There can be various reasons why a learner might face challenges with fractions, including difficulty understanding the concept, complex operations, lack of visual representation, and more. These difficulties can hinder with the student’s potential and understanding of the concept. Hence, in this article, we will debunk some common misconceptions about fractions that students may develop and why they struggle with them

Fraction Folly: Clearing up common misconceptions

Fractions are an important concept in mathematics, however, students may have misconceptions that can lead to confusion and misunderstandings while solving problems. Some of the common misconceptions include

1. Leaving the denominator unchanged in fraction addition and multiplication problems. 

A common misconception small children make when working with fractions is that the denominator does not need to change when adding or multiplying fractions. In reality, the denominator is an essential part of a fraction and needs to be considered when adding or multiplying fractions. When adding fractions, the denominators must be the same in order to add the fractions together. When multiplying fractions, the numerators are multiplied together and the denominators are multiplied together

2. Failing to find a common denominator when adding or subtracting fractions, unlike denominators. 

it is not necessary to find a common denominator when adding or subtracting fractions with unlike denominators. In reality, finding a common denominator is a crucial step when adding or subtracting fractions with unlike denominators. The denominator of a fraction represents the total number of parts in the whole, and in order to add or subtract fractions together, they must have the same total number of parts.

3. Cross multiplying instead of row multiplying

One of the most common misconceptions is that students will cross multiply instead of row multiply when working with fractions. This is often because students do not understand the reasoning behind why we use row multiplication instead of cross multiplication. To avoid this misconception, it is important to not just teach students the algorithm for row multiplication but also to explain the reasoning behind it.

4. Multiplying fractions by whole numbers

Sometimes students also multiply both the numerator and denominator when working with fractions and whole numbers. This is because students may not understand why we only multiply the numerator by the whole number.

5. Whole numbers vs. non-whole number rational numbers

Students may believe that whole numbers are fundamentally different from non-whole number rational numbers. An example of this would be believing that 2 is different from three-fifths. It is important to help students understand the relationship between whole numbers and non-whole number rational numbers.

6. Failing to understand the invert-and-multiply procedure

Other misconceptions include treating numerators and denominators as separate whole numbers, failing to find a common denominator when adding or subtracting fractions with unlike denominators, believing that only whole numbers need to be manipulated in computations with fractions greater than 1, leaving the denominator unchanged in fraction addition and multiplication problems, and failing to understand the invert-and-multiply procedure for solving fraction division problems. For the same, teachers can indulge kids in some fraction-dividing games.

7. Fractions with larger denominators are always larger:

Fractions are often compared by finding a common denominator, but this is not the only method to compare them. Fractions can also be compared by cross-multiplying or converting them to decimals. For example, 3/4 is larger than 1/2 by cross multiplying and converting them to decimal but it is smaller than 9/8 by comparing with a common denominator. Students can also explore some multiplying fractions games for better understanding. 

8. The numerator and denominator of a fraction must be real numbers: 

This is not always the case as numerators and denominators can be any real numbers. For example, the fraction 3.5/2 is a valid fraction, even though the numerator and denominator are not whole numbers.

Why do students struggle with fractions?

Students struggle with fractions for many reasons. The first one is that fractions make students think hard about whole numbers^[1] and the division of these numbers can be challenging to grasp. With the introduction of a concept like fractions, the students need to establish an understanding where they can establish a relationship between the numerator and the denominator, which can often turn out as a struggle for many students as they might have to perform addition, subtraction, multiplication, and division at various steps. 

Since these steps come with their own rules and regulations, they can be complex. Lastly, fractions can be abstract, and connecting the visual representation^[2] of a fraction to its conceptual understanding can also be a bit of a task hence it is common that students might face misunderstandings and challenges, especially when they are holy and solely dependent on rote memorization of procedures.

Furthermore, fractions can also become tough for students who have learning disabilities, like dyscalculia. 

Conclusion

It is essential to teach students algorithms for working with fractions and help them develop a conceptual understanding of the underlying principles behind these algorithms. This will help them avoid common misconceptions and develop a deeper understanding of fractions.

As students struggle with fractions for various reasons, it is essential for educators to provide students with multiple teaching methods, including visual aids and hands-on activities, to help them better understand and apply the concepts of fractions. Additionally, providing students with ample practice opportunities and real-world examples can help them see the relevance and importance of fractions in their everyday lives.

Reference

1. Gabriel, F., Coché, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2013). A componential view of children’s difficulties in learning fractions. Frontiers in Psychology, 4. https://doi.org/10.3389/fpsyg.2013.00715
2. Singh, Parmjit & Hoon, Teoh & Md Nasir, Nurul & Cheong, Tau Han & Rasid, Syazwani & Hoong, Joseph. (2021). Obstacles Faced by Students in Making Sense of Fractions. The European Journal of Social & Behavioural Sciences. 30. 3282-3299. 10.15405/ejsbs.287. 

Manpreet Singh

An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having  just a simple math fear. This is still an underresearched and understudied subject. I am also the founder of Smartynote -‘The notepad app for dyslexia’, 

The post 8 Common Misconceptions About Fractions appeared first on Number Dyslexia.

In math, every digit in a number has a place value. Place value can be defined as the value represented by a digit in a number on the basis of its position in the number. For example, the place value of 7 in 3,743 is 7 hundred or 700. However, the place value of 7 in 7,432 is 7 thousand or 7,000. Here, we can see that even though the digits are the same in both the numbers, their place value changes with the change in their position. 

Understanding the place value of digits in numbers helps compare numbers. It also helps in writing numbers in their expanded forms.

Through this article, let us have an insight into the misconceptions that can occur while learning about place value.

Place value: A concept inescapable

Why is place value so important? Learning place value is essential because it provides the foundation for regrouping, multiple-digit multiplication, and more in the decimal system, as well as a starting point for the understanding of other base systems. Almost all mathematical concepts build on the understanding of Place Value making it one of the key concepts in mathematics. 

Math is all about patterns, and our number system is based on patterns of ten. Once a child truly understands this, they can work with any number, including negative numbers and including decimals.

Place value finds significance in almost every aspect of daily life. All these factors make learning place value important. 

Place value misconceptions

There is an interplay of various factors in understanding place values. If a child is struggling with place value it means that there exists a misconception in either of these.

1. Base 10 structure

Hindu-Arabic system uses symbols for the numbers 1 to 9 and then a place value system with a place holder of 0. Using these ten symbols we are able to represent large numbers just by their value in different positions. It is very difficult for the child to initially understand how the numbers change from 9 to 10. The child may not be able to put zero and one in the right places. 

2. Zero 

Learners need to recognize 0 as a label for an empty set, or nothing, as well as a placeholder for numbers in our base-10 structure. This might seem very confusing for a child. Children tend to get confused and are in a fix about what to denote the empty ‘place value’ with.

3. Partitioning 

Partitioning involves separating out numbers so that the value of each digit can be seen, for example, 385 = 300 + 80 + 5.  The child may denote this as 3+8+5. Here, Multi digits are seen as digits independent of place value by the child, which is a misconception. – 

4. Transposition

The child misapplies the rule for reading numbers from left to right (this difficulty is often caused by teen numbers) as in 71 could be read as 17; 41 as 14.

5. Exchange

This is related to the value and position of each digit. Ten in any ‘place’ in a number can be exchanged for one in the next place to the left, so, for example, 10 hundred can be exchanged for 1 thousand. Conversely, one in any ‘place’ can be exchanged for ten in the next place to the right. Kids generally get confused about this concept and end up writing the wrong place value of a digit.

Can misconceptions make the concept harder for young learners?

The biggest misconception about Math is that the concepts are not applicable in real life. But this is a myth. Preconceived fears and misconceptions make learning less enjoyable and induce stress. 

Other misconceptions like Math is difficult and one needs an inborn talent for it has always ruled the minds of children and made the subject intimidating. 

Debunking these myths could play a large role in furthering math education. Any one of these misconceptions could stop a student from embracing the subject and possibly end their attempt at pursuing a career in it. When studying math it can be easy to get frustrated. Math concepts can be hard to understand at first and often it just takes a little bit of time and effort to get a student to the point of understanding. 

Concepts like place value need a thorough understanding of numbers and counting. Although it might seem tricky at first, the misconceptions that surround it can make it seem more difficult to master. Activities in counting or games help foster an enjoyable learning atmosphere.

Conclusion

To wrap up, a misconception is the result of a lack of understanding or in many cases the misapplication of a rule‟ or mathematical generalization (Spooner, 2002);

 Any misconception in learning has to be tackled as it causes a gap in knowledge. It becomes imperative to overcome this in order to proceed with the learning. 

Misconceptions in mathematics are inevitable and often predictable. But they can be used to support deep learning by helping students recognize their misconceptions and revise their thought processes. These Misconceptions are as common as any other phenomena in real-life situations. Just about any concept, regardless of how well it is taught, could be misunderstood.

Misconceptions are therefore a problem with understanding, not with fluency or memory.

Manpreet Singh

An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having  just a simple math fear. This is still an underresearched and understudied subject. I am also the founder of Smartynote -‘The notepad app for dyslexia’, 

The post 5 Common Place Value Misconceptions appeared first on Number Dyslexia.