Introduction
Fraction, as a fundamental mathematical concept, plays a significant role in education. At its core, it represents a division of a whole into smaller, equal parts. They have the numerator at the top and the denominator at the bottom to articulate how much of a whole is being considered (Van de Walle et al., 2019, p. 344). Understanding fractions relies on two fundamental concepts: partitioning and equivalence (Švecová et al., 2022, p. 1).
Partitioning is a key idea in fractions that involves dividing a whole into equal parts or groups. This process allows for a clear understanding of how fractions represent portions of a whole. Equivalence is another important concept in fractions, referring to different fractions that represent the same value, such as 1/2 and 3/6.
In the Australian curriculum, teaching this concept is always done through a constructive approach (Australian Curriculum, Assessment, and Reporting Authority (ACARA), n.d.). It is based on the idea that students should actively construct their own understanding of fractions through hands-on experiences and exploration. This methodology emphasizes the use of manipulatives, visual representations, and problem-solving activities to help them develop a deep conceptual understanding of fractions. It begins in Year 1, where students are expected to recognize and describe one-half as one of two equal parts of a whole (Van de Walle et al., 2019, p. 344). The complexity increases as a student progresses from one study level to another – comparisons of fractions with different denominators, arithmetic operations, percentages, and decimals are gradually introduced.
In daily life, each of these aspects is applied in recipes, interpretation of nutrition labels, scheduling activities, measuring construction materials, and more (Švecová et al., 2022, p. 1). Dividing wholes, evaluating discounts and finances, and also developing critical thinking through basic mathematical reasoning – all of these practical implementations play an important role in the need to learn fractions. Therefore, this paper analyzes fractions and attempts to comprehend the constructs and models of fractions.
Part-Whole Construct
Description
To effectively understand fractions, a student must comprehend all possible concepts that they can represent. The most common and fundamental way learners use to explore the meaning of fractions is the part-whole construct (Van de Walle et al., 2019, p. 344). According to Martinez and Blanco (2021), this construct involves using fractions to represent dividing a whole number into equal parts (p. 1). This is the initial method used in teaching learners about this concept, as it is assumed that their first experience with fractions is often derived from fair sharing.
Teaching Strategies
One strategy for teaching fractions using the part-whole construct involves using coin pictures to show how a learner can divide them into equal parts. For example, the teacher will start with a 20-cent coin and ask students to divide it into two equal parts to have 2-10-cent coins. Each of the 10 cents will represent half of the 20-cent coin that the instructor initially has. When divided into four parts, the teacher will show the students that they will have four coins, each worth 5 cents (1/4).
Advantages/Disadvantages
The first advantage is that the part-whole construct introduces fractions to young students who are starting to learn. Learners gain a tangible and intuitive understanding of the concept by visualizing fractions and recognizing their features and values (Top Drawer Teachers, n.d.). Once a student has understood the foundational idea of this concept, they become equipped to proceed to a more complex construct. However, the construct has drawbacks, including its inability to explain more complex fractions and its limitation to addition and subtraction operations (Witherspoon, 2019, p. 3). It may fail to address other fraction aspects, such as comparisons and equivalence.
Measurement
Description
The measurement construct is the second way that teachers use to help students understand fractions. It involves interpreting fractions as measurements, which can be ordered on a number line. It views fractions as a point in the line, making connections to measurements and understanding (Top Drawer Teachers, n.d.). A student identifies a length and then uses it to determine the value of an object (Wilkie & Roche, 2022, p. 2). For example, one can use the unit fraction 1/8 as the selected length in the fraction 5/8 and then count how many times it takes five to be a whole number.
Teaching Strategies
A teacher can use the Australian coins to illustrate the measurement construct to their students. The instructor will start by drawing a number line on the paper or a whiteboard and then label the values as 5 cents, 10 cents, 20 cents, and so on. They will explain to the learner that when a 5-cent coin is placed in the “5 cents” label, it represents 1/1. However, when the 5-cent coin is put in the “10 cents” mark, it becomes 1/3. When the 5-cent coin is placed in the “20-cent” point, it becomes 1/4.
Advantages/Disadvantages
The first advantage is that the measurement construct helps students understand the concepts of measurement and fractions, as well as their connections (Top Drawer Teachers, n.d.). Moreover, using Australian coins in this context makes fractions more relatable to real-world quantities. However, this construct is associated with a lack of accuracy and consistency, as well as an inability to address advanced levels. Fractions in this construct must have the same units to be added or subtracted.
Division/Quotient Fractions
Description
The notion of fractions in the division/quotient relates to sharing equally. It views fractions as the outcome of dividing one quantity by another (Martinez & Blanco, 2021). This construct views a fraction as a quotient and involves dividing two whole numbers (Van de Walle et al., 2019, p. 344). It can be represented in two different ways. For example, if 50 Australian cents are divided among five people, the value becomes 50/5 or 50 ÷ 5.
Teaching Strategies
With the Australian coins, students can effectively understand the division/quotient construct. For instance, the learner will begin with a 50-cent coin and one student. The first process will be to give it whole to one student and indicate that it represents 1/1. In the next step, the learner will be given a 10-cent coin and told that it is one-fifth (1/5) of a 50-cent coin. This will continue by adding more students and coins to represent other fractions.
Advantages/Disadvantages
The division/quotient construct can help students understand various fraction operations, including subtraction, addition, division, and multiplication. It also bridges the gap between simple and complex fractions (Top Drawer Teachers, n.d.). However, students new to fractions or younger ones may not be intuitive to this concept as it introduces new challenging operations like division (Van de Walle et al., 2019, p. 344). It also focuses more on arithmetic operations and does not offer a better understanding of fractions.
Operator
Description
An operator indicates a function or an operation on one or more quantities. In this approach, fractions are viewed as more than just static values. In some instances, fractions can be used to indicate an operation, such as 2/5 of the audience carrying banners or 4/5 of 20 square meters (Van de Walle et al., 2019, p. 344). These situations involve a fraction of a whole number resulting from a combination of two multiplicative operations or as two discrete, related functions.
Teaching Strategies
To illustrate the operator construct, the instructor can use Australian coins of different values. The instructor will begin with a 20-cent coin and explain to the learners that it represents 1/1 because it is whole. They will then introduce a 10-cent coin and explain that it represents an operation on the 20 cents (1/2). This will help them understand how fractions can act as operators.
Advantages/Disadvantages
The operator constructor enables learners to connect the fractions taught in class with real-world scenarios, making the material more practical and relevant. By consistently applying this concept through addition, subtraction, division, and multiplication, they develop an understanding of fractions and learn how to apply them effectively (López-Martín et al., 2022, p. 4). However, it is linked to drawbacks, such as failing to effectively teach basic fraction concepts, including part-whole constructs (Top Drawer Teachers, n.d.). Moreover, some students may find complex fractions challenging, hence making learning hard.
Ratio
Description
At higher levels of study, as a child grows, fractions can be used to express ratios. This concept refers to a comparison or relationship between two different quantities in a specific order, rather than being a number in isolation (Van de Walle et al., 2019, p. 345). The approach will enable students to comprehend scaling and proportions. For example, the fraction 1/4 can mean that the probability of an event happening is one in four.
Teaching Strategies
To teach the ratio concept and help students understand fractions, the educator will use a set of Australian coins. They will explain that ratios involve comparing two quantities, such as 10 cents and 5 cents. They will then emphasize that the ratio 5:10 equals the fraction 5/10, which represents 1/2 in the simplified form. The educator will also show the learners how equivalent fractions can be expressed as ratios.
Advantages/Disadvantages
This construct helps them understand proportions and scaling, as well as how to apply these concepts to the real world. However, it also has drawbacks, such as complexity for some learners, especially those new to the concept (Top Drawer Teachers, n.d.). It focuses more on comparison than on basic concepts, such as the part-whole construct.
Fraction Models
Area/Regional Model
Fraction models are essential tools that educators often use to enable their students to engage tangibly with numbers and comprehend, as well as work with, fractions. The first model is an area/regional model, which represents fractions using visuals. It utilizes two-dimensional shapes and regions based on the idea that fractions often represent a part of a whole region or area (Van de Walle et al., 2019, p. 347). The numerator is often the large coin, while the denominator represents the whole section. In the Australian coins, this model can prove very significant to students. Say the teacher has four coins (5 cents each) representing 1/1. With the coins being 5 cents each, the smaller coin will represent 1/4 of the total value of the four coins.
Linear/Length Model
The linear/length model is the second approach that educators often utilize in teaching fractions. It measures lengths and other dimensions, comparing them rather than areas. They are important in helping learners develop a better understanding of fractions (Van de Walle et al., 2019, p. 348). In this model, fractions are represented as positions on a number line, with the denominator serving as the whole linear scale. 10 cents is placed at mark 10. At point 5, 5 cents indicates half (1/2) of 10 cents.
Collection/Set Model
The third fraction model is a collection/set that offers a different perspective on fractions, especially when dealing with measurements and comparisons. It is a set of objects, and the fraction parts comprise its subsets. In the collection/set model, the numerator represents a part of the denominator value, indicating the whole set (Van de Walle et al., 2019, p. 348). In the theme context, an educator will have three-cent coins representing 3/3. The numerator indicates that the teacher has three of the available coins. However, when a student is given 2 of the coins, the teacher remains with one (1/3).
Student Misconceptions
The first misconception that children face with fractions is that numerators and denominators are separate values. For example, the fraction ¾ may not be understood as 3 is part of the 4. This can be overcome by finding fraction values on a number line (Van de Walle et al., 2019, p. 345).
The second issue is that learners need help understanding equivalent fractions. For instance, they can struggle to understand that 1/2 and 2/4 represent the same qualities (Jarrah et al., 2022). This can be addressed through the use of visual models, such as a set or area, to emphasize the concept of partitioning.
The third misconception that students face is that they mistakenly apply operation rules for fractions in the same way they do for whole numbers. For example, they can assume that 1/2 + 1/2 = 2/4. This can be addressed by using visuals and context to emphasize estimation (Van de Walle et al., 2019, p. 345). The last one is that many learners face difficulty comparing complex fractions, such as 13/14 and 3/11 (Jarrah et al., 2022). The reduction to a common denominator method can overcome this.
Role of Cultural Diversity in Learning Fractions
Leveraging the diverse cultural, ethnic, and religious backgrounds of students plays a significant role in enhancing their experiences. While learning fractions, the teacher can incorporate culturally relevant examples in their teaching materials and activities. Secondly, educators can learn to acknowledge that learners are different and offer fraction-related materials in multiple languages if possible (Iversen, 2023). Other things they can do include encouraging students to share their cultural practices and perspectives, fostering a collaborative environment, involving students’ families, and incorporating cultural art forms. This will help make fractions more meaningful, relevant, and inclusive.
Conclusion
In conclusion, fractions play a significant role in mathematics and can be applied in various aspects of life. This discussion highlights the importance of fractions and how they can be taught more effectively through constructivist approaches. It has explored key fraction constructs, model issues, and misconceptions, offering a unique insight into this concept. By employing visual aids, hands-on experience, and real-life scenarios, students can gain a deep understanding of fractions.
References
Australian Curriculum, Assessment, and Reporting Authority (ACARA). (n.d.). Australian Curriculum.
Iversen, R. L. (2023). The convivial concealment of religion: Navigating religious diversity during meals in early childhood education–A Norwegian case. British Journal of Religious Education, 45(3), 263-276.
Jarrah, A. M., Wardat, Y., & Gningue, S. (2022). Misconception on addition and subtraction of fractions in seventh-grade middle school students. Eurasia Journal of Mathematics, Science and Technology Education, 18(6), em2115.
López-Martín, M. d., Aguayo-Arriagada, C. G., & López, M. (2022). Preservice elementary teachers’ mathematical knowledge on fractions as operator in word problems. Mathematics, 10(3), 423.
Martinez, S., & Blanco, V. (2021). Analysis of problem posing using different fractions meanings. Education Sciences, 11(2), 65.
Royal Australian Mint. (n.d.). Circulating coins.
Švecová, V., Balgova, M., & Pavlovičová, G. (2022). Knowledge of fractions of learners in Slovakia. Mathematics, 10(6), 901.
Top Drawer Teachers. (n.d.). Fractions.
Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. (2019). Elementary and middle school mathematics: Teaching developmentally. Pearson.
Wilkie, K. J., & Roche, A. (2022). Primary teachers’ preferred fraction models and manipulatives for solving fraction tasks and for teaching. Journal of Mathematics Teacher Education, 1-31.
Witherspoon, T. F. (2019). Fifth graders’ understanding of fractions on the number line. School Science and Mathematics, 119(6), 340-352.
Wu, H.-H. (2021). Misconceptions about the long division algorithm in school mathematics. Journal of Mathematics Education at Teachers, 11(2), 1-12.