Examining Preservice Elementary Mathematics Teachers' Understandings About Irrational Numbers

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PRIMUS, 21(5): 401–416, 2011Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970903256928

Examining Preservice ElementaryMathematics Teachers’ Understandings About

Irrational Numbers

Bulent Guven, Erdem Cekmez, and Ilhan Karatas

Abstract: The purpose of this study is to provide an account of preserviceelementary mathematics teachers’ understandings about irrational numbers. Threedimensions of preservice mathematics teachers’ understandings are examined: definingrational and irrational numbers, placing rational and irrational numbers on the num-ber line, and operations with rational and irrational numbers. The study was conductedwith 80 first- and fourth-year preservice elementary mathematics teachers with the useof an open-ended test consisting of 10 questions prepared through a literature review.The findings showed that participants hold misunderstandings about the definitions ofirrational numbers, operations between rational and irrational number sets, and theirrelationship on the number line. Explanations used by the majority of participants werebased on representations and intuitions.

Keywords: Irrational numbers, students’ understandings, preservice mathematicsteachers.

1. INTRODUCTION

It is evident that having an adequate knowledge regarding real numbers hasa significant role in understanding and learning advanced mathematical con-cepts. The level of students’ knowledge about real numbers is directly relatedto the level of their knowledge about rational and irrational numbers. Therefore,understanding of irrational numbers is crucial for the conceptualization of thenumber concept in a broader sense than just understanding the concept of ratio-nal numbers. To help students extend their concept of number from rationalnumbers to real numbers, teaching should be designed to make them aware ofthe strict hierarchy within the real number system. It is known that at some

Address correspondence to Bulent Guven, Fatih Faculty of Education, SecondaryScience and Mathematics Education Department, Karadeniz Technical University,61335, Trabzon, Turkey. E-mail: [email protected], [email protected]

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402 Guven, Cekmez, and Karatas

stages of this hierarchy, which begins with natural numbers and ends with realand complex numbers, some students experience conceptual difficulties [2,4,9].Within this hierarchy, students are taught natural and rational numbers beforeirrationals. Students can profit from real-world models and picture represen-tations to develop an understanding of the operations with rational or naturalnumbers. However, they do not have the same opportunity when dealing withirrational numbers, which are more abstract.

1.1. Students’ Understanding of Irrational Numbers

It is not surprising that students experience difficulties in understanding irra-tional numbers. As we know, the discovery of irrational numbers by earlyGreek mathematicians is one of the most important milestones in the history ofmathematics, and the establishment of the theory of irrationals had to wait untilthe 19th century [3]. During this long time period, mathematicians had strivedto give a full explanation of the concept of irrationals, and it became possi-ble after the contributions of Dedekind, Cantor, and Weierstrass. Therefore, itwould not be plausible to expect students to grasp the concept of irrationalitywith ease.

Earlier studies that focused on the concept of irrational numbers reportedthat not only high-school students, but also prospective teachers, experiencedifficulties in understanding irrationals. Fischbein, Jehiam, and Cohen [2]investigated the knowledge that prospective teachers and high school studentspossess regarding irrational numbers. This study reported that some of the par-ticipants in both the groups were neither able to explain the concepts of rationaland irrational numbers, nor able to identify the numbers as being rational,irrational, or real.

Another study that was conducted on 84 in-service teachers by Arcavi et al.[1] found similar results. This study reported that many teachers were not ableto define numbers as being rational or irrational, and brought to light the factthat there is a common belief among teachers that irrationality relies on dec-imal representations. Similarly, Zazkis and Sirotic [4] investigated preservicesecondary teachers’ understandings of irrational numbers, and reported thatstudents have a preference toward decimal over the common-fraction represen-tation. A study by Sirotic and Zazkis [5], which involved 46 preservice teacherswith diverse mathematical backgrounds, focused on the prospective secondaryteachers’ knowledge about the relationships between rational and irrationalnumber sets. Their study adopted the conceptual framework put forward byTirosh et al. [7]. In this framework, a learner’s mathematical knowledge isbelieved to be embedded in a set of connections among algorithmic, intuitive,and formal dimensions. It was reported that there were inconsistencies amongparticipants’ intuitive dimension knowledge and the other two dimensions ofknowledge.

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Understandings About Irrational Numbers 403

1.2. Irrational Numbers in the Turkish Educational Setting

In Turkey, students’ first encounter with the concept of irrationality takes placeduring the last year of elementary education (Grade 8). Students learn the ratio-nal number concept, and how to perform operations with rational numbersduring the previous years. In fact, students encounter the well-known irrationalnumber π for the first time when they learn the formulas for finding the perime-ter and area of a circle and for finding the volume and area of a cylinder in thefifth grade, and often students are told to use 3, 3.14, or 22

7 in place of the exactvalue of number π in problems.

No explanation regarding the irrationality of the number π is givento students until the eighth grade. Regarding irrational numbers, the eighthgrade mathematics curriculum includes the following statement as a learningoutcome: “. . . explains the difference between an irrational and a rational num-ber,” and it demands teachers stress the fact that while rational numbers can bewritten as a fraction of integers, irrational numbers cannot. Teachers discusswith the class whether every decimal expansion can be written as a fractionof integers. Also, the eighth-grade mathematics curriculum refers to the rela-tion between irrational and square root numbers, and gives examples of how tofind the location of an irrational number on the number line with the help ofthe Pythagorean Theorem. The eighth-grade curriculum requires that teachersconduct class activities in which students evaluate the decimal expansions ofsome square root numbers by using calculators to see that they neither termi-nate nor become periodic. Regarding finding the location of irrational numberson the number line, the curriculum presents an activity shown in Figure 1.

Figure 1. Finding the location of√

34 on the number line (color figure availableonline).

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Students’ next encounter with the irrational number concept takes placeduring their first year of secondary education (Grade 9). Regarding irrationalnumbers, the ninth-grade mathematics curriculum includes the following state-ment as a learning outcome: “States the existence of numbers that are notrational (irrational).” Instructions for this outcome in the curriculum state thatteachers should stress that the decimal expansion of irrational numbers nei-ther terminates nor becomes periodic, and also that they should prove that thenumber

√2 is irrational by giving a proof by contradiction.

In the first year of an elementary mathematics teacher education program,students enrolled in a course called “General Mathematics.” During this course,students first define the natural numbers via Peano Axioms, and then graduallyconstruct the other number sets.

1.3. The Purpose and Research Questions of the Study

Undoubtedly, it is the teachers who make students grasp the concept ofirrationality during the process in which students extend their notion of num-ber from rationals to irrationals. As we mentioned earlier, some studies ineducational research reported that both students and teachers possess miscon-ceptions about irrational numbers. The overall aim of this study is to provide anaccount of Turkish elementary mathematics student teachers’ understandingsof the concept of irrational numbers. Specifically, this study seeks answers tothe following questions:

● Can preservice mathematics teachers give formal definitions for rational andirrational numbers? Can they classify a given set of numbers as either rationalor irrational?

● What knowledge do preservice mathematics teachers have about the rela-tionship between irrational numbers and the number line?

● What understandings do preservice mathematics teachers have about theoperations with rational and irrational numbers?

● Is there a conceptual difference between first- and fourth-year preserviceelementary mathematics teachers’ understandings of irrationality?

2. METHODOLOGY

2.1. Sample

The population sample for this study consists of 80 (53 men and 27 women)preservice elementary mathematics teachers. Forty-one students were in theirfirst year at the university (24 men and 17 women), while others were in their

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last year (29 men and 10 women). Our aim in selecting the students from twodifferent levels was to determine the effect of the university instruction onthe conceptual development of the students with regard to irrational numbers.At the time this study was conducted, first-year students had been studyingat the university for two weeks, while others (fourth-year elementary math-ematics teachers) had taken courses such as general mathematics, analysis,linear algebra, differential equations, and algebra. Further, the general mathe-matics course included irrational numbers as a topic. However, the first yearpreservice mathematics teachers had not studied this topic when this studywas conducted. As a result of having taken different courses, the fourth-yearstudents were expected to have a reasonable understanding of the concept ofirrational numbers prior to the study.

2.2. Data Collection

In this study, an open-ended test consisting of ten questions regarding irrationalnumbers is used as the data collection tool. We took into consideration theearlier studies in the literature [2, 6, 8], and the suggestions of the experts inthe field in selecting the questions included in the questionnaire. The questionsare categorized into three categories. These categories are shown in Table 1.

3. RESULTS

3.1. Defining Rational and Irrational Numbers

The aim of the first question used in the open-ended test was to determinestudents’ knowledge about the definitions of rational and irrational numbers.With this aim, we asked the students to give definitions for rational and irra-tional numbers. We used the following definitions as the criteria to determinewhether an answer is true: For the rational number: “ a

b : a, b ∈ Z ∧ b �= 0” or“A number is said to be rational if its decimal expansion either terminates oreventually becomes periodic.” For the irrational number “A number is said to beirrational if it cannot be expressed as a quotient of two integers” or, “A numberis said to be irrational if its decimal expansion neither terminates nor becomesperiodic.”

As seen from Table 2, 17% of first-year students and 44% of fourth-yearstudents gave correct definitions for rational and irrational numbers. About49% of first-year students and 15% of fourth-year students either gave anincomplete definition or did not answer at all. Within the incorrect defini-tions given by students, two of them are prevalent in both groups. One of

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Table 1. Categories of Questions

Category Questions

Defining rationaland irrationalnumbers andclassifying agiven set ofnumbers aseither rationalor irrational

There are two questions in this category. In the first question, weasked the students to determine which set (rational, irrationaland real) the numbers in the following set belong to{−4, 22

7 , π , 5√

2, 0.1923712 . . . , 2 + √3, 2.0777.., 5√5,

√16, 2

π, 3.14

}.

In the second question we asked the students to write thedefinitions of rational and irrational numbers.

Placing numberson the numberline (knowledgeabout therelationshipbetweennumbers andthe numberline)

In this category, our aim was to reveal the students’ knowledgeabout the relationship between rational and irrational numberson the number line with respect to the density of the twonumber sets. For this purpose, we asked students the followingquestions.

• Can we always find an irrational number between any twodifferent irrational numbers on the number line? Justify youranswer.

• Can we always find a rational number between any twodifferent irrational numbers on the number line? Justify youranswer.

• Can we always find a rational number between any twodifferent rational numbers on the number line? Justify youranswer.

• Can we always find an irrational number between any twodifferent rational numbers on the number line? Justify youranswer.

Operations withrational andirrationalnumbers

In this section, we aimed to investigate how students answer toquestions that are about the effects of operations betweennumbers of the same kind.

• Is the result of adding two irrational numbers alwaysirrational? Justify your answer.

• Is the result of adding two rational numbers always rational?Justify your answer.

• Is the result of multiplying two irrational numbers alwaysirrational? Justify your answer?

• Is the result of multiplying two rational numbers alwaysrational? Justify your answer.

the responses that serve as an example for the first prevalent false definitionis: “Rational numbers are the numbers that can be presented on the numberline; on the contrary, irrational numbers are the numbers whose exact valuecannot be determined; therefore, they cannot be located exactly on the numberline.” Another response is: “Numbers that exist are called rational numbers,numbers whose existence is assumed are called irrational numbers.”

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Table 2. Frequencies and Percentages of Correct, Incorrect,and Incomplete Answers

True False Incomplete

Year f % f % f %

First year 7 17 14 34 20 49Fourth year 17 44 16 41 6 15

What we see from these statements is that some of the students in bothgroups see the existence of irrational numbers differently from that of ratio-nal numbers, and think that the exact location of an irrational number cannotbe determined on the number line. The second definition that is incorrect andprevalent among the students in both groups indicates that some students con-fused the concepts of rational and complex numbers. An example of suchconfusion is in one of the participants’ response: “An irrational number is theeven root of a negative number, such as

√−1.” Another student response illus-trative of this confusion is “irrational numbers are defined as x + iy; i2 = −1,x, y ∈ �”

In responses that we classified as incomplete, students gave examples forrational and irrational numbers instead of a formal definition. For example, afourth-year student wrote “Numbers like 1

2 , 6 are called rational, numbers like√5,

√3 are called irrationals.”

3.1.1. Recognizing Numbers as Either Rational or Irrational

In this section, 11 numbers of various types were presented to students whomwere asked to determine which set these numbers belonged to. Here, we referto the most arresting results. Percentages of the results are shown in Table 3. Inthe table, grey cells indicate the correct sets for each number.

As seen in Table 3, more than half of the students in the first-year groupdo not identify π as an irrational number. In addition, among the students inthe first-year group, 27% do not know that π is a real number. As we expected,fourth-year students showed better performance than first-year students, butthere were still 26% who identified π as a rational number.

To determine whether the students could correctly identify the decimalrepresentations of numbers as either rational or irrational, we asked them toidentify the nature of 0.1923712. . . and 2.0777. . . . Nearly two-thirds ofthe first-year students identified 0.1923712. . . as rational, and also a signif-icant portion of them did not recognize it as a real number. In identifyingnumber 0.1923712.., again, fourth-year students showed better performancethan first-year students. But with regard to number 2.0777. . . a surprisingresult emerged. The percentage of the first-year students’ correct answers is

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Table 3. Percentages of Correct and Incorrect Answers

Number Year Rational (%) Irrational (%) Real (%)

π First Year %56 %37 %73Fourth year %26 %74 %92

0,1923712. . . First Year %63 %29 %68Fourth year %28 %69 %95

2,0777. . . First Year %73 %22 %61Fourth year %49 %46 %92

2

πFirst Year %63 %29 %80Fourth year %31 %69 %85

3.14 First Year %83 %12 %56Fourth year %90 %10 %100

larger than that of those in fourth year. Nearly half of the fourth-year studentsidentified 2.0777. . . as an irrational number.

Another question to which we tried to obtain an answer in this section was,“Do the students identify correctly a given number represented in the formatab when one of its components (numerator or denominator) is irrational?” Toanswer that question, we asked the students to determine to which set or sets2π

belongs. The results show that in both groups, the percentage of incorrectanswers to this number is bigger than the percentage of incorrect answers tonumber π . From this result, it is understood that in both groups some studentswho identified πcorrectly were not able to identify 2

πcorrectly.

During their school life, students are asked to solve mathematical problemswith a note at the end of it saying “Take π equal to 3.14.” We think that studentswho experienced such problems may develop a misconception that they wouldconsider the number 3.14 as the exact value of π . To determine whether thestudents possessed such a misconception, we asked them to identify the number3.14. The results showed that most of the students in both groups did not holdsuch a misconception. Nevertheless, 12% of the first-year students and 10% ofthe fourth-year students identified 3.14 as irrational. While we examined thestudents’ responses to that number, we discovered that some of the studentswho classified the number 3.14 as irrational, scribbled the number π next to3.14, and put the equal sign between these two numbers. Having discoveredthis fact, we then looked into these students’ answers to number π , and noticedthat all of them classified π as irrational.

3.2. Placing Numbers on the Number Line (Knowledge Aboutthe Relationship Between Numbers and the Number Line)

Another aim of our study was to reveal students’ ideas about how rationaland irrational numbers fit together on the number line. To this end, we asked

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Table 4. Percentages and Frequencies of the Student Responses

Yes NoNo

answer

Questions Year f % f % f %

Can we always find anirrational number betweenany two different irrationalnumbers on the number line?


Can we always find a rationalnumber between any twodifferent irrational numberson the number line?


Can we always find a rationalnumber between any twodifferent rational numberson the number line?


Can we always find anirrational number betweenany two different rationalnumbers on the number line?


students four questions in the second part of the open-ended test. Table 4shows the questions as well as the percentages and frequencies of the students’answers. In the table, grey cells indicate the correct answers. In the question-naire, we asked students to justify their answers for each question. We referredto the most striking results in the separate sections for each question.

3.2.1. Finding an Irrational Number Between Any Two Different IrrationalNumbers on the Number Line

The majority of students in both groups answered this question correctly. Aswe mentioned earlier, we asked students to justify their answers. Sixty per-cent of the students who answered correctly justified their answers, and noneof the students whose answer was incorrect provided a justification for theiranswer. A first-year student made the following statement to justify his answer“Let a and b be two different irrational numbers and a>b. we can always writea > a+b

2 > b, therefore there is always an irrational number between two dif-ferent irrational numbers.” Some students in both groups used the same way intheir justification, but instead of using variables they used numbers, for exam-ple, a first-year student wrote “Yes it is possible, for example, between 4

√2

and 4√

3 there exists (4√

2 + 4√

3)/2.” These statements indicate that thereis a common belief among students that the set of irrationals is closed underaddition.

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There is another misconception lurking behind these statements. Some stu-dents misinterpret the mathematical meaning of “any” and “always” becausethey found it possible to generalize a mathematical statement on the basis of afew cases. Taking the average of two numbers is not the only way that studentsjustified their answers. Some students based their justifications on the fact thatthere is an infinite number of irrationals. For example, one of the participantswrote “There exists an infinite number of real numbers between two differ-ent points on the number line, so one of these reals must be irrational.” Noneof the justifications refer to the density of irrational numbers. From this fact,it is understood that the concept of abundance of irrationals is based on theirintuitive knowledge rather than formal knowledge.

3.2.2. Finding a Rational Number Between Two Different Irrational Numberson the Number Line

The results in Table 3 show a decrease in the number of correct answers tothis question when compared with that of those to the earlier question. Fiftypercent of the students who answered the question correctly justified theiranswers. Among students who responded incorrectly, only two of them jus-tified their answer. One of these two wrote “No, we cannot because we canselect small intervals on the number line which do not contain any rationalnumbers.” The other one just stated “No, not always.” All the students whoresponded correctly based their justifications, as in the earlier question, on thefact that there are an infinite number of irrationals. Students who used the meanvalue of two numbers in their justifications to show the previous statementto be true did not use the same method for this statement. Although it can-not be regarded as a sufficient justification, the same method can be used asone of the possible ways to obtain a rational number between some irrationalnumbers (e.g., −√

2+√2

2 = 0). This finding supports our conclusion that thereis a common belief among students that the set of irrationals is closed underaddition.

3.2.3. Finding a Rational Number Between Any Two Different RationalNumbers on the Number Line

As seen in Table 3, most of the students determined this statement to betrue. The justifications of some students among those who answered cor-rectly showed that they have misunderstandings about the relationship betweenrational numbers and number line. Although they answered correctly, they pro-vided faulty justifications for their answers. For example, a first-year studentwrote “Yes, always, we can. It is so because every point on the number line

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corresponds to a rational number.” Besides faulty ones, some justifications werenot clear as to what they meant. For example, a fourth-year student wrote “Surewe can, it is possible to change the denominator of a rational number.” Wethink that this implies the following: suppose we want to find a rational num-ber between 4

6 and 56 , if we multiply both parts of these numbers by two 2, we

get the numbers 812 and 10

12 , respectively, and 912 lies between these two num-

bers. Some students were able to develop mathematically correct justifications,for example, a fourth-year student wrote “Let a

b and cd be two rational num-

bers; then the numberab + c

d2 is between these two numbers, and it is rational.”

None of the students who determined that this statement was false justifiedtheir answers.

3.2.4. Finding an Irrational Number Between Two Different Rational Numberson the Number Line

The results in Table 3 show that students’ performance on this question is lowerthan on both the first and the third questions. All the students who answeredcorrectly and justified their answers based their justification on a single exam-ple or on the fact that there are an infinite number of irrationals. For example,a first-year student wrote “Yes, for example, 2

√2 is between 2 and 4.”

The following justification, given by a fourth-year student, can serve as anexample for the justifications that were based on the fact that there are an infi-nite number of irrationals: “Yes we can, because there is always a gap betweentwo different numbers, and the set of irrational numbers is infinite. Therefore,one of them can lie in this gap.” A fourth-year student who answered incor-rectly used the decimal representation of rational numbers in his justification,and said “we may not, because in some cases it is not possible to find a num-ber whose decimal expansion does not become periodic between two numberswhose decimal expansion becomes periodic.”

3.3. Operations with Rational and Irrational Numbers

The final aim of our study was to determine how students answered questionsthat ask whether the rational and irrational number sets are closed under addi-tion and multiplication. To this end, we present students with four mathematicalstatements about the effects of the operations of numbers of the same kind,and asked them to determine the truth value of these statements, and to justifytheir answers. Table 5 shows the percentages and frequencies of the students’answers. In the table, colored cells indicate the correct answers. FollowingTable 5, we referred to the most striking results in the separate sections foreach statement.

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Table 5. Percentages and Frequencies of the Student Responses

True FalseNo

Answer

Statements Year f % f % f %

The result of addingtwo irrationalnumbers is alwaysirrational.


The result of addingtwo rationalnumbers is alwaysrational.


The result ofmultiplying twoirrational numbersis alwaysirrational.


The result ofmultiplying tworational numbers isalways rational.


3.3.1. Adding Two Irrational Numbers

The result of adding two irrational numbers can be either rational or irrational.As seen in Table 5, 54% of first-year students and 77% of fourth-year studentsdetermined that this statement was true. This supports our conclusion that wereached in the Placing Numbers on the Number Line section that some stu-dents believe that the set of irrationals is closed under addition. Some of thestudents who determined the statement to be true based their justifications ona single example, and then made generalizations. For example, a first-year stu-dent wrote “It is true,

√3 + √

3 = 2√

3 , so irrational + irrational =irrational.”Some others, who did not infer from a single example, used the decimal expan-sion representation of irrationals to show the statement to be true. For example,a fourth-year student wrote, “it is true, because if we add two numbers whosedecimal expansion neither terminates nor becomes periodic, then the decimalexpansion of the result neither terminates nor becomes periodic either.” But aswe know, the decimal expansion of the result of adding two irrational numberscan repeat (e.g., 0,122112. . . + 0.211221. . .= 0.333. . .).

As we mentioned earlier, the discovery of irrational numbers by earlyGreek mathematicians is one of the most important milestones in the historyof mathematics. Mathematicians of that time could not make anything of the

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existence of incommensurable line segments. The findings showed that somestudents experience the same conceptual difficulties in understanding irra-tionals. That kind of difficulty was seen in a fourth-year student’s justificationas, “the statement is true because if we add two numbers which we cannot mea-sure, we cannot measure the total either.” All the students who determined thestatement to be false falsified the statement by giving a counter-example.

3.3.2. Adding Two Rational Numbers

As seen in Table 5, 80% of first-year students and 72% of fourth-year stu-dents gave the correct answer by acknowledging the statement to be true. Mostof these students pointed out in their justifications that the set of rationals isclosed under addition. The justification of the students who acknowledged thestatement to be false revealed that two misconceptions exist among students.Some students do not include integers in the set of rational numbers. For exam-ple, one of the students in fourth year stated, “ 1

4 + 34 = 1 no it is not always,

sometimes, can be an integer.” The students who possessed the second mis-conception believed that the exact value of π is 22

7 . For example, one of thestudents in first year stated, “it is not true because 21

7 + 17 = 22

7 , and this isequal to number π .”

3.3.3. Multiplying Two Irrational Numbers

As seen from Table 5, the majority of the students gave the correct answer byacknowledging the statement to be false. All the students who gave the correctanswer used the same method in which they multiplied two irrational numbersand got a rational number to falsify the statement in their justifications(e.g.,√

3 · √3 = 3). Students who gave a wrong answer stated in their justifications

that the set of irrational numbers is closed under multiplication.

3.3.4. Multiplying Two Rational Numbers

It is relatively easy to prove this statement. Most of the students in both groupsgave a correct answer, and their justifications are more like a formal proofrather than based on a single example. For example, a fourth-year studentstated, “ a

b ∈ Q ∧ cd ∈ Q ⇒ a

b · cd = ac

bd ∈ Q , therefore the statement is true.”Among the students who acknowledged the statement to be false, only onestudent provided a justification for her answer. That student wrote, “the multi-plication of two numbers whose decimal expansion repeats can yield a resultwhose decimal expansion neither terminates nor becomes periodic.”

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4. DISCUSSION AND CONCLUSIONS

As a result of this study, it is seen that a significant portion of the participantswere not able to define rational and irrational numbers correctly. Parallels withour work are found in some studies in the literature reporting that both prospec-tive mathematics teachers and high school students failed to give a completedefinition of these numbers [1,2]. In this study, although the number of fourth-year students who gave correct definitions for rational and irrational numbersare more than that of first-year students, surprisingly, the situation is the samefor incorrect definitions. First-year students generally provided incomplete def-initions. This fact shows that an important portion of the first-year students donot have a clear conceptualization regarding irrational numbers. This showssome students’ pre-university knowledge about numbers to be insufficient,resulting in an inability form an adequate knowledge base about numbers dur-ing their university studies. We think that the factors that cause students to formand retain these misunderstandings about rational and irrational number con-cepts are worth investigating further. One of the most important results of thisstudy is that some students confused the concepts of irrational and complexnumbers. We have not found any study in the literature that reported such afinding. One of the reasons for this might be that students do not see irrationalnumbers as a member of a real number set.

The findings also demonstrated that some students think that the irrationalnumber set only consists of square root numbers. In our view, this might bedue to the examples that were given to students during their school years.Therefore, more examples, including different forms of irrational numbers,must be given to students in high school mathematics courses.

The findings showed that more than half of the first-year students identifiedπ as rational. This finding is consistent with other studies in the literature [2].The reason why students identify π as rational might be due to the fact that theywere sometimes told to use numbers 3, 3, 14 or, 22

7 instead of π in elementaryand high school mathematics courses. In university mathematics courses, thosenumbers are not used for π . This might be the reason why fourth-year studentsout-performed first-year students in identifying the nature of π .

In our study, we conclude that representation of a number has an effecton students’ decisions about the nature of that number. Some of the studentswho identified π as irrational identified 2

πas rational. We think that the reason

why these students could not identify 2π

as irrational is because it is written as aquotient, which makes it look like a rational number. A significant proportion ofthe students identified 2.0777. . . as irrational. This suggests that these studentsidentify all the numbers whose decimal expansion has an infinite number ofdigits as irrational; in other words, they do not take into consideration whetherit is periodic.

The findings showed that the majority of the students gave correct answersto the questions about placing numbers on the number line. But when they were

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asked to justify their answers, they were not able to provide formal justifica-tions for their answers. This suggests that students’ understandings about thisissue are based on their intuitive knowledge.

4.1. Educational Implications

The results of this study showed that a significant proportion of the preserviceteachers could not form strong conceptual understandings regarding irrationalnumbers. The results also showed that most of the preservice teachers basedtheir understandings about irrational numbers on intuition, rather than on for-mal knowledge. Therefore, some changes to the content of the courses thatcover irrational numbers must be made. The results of this study show thatmerely defining irrational numbers as numbers that are not rational, and per-forming arithmetic operations on irrational numbers do not guarantee students’conceptual understandings. We think that integrating the historical develop-ment of this concept into teaching will help preservice teachers understand thisconcept better and understand the formation of this concept within a learner’smind. This kind of approach to the teaching of this concept will not only helpstudents to gain a better understanding of this concept but also enrich theirpedagogical content knowledge. The results showed that the representation ofa number has an effect on students’ decision about the nature of that number.Therefore, in the teaching of a particular number set, it would be beneficial forstudents to be introduced to all possible kinds of representations, which maystand for the elements of that number set.

REFERENCES

1. Arcavi, A., M. Bruckheimer, and R. Ben-Zvi. 1987. History of mathe-matics for teachers: the case of irrational numbers. For the Learning ofMathematics. 7(2): 18–23.

2. Fischbein, E., R. Jehiam, and D. Cohen. 1995. The concept of irrationalnumbers in high school students and prospective teachers. EducationalStudies in Mathematics. 29(1): 29–44.

3. National Council of Teachers of Mathematics (NCTM). 1969. HistoricalTopics for the Mathematics Classroom. Washington, DC: NCTM.

4. Peled, I., and S. Hershkovitz. 1999. Difficulties in knowledge integration:Revisiting Zeno’s paradox with irrational numbers. International Journal ofMathematics Education, Science and Technology. 30(1): 39–46.

5. Sirotic, N., and R. Zazkis. 2007a. Irrational numbers: The gap between for-mal and intuitive knowledge. Educational Studies in Mathematics. 65(1):49–76.

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6. Sirotic, N., and R. Zazkis. 2007b. Irrational numbers on the numberline—Where are they? International Journal of Mathematical Educationin Science and Technology. 38(4): 477–488.

7. Tirosh, D., E. Fischbein, A. Graeber, A., and J. Wilson. 1998. Prospectiveelementary teachers’ conceptions of rational numbers. http://jwilson.coe.uga.edu/Texts.Folder/Tirosh/Pros.El.Tchrs.html. Accessed 14 August2008.

8. Zazkis, R. 2005. Representing numbers: Prime and irrational. InternationalJournal of Mathematical Education in Science and Technology. 36(2–3):207–218.

9. Zazkis, R., and N. Sirotic. 2004. Making sense of irrational num-bers: Focusing on representation. In M.J. Hoines and A.B. Fuglestad(Eds.), Proceedings of 28th International Conference for Psychology ofMathematics Education, vol. 4, pp. 497–505. Bergen, Norway: BergenUniversity College.

BIOGRAPHICAL SKETCHES

Bulent Guven is an associate professor in the Secondary Science andMathematics Education Department in Karadeniz Technical University,Trabzon, Turkey. He received his Ph.D. from Karadeniz Technical University.He is interested in teaching and learning mathematics, geometry instruction,and problem-solving with technology.

Erdem Cekmez is a doctorate student in Secondary Science and MathematicsEducation Department of Karadeniz Technical University. He is interestedin computer-based mathematics education, mathematical understandings, andcalculus instruction.

Ilhan Karatas is an assistant professor at Zonguldak Karaelmas University,Eregli Faculty of Education, Zonguldak, Turkey. He received his Ph.D. fromKaradeniz Technical University in Trabon, Turkey. His research areas areproblem-solving and learning mathematics.

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Documents

Examining Preservice Elementary Mathematics Teachers' Understandings About Irrational Numbers