Influence of Number Line Approach on Learning Fractions: A Case of Basic Six Learners in the Cape Coast Metropolis

doi:10.21203/rs.3.rs-3115078/v1

While the number line approach is successful for fractional learning, there is a dearth of study on its usage in Ghanaian upper primary schools. The study explored the influence of the number line approach on learning fractions. Learners’ achievements, attitudes, and learner’s challenges when using the number line were explored. A quantitative research method embedded with a pretest-posttest non-equivalent design was used. The Fractions Achievement Test (FAT) and a structured questionnaire were the instruments used for data collection. Eighty-one basic six learners with two intact classes (39 in control and 42 experimental groups) were purposively selected within the Cape Coast Metropolis. The experimental group was exposed to learning fractions using the number line, while, in the control group, fraction lessons were carried out through the use of set models. An independent sample t-test was employed to analyse the differences between the pre-test and post-test of the control and experimental group. The findings revealed that there was a significant difference in performance between the post-test scores for the control group (M=7.36, SD=1.78) and the experimental group (M=9.21, SD=2.83); t(69.77)= -3.55; p=0.001 respectively. This implies that the experimental group outperformed the control group due to the influence of the number line approach. Learners in the experimental group had better, understanding, application, and positive attitude toward learning fractions due to the use of number line. Finally, learners however had some challenges, such as the inability to recognise that zero is part of the number system, poor estimation, and counting the tick lines on the number line instead of the intervals between them.

Pure Mathematics

Fractions

Experimental and Control group

number line

Learners attitudes

The importance of mathematics education cannot be overemphasised. In support of this, the International Mathematics Union (IMU) and the African Mathematical Union (AMU) support research into the importance of mathematics education, emphasising that every school going child, without exception, should possess fundamental mathematical skills (Hasler & Akshoomoff, 2019). Notwithstanding the importance of mathematics, most grade four learners are not proficient in mathematics (National Assessment of Education Progress, 2019). Most learners face learning difficulties, and most teachers makes it difficult in making learners understand the subject (Hidayat & Prabawanto, 2018; Copur-Gencturk, 2021).

It is a common misconception in mathematics education that algebra is the foundation of all mathematics (Hoon et al., 2016); however, learners do not dive immediately into the abstractions and theories of algebra unprepared; fractions set the pace for algebraic thinking and proportional justification and is a vital element for the development of mathematical understanding and a stepping stone to a variety of desirable careers (Wijaya, 2017; Teoh et al., 2020). Learners who struggle in mathematics at the elementary, middle, high school, and college never gained fractional arithmetic skills (Bentley & Bossé, 2018).

According to Braithwaite et al. (2017), learners who are unable to learn advanced mathematics are less likely to succeed in the workforce because they are less likely to learn the fundamentals of fractions and are more likely to make mistakes (Van Steenbrugge et al., 2014). Given this, Wilkings and Norton (2018) argued that these difficulties provide a significant problem for students. For instance, evaluations of textbooks show that learning opportunities usually focus on procedural information, which exacerbates learners’ conceptual knowledge deficiencies (Lenz & Wittmann, 2020).

In addition to general cognitive abilities and whole number skills, the knowledge of fractional magnitude predicts overall and more specific results for mathematics (Karamarkovich & Rutherford, 2019).

Teachers must understand how their learners have demonstrated their fractional abilities with a number line since the number line is an excellent representation of increasing learners ability in fraction (Hwang et al., 2019). Barbieri et al. (2020) recommended that using the number line approach increases learners’ attention and attitude because it allows more discussion and strategies to be demonstrated. While Yu (2018) opined that fractions on number lines could help learners overcome their whole number bias and teach fractional magnitudes.

Learning fractions demands not only the familiarity with fractional concepts, definitions, and properties but also the capacity to build cardinal and ordinal numbers, compare and organise fractions, and use them to find fractions of shapes and numbers. Upper Primary learners are to simplify fractions, utilise them as operators, and find fractions of integers and quantities (NaCCA, 2019). However, many learners struggle with fractions despite years of instruction (Hwang et al., 2019; Roesslein & Codding, 2019). The Trends in International Mathematics and Science Study [TIMSS, 2007–2011], the (Program for International Student Assessment) [PISA] (2015), and the (Early Grade Mathematics Achievement) [EGMA] (2015) all revealed that Ghanaian learners could not perform above curricular expectations (Mereku, 2016), highlighting the critical role of educators.

A similar scenario exists in the Basic Education Certificate Examination (WAEC, 2015, 2016, and 2017), where the Chief Examiner has repeatedly reported learners’ deficits in the fractional concept.

Nevertheless, failure to progress from Basic Education Certificate Examination and West African Senior Secondary Certificate Examination (BECE and WASSCE) will deny candidates access to the next level of education (Ntow, 2009). A number line is suggested to represent part-whole, quotient, measure, ratio, and operate among a schemata (Morano, Riccomini & Lee, 2019; Barbieri et al., 2020), which comprises rote knowledge, relationship knowledge, and visualisation abilities. Despite its effectiveness, most teachers and researchers do not employ the number line approach, which serves as one of the common reasons for learners’ difficulties (Gersten, Schumacher & Jordan, 2017; Dyson, Jordan, Barbieri, Rodrigues & Rinne, 2018). Anecdotal evidence from the researcher indicates that most Ghana Education Service (GES) approved textbooks did not consider number lines to solve fractions, even though the syllabus suggests number lines on a fractional chart.

While the number line approach is successful for fractional learning (Gersten et al., 2017; Dyson et al., 2018), there is a dearth of research on its usage in Ghanaian upper primary mathematics. However, interventions in Ghana have primarily focused on either area or set models, with linear models being neglected (number line) (Ametepeh, 2018; Amuah, Davis & Fletcher, 2019; Bernard, Golbert & Gabina, 2020). The current study employed a pretest-posttest non-equivalent quasi-experimental design embedded with a quantitative research approach to examine basic six learners’ performance, attitude, and challenges when using the number line in learning fractions. In addition, differences between the learners’ pre-test and post-test scores were examined to explore the influence of the number line approach on learners learning of fractions.

Research questions

The following research questions underpinned the study

What are basic six learners’ level of performance through the use of the number line in learning fractions?
What are the basic six learners’ attitudes toward the learning of fractions using the number line?
What are the basic six learners’ challenges in using the number line in learning fractions?

Research hypotheses

The following hypotheses were formed to guide the study;

H₀: There is no significant difference in performance between the control and experimental groups on the pre-test scores
H₀: there is no significant difference between the control and experimental groups on the post-test scores.

Theoretical Underpinnings

This study was underpinned by the constructivist theory, which asserts that humans can better understand the information they have created (Mohammed & Kinyó, 2020). The learner is regarded as the focus of the instructional process. Prejudices, experiences, the period in which we live, and physical and mental maturity all influence how we learn. When a learner is motivated, he or she uses willingness, determination, and action to gather, convert, formulate hypotheses, use applications, interactions, or experiences to test these assumptions and draw accurate conclusions. As constructivism implies, learners are not “blank slates” devoid of ideas, concepts, or brain structures. Furthermore, constructivist recognises that learners are not empty vessels or blank slates waiting for knowledge (Noureen et al., 2020). Instead, learners build new knowledge from diverse past experiences, acquaintances, and beliefs (Noureen et al., 2020). Gupta and Gupta (2017) agreed that, similar to how all cells develop from pre-existing cells in cell theory, information already exists in the human body, and all that is required is methods for investigation. This demonstrates that learners have a sense of self-awareness when they come to class. They do not absorb ideas presented by teachers but instead, create their knowledge.

This study looked into the effectiveness of the number line approach in learning fractions. As a result, constructivism is relevant in this study because when learners construct knowledge independently, misconceptions may occur as they seek to form new ideas. Although misconceptions can never be avoided entirely, teachers can intervene before they become deeply rooted. Before addressing errors or developing interventions to promote understanding, teachers need to determine why their learners make mistakes or how misunderstandings have developed (Harbour et al., 2016).

Importance of Learning Fractions

A numerical line is an essential tool to use regularly in fractions during the lesson. A number line is a helpful tool for seeing which fraction is smaller and which is bigger and reinforces that those two fractions always have another fraction (Van De Walle et al., 2013). A number line is a graph showing the range of infinity-to-infinity integers. The number line is a great pedagogical tool, according to Skoumpourdi (2010), particularly as it allows learners to view mathematical concepts directly: the number line is utilised for counting, estimating, and time representation and for presenting distinct number sets. In addition, the number line can give geometric representation and measure and compare arithmetic processes.

According to Lamon (2012), there are three primary reasons why fractions must be taught:

Fractions have a significant impact on learners’ attitudes toward mathematics.
Fractions are a necessary part of school mathematics and daily life. Fractions are not just important in mathematics; they provide the basis of more advanced notions like ratios, rates, percent, proportions, proportionality, linearity, and slope. The ability to work with fractions is useful in many areas of life, including cooking, discount calculations, rate comparisons, unit conversions, reading maps, and financial planning.
In order to be mathematically competent, a solid foundation in fractions is required. The final report, Foundations for Success, written by the National Mathematics Advisory Panel in 2008, concluded that algebra is the most important subject for students to learn in high school and college. The primary reason American learners struggle in algebra is a deficiency in fraction fluency. The challenging “algebra for everyone” task will remain unachievable until “fractions for everyone” is achieved.

Importance of the Number Line as an Approach in Teaching Fractions

Additionally, Gray (2018) lists the following benefits of utilising a number line to teach fractions:

Number lines assist learners in visualising fractions as portions of a whole or a set and as a fraction of distance or time.
Number lines aid in the comparison of fractions.
Number lines are more successful than traditional visual models for teaching fractions.
Number lines assist us in determining equal fractions.
Number lines assist us in visualising a fraction as a number between two whole numbers.

According to Van de Walle, Karp and Bay-Williams (2013), number lines help learners compare numbers and acknowledge fractions as quantities rather than as “one number over another.” Additionally, number lines can broaden learners’ understanding of fractions by including negative fractions and fractions with values higher than 1, decimals, and per cent. Number lines are effective for illustrating the concept of fraction density.

Learners’ Misconceptions in Learning Fractions

Numerous studies have been conducted on learners’ mathematical misconceptions and errors (e.g., Mohyuddin & Khalil, 2016; Burgoon, Heddle & Duran, 2017; Aliustaoglu, Tuna & Biber, 2018). These misconceptions and errors might result from a number of causes, such as students' attitudes towards mathematics (Kusmaryono, Suyitno, Dwijanto & Dwidayati, 2019), teaching framework (Skott, 2019), teaching skills (OECD, 2019), learners’ preconceptions (Diyanahesa, Kusairi & Latifah, 2017), limited understanding in fractions(Saputri & Widyaningrum, 2016), and a lack of appropriate mathematical misconceptions appear to be connected with incorrect concepts developed by learners in mathematics as a result of lack of clarity in concept learning. As a result, individuals may feel that what they are doing is right or that they are unclear of what they are doing (Neidorf, Arora, Erberber, Tsokodayi & Mai, 2020). Such misconceptions may be the result of their past knowledge, which they erroneously generalised (Im & Jitendra, 2020). A mistake could be made as a result of incompetence or a lack of knowledge on the need to confirm the responses given (Hansen et al., 2015). Learners' understanding of mathematical concepts might be hampered by lingering misconceptions, leading to an abundance of mistakes (Im & Jitendra, 2020). An inaccurate conception of mathematics may lead to unfavourable attitudes and an incorrect perspective on the subject (Belbase, 2013).

Misconceptions are logical errors. At any level of fraction knowledge, there is the possibility of making a mistake. Makonye and Fakude (2016) define misconceptions as misguided beliefs and concepts that underpin a person’s state of mind, resulting in a cascade of errors. Failure to recognise that components of the whole are of equal size is an example of a misperception in the early stages of fraction learning (i.e., \(\frac{2}{3}\)would represent 2 of 3 equal parts). Siegler and Lortie-Forgues (2015) believed that misconceptions, such as learners not understanding an infinite number of fractions referring to the same magnitude, still support misconceptions. However, Fazio et al. (2011) posit those problem-solving errors are caused by inadequate confidence when dealing with fractions. Given this, Ramadianti, Priatna, and Kusnandi (2019) pointed out that this error occurs due to learners’ lack of context for recognising fractions. Furthermore, learners sometimes avoid the fractional parts of operations when doing arithmetic with mixed numbers. Learners’ misconceptions about fractions and the avoidance behaviours that result from them are common throughout their schooling. Fraction problems can last well into adolescence and adulthood (Siegler & Lortie-Forgues, 2015).

Fitri and Prahmana (2019) concluded from a sample of 30 seventh-grade learners from SMP Negeri 1 Piyungan that “learners continue to make mistakes when they recruit the unknown components of the problem and cannot use fractional concepts in counting and incorrectly convert mixtures into ordinary fractions.” In addition, learners make mistakes when converting integers to fractions and are less careful when counting. Finally, learners sort fraction numbers incorrectly (p. 8).

Widodo and Ikhwanudin (2018) reached a similar conclusion after interviewing, recording, observing, and using paper and pencil measures on 31 grade six learners about the challenges they encounter when dealing with fractions on the number line; they described four common student blunders: misunderstanding units, misinterpreting tick marks, incorrectly partitioning, and guessing. When examining learners’ answers, it was found that the answer was incorrect because of factors such as the ranking by the numerator and denominator proximity and the ranking of the minority or the majority, among other natural numbers. Thus, they proposed that, when teaching fractions, teachers should focus on unit understanding, clarify tick mark interpretation, remind learners of the need for partitioning and un-partitioning operations, and teach good estimate techniques.

Alkhateeb (2019) highlighted fifth-graders common mistakes in fractions and their associated thinking strategies in Zarqa (Jordan). Using a mixed-method approach with the diagnostic test and individual interview as an instrument, 240 learners were randomly selected, while 30 were interviewed. The outcome of the study showed various mistakes made by learners, which are as follows: the common mistakes were learners’ relations with fractions as integers, errors about basic concepts of the fractions such as taking into account that the fractional number is always higher than the figure \(\frac{A}{B}\) and that figure \(\frac{A}{B}\) is always less than one; another misconception was that learners misinterpret the numerator and the denominator with the actual value of the fraction without paying much attention to the integer in the fractional number. The results further show that more than 50% of the learners made mistakes associated with finding solutions to fractions issues regarding learners’ thinking and associated errors. The most apparent error was stating the fractions without prioritising the equal parts.

According to Siegler et al. (2011), learners can learn estimation fraction magnitudes between 0 and 1 (\(\frac{1}{2}\),\(\frac{1}{3}\),\(\frac{2}{3}\),\(\frac{1}{4}\),\(\frac{3}{4}\)) to help and support them in generalising their fraction magnitude knowledge. Learners will be able to reject irrational solutions if they have a sense of how near the answer might be, based on fraction magnitude. For instance, learners may refuse the approach that results in arithmetic errors of type \(\frac{1}{2}\)+\(\frac{1}{3}\)= \(\frac{2}{5}.\)This may prompt them to experiment with alternative approaches and see whether their response made sense. To support a more general knowledge of fraction magnitude, possessing a feeling of learners will be able to reject implausible answers by knowing what the answer might be near, depending on fraction magnitude (Siegler et al., 2011).

Trivena, Ningsih, and Jupri (2017) also observed: “how primary five learners understand fraction addition and subtraction.” Both learners and teachers were subjected to a test that included the Certainty Response Index (CRI) and an interview. In analysing student responses, both the CRI and interviews with both learners and teachers were used. The findings revealed that learners’ mastery of addition and subtraction concepts was dominated by the category “misconception.” These data revealed that the mastery concept of fraction addition and subtraction in fifth grade remained low. The learners, in particular, are unaware that addition and subtraction operations must equalise the denominator.

Hıdıroğlu (2016) argued two reasons why targeted results in the fractional unit were of fragile accessibility: learners’ misconceptions and the teachers’ learning-teaching process, which does not consider the learners’ prior knowledge. Learners’ thinking is transformed when they learn about fractions. Learners experience difficulties moving from whole numbers to fractions because they do not focus on “numeric entities” (Siegler et al., 2011, p. 274). Even if fractional education starts at primary school, even secondary and school learners often confuse fractions and entire properties (Siegler et al., 2011; Vosniadou, 2014).

While Lewis, Matthews, and Hubbard (2016) confirm that this mistake is not unique to learners when university undergraduates were asked which sum of \(\frac{12}{13}\) and \(\frac{7}{8}\) was closest to 1, 2, 19, or 21, and 15% chose 19 or 21. This incorrect response indicates that learners focused on the fraction’s components (numerator and denominator) rather than on its overall meaning and added the numerator (to get 19) or denominators (to arrive at 21). Due to their inability to process fractions holistically, individuals may wrongly apply their knowledge of whole number properties to fraction tasks, resulting in a “whole number bias” (Ni & Zhou, 2005; Siegler & Pyke, 2013; DeWolf & Vosniadou, 2015). For example, because the entire number 9 is higher in magnitude than the number 2, this prejudice may mislead someone to perceive the number 9 as more important than the number 2 and regard \(\frac{1}{9}\) as larger than \(\frac{1}{2}\).

Eroğlu (2012) discovered that Moss and Case conducted a study to determine whether prospective primary and secondary school mathematics teachers knew their learners’ fractional errors. The future teachers were cognisant of their learners’ errors but limited their explanations. They suggested using verbal descriptions, area models, real-world examples, preliminary knowledge replicas, standard teaching solutions, questions, simple examples, and exhibitions to assist learners in resolving their errors. They proposed verbal explanations, area models, real-world examples, standard teaching solutions, and leading questions, straightforward examples, opposite examples, exercises, and practises to help learners recognise and correct their errors. These earlier syntheses’ results helped develop a sense of useful teaching components for challenging learners in fractions.

However, these studies reviewed by Eroğlu (2012) and Zhang, Clements, and Ellerton (2015) focused on learners’ conceptual misunderstanding from area models to multiple representations using 40 respondents. Fitri and Prahmana’s (2019) study focused on the problems of learners in solving fractions using a descriptive research approach and a sample of 30 learners; learners’ errors were detected without emphasis on how those learners overcame their issues. Cramer, Ahrendt, Monson, Wyberg, and Miller (2017) also looked at the challenges that third-grade learners encounter using number lines as a model for a fraction using interview and qualitative research design.

Mitchell and Horn (2008) conducted a study to discover learners’ misconceptions regarding number lines. Twenty-nine grade six learners worked on eight number line tasks using an interview from two schools in metropolitan Melbourne. The fraction number line task was chosen to examine learners’ reasoning comprehension to measure the fraction sub-construct. An interview from year five learner completed question 11 during his interview on how to place \(\frac{1}{2}\) on the number line. “Put a cross where the number half would be on the number line,” he read aloud. He drew a cross halfway between 2 and 3. “Half of it,” he said when asked how he figured it out. Because zero is not a number in the middle [indicating the 1 and 4 on the number line]. This is the middle “[Counting in from both sides]” He did not count zero because he did not take into account the number, so 2\(\frac{1}{2}\) was the halfway point between 1 and 4. His answers during the interview and 24.4% of learners who completed the question on paper support the idea that it suggests procedure rather than a quantity or a distance from zero.

Learners’ Attitude Towards the use of Number Line Approach in Learning Fractions

Karika and Cskos (2022) used the number line approach study how well learners can conceptualise fractions in their heads. High reliability (alpha = .95) was found for the test among a sample of 124 fifth graders. According to the results, the correlation coefficients between learners’ overall performance and their attitude factors range from 0.21 (the usefulness of learning fractions) to 0.62 (the importance of studying fractions) (attitude towards fractions). Each of these R-values is statistically significant at the p < 0.05 level. Each group of items investigated thus far showed correlation coefficients of around the same size and significance. Performance was significantly correlated with learners’ attitudes toward learning fractions using a number line.

The fundamental objective of the research conducted by Govindarajan and Choo (2022) was to enhance the performance and attitudes of elementary school students in mathematics by introducing them to an effective learning approach (the number line). A quasi-experimental time-series design was used. Forty students were assigned to the treatment group, which used a blended learning platform (Moodle), whereas the same number of students in the control group traditionally received their education. Pre-Test, Tests 1 and 2, Post-Test, and Attitude Questionnaires and Interviews were utilised to collect data. One-way analysis of variance (ANOVA) was used to compare the experimental and control groups to assess the data analysis strategy. The data shows a noticeable distinction between the two approaches at the p < 0.05 level. The study’s results confirmed that blended learning effectively raises students’ academic performance. It was also discovered that students’ attitudes improved due to their time in blended learning.

Hensberry, Moore, and Perkins (2015) investigate how using a simulation to teach mathematics impacts students’ motivation and performance. During four days, two groups of fourth graders used the simulation to practice basic fraction skills. Pre-test and post-test, a survey of students’ attitudes, and in-depth interviews with a sample of students all contributed to the data collected. Both procedural and conceptual understanding of fractions improved significantly between the pre-and post-tests. The focus group interview data corroborated the survey results showing that most students felt the interactive simulation helped teach them about fractions. These findings show that interactive simulations can be powerful instruments for fostering procedural and conceptual comprehension when combined with good instruction.

This study was further concurred by Barbiere et al. (2020), whose preliminary analyses indicated a statistically significant interaction between classroom attentive behaviour and intervention group on fraction concepts on the posttest, implying that there was a moderating effect of the experimental intervention on the detrimental effect of low attentive behaviour on learning. Students who struggle with fractions benefit greatly from being taught using a number line-based method that combines research-based learning strategies.

A quantitative research approach embedded with a pretest-posttest non-equivalent design was used. This experimental research design determined whether the number line could influence learning fractions and the challenges associated with the number line approach. The researchers manipulated one independent variable (the number line) to verify if it would affect a pupil’s performance toward the dependent variable (fractions) under a controlled condition in this research design. The post-tests were administered to examine the treatment effect after the experimental group received the number line approach instruction and the control group received the conventional instruction.

Sample and Research Procedures

The study comprised an experimental and a control group. The experimental group was exposed to using the number line to learn fractions. Two schools with an intact class of 42 (Experimental) and 39 (Control) were selected purposively for the study. A pre-test was administered to the two schools during the first week; the mean scores were established. With the mean scores, the learners from the school with low mean were placed in the experimental group using the number line model. The learners from the school with a high mean were placed in the control group, which experienced fractions using the set and area models. The teacher introduced the topic according to the guidelines used by the basic six curricula. Shaughnessy et al. (2014) posited that this design is appropriate if the purpose is to analyse the pre-test mean scores between the experimental and control groups and the post-test mean scores of the group. In the pre-test-post-test control group, one of the groups formed previously for some other purpose was randomly selected as the experimental group.

The selection of these schools from the two towns was influenced by geographical proximity, time constraints, and the desire to avoid treatment contamination. The two groups were decided after the mean scores were established. Participants for this study comprised basic six learners within the Cape Coast Metropolis, and these learners were considered ideal for the study. After all, these classes were targeted because fraction starts to become a focus in primary four, and by primary six, learners should have foundational fraction knowledge.

Instruments

A self-constructed questionnaire was used. The purpose of the structured questionnaire was to collect data on learners’ attitudes before and after the number line instructions. The questionnaire elicited information on learners’ attitudes and whether they embrace fraction lessons using the number line. It was divided into three sub-construct, “Behavioural Engagement,” “Interests,” and “Confidence.” Questions like, “I can locate fraction problems faster when I use the number line,” I now see the number line as part of a fraction. There were 15 questionnaires on learners’ attitudes towards learning fractions with the number line.

The researchers adopted the Fraction Acheivement Task (F.A.T.) on fractions given to Basic 6 learners of similar characteristics but in different Municipalities (KEEA) in the Cape Coast Region of Ghana. A questionnaire on learners’ attitudes toward learning fractions with the number line was scored immediately. Those items which were ambiguous were taken out. The data was evaluated, and the instrument’s reliability coefficient was determined using Cronbach alpha, which was estimated to be α = 0.898. After that, the reliability of the achievement test was established by applying the Kuder-Richardson formula 20 (KR20). The KR₂₀ value was 0. 96. Therefore, the F.A.T. assessment adopted in the Ghanaian context was reliable.

Data Analysis

The learners’ responses to the after-instruction questionnaire and the lesson test scores were entered into Microsoft Excel and then transferred to SPSS (version 26).

Research question one

What are the basic six learners’ attitudes toward learning fractions with the number line? Data were analysed using a paired samples t-test, percentages, and mean to explore their attitude before and after the intervention.

Research question two

What are the basic six learners’ challenges in using the number line in learning fractions? Percentages, excerpts, and snapshots of both groups’ challenges were identified, and explanations were provided.

Hypotheses

An independent sample t-test was used to test if there was a significant difference between the pre-test and post-test scores at the 5% significance level between the two groups.

An independent samples t-test was conducted. At an alpha 𝛼 level of 0.05. A Levene’s test of equal variance was also performed. The results show that the sig. value for Levene’s test was below 0.05 (0.091), indicating that the assumption has been violated, and the results on the second row were reported. Table 1shows the outcome of the pre-test scores.

Table 1

Difference in pre-test scores
Schools Categories	N	M	SD	t	Df	Sig.
				-0.11	63.29	0.91
Control	39	4.97	2.413
Experimental	42	5.02	1.522
Source: Fiend Data (2021)

From Table 1, the Control group had a mean score of \(\left(M=4.97, SD= 2.41\right)\). The maximum score was 10, with the minimum score being 2 compared to the Experimental group \(\left(M=5.02, SD= 1.52\right)\) where learners had a maximum score of 9, with 2 being the minimum score. The results from the t-test, as shown in Table 1, indicated that there was no statistically significant difference in the pre-test scores for the Control \(\left(M=4.97, SD=2.41\right)\) and Experimental group \(\left(M=5.02, SD=1.52\right)\); \(t \left(63.29\right)= -0.11;p=0.91\). This implies that the two groups performed almost equally at the start of the intervention. Therefore, the researchers failed to reject the null hypothesis, which states that “there is no significant difference between the control and the experimental group on the pre-test.”

After the intervention, a post-test was conducted. The research hypothesis was formulated as follows.

H₁

There is no significant difference between the two groups on the post-test scores in the learning of fractions.

Table 2

Difference in post-test scores
Schools Categories	N	M	SD	T	Df	Sig.
				-3.55	69.77	0.001
Control	39	7.36	1.78
Experimental	42	9.21	2.83
Source: Field Data (2021)

From Table 2, the Control group had a mean score of \(\left(M=7.36, SD= 1.78\right)\). The maximum score was 13, with the minimum score being 4, compared to the Experimental group \(\left(M= 9.21, SD= 2.83\right)\) where learners had a maximum score of 16, with 5 being the minimum score. The results from the t-test, as shown in Table 2, indicated that there was a statistically significant difference in the post-test scores for the Control \(\left(M=7.36, SD=1.78\right)\) and the Experimental group \(\left(M=9.21, SD=2.83\right)\); \(t \left(69.77\right)= -3.55;p=0.001\). This indicates that the experimental group outperformed the control group following the intervention.

This study confirmed Soni and Okomoto (2020) findings which support that applying the number line improved learners’ fraction-related concepts as measured using knowledge, understanding, and application. The findings also support Hamdam et al. (2017), who revealed that learners who learned fractions through the number line were more adept at performance level than those using area models.

Research Question 1

What are basic six learners’ attitudes toward learning fractions with the number line?

A four-Likert scale was used to examine the experimental learners’ attitudes toward learning fractions with the number line. Fifteen questions were subdivided into three sub-constructs, with five statements each. The first construct, “Behavioural Engagement,” meant how learners acted when learning fractions with the number line. The second factor was “Interest” regarding how much they enjoyed the lesson and worked extensively on the tasks. “Confidence” looked at learners’ confidence in learning and understanding fractions with the number line. This was based on learners’ ability to explain their thinking and justify their answers on FAT tasks. The result is shown in Table 3.

Table 3

Experimental Learners’ Attitude toward Learning Fractions with the Number Line
Items	Statement		S.A	Agree	Disagree	SD
		Mean	%	%	%	%
Behavioural Engagement
1	I can locate fraction problems faster when I use the number line	1.64	42.9	50.0	7.1	0
2 3	For some reason, the use of number lines is really hard for me It was easier to learn and understand fraction concepts using the number line	2.50 1.40	19.0 71.4	28.6 16.7	35.7 11.9	16.7 0
4	I can now understand how to locate \(\frac{1}{2}\)with the number line	1.29	81.0	9.5	9.5	0
5	I now see the number line as part of a fraction	1.64	59.5	23.8	9.5	7.1
Confidence
6	I have much self-confidence when using the number line in solving fractions	1.43	64.3	28.6	7.1	0
7	I can now apply and explain my thinking on how to locate fractions using the number line	1.62	45.2	47.6	7.1	0
8	Once I start working with the number line, it is hard to stop	1.86	50.0	23.8	16.7	9.5
9	The individual tests helped me to check my understanding of the lessons	1.50	66.7	19.0	11.9	2.4
10	I have a few problems on how to explain how to locate \(\frac{1}{2}\) using the number line	2.40	19.0	33.3	35.7	11.9
Interest
11	The examples and exercises given in the lessons enhanced my understanding of fraction concepts	1.48	59.5	35.7	2.4	2.4
12	The activities in the number line helped me to follow the lessons and think critically in understanding the concept	1.62	57.1	28.6	9.5	4.8
13	From the teacher’s examples during the lesson, I can relate the concepts learned to real-life situations.	1.43	69.0	21.4	7.1	2.4
14	The use of the number line helped to identify patterns, compare and relate ideas to make generalisations	1.45	66.5	21.4	11.9	0
15	I prefer that my teacher use this approach to teach us different concepts of fractions.	1.19	81.0	19.0	0	0
Source: Field Data (2021)

From Table 3, it was indicated that twenty-one learners representing (50%) agreed with the statement that they could locate fraction problems faster on the number line, with 18 (42.9%) strongly agreeing, while 3 learners (7.1%) disagreed, indicating a mean of 1.64. fifteen learners (35.7%) disagreed with the statement that “for some reason, the use of number lines is hard for them,” with 12 learners (28.6%) agreeing, while the least strongly disagreed (16.7%), given a mean of 2.50. Again, 30 (71.4%) strongly agreed that using the number line made learning and understanding fraction concepts convenient, with 16.7% representing 7 learners agreeing and only 5 (11.9%) disagreeing.

Despite item number 4 being one of the most strongly agreed (81%) statements, “I can now understand how to locate \(\frac{1}{2}\)with the number line.” 4 (9.5%) support the assertion, and 4 (9.5%) disagree. Twenty-five learners (59.5%) with a mean of 1.64 (SD = .932) strongly agreed that they can now see the number line as part of a fraction, while (23.8%) agreed on the motion, with 9.5% and (7.1%) disagreed and strongly disagreed respectively. Also, 47.6% of learners agreed that they can now apply and explain their thinking on locating fractions using the number line, with 45.2% strongly agreeing, with the least being 7.1%, representing a mean of 1.62.

Most learners support the statement, “Once I start working with the number line, it is hard to stop,” with 50% strongly agreed and 23.8% agreed. While a minimum of 16.7 disagreed, providing a mean of 1.86. Most learners, 66.7%, strongly agree that the tests helped them check their understanding, whereas only 11.9% disagree. Furthermore, the majority, 35.7%, disagreed that they have few problems how to locate \(\frac{1}{2}\) using the number line, with 33.3% agreed, given a mean of 2.40. Most learners (59.5%) thought that the examples and exercises provided in the lessons helped them better understand fraction concepts. Most learners (57.1%) strongly agreed that the number line exercises assisted them in following the lessons and thinking critically about the concept. Most learners (69%) strongly agreed that the teacher’s examples helped them apply the principles learnt in class to real-life circumstances. Last but not least, 66.5 percent of learners strongly believed that using the number line assisted them in identifying patterns, comparing and relating ideas, and making predictions. Lastly, it was observed that most learners representing 81%, strongly agreed that they prefer their teacher to use this approach to teach them different concepts of fractions. In addition to research question two, a paired-sample t-test was performed to compare experimental group learners’ attitudes toward learning fractions with the number line before and after the intervention. The data is presented in Table 4. A paired-sample t-test was performed to compare experimental group learners’ attitudes toward learning fractions with the number line before and after the intervention.

Table 4

Paired Samples t-test on Experimental Learners’ Attitude toward Learning Fractions with Number Line (N = 42)
	Before Intervention		After Intervention		p value	Effect Size(d)
Attitude	M	SD	M	SD
Behavioural Engagement	1.13	0.227	3.22	0.612	0.0001	3.26
Interest	1.36	0.350	3.56	0.305	0.0001	5.13
Confidence	1.42	0.363	1.43	0.244	0.0001	4.82
Overall attitude	1.303	0.223	3.736	0.247	0.0001	6.08
Source: Field Data (2021)

Table 4 shows that the learners’ overall mean attitude on learning fractions using the number line improved significantly after the intervention on all the sub-constructs. The overall mean attitude before the intervention for the experimental group \(\left(M=1.303, SD= 0.223\right)\) and after the intervention \(\left(M=3.736, SD= 0.247\right)\), \(t \left(41\right)= 31.57;p <=0.001\) indicated a significant difference. Cohen’s d was estimated at 6.08, which is a large effect. This suggests that the experimental group of learners acknowledged that they developed a positive attitude toward learning fractions with the number line.

This study’s findings on learners’ attitudes toward learning fractions with the number line support earlier findings from Erlinda and Surya (2017), who revealed that learners find the number line intervention more interesting. This finding aligns with Morrison and Hamshaw (2015), who revealed that learners’ confidence before the study trial was the lowest compared to after employing number line instructions. This study supports Karika and Csíko’s (2022) findings, revealing that learners have a positive attitude toward learning fractions with a linear model. In addition, this study supports Govindarajan and Choo’s (2022) findings, which indicated that learners’ attitudes regarding fractions improved due to the number line. Also, Hensberry et al. (2015) support that learners’ attitudes toward learning fractions with the number line were overwhelmingly positive, and these findings were supported by the experimental group attitude during the questionnaire. Barbieri et al. (2020) concur that using the number line approach increases learners’ attention and attitude since it allows more discussion and strategies. Finally, Lamon (2012) also believes fractions on the number line significantly impact learners’ attitudes about mathematics.

It was observed that learners encountered some challenges when using the number line. We present to you the last research question below.

Research Question 2

What are the basic six learners’ challenges in using the number line in learning fractions?

Table 5 presents the results of incorrect and correct answers between the experimental group.

Table 5

Incorrect and correct responses from the Experimental group
Item No.	Incorrect Answers (%)	Correct Answers (%)
1	9(21.4)	33(78.6)
2	12(28.6)	30(71.4)
3	7(16.7)	35(83.3)
4	14(33.3)	28(66.7)
5	5(11.90)	37(88.1)
6	9(21.4)	33(78.6)
7	18(42.9)	24(57.1)
8	15(35.7)	27(64.3)
9	33(78.6	9(21.4)
10	33(78.6)	9(21.4)
Total	42	100
Sources: Field data (2021)

Results from Table 5 indicate that 9(21.4%) learners incorrectly answered the question. In item two, 12(28.6%) learners had it wrong from the group. It was also observed that only seven learners representing 16.7%, incorrectly answered the question. From item four, 33.3% of learners wrongly answered the question. Only 5(11.9%) from the experimental group incorrectly answered item five. From item 6, it was observed that nine 9(21.4%) learners from the experimental group incorrectly answered it. Results from item eight indicate that 15(35.7%) incorrectly answered the question from the experimental group. 33 (78.6%) learners incorrectly answered questions 9 and 10, respectively. Six challenges were identified, which are explained and shown below.

1. Counting of tick marks instead of the interval between them

It was observed that 19 learners’ faced this challenge. A learner, as indicated in Fig. 1, wrote \(\frac{4}{8}\). The learner incorrectly counted the tick marks instead of the intervals between the unit length. The learner believed that since the first indicated point, “P” is \(\frac{2}{8}\), the next indications should be \(\frac{3}{8}\) and\(\frac{4}{8}\) respectively for both Q and R.

1. Difficulty differentiating fractions from the whole number

It was observed that 18 learners had difficulty differentiating fractions from the whole. An example of an incorrect response was that the learner failed to realise that the point was between one and two on the number line. Given less regard for the unit in which the point was situated.

2. Considering only the numerator in locating fractions with less regard to the number of partitions.

It was observed that nine learners’ faced this challenge. For example, a learner believed that since the denominator of \(\frac{3}{4}\) was not noted within the fractions on the number line; it was ignored, thus locating the fraction with only the numerator, as shown in Fig. 4.

3. Difficulty naming or locating fractions in unit lengths other than 0–1

More than half (33) of the learners find it difficult to locate fractions in unit lengths other than 0–1. An excerpt is shown in Fig. 5.

4. Poor estimation:

The result showed that thirty-three learners failed to recognise that \(\frac{1}{3}\) cannot be located between one and two, neither can \(\frac{2}{3}\)be located between two and three on the number line. The result is shown in Fig. 6.

5. Unable to recognise that 0 is part of the number system.

It was observed that 19 learners failed to recognise 0 as a number during estimation, as shown in Fig. 7.

In the literature, it was evident that learners faced challenges when using the number line in learning fractions. This study’s findings are consistent with Widodo and Ikhwanudin (2018), who posited that learners have difficulty interpreting tick marks. When a number line contains multiple units, such as 0 to 3, learners find locating it on the number line challenging, as confirmed by Alkhateeb (2019). The challenges learners face align with Aliustaoğlu et al. (2018), who opined that learners encounter difficulties concerning part-whole, determining fraction units incorrectly, and misinterpreting fractions as whole numbers.

This study aligns with Mitchell et al. (2008) after interviewing five basic six learners. Their result was that 24.4% of learners do not consider zero when working with the number line. This misconception in this study could account for what Sarwadi et al. (2014) posited as “inaccurate prior knowledge.” While Makhubele (2021) also sees it as inadequate knowledge of the basic concepts.

Findings

The study revealed the following findings:

It was revealed that, after the instruction, the experimental group performed higher than the control group due to the intervention. It was observed that learners’ performances were significant as measured at the Knowledge, Understanding, and Application.
It was revealed that learners’ attitudes regarding number lines in learning fractions were positive.
Experimental learners encountered challenges such as the following.
Counting the tick marks instead of the intervals between them.
Difficulty in differentiating fractions from whole numbers.
Consideration of only the numerator in locating fractions with less regard for the number of equal partitions.
The misconception of the unit length: thinking of all unit lengths as 0 – 1, thus difficulty naming or locating fractions in unit lengths other than 0 – 1.
Poor estimation: failure to recognise that cannot be located between one and two, neither can be located between two and three on the number line.
Unable to recognise that 0 is part of the number system

Conclusions

It can be concluded from the findings that learners’ level of performance was high due to the number line intervention. Most learners understood the unit length 0–1, but differentiating that from other units was challenging to them as they found it difficult to name fractions located in other unit lengths aside from 0–1. Learners found the number line lessons enjoyable; their attitude and interest aroused; they became familiar with the instruction, making them less anxious. Learners gained confidence in responding to questions and engaging in conversations.

Though the denominator sometimes tells the number of equal partitions of the unit, it cannot always be the case, as some fractions may be renamed with their equivalents. The fact that you want to model \(\frac{4}{8}\) does not necessarily mean the unit should be partitioned into eighths. It can be sixths or quarters or even tenths, yet \(\frac{4}{8}\) can be modelled by using the idea of equivalents. An inadequate understanding of this probably made some learners locate fractions using only the numerator. Partly, poor estimation strategies culminated in this failure, as good estimation techniques could have given them the idea that \(\frac{2}{3}\) is not the same as 2.

Recommendations

Based on the significant findings of the study and its conclusions, the following recommendations are given:

The number line has proven to be a key model in developing knowledge, understanding, and application when learning fractions. Learners develop a positive attitude toward learning fractions with the number line. Stakeholders should organise in-service training for mathematics teachers to make use of the number line in addition to other models for teaching fractions.

Mathematics teachers should take time to explain the various terms in fractions to learners.

The Ghans Education Service-approved textbooks should consider the number line to solve fractions, and other stakeholders, such as mathematics teachers, should also give attention to the tick marks and the spaces between them to help learners relate to the tick marks on the number line.

The study was limited to two different public upper primary six schools in the Cape Coast Metropolis; therefore, the study conclusions may affect the generalisability of the findings. It is possible that the questionnaire would be interpreted differently due to inconsistency and respondent unfairness. The study also looked into the location and explanation of learners’ thinking regarding fractions; nevertheless, replicating the study in other locations and focusing on fraction operations would add to the study’s conclusions. It is recommended that further studies should focus on teachers’ understanding of using the number line in teaching fractions at the upper primary school level.

Funding: The authors received no direct funding for the research.

Author details

Maxwell Adu Sarpong¹

E-mail: [email protected]

ORCID ID: https://orcid.org/0000-0002-3545-6329

Forster D. Ntow²

ORCID ID: https://orcid.org/0000-0001-5127-2709

Department of Mathematics and ICT Education, University of Cape Coast, Ghana.

Ethical clearance: The study was approved by the University of Cape Coast Institutional Review Board (UCCIRB) on November 3, 2021 (Protocol ID: CES-ERB/UCC.EDU/V5/21-91).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Citation information

Cite this article as: Influence of Number Line Approach on Learning Fractions: A Case of Basic Six Learners in the Cape Coast Metropolis. Maxwell A. Sarpong and Foster D. Ntow (2023). International Journal of Science and Mathematics Education.

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Influence of Number Line Approach on Learning Fractions: A Case of Basic Six Learners in the Cape Coast Metropolis

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Introduction