The Fraction Magnitude Knowledge Through Representations at Students with Mathematics Difficulties

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Advances in Computer Science Research, volume 95 Mathematics, Informatics, Science, and Education International Conference (MISEIC 2019) The Fraction Magnitude Knowledge through Representations at Students with Mathematics Difficulties

Yusuf Fuad Ijtihadi Kamilia Amalina Universitas Negeri Surabaya Universitas Negeri Surabaya Surabaya, Indonesia Surabaya, Indonesia

[email protected] [email protected]

Abstract—Elementary school students’ knowledge of Fraction magnitude knowledge is important to master fraction magnitude may predict their mathematics higher mathematics concept for example algebra [1, 2, 3]. achievement. Students’ mathematics achievement is influenced Moreover, fraction magnitude knowledge is linked to student by their mathematics difficulties. This study exposes students’ mathematics achievement or support each other [12, 13]. representations in estimating fractions to indicate their fraction Poor mathematics achievement can be affected by learning magnitude knowledge. This study is conducted on the 4th grade of elementary students in Surabaya. A class which consists of 10 problems. Two types of learning problems can be girls and 18 boys, had chosen purposively from 3 available distinguished into a learning disability is situated in the classes. All students had to answer two tests, namely the child’s own cognitive, while learning difficulty is situated diagnostic test and the fraction magnitude test. Four volunteer outside the child [14]. Because of these reasons, mathematics students were selected as research subjects which have different difficulty in fraction becomes special attention [15]. and interesting mathematics difficulty in fraction. Semi- structured interview was conducted to all subjects to reveal Although many students understand fractions easily, students’ thinking process when solving diagnostic and fractions fraction magnitude is challenging for students with magnitude tests. Results suggest that 46% of 26 students who mathematics difficulty [16]. Research investigation to have bad score (low ability) have mathematics difficulties in fraction magnitude knowledge with mathematics difficulty representing and estimating simple fraction because they students and their typically achieving peers have shown considered nominator and denominator as independent significant and widening gaps in fraction achievement numbers (i.e. 1/3 is represented as a bar model which is divided between two groups at the upper elementary level [11, 17, 18] by 4 equally parts and a part is in shaded). More than 20% of and the middle school level [7, 19]. Mathematics difficulty 26 students had mathematics difficulties which were not able to estimate non-simple fraction but they were able to represent it. students in the 3rd and 6th grades are slower in understanding Their ability is influenced by difficulties in fraction magnitude. the concepts and procedures of fraction than typically achieving peers [18]. It contributes to mathematics Keywords—fraction magnitude knowledge, representation, achievement gap [16]. students’ difficulties One factor which affects poor fraction performance for I. INTRODUCTION mathematics difficulty students is weak fraction magnitude Fraction is important not only in mathematics but also in knowledge. Fraction magnitude knowledge includes learning other subjects. Understanding fraction is the gateway to about the way in which properties of whole number and understand the higher concept in mathematics [1, 2, 3]. fractional number are similar and the ways in which they Fraction is defined as part of whole [4], and also defined as diverge [13]. Fraction magnitude knowledge is the basic numbers that were represented by the ordered pair of a/b, fraction concept [13] and weak fraction magnitude where a and b are natural numbers with b ≠ 0 [5]. understanding is one of characteristics of student with mathematics difficulty [18]. The concept of fraction is still difficult for Indonesian students in particularly for elementary students. This II. METHOD difficulty emphasizes that students did not understand about This study is conducted on the 4th grade of elementary fraction magnitude [1, 3, 6, 7]. Fraction magnitude students in Surabaya A class consists of 10 girls and 18 boys, knowledge is the ability to estimate, to represent, and to had chosen purposively from 3 available classes. All students compare fractions [8, 9, 10]. A mistake which Indonesian had to answer two tests, namely the diagnostic test and the students had are comparing fractions, both unit and non-unit fraction magnitude test. fractions [6]. Another mistake is from 28 students in Surabaya, 11 students had low ability in estimating fractions The diagnostic test investigates student mathematics because they do not know the concept of fraction and fraction difficulty gradually. It examines from the easiest to the magnitude [11]. hardest concept of fraction magnitude. There are 5 topics of fractions in this test [10]. Each topic consists of 4 questions about fraction comparison through representation. The

Copyright © 2019, the Authors. Published by Atlantis Press SARL. This is an open access article under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/). 90 Advances in Computer Science Research, volume 95 representations used are bar and circle models of Results of fraction magnitude test, 95% of 26 students representations. The fraction magnitude test is focused on the with mathematics difficulty were not able to represent and to ability to estimate fraction through representation. There are estimate fractions, for example in estimating 3/4, they did not 2 questions about fraction estimation into number line [8, 10]. know the relationship between numerator and denominator. There are 5 fractions of each question and range of number They considered numerator and denominator as an line is 0 to 1. For example “put and represent these fractions, independent number. Meanwhile, a student with medium 1/8,1/2,2/4,5/8,3/4 into number line 0 to 1. All these tests are ability could not take fractions appropriately into number validated by an expert. line. Two tests are analysed and categorized into high, medium This study underlined that students with mathematics and low abilities in which take into consideration of students’ difficulties were not able to represent fractions with different document scores test. Based on these results, will be further denominators and estimate them. Most students had analysed students’ mathematics difficulties in fraction. There difficulties in understanding simple fractions and fractions are many difficulties in fraction which are conceptually and with different numerator and denominator (non-unit procedurally in estimating and representing simple or non- fractions). A student who had difficulties in understanding simple fractions. Four volunteer students were selected as simple fractions (SF), a student with mathematics difficulty research subject which have different and interesting in representing similar fractions and fractions with same mathematics difficulties in fraction. Semi-structured numerator (SNF), a student with difficulty in represent interview was conducted to all subjects to reveal students’ equivalent fractions (EF), and a student who had difficulties thinking process when solving diagnostic and fractions in fractions that have common multiple (CM) were selected magnitude tests. Table I shows the topics to examine as a research subjects. Figure 1 shows the example of student mathematics difficulty in fraction magnitude. There are 5 SF who had difficulty in simple fractions and some dialogs topics about simple fraction and non-simple fraction. between SF and interviewer (I). I : How could you represent TABLE I. THE TOPICS AND EXAMPLE TO EXAMINE MATHEMATICS these fractions? DIFFICULTY IN FRACTION MAGNITUDE SF: I represented ½ as a bar which is divided into 3 Topics Examples equal parts and 1 part is 1 1 1 The simple fraction , , I : shaded 2 3 4 SF: How it could be? Fractions with the same denominator 2 4 5 , , the upper side is 1 so I (similar fraction) 6 6 6 shaded a part and the 3 3 3 lower side is 2 so I did not Fractions with the same numerator , , 6 7 8 I : shade 2 parts of a bar 1 2 3 What is the name of the Equivalent fractions , , 2 4 6 SF: upper and lower sides? Fractions that have common multiple 3 5 1 I forget, one of them is , , 4 12 6 denominator but I do not know which one is the I : denominator. So did you think that 1/5 III. RESULTS AND DISCUSSION SF: is the greatest fractions? The diagnostic test has 5 topics which investigate Yes because 1/5 has 6 students’ difficulties in fraction. The result of the diagnostic parts and a part is shaded which is greatest than all test showed that from 28 students 1 student has high ability, fractions here 1 student has medium ability, and 26 students have low Fig. 1. Student SF fractions representation and estimation. mathematics abilities in fraction. Students with low mathematics ability had mathematics difficulties in Student SF with mathematics difficulties represent numerator understanding fractions. Question number 1 is about simple and denominator as an independent number, for instance, ½ fraction in which 12 students did not know how to compare is represented as a bar which is divided into 3 equal parts simple fractions and represent them. Therefore, they were not while one part is in shaded. He considered 1 and 2 (in the able to compare 4 other topics (similar fractions, fractions number ½) as independent numbers. Therefore he concluded with same numerator, equivalent fractions, and fractions that that 1/5 is the greatest fraction. have common multiple). Question number 2 is about similar fraction in which 3 students had difficulty in compare and Student SF had difficulty in understanding simple fraction, represent them. Question number 3 is about fractions with hence he also could not represent and estimate simple same numerator; 4 students had difficulty in represent them. fractions in number line. This invention is supported by Question number 4 is about equivalent fraction; 2 students statement that fraction magnitude knowledge is linked to had difficulty in represent them. Question number 5 is the student mathematics achievement or support each other and hardest and 5 students had difficulty in decide the greatest poor mathematics achieving can be affected by learning fractions with different numerator and denominator. problems, one of them is learning difficulties especially in Meanwhile, there are a student categorized into high ability this case is mathematics difficulty [11, 12, 13]. had not mathematics difficulty, and a student with medium Student SNF was able to represent and compare similar ability had difficulty to understand equivalent fractions. fraction but she did not know conceptually because she was not able to apply this concept into fraction with same

91 Advances in Computer Science Research, volume 95 denominator. While represented 2/6 and 4/6 into bar she got of these fractions in number line. She put 2/4 after ½ because true representation and she said that 4/6 is greater than 2/6 numerator and denominator of 2/4 are greater than ½, just 1/8 because has same denominator so she just take consideration is located before ½. Because of these argument, there is a of numerator. At the otherwise she confused to represent finding that student EF did not understand equivalent fraction with same numerator. In compare 3/6 and 3/7, she fraction. This error is influenced by he had difficulty in said 3/7 is greater than 3/6 because these two fractions have fraction magnitude knowledge. same numerator. In this case she decided fraction with greater Student CM had difficulty in concept of fraction comparison denominator is bigger than fraction with smaller especially fractions that have common multiple. He consider denominator. Because of it, she could not estimate these that fractions which have greater numerator and denominator fraction into number line. Figure 2 shows the example of is the greatest. In case of fraction comparison between ¾ and student SNF work. 5/12, he stated that 5/12 is greater than ¾ because it has I : How could you represent greater numerator and denominator than ¾. He also did these fractions? wrong in a procedure of cross multiple. He argued that in SNF: I represented 2/6 as a bar which is divided into 6 equal fraction comparison we allowed to use cross multiple i.e. 3 is parts and 2 parts are shaded, multiplied by 12 and 5 is multiplied by 4, but he did wrong in 4/6 is 6 parts 4 of them are put these results. Figure 4 illustrates the example of student shaded CM work. I : How about 3/6 and 3/7? SNF: I confused because the I : How could you represent denominators are different. these fractions? Maybe 3/6 is 3+6 so I can CM: I represented using circle concluded that a bar is divided model, 3/4 as a circle which by 9 parts and 3 parts are is divided into 4 equal parts unshaded. So 3/7 is greater and 3 parts are shaded. And than 3/6 because it has greater I did the same way in 5/12 denominator I : Could you decide which I : So, could you order these one is the greater? SNF: fractions? CM: Ya, I used cross multiple, I Ya, from the lowest to the multiplied 3 and 12 then I greatest are ½, 3/6, 3/7, and put the result under 12. I 3/8. also multiplied 5 and 4 then Fig. 2. Student SNF fractions representation and estimation. I put the result under 4. So 5/12 is greater than 3/4 Student EF had difficulty in estimating equivalent fractions. She was able to represent them in bar models and circle Fig. 4. Student CM fractions representation and estimation. models, but from depth interview she was not able to estimate these fractions in number line. She could represent fractions IV. CONCLUSION into bar or circle models procedurally but not conceptually. Fraction magnitude knowledge is linked to student’s Figure 3 shows the example of student EF fractions mathematics achievement or support each other, while poor representation in estimating fractions. mathematics achievement in fraction can be affected by I : How could you represent learning difficulties, one of them is mathematics difficulty in these fractions? fraction. Student fraction difficulty can influence student EF: I represented 1/8 as a bar fraction magnitude knowledge. Student with mathematics which is divided into 8 difficulty in understanding simple fraction for example 1/4 is equal parts and 1 part is shaded and I also represent stated by a bar is divided into 5 and a bar is shaded, he could it into circle model. I did not estimate and represent simple fraction. Likewise student the same ways in others. with mathematics difficulty in understanding similar I : So, could you order these fractions, he could represent fraction but could not estimate fractions? EF: Ya, from the lowest to the them. Student with mathematics difficulty in fractions with greatest are 1/8, 1/2, 2/4, different numerator and denominator, or fractions that have 5/8, and 3/4 common multiple they cannot estimate these fractions in I : Did you think that ½ is is number line. These problems is because they did not smaller than 2/4? EF: Yes because ½ has 1 understand fraction magnitude therefore they applied shaded part while 2/4 has 2 procedures without knowing the concept. It had shaded parts. And I can consequences they did wrong in these procedures. concluded that 2/4 has greater numerator and REFERENCES denominator so it is greater than 1/2 [1] F. Gabriel, “Understanding Magnitudes to Understand Fractions,” APMC, 21 (2), pp. 36-40, 2016. Fig. 3. 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