A Case Study on Students’ Performance in Fraction Number Sense
Nur Syamimi Azmi1, Teh Faradilla Abdul Rahman2*, Raisnee Lumbihan2, Nurul Farhana Zolkipli2, Sharifah Norasikin Syed Hod2
1 Faculty of Education, Universiti Teknologi MARA, Cawangan Selangor, Kampus Puncak Alam, Selangor, Malaysia
2 Centre of Foundation Studies, Universiti Teknologi MARA, Cawangan Selangor, Kampus Dengkil, Selangor, Malaysia
*Corresponding Author: [email protected] Accepted: 1 November 2020 | Published: 15 November 2020
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Abstract: Student with good number sense tends to perform better mental computations, sense-making approach, planning and control, flexibility and appropriateness sense of reasonableness. While it is undeniable that number sense is important to be used every day, most students failed to make relation between mathematics and their daily life activities.
Thus, the main purpose in this study is to identify the students’ ability of number sense in fraction. The second aim was to determine the relationship between students’ performance in mathematics in general and fraction number sense. Mixed method was used in this study which included a set of midterm test, a set of fraction number sense questions and 13 interview items. The respondents’ number sense ability toward fraction was analysed using Mann Whitney U test. Pearson Correlation test was used to determine the relationship between students’ overall performance in Mathematics with their ability in fraction number sense. The result indicated that respondents had low level of number sense ability and there was almost no significant correlation between Mathematics achievements and fraction number sense. The study concluded that number sense performance must be improved by considering visual representation such as number line and other related to real life activities in pedagogy.
Keywords: number sense, fraction number sense, fraction
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1. Introduction
The ability of understanding numbers intuitively is not only attained by human but also in animal species (Merritt et al., 2012). According to Rezat and Ejersbo (2018), there are two different notions of number sense, the first one is the number sense from the biological/cognitive area and the second is from educational/pedagogical perspective. Even though the two perspectives somehow are related to each other but in this article, the term of number sense is focus more on the educational views. Research on the number sense has been widely studied for many years and the term of number sense also has been defined differently by many researchers.
Akkaya (2016) defined number sense as the understanding of different relationships between numbers and operations and the flexible use of these relationship while Maghfirah and Mahmudi (2018) stated that number sense refers to good intuition and general understanding of numbers, operations and relationship between them. Number sense relies on the fundamental of understanding what numbers mean, how the numbers are represented, how
the numbers are correlated to one another and how the computations of these numbers take place (Singh et al., 2019). Based on these definitions, the common essence of number sense is about understand the numbers, the relationships and the operations applied on them. Many studies showed that the capability of students’ number sense is still weak even though many researches and suggestions of improvement have been done including the study about the number sense of fraction. After more than three decades, numerous rounds of education reforms, hundreds if not thousands of research studies on mathematics teaching and learning, and billions of dollars spent to effect educational change, little improvement was evident in students’ understanding of fraction arithmetic (Lortie-Forgues et al., 2015).
A proper fraction is a number between 0 to 1 and consists of two whole numbers as numerator and denominator (where the denominator is not 0). Although it is a very small number, it plays a key role in learning mathematics. Students who do not have good number sense in fraction are not able to make estimation such as ‘which fraction is bigger; 5/9 or 6/11?’ or ‘how to arrange fractions 1/3, 3/5, 2/7, 7/10 in order’. Lortie-Forgues et al. (2015) identified seven sources of difficulties in fractions; notation, accessibility of fraction magnitude, opaqueness of standard fraction, complex relation between rational and whole number arithmetic procedures, complex relation between rational number arithmetic procedures to each other, opposite effects of multiplying and dividing positive fractions and sheer number of distinct components of fraction.
Kor et al. (2019) found out that the low and intermediate achievers in fraction sense applied the rules of fraction blindly most of the times when encountered with unfamiliar fractions and show a lack of understanding of how and why a procedure works. Study among two levels of secondary school students by Singh et al. (2019) shows that there is a significant gap between students’ mathematics subject grade and number sense test (NST) score, where students’
sense of numbers is very much lacking. In their study, the NST score were compared with their mathematics grade.
According to Ministry of Education Malaysia (2014, 2015, 2016, 2017, 2018), through Primary School Curriculum Standard (Kurikulum Standard Sekolah Rendah) (KSSR), fraction is being teach formally from standard 1 (age 7) until Standard 6 (age 12) in primary school under the learning area of Number and Operation. The syllabus covers the very basic concept of fraction with denominator of 2 and 4, then to denominator of 10 and gradually increasing in year with operations of addition, subtraction, improper fraction to the operation of multiplication and division of fractions. Once the students finish primary school, with good number sense in fraction, they will continue to apply the conceptual understanding and the knowledge of procedures of fraction in secondary school such as in topics of rational numbers, sequence and indices to name a few. It is troublesome if the students do not have this sense from the primary school in order to advance in the next level of mathematics.
Lortie-Forgues et al. (2015) emphasized the importance of knowledge of fraction in predicting success in more advanced mathematics and the subsequent academic and occupational success. Unfortunately, many students struggle with fractions during this foundational period (Fuchs et al., 2013; Hecht & Vagi, 2011). Hence, our main purpose in this study is to identify the students’ ability of number sense in fraction. The second aim was to determine the relationship between students’ performance in mathematics in general and fraction number sense.
2. Literature Review
Number sense refers to the understanding of different relationships between numbers and operations and the flexible use of these relationships (Akkaya, 2016). Individuals with good number sense tend to exhibit the following characteristics when performing mental computations; sense-making approach, planning and control, flexibility and appropriateness sense of reasonableness. This is very important skill to be mastered by every individual to enable them to handle numerical problems in daily life. In Malaysia, many research and studies have been focused on number sense for the past ten years, even though the term number sense is not widely used in the education system (Mohamed et al., 2010). One of the components in number sense is fraction knowledge. Fraction knowledge plays a vital role in mathematics education particularly and daily life generally. It promotes everyday skills, such as calculating expenses and paying debts. Procedural knowledge of fractions consists of fluency with the four fraction arithmetic operations: addition, subtraction, multiplication and division (Bailey et al., 2014). While the conceptual knowledge in this area suggests that students must understand how the numerator and denominator work together to determine magnitudes and that two or more fractions can be ordered on the number line (Fazio et al., 2016).
The study which focuses on secondary students is essential to the education system in Malaysia as the foundational role fractions play in algebra, advanced mathematics and technological careers. Bailey et al. (2014) also reported that fraction knowledge was consistently related to mathematics achievement in the United States and Chinese children.
In Malaysia, fractions were introduced in the elementary schools. This makes fractions the first numbers that are relational expressions to which students are exposed (DeWolf et al., 2015). Many students come to secondary school after finishing elementary level without strong foundations of mathematics which includes fraction knowledge. If one cannot master the fraction knowledge at such the young age, he will face problems in grasping algebra at the next level which are the secondary and tertiary education. For example, students cannot estimate solutions to arithmetic problems as the result of failure in understanding fraction.
They are not able to come out with the proper solutions since they are only guessing the answers based on the numbers given in the problems. Research by Bailey et al. (2015) suggested that the United States and Chinese middle children had high knowledge of fraction concepts and procedures with mean accuracy of each task exceeded 90%.
We need to overcome this issue by observing into the deep root of fraction knowledge problems thoroughly. The role of mathematics teacher at the elementary level is important.
They are responsible to deliver fractions conceptual and instruction. Classroom instruction might differ from one grade to another grade. For some students, they prefer to learn conceptually because they are able to see how mathematics is used to describe the world.
However, students who dislike conceptual instruction may dislike it in part because it does not make calculus useful or sensible to them. They feel more confident in their ability to memorize and reproduce mathematics (Byerley, 2019). Teachers also may highlight the relation between fractions and algebra. Besides that, the pedagogical knowledge of teachers can be enhanced with the provision of in-service seminars on topics such as number sense and its importance, number sense strategies and how to help students develop number sense.
Moreover, teacher training programs covering similar topics should be considered for teacher candidates (Akkaya, 2016).
3. Purpose of the Study
The purpose of this study is to have a better understanding towards the sense of fraction that students possessed. This study focused on to examine the ability of number sense in fractions among secondary students, the students understanding in the basic meaning of decimal numbers, students’ performance of number sense and the part of number sense in fraction that students are weak in.
The specific objectives of this study are:
1) To identify the students’ ability of fraction number sense
2) To determine the relationship between students’ performance in Mathematics in general and fraction number sense.
Research Question
1) How students possess to fraction number sense?
2) What is the relationship between students’ performance in Mathematics in a class and their total score in fraction number sense?
4. Methodology
There were 163 samples used in this study and they were tested for their number sense towards fraction. Samples were selected using stratified random sampling. Stratified sampling is a method of sampling where population is divided into smaller groups called strata, based upon a similar attributes or strength. The samples were secondary students from schools under Kota Bharu zone in Kelantan. There are 174 schools in Kelantan under the jurisdiction of the State Department of Education and they are divided into 10 zones; Kota Bharu, Pasir Mas, Pasir Puteh, Bachok, Tanah Merah, Kuala Krai, Gua Musang, Tumpat, Machang and Jeli.
This study was a mixed-method research which consists of both quantitative and qualitative data. For quantitative approach, data was collected through a set of questionnaires consists of demographic profile of the respondent and several questions to test the respondent’s number sense towards fraction. The questionnaires were distributed to 16 years old students during a double-period class with a duration of 1 hour. All the samples had been taught about the concept of fraction in the similar way, similar environment and similar level. Before the questionnaires were distributed, respondents were given an explanation about the purpose of the questionnaires and a proper instruction about the test was given. Mark was allocated to each question in the questionnaire and midterm test score obtained by each respondent was recorded to become the raw data in this study. In a quantitative research, reliability refers to the capability of all the items in the research instruments to consistently measure the concept.
The method used in this study is considered reliable because the same instrument was applied to all samples and all samples were deemed to have a consistent knowledge of the concept being investigated.
After the questionnaires were collected, respondents were interviewed in person to find out their specific problem area pertaining fraction in mathematics. It was carried out immediately after the test score were analysed and took about 40 minutes each. The interview was a two- way discussion where respondents were given chance to ask questions and to give their opinions pertaining to the topic of discussion. The questions and topics discussed in the interview were set in a structured way to ensure questions and topics being discussed were
asked and discussed in a similar way from one student to another. Next, the interview session was recorded for further analysis.
The midterm test scores were analysed using SPSS software. The respondents’ number sense ability toward fraction was analysed and concluded using Mann Whitney U test. Pearson Correlation test was used to answer the second research question which to determine the relationship between students’ overall performance in Mathematics with their ability in fraction number sense. The overall performance in Mathematics was collected from midterm test conducted at the school level.
5. Result
Table 1: Respondents from Two Different Schools Frequency Percent School A
School B Total
47 39 86
54.7 45.3 100.0
The data collections were carried out at two different schools in Kelantan which researcher labelled them as School A and School B throughout this paper. The number of respondents in both schools are not equal but the difference is not huge and this had no effect on the findings as the purpose is not to find differences between the two schools. The administration of both schools allowed only one class to participate in this research which made up as many as 47 students from School A and 39 students from School B. The selection of class was up to the schools’ administration choice which made researcher could not control the number of respondents in this study.
5.1 Interview
A total of five students were randomly selected for the interview. The interviews were conducted to examine the students’ ability in number sense including the difficulties they faced while solving fraction and ability to connect fraction with real-life problems.
Table 2: Respondents’ interview responses on Number Sense in Fraction
No. Questions Code Respondents ID
1. What do you think about fraction number?
Easy R7, R8
Hard R6, R9, R10
2. When you see a fraction number, how do you imagine what it represents?
A pieces (such as pieces of pizzas)
All respondents A situation (such as 2 of 3
shirts red)
None 3. Did you apply fraction number in
your daily life?
Yes All respondents
No None
4. If you were given a 2 1/2 cakes, how do you divide it to five people?
Using calculator to calculate it R7, R8
I don’t know R6, R9, R10
5. Where else can you find fraction numbers in your daily life?
Mathematics subject All Respondents
Food Portion None
6. Can you simplify this fraction number, ?
Using calculator to calculate it R7, R8
I don’t know R6, R9, R10
7. What are the challenges when you faced to solve fraction number?
Difficulty to divide a whole into equal parts
All respondents
Difficulty to compare the denominator and numerator numbers
None
8. Can see any pattern of fraction number before you solve it?
Yes R7
No R6, R8, R9, R10
9. Can you notice that fraction number relate with decimal number?
Yes All respondents
No None
10. How can you apply decimal number in you daily life?
Money All Respondents
11. If you were given a fraction number, can you convert the number into decimal number?
Yes R7, R8, R9
No R6, R10
12. Did you have any problem to convert it?
Yes R6, R10
No R7, R8, R9
13. Is number fraction is easiest topic for you?
Yes All respondents
No
Table 2 shows the recorded responses using ruled-based strategy. Rule-based strategy allows researcher to identify number sense difficulties and students’ ability easily during the interview. The students were asked to answer two simple fraction questions during the interview, and it was found that two of the respondents (R7, R8) were using calculator to solve the problems while the other three respondents were unable to find a way to solve them.
On the other hand, the respondents who used calculator to solve the problems were unable to show different ways of mathematical solutions, they also admitted that they faced difficulties solving mathematical problems without calculator. Moreover, all respondents expressed low confidence while answering the two questions. Despite of the fact the fractions seems an easy topic for all respondents, four of the respondents were unable to imagine any pattern of the fraction number before solving the problem.
5.2 Pearson correlation between students’ performance in Mathematics in a class and their total score in fraction number sense
A set of fraction questions was distributed to the respondents in order to identify the mathematics performance. Table 3 reports the respondents in school A had better achievement than respondents in school B. By using this data, the next step is to find the relationship between respondents’ mathematics performance and respondents’ score in fraction number sense.
Table 3: Mean rank on students’ achievement in number sense N Mean Rank Sum of Ranks
School A 47 39.38 1851.00
Total score School B 39 48.46 1890.00
Total 86
Pearson correlation was found to be the most appropriate to determine the relationship of the two variables; mathematics performance in general and specific score in fraction number sense.
Table 4: Pearson correlation between students’ performance in Mathematics in a class and their total score in fraction number sense
Total score Mathematics performance score
Pearson Correlation 1 0.066
Total score in fraction number sense
Sig. (2-tailed) 0.546
N 86 86
**Correlation is significant at the 0.01 level (2-tailed)
The result shown in Table 4 indicates there was a weak correlation between students’
performance in Mathematics and their total score in fraction number sense. The increase or decrease of Mathematics performance was not correlated with changes in score of fraction number sense. Moreover, the significant value (0.546 > 0.05) also indicated there was no statistically significant correlation between performance in Mathematics and their total score in fraction number sense.
6. Discussion
The aim of this study is to identify the components that they are weak in fraction which had been collected from interview. It is worth to note that these sample of respondents were having difficulties in fraction topic related to part-to-whole relationship or partitioning fraction. Part-to-whole relationship is a topic where one of the questions ask the students to identify the parts of a given whole number. For example, question 4 stated “If you were given a 2 cakes, how do you divide it to five people?”. This question drove the students to think of a way on how to cut the given fraction number into five equal smaller unit. Apart from that, the content of this question is to test the students’ understanding on the fundamentals of denominator and numerator of a fractions. However, the respondents were unable to solve the questions without using calculator and three of the sample interviewees were failed to find the solution. It is worth to note that question 4 included whole number instead of natural number where from the interview, all respondents expressed low level of knowledge on how the whole number can be divided into equal parts.
Other than that, the respondents were also showed limited skills in fraction topic which related to magnitude and equivalence. Fraction magnitude is an important topic and inevitable steps of mastering fractions (Amalina et al. 2018). In this study, sample respondents confused on how a fraction could represents an amount when given cases like slices of a pizza and proportion of male and female students in a class which related to their daily life. It was found that the respondents were unable to locate in the number line given because to their understanding represented 3 out of 5 pieces and not a number. From the basic, the respondents first need to know about magnitude of numbers. Next, use a strategy to identify fraction numbers without given magnitude numbers. Literatures had suggested fraction magnitude numbers can be identified through number line (Gersten et al., 2016;
Fazio et al., 2016; Amalina et al., 2018), which then simplify the process of learning number quantity and comparing fraction numbers. This had relation with question 8, “Can you see any pattern of fraction number before you solve it?” where more than seven of the respondents answered ‘No’. Based on the interview, it was identified that the respondents were having problem in examine the denominator and numerator of a fraction, which failed them to identify which fraction is smaller or bigger in amount.
Another findings from question 6 (“Can you simplify this fraction number, 1/4+ 1/3?”) also stand on the same conclusion that the respondents were found had limited understanding on the meaning of denominator and numerator, where the respondents were unable to manage the concept of solving fraction when different units ( and ) were given. They simply produced the addition of whole numbers , without take into consideration about the concept of fraction and failed to apply strategy in solving fraction problems.
The second aim of this study is to examine the relationship between students’ performance in Mathematics in a class and their total score in fraction number sense. The overall performance in Mathematics was collected from midterm test conducted at the school level.
Based on the Pearson correlation coefficient r value, it was found that there was almost no correlation and no significant relationship between students’ performance in Mathematics in a class and their total score in fraction number sense. This finding indicated that inability in fraction number sense had no relation to changes in overall Mathematics performance.
Therefore, it can be said that the respondents might be good in other Mathematics topics. In addition, fraction is a small topic in the whole syllabus, so respondents will have other topics to score to get higher marks in overall Mathematics.
The findings showed the respondents achieved higher score in overall Mathematics than specific fraction number sense test. This result match with responses from interviews where they were identified weak in number sense of mathematics, specifically in fractions. The respondents could not recognize the relationship of part to-whole, concept of magnitude and equivalence, pattern of fractions and strategy to solve fraction. Although the overall Mathematics score is higher, this might be caused by drill and practice techniques emphasized by the teacher without realizing the importance of number sense. Teacher should encourage students to use different strategies and create more experiences in understanding on numbers and operations to improve number sense ability. Woods et al. (2017) suggested some features of explicit and systematic instruction such as broad idea and mathematical conceptual, well defined lesson objective, resource for content and sample questions as well as provides students with visual representation of the order and magnitude of numbers using number line. This number line can be used later to expand their understanding to make comparisons.
7. Conclusion
Number sense is a skill that should be taught and developed at the primary school level so that students know the meaning of those numbers rather than just memorize formulas and computation techniques without understanding the beauty behind it. The present investigation addressed how sample students response to fraction number sense and the relationship between students’ performance in Mathematics in a class and their total score in fraction number sense. The result indicated that respondents had low level of number sense ability. It is important to note that the respondents scored low in fraction number sense test because they failed to analyse the pattern of a fraction whether it is part to-whole relationship, magnitude and equivalence or other pattern of fractions.
This study only used fraction number sense topic in the set of questions which cannot be generalized to the whole number sense ability of a student. Future studies are suggested to use more topics to examine number sense ability among students. In addition, it could become a comparative study to measure which topic the students are good at. Having sense when look at number and able to imagine the pattern is vital for individual to relate mathematics concept with real life activity. The skills in a person have in number sense could further increase other thinking skills at the same time such as creative thinking, logical thinking and critical thinking. This study suggests the content of Mathematics syllabus in Malaysia should focus more on number sense skills to nurture other thinking ability which could be beneficial in many parts of life. Therefore, the area of number sense in Mathematics should be paid attention by the curriculum development department, Ministry of Education and teachers.
In conclusion, this study had highlighted the causes of students’ low level in number sense.
Although there was almost no significant relationship between Mathematics performance and number sense ability, the skill of understanding relevancy of numbers in life is crucial.
Students should be well prepared for rapid life changing of technology demand, numeracy and computational sense requirements in the future down the road. The Mathematics Curriculum for secondary school which aims to develop individuals who are able to think critically in mathematics and can apply the knowledge of mathematics effectively and responsible in solve the problematic and can make a good decision (Ministry of Education, 2012). To achieve this, educator must be well trained to use a number of different teaching and learning tools in classroom pedagogy.
Conflicts of Interest
The authors declare that there is no conflict of interest.
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