble
5x4� 34 0, x4� 34 0] No. . .
Interviewer: What are you looking for?
Abby: Ehh. [pause] A three, like I need to have a three. It is the solution.
And. . . and the expression needs to be positive . . . . Not negative
[writes (x� 3)4 4 0]. Yes. . . yes (x� 3)44 0 is a case where the
solution of a given inequality is x¼ 3.
It is clear that this question served as an anchoring task. Abby immediately
identified the similarity between the ‘x¼ 3 task’ and the ‘x¼ 0 task’. Then she
successfully connected her correct response to the previously discussed ‘solve task’
(Task 9) and the task, ‘Can x¼ 0 be the solution of an inequality?’ At this stage
she was using pen and paper, looking for a format where the given solution 3, and
the even power, which she regarded as necessary, would be implemented in
creating a suitable inequality. She first tried to design an inequality whose right
side consisted of a non-negative expression E(x), and an E(x)4 0 situation. She
assumed that since 3 was the required solution, there should be a 3 in the new
inequality, and she therefore started with 5x4� 34 0 which included the even
power to ensure the non-negative demand, the solution 3, and the ‘4 0’ rela-
tionship. In her second attempt, she ignored the coefficient 5, but the expression
x4� 3 was still not a non-negative expression. Her third attempt was successful
and she managed to formulate the example (x� 3)44 0, which satisfied all the
requirements, and presented it.
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To make sure that x¼ a had been grasped by Abby as the possible solution of
an inequality we also asked whether x¼ 7 could be the solution. Upon her correct
response, the interviewer once more challenged her grasp of the role of an example
in proving existence statements. As before, she exhibited a good understanding:
Interviewer: Does your example prove that x¼ 7 can be the solution of an
inequality?
Abby: Of course. If I provide a case, an inequality, to which the solution
is x¼ 7, then I have shown that this situation is possible. I mean
that x¼ 7 can be the solution of an inequality. I even showed what
type of inequality fulfills this condition.
Interviewer: Yes?!?
Abby: Ehh . . . (x� 7)44 0,
In conclusion, Abby was asked to comment on her previous response that ‘the
solution of an inequality has to be a range’.
Abby: This was really a strong impression I had. To be honest, it seems
very logical even now. [mumbles: an inequality should have a
solution in the form of an inequality]. But I know for sure that
x¼ a can be a solution of an inequality, because I can show such an
inequality. I just didn’t have the necessary examples and the
general impression was that the sign [inequality] in the task should
match the sign in the solution.
In her reaction, Abby related to her continuing intuitive tendency to regard
only an inequality as the solution of an inequality. However, she also exhibited
certainty in the correctness of her counterintuitive response as she valued the role
of the examples she found in proving its correctness.
Betty’s interview—consistently rejecting a single-value solution
Betty was the second consistent student but unlike Abby she wrote two
incorrect solutions. Given such a starting point, it was difficult to have a ‘positive
opening’. Thus the interview began with clarifying questions to better understand
Betty’s ideas, while the interviewer sought opportunities to pose guiding ques-
tions. The first step was to show Betty the two solutions she had provided to Tasks
1 and 9, and ask her to explain her ideas. She chose to start with her x4 0 solution
for the 5x44 0 inequality:
Betty: I divided both sides of the inequality [5x44 0] by five and reached
x44 0. [pause]
Interviewer: Is it OK to divide both sides of an inequality by 5?
Betty: Yes. I can divide both sides of the inequality by a positive number.
Interviewer: And?
Betty: I calculated the fourth root of both sides, and got x4 0.
Interviewer: Is it OK to calculate the fourth root of both sides of an inequality?
Betty: Sure. The fourth root is a root of an even order, so we can calculate
it when the given expressions are not negative. This is exactly the
case here. Neither 5x4 nor zero are negative, so it’s OK to perform
this calculation for which I got x4 0.
Interviewer: Ahha. . . sure?
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Betty: Sure. These are all procedures I know very well from solving
equations.
Betty performed a valid manipulation of ‘dividing both sides by 5’, but
incorrectly defended the ‘calculation of the fourth root’ of both sides. She
explained that her certainty in the correctness of her solution derived from her
experience with such procedures for solving equations. In order to probe beyond
the algorithm, the interviewer asked Betty to explain the meaning of her solu-
tion in terms of ‘positive’, ‘zero’ and ‘negative’ values. When Betty explained that
‘x smaller or equal to zero’ means that ‘x can be either a negative number or zero’,
the interviewer asked her to examine specific values:
Interviewer: Is (�2) a possible value for x?
Betty: Yes. It is smaller than zero [pause]. �2 is smaller than zero. Yes.
It’s OK.
Interviewer: Ahha. . . Is it?
Betty: Yes. I think I gave a good explanation [pauses looks at the task]
may be I’ll show you here [points to the original inequality; writes]
5x(�2)44 0 [pause] ehh...
Interviewer: Yes???
Betty: Actually, no. If I substitute (�2) in the given inequality, then (�2)4
is 16, then times 5 is [pause] positive. Something’s wrong here.
The power is 4. It’s even, so when we substitute (�2) the result is
positive. It [x] cannot be a negative number. [long pause] The
solution is x¼ 0.
Betty’s examination of (�2) started by substituting it in the x4 0 expression
of her solution, where it was found to be adequate. Then, she substituted (�2) in
the original inequality and realized there was a contradiction. She continued with
the qualitative analysis of the role of the even power in determining the solution,
and reached the conclusion that she had been wrong initially, and that the solution
should be x¼ 0.
Betty: It’s very strange. I can’t find what’s wrong with my original
solution. If I go step by step, I reach x4 0, [pause] which is an
error. Only if I examine the expression in general terms . . . . ehh
like what’s positive. . . and . . . and what’s negative. . . then I reach a
correct solution. [pause] Why? How can I know when to solve it
this way? [pause] It’s also amazing that I expected the solution of
an inequality to be an inequality. . . [pause] a range of numbers,
like x larger than something or x smaller than something. I feel so
confused. . . actually frustrated [pause] like I thought I knew how to
go about such inequalities.
The x¼ 0 solution that Betty reached surprised her and it shattered her
intuitive assumption that an inequality must result in an inequality. She could
not reach the same solution by algebraic manipulations. However, in spite of
these peculiarities, she was convinced by her verbal analysis rather than by the
algorithmic procedure, and she very clearly pointed to x¼ 0 as the solution of the
inequality. She then recalled with no further probing of the interviewer that she
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responded that x¼ 3 could be the solution of an inequality, which faced her with a
dilemma:
Betty: Since x¼ 0 is the solution of the inequality 5x44 0, it is clear that
x¼ 0 can be the solution of an inequality. The question is can x¼ 3
also be the solution of an inequality?
Interviewer: What would you say?
Betty: [long pause] No. I would say that this is impossible. In general,
I still think that a solution of an inequality should be an
inequality. I thought that it always had to be so, [pause] but I
now see that x¼ 0 is the solution of the inequality 5x44 0. Now
I see a case where a single-value can be the solution of an
inequality, but I only found an example for zero, and zero may
be a special case.
Interviewer: So?
Betty: The form of the solution x¼ 0 and x¼ 3 seems similar, and the
general question, can this ‘expression of equality’ be the solution
of an inequality, also seems similar, but somehow I have the feeling
that it’s not similar.
Interviewer: What do you mean?
Betty: [Takes the pen and writes 3x44 0, x4�4 3, x4 � 34 0, x4þ 34 0]
Zero is special. It’s neither positive nor negative. It’s different.
So I believe that this [x¼ 0] is the only case where a single-value
can be the solution of an inequality.
Betty asked herself whether it was possible to deduce the case of x¼ 3 from the
case of x¼ 0. She decided that x¼ 0 was the solution of an inequality, but x¼ 3
could not be such a solution, since she could not find a suitable example. This led
her to jump to the conclusion that there was no such example.
Interviewer: Can we be sure that there is no such example of an inequality for
x¼ 3? I mean, perhaps you just haven’t found an example yet, but
it does exist.
Betty: You’re right. The fact that I didn’t find an example doesn’t prove
that such an example doesn’t exist. That’s why I also gave a general
explanation about the special nature of zero. Inequalities result in
ranges, in inequalities, and only zero can be a single-value solution
of an inequality.
Betty was aware that her lack of success in finding an example did not prove
that one did not exist. What did seem conclusive to her was the intuitive
explanation that an inequality has to result in an inequality and that zero is special.
At this stage, the interviewer decided to tell Betty that x¼ 3 can be the solution
of an inequality, and see where she would look for evidence.
Interviewer: What if I tell you that x¼ 3 can be the solution of an inequality?
Betty: [bursting into the interviewer’s words] Then you have to convince
me, because I believe that it’s impossible.
Interviewer: Will an example be good enough?
Betty: Sure.
Interviewer: What will you then do with your ‘zero-explanation’?
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Betty: Nothing. You haven’t given an example yet, and I don’t believe youhave one.
Once more we can see how confident Betty was in her solution.We would like to note that following this interview, Betty was asked to solve
10 inequalities at home, including the following: (5x� 15)44 0 x2�6xþ 94 0 and|x� 3|4 0. She managed to reach the correct x¼ 3 solution to the first and to thelast tasks, and was interviewed for about half an hour once more. The intervieweragain discussed with her the possibility of x¼ 3 as a single-value solution to aninequality. Again, she applied the ‘starting positive’ approach, using Betty’scorrect response as a springboard for leading her in the correct direction duringthe interview.
Inconsistent reactions
Carmel and David gave different responses to Tasks 1 and 9. Carmel correctlywrote that x¼ 3 could be the solution of an inequality and justified her claim by theexample (x� 3)24 0. However, she did not identify the underlying mathematicalsimilarity between Tasks 1 and 9, and instead of x¼ 0, she incorrectly marked ‘�’as the solution to Task 9. On the other hand, David inconsistently claimed thatx¼ 3 could not be the solution of an inequality, ‘because the solution of aninequality is an inequality’; nevertheless he still reached the solution x¼ 0 toTask 9.
Carmel’s Interview—inconsistently accepting the option of a single-value solutionbut not reaching a single-value in the ‘solve’ task
The interviewer started by telling Carmel that she had correctly claimed thatx¼ 3 could be the solution of an inequality, and asked her to elaborate on herjustification.
Carmel: I gave the example (x� 3)24 0 to show a case where x¼ 3 is thesolution of an inequality.
Interviewer: What do you mean?Carmel: This [(x� 3)24 0] means that (x� 3)2 should be either negative
or zero. But (x� 3)2 is an expression that can’t be negative,because of the even power. So, it can only be equal to zero.[pause] (x� 3)2¼ 0 means that (x� 3)¼ 0, and that is true whenx¼ 3. That’s how I showed that x¼ 3 can be the solution ofan inequality.
Carmel expressed a good understanding of the role of the even power in makingthe expression non-negative, and also a broad understanding of the role that anexample plays in proving an ‘existence statement’. The following questions posedby the interviewer were aimed at directing her from her present example(x� 3)24 0 to correctly solving 5x44 0.
Interviewer: Can x¼ 1 be the solution of an inequality?Carmel: Yes. Like . . . as . . . instead of 3 you put 1 . . . so . . . I mean . . . ehh. . .
(x� 1)24 0.Interviewer: Can x¼ 0 be the solution of an inequality?
Carmel: [thinking] Yes. Like it’s (x� 0)24 0. . . [pause] it’s actuallyx24 0.
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Interviewer: Can the power be anything but 2?Carmel: No. Yes. I mean that it can be any even number . . . like . . . 2, 4,
100 [pause] x44 0, x64 0.Interviewer: Can it also be 3x64 0, 5x44 0?
Carmel: Yes.
Interviewer: How would you convince me that x¼ 0 is the solution of 5x44 0?Carmel: Any non-zero number to the power of 4 results in a positive
number, so it can’t be either negative or zero. Zero to the powerof 4 is zero then multiplied by 5 still remains zero. This is theonly solution.
Carmel was indeed led to use her correct solution to (x�3)24 0 as a basisfor finding validating examples for her claims that x¼ 1 and x¼ 0 could bethe solutions of inequalities. She then successfully generalized from a square-expression to any even power with a positive coefficient expression. After hercomprehensive explanation why x¼ 0 was the solution of 5x44 0, she wasreminded of her original ‘�’ solution to the same task.
Carmel: Wow . . . [pause] I remember thinking that 5x4 consists of 5—apositive number, multiplied by x4, which is also an expression ofa positive number, because of the even power. So I concludedthat the expression is always positive [pause] and that it can neverbe negative. I completely forgot about the zero-possibility [longpause].
Interviewer: I see.Carmel: What troubles me is that when I solved the inequality I was
sure . . . like really sure . . . that I had covered all possibilities. Iask myself how I can guarantee that I won’t make a mistake likethat again?
Interviewer: What would you suggest?Carmel: I dono’ perhaps . . . like . . . It may be a matter of getting organized
[thinking]. Maybe I should write a ‘smaller or equal task’ like5x44 0 as a system of an inequality and an equation [writes(a) 5x4<0 or (b) 5x4¼ 0] and . . . and here the solution of inequal-ity (a) is �, and the solution of the equation (b) is zero. Thereforethe solution of the system is zero.
In her analysis, Carmel presented the line of reasoning that had led her to mark‘�’ as the solution to 5x44 0, finally realizing that she had related to the evenpower as necessarily resulting in a positive value, while neglecting the option ofzero. Even more interesting was her analysis of her intuitive certainty about herincorrect solution and her need to find a systematic way to prevent such mistakes.Her suggestion was to present any ‘smaller or equal’ task as an ‘or-system’ andto solve it as such.
David’s Interview—inconsistently reaching a single-value when solving, but gen-erally rejecting the option of a single-value solution
As mentioned before, David, in spite of the x¼ 0 example that he himself gave,made a general, negating statement that expressed the view that x¼ a could notbe the solution of an inequality. The interviewer started by telling David that he
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had correctly marked x¼ 0 as the solution of 5x44 0, and asked him to elaborateon his response.
David: The question was for what x’s is 5x4 not positive, [pause] but thisexpression is positive for any x except . . .but . . . like not for x¼ 0.Therefore the solution is x¼ 0.
Interviewer: What is the solution of x44 0?David: The same; x¼ 0.
Interviewer: What is the solution of x24 0?David: Also zero. The same.
Interviewer: What is the solution of (x� 1)24 0?David: One.
Interviewer: Sure?David: Yes. Here too it’s that (x� 1)2 has to be non-positive. But it’s
always positive, except for the case where x� 1¼ 0, then x¼ 1.
David provided a good analysis of his solution to the 5x44 0 inequality andthen correctly solved the following x44 0, x24 0 and (x� 1) 24 0 inequalities.At this stage, the interviewer moved from the format of ‘solve’ tasks to ‘Can youprovide an inequality where the solution is a single-value ‘a?’ tasks.
Interviewer: Can you find an example, an inequality where the solution is x¼ 2?David: (x�2)24 0 seems to be OK. [pause] Yes, it is.
Interviewer: Can x¼ 3 be the solution of an inequality?David: Yes. Change the 2 to 3. . . here. . . I mean, it’s the solution of
(x� 3)24 0. [Looks at his original response on the table] Thisquestion was given in the questionnaire. [pause] I was wrong here[in the written response]. I tried to find a suitable example, butthen I couldn’t find one. All the examples that I did find hadinterval-solutions. They had the form of an inequality. So itseemed like a general behavior that the solutions of inequalitiesalways have the form of inequalities.
Interviewer: What would you say now?David: x¼ 3 can be the solution of an inequality. What this actually
means is that my impression that an inequality must result in aninequality was wrong, [pause] even though it seemed convincing[pause] seems convincing. But. . . an inequality can result in asingle-value.
David made a smooth shift from the ‘solve’ task to the ‘find a suitableinequality’ tasks. The question ‘Can x¼ 3 be the solution of an inequality?’reminded him of his initial response to the written questionnaire. He immediatelyunderstood that the example he had found refuted his general claim that ‘thesolutions of inequalities are intervals’. He also mentioned that his initial impres-sion of ‘the general behavior’ of inequalities resulting in the form of inequalitiesseemed convincing but he was now certain of its incorrectness.
4. DiscussionAmong the questions evolving from this study are: What have we learned from
this study about students’ solutions to inequalities that result in a single-value?
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What are possible reasons for students’ incorrect solutions? And what can weconclude regarding related instruction? In this section we will address thesequestions.
4.1. What have we learned from this study?In this paper, we focused on secondary school students’ reactions to single-
value solutions to inequality tasks. The main research questions posed in theintroduction were: (1) Do students accept the expression {x | x¼ a} as the solutionof an inequality when it is (a) presented in a multiple-choice task? or (b) presentedin a ‘reversed’ task, asking whether a given set can be the truth sets (the solution) ofan equation or of an inequality? and (2) Are the students’ reactions to these twotasks usually consistent?
The examination of students’ reactions to the non-standard Task 1 indicatedthat, as expected, many of the students encountered difficulties in identifying thepossibility of {x | x¼ 3}as the solution of an inequality. About half of them claimedthat this cannot be the solution and their justifications were either that ‘thesolution to an inequality has to be a range,’ or that ‘solutions to inequalitiesmust be inequalities’. Even among students who declared the existence of aninequality for the single-value solution, only a very limited number gave asatisfactory example to match this claim. Most prevalent were irrelevant justifica-tions and examples providing either a system of linear inequalities or an inequalitylike x� 1>0, where x¼ 3 was included in the truth set. In the oral interviews, afterclarifying the difference between ‘x¼ 3 being a solution, i.e. one of many solu-tions,’ and ‘x¼ 3 being the only solution’ to an inequality, students withdrew theirinitial correct claim that x¼ 3 could be the solution of an inequality.
Moreover, quite surprisingly, also in their solutions to the standard, ‘solve’task, only about half of the participating students identified x¼ 0 as the solutionof the inequality 5x44 0. As in previous studies that reported ‘strange’ solutionslike � and R as problematic for students [11], this study pointed to the ‘single-value’ type of solution as another problematic case of inequality tasks. A closerexamination of students’ solutions of 5x44 0, reveals connections betweenstudents’ ways of solving this task and their tendency to solve it correctly.
Two main approaches were found in students’ solutions—the algorithmic,algebraic manipulations approach, and the verbal examination of the givenexpressions. When using the algorithmic approach, students performed a sequenceof algebraic manipulations consisting of both valid and invalid manipulations.In the verbal examination of the task, students related to the influence of aneven power on the sign of the solution and to the meaning of ‘smaller than or equalto zero’. In their verbal analysis, many of them were able to conclude that ‘thesolution can never be a positive one’ or that ‘the solution must be either smallerthan or equal to zero’. The correct solution was usually reached by students whoapplied the verbal analysis method.
Furthermore, each of the approaches contained specific pitfalls leading totypical errors. When applying the algorithmic approach, students usually reachedthe incorrect solution x4 0, exhibiting a limited ability to distinguish betweenvalid manipulations and invalid ones. The incorrect solution � was commonlyreached when students misused the verbal analysis of the expressions given in thetask. In these analyses, they assumed that the expression 5x4 was ‘always positive’,ignoring the ‘zero option’.
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Most interesting are the findings regarding the consistency of students’
reactions to the two tasks. We should remember that both tasks were included
in the same questionnaire and students were free to move back and forth among
the different tasks. In this manner, students’ correct solutions to Task 9 could
have served as an example for correctly solving Task 1. However, no student gave
an explicit sign of returning to Task 1 after correctly solving Task 9. Furthermore,
a non-negligible number of students responded to Task 1 and to Task 9 in
a contradictory manner. They wrote, ‘An inequality can only be the solution of
an inequality’ (Task 1), and then that x¼ 0 was the solution of the inequality
5x44 0 (Task 9). These students addressed each problem separately, as an isolated
target, failing to notice even contradictory responses that they themselves had
given to different tasks related to the same issue [22]. They exhibited disconnected
fragments of knowledge, probably derived from partially memorized ‘recipes’ of
rule-based procedures for solving certain inequalities [23].
In several cases, where students marked the single-value, zero-solution to
Task 9 but ruled out any non-zero value from being a single-value solution to an
inequality in Task 1, they explained that zero was a ‘special case’. In the oral
interviews, some elaborated on this issue. Students who were able to grasp the
similarity of the {x | x¼ 0} solution to all other single-value solutions were
encouraged by this understanding to search for an example where x¼ 3 was the
solution of an inequality. For example, while Betty regarded the issues of an ‘x¼ 0
solution’ and an ‘x¼ 3 solution’ for inequalities as two distinct cases, Abby
identified them as similar ones. Abby explicitly said that ‘the case of x¼ 0 is
similar to that of x¼ 3’, and continued, ‘it is basically the same question’. The
identification of this similarity led her to search for an example where x¼ 3 was the
solution of an inequality, and then to construct such an example.
Occasionally, we also found inconsistencies in students’ solutions to Task 9
(5x44 0), when during individual interviews they shifted from one way (e.g. the
verbal examination) to the other (e.g. the algorithmic approach). Since it was the
same task to which these students reached different solutions, a mere confrontation
of the students with their solutions was usually sufficient for raising their aware-
ness ‘that something is wrong’, and for encouraging them to examine and resolve
the incompatibilities. These cases raised no doubts regarding the need to reach the
same solution.
In conclusion, our findings showed that students encounter difficulties in
reaching single-value solutions to inequality tasks and that they tend to reject
a single-value option when directly asked to consider it. In the next sub section
we examine possible reasons for students’ erroneous solutions.
4.2. What are possible reasons for students’ incorrect solutions?
Our data indicated that when solving inequalities—as in many other cases in
which formal, mathematical performance contradicts intuitively-based beliefs—
intuition interferes with students’ mathematical decisions [24]. Our study
revealed two intuitive beliefs: (a) inequalities result in inequalities (i.e. the solution
(truth set) of an inequality must have the format of an inequality), and (b) solving
inequalities and solving equations are the same process and these beliefs have an
overpowering influence on students’ solutions to algebraic inequalities that result
in a single-value.
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The influence of the intuitive belief inequalities result in inequalities wasextensively evident and seemed to influence many solutions to Task 1. Abouthalf of the students absolutely rejected the possibility of a single-value solution toan inequality and expressed their intuitive belief not only by stating that {x | x¼ 3}cannot be the solution of any inequality but also in their justifications, where theyexplicitly wrote or said that ‘the solution of an inequality must be an inequality’.Moreover, even a substantial number of those whose judgment was that {x | x¼ 3}could be the solution of any inequality, and thus appeared to accept the single-value option, revealed the same intuition when asked to explain their solutionsduring individual interviews. That is to say, the impact of the inequalities mustresult in inequalities belief was evident even among students who first argued thatx¼ 3 could be the solution of an inequality. Students like Abby, who seemed torespond correctly that x¼ 3 could be the solution of an inequality, were actuallytrapped in the intuitive belief that inequalities result in inequalities. In fact, theyregarded the value 3 as one of many values included in the range of the solution.When confronted with the requirement to consider a single-value as the solutionof an inequality, their intuitive ideas were revealed. Consequently, some of themwithdrew their initial acceptance of a single-value solution to an inequality,replacing it by: ‘the single-value ‘3’ belongs to the truth set, rather than beingthe entire truth set’.
In addition, the intuitive belief inequalities must result in inequalities was evidentnot only in the students’ reactions to Task 1 but also in their reactions to Task 9.The findings indicated that a number of students who reached the incorrectsolution x4 0, expressed confidence in the correctness of their solution andsubstantiated their acceptance by the ‘solutions to inequalities must be inequali-ties’ belief.
The influence of the intuitive belief solving inequalities and equations are thesame was extensively evident in many students’ solution to Task 1. Students whosolved 5x44 0 in an algorithmic manner usually applied a sequence of algebraicmanipulations they had learned for the solutions of equations. Some explicitlysaid that equations and inequalities are similar and that they have to be solved bythe same procedures. These students usually expressed confidence in the correct-ness of their (incorrect) solution, because all the procedures had been successfullyapplied ‘a million times’ when solving equations [12].
4.3. What can we conclude regarding related instruction?As we mentioned in the introduction, it has been recommended time and again
that students’ ways of thinking be considered in planning and in carrying outinstruction [2, 3]. Most prominent amongst our findings were the intuitive beliefsthat inequalities must result in inequalities and that solving inequalities and equationsare the same process. The question that naturally arises is how can this informationbe implemented to instruction?
It is our view that teachers should be familiarized with these intuitive beliefsand with their impact on students’ reasoning when solving inequalities. Theyshould, subsequently, be encouraged to look for ways to promote students’awareness of various intuitive obstacles and of the need to be ‘on guard’ whensolving mathematical problems [24]. Teachers may address students’ beliefs thatinequalities must result in inequalities by including more, and a greater variety of,inequality tasks that result in single-value solutions, as well as by explicitly relating
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to this intuitive belief as one of the issues discussed when teaching inequalities.More specifically, reverse order tasks, i.e. Here is the solution, if possible, providea suitable task that results in this solution, or What if Not tasks [25], i.e. Examinethe inequality 5x44 0—what if the coefficient is not 5 or if the power is not 4?; maybe challenging to students. These tasks will allow students to concentrate on themore general mathematical aspects and will also provide them with an additionalopportunity to review and reflect upon materials previously studied.
Likewise, teachers may address students’ beliefs that solving inequalities andequations are the same by explicitly discussing in class our intuitive tendenciesto reason in this way, and then encouraging a mathematical analysis of thedifferences between ‘equal’ and ‘unequal’ relations (e.g. ‘¼’ being a symmetricrelation while ‘<’ being asymmetric). The What if Not approach that wassuggested for treating the inequalities must result in inequalities belief may provebeneficial in addressing this intuitive solving inequalities and equations are the samebelief too—for example, examining the inequality 5x44 0, and asking What if it isNot a ‘4 ’ relation? Among the possible options for discussion are ‘5x44 0’,‘5x4>0’, ‘5x4 5 0’, ‘5x4 6¼ 0’ and also ‘5x4¼ 0’, i.e. the related equation. Thesetasks grant the opportunity to examine similarities and differences between the twoentities, that of inequalities and the previously studied one equations.
Moreover, our findings showed that the verbal analysis method may lead tobetter results when dealing with inequalities of the type ax2n4 0 (a>0, n " N),and it is therefore recommended to suggest this method in class and promotestudents’ awareness of their correct, single-value solutions.
Another interesting finding indicated two types of inconsistencies in students’performance with inequalities that result in single-value solutions. One type isincompatible solutions to the solve 5x44 0 task yielded by different solvingmethods, and the other type is incompatible solutions to the standard ‘solve’task and the reversed non-standard task. With regard to the first type ofincompatibilities, during interviews we found that students who addressed theinequality 5x44 0 using both the algorithmic as well as the verbal approachspontaneously identified contradictory solutions, if such existed. However, inour study, only students who were somehow encouraged in the interview stageto use the two approaches were troubled by the question of how to examine thecorrectness of their own solutions to inequalities. We therefore suggest—at leastwhen relating to inequalities of the type illustrated in this paper—applying boththe algorithmic and verbal approaches, examining together with the students anyinconsistencies that may arise, and analyzing these inconsistencies with referenceto relevant formal considerations. It should be noted that since the examinationof the validity of a solution to an inequality is quite complex, solving it in differentways may be useful for checking the already reached solution.
Addressing the second type of inconsistencies found here, i.e. betweenstudents’ solutions to the standard, ‘solve’ task and to the reversed non-standardtask, was revealed as more problematic. Even though for us (teachers or research-ers) both tasks essentially dealt with the same issue of inequalities resultingin a single-value solution, this was not necessarily the case for the students. Ourstudy has illustrated how students’ notions of what is similar or different influencetheir solutions, and it is therefore extremely important to consider this whendesigning instruction. While, as we mentioned, equations and inequalities wereregarded as totally similar, zero and non-zero solutions were regarded as different.
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Students’ tendency to regard zero as being different from all other numbers ledthem to conclude that only zero could be a single-value solution of an inequality.For example, Betty said that ‘zero may be a special case’, and Carmel mentioned‘I completely forgot about the zero possibility’. We therefore suggest discussingcases where zero plays a different role. For example, the inequalities 5x44 0 where{x | x¼ 0}, 5x44 5 where {x|�14 x4 1} and 5x44 (�9) where the truth set is �,all have different types of solutions. But inequalities like 5x44 0, 5(x� 8)64 0 and(xþ 5)24 0, all result in a single-value, and it is of no significance whether thesingle-value is zero, eight or minus five. In general, we recommend that on variousappropriate occasions, teachers refer in their teaching to what is similar and whatis dissimilar in zero vs. non-zero cases, in equations vs. inequalities, etc.
The presentation of the two tasks on which we focused in this paper also grantsan opportunity for discussing the role of consistency in any mathematical perform-ance, and issues such as, ‘Is it reasonable to accept two different solutions to thesame inequality?’ or ‘How can we verify our solutions to inequalities?’
While these are only a few of the implications, it is notable that even a mini-study like the present one can suggest a number of implications for teachingmathematics in general and for the teaching of inequalities in particular. Ourfindings call for interventions that deal with the specific issue of algebraic inequali-ties and with the general issue of consistency in mathematical reasoning. Generalquestions arise such as ‘How can we promote students’ awareness of their intuitivetendencies?’ and ‘How can learners be led to feel the need to examine thecorrectness of their solutions?’, as well as specific questions such as how tointroduce inequalities, how to cope with inconsistencies in students’ reactions toinequalities, how to validate the correctness of specific solutions to inequalities,and what role can computer-based instruction play in promoting students’ sol-utions to inequalities? All these questions call for further research. Moreover,relevant interventions and their application should also be investigated in order toexamine their impact on students’ performance. In this way, the cycle of examin-ing, designing, developing, applying, and reexamining the impact of research-based instruction goes on widening and being constantly evaluated.
AcknowledgmentA shorter version of this manuscript was presented at the 25th Annual
Meeting for the Psychology of Mathematics Education in Utrecht, Holland in2001.
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