According

to Dr. Catherine Bruce, Diana Chang and Tara Flynn, Trent University Shelley

Yearly, Trillium Lakelands DSB, on assignment with Ontario Ministry of

Education.

The mathematics education literature is

responding on finding the understanding of fractions which happened to be that

challenging on the student grasp of North America. Based on the fraction

concepts of (Lipkus, Samsa, & Rimer, 2001; Reyna & Brainerd, 2007), even

adults continue to struggle when fractions are important to their daily work related

tasks.

Fractions involve difficult-to-learn for

which students are present ongoing on pedagogical challenges in Mathematics

subject. The said difficulties begin early in the primary years according to Empson

& Levi, 2011; Moss 7 Case, 1999 and continue through middle school based on

(Armstrong and Larson, then into secondary and even tertiary education. These

challenges and misunderstandings that the students face in understanding

fractions persist into adult life and even into the present life. The science,

technology, engineering and mathematics (STEM) that is demanding for

considerable fractions knowledge; that it happened to prevent an advanced

knowledge about the other Mathematics topic without the present of knowing

fraction first. The said understanding of fraction can be implied in different

purposes like in medicine, which the implication of it is severe. The

mathematics education together with the research communities have working to

resolve the challenges that the student presented on the said learning of

fractions. The understanding of fractions needed to be underpinned by a large

willingness of wanted to learn and to understand the processes itself. It is

clear that without the strong foundation in fractions can eventually cause the

students to have a face a conflict on mathematics. However, this problem is complex

one and needed a long-term commitment to have gain a greater understanding on

what would going to do on how to support the students to build the said solid

foundation.

A fraction is a number which can tell us

about the relationship between two quantities. These two quantities provide

information about the parts, the units we are considering and the whole.

Moseley and Okamoto (2008) found that,

unlike top achievers, average and high achieving students are not developing

these multiple meanings of rational numbers, resulting in a student focus on

surface similarities of the representations rather than the numerical meaning.

Furthermore, Moseley (2005) demonstrated that students who were familiar with

both the part-part and part who interpretations had a deeper understanding of

rational numbers.

Fractions are difficult to learn because

they require deep conceptual knowledge of part-whole relationships (how much of

an object or set is represented by the fraction symbol), measurement (fractions

are made up of numbers that can be ordered on a number line) and ratios (Hecht,

Close & Santisi, 2003; Moss & Case, 1999).

The following specific challenges faced by

learners are discussed in this section: difficulties understanding and

representing fraction relationships, confusion about the roles of the numerator

and the denominator and the relationship between them, use of a ‘gap thinking’

approach, lack of attention to equivalence and equi-partitioning.

Students gasp and misconceptions are

powerfully revealed through their drawn representations of fractions, and

studies in this area provide evidence to suggest that the multitude of

representations used, some of which are potentially distracting representations,

do not help students build deep

understanding (Kilpatrick, Swafford, & Findell, 2001). Circular

representations are problematic because partitioning circles equally is more

difficult for odd or large numbers.

Students frequently conceive a fraction as

being two separate whole numbers (Jigyel & Afamasaga-Fuata’i, 2007) and

consequently apply whole number reasoning when working with fractions. Huinker

(2002), as cited in Petit et al., (2010) states that ‘students who can

translate between various fraction representations “are more likely to reason

with fraction symbols as quantities and not as two whole numbers” when solving

problems’. Students must also understand that the numerator and denominator

have different roles within the fraction and that the interpretations vary

depending on the role. Further confusion about the role of the numerator and

denominator arises with a premature introduction to fraction notation and/or

the inadvertent use of imprecise language.

Without the requisite conceptual understanding

such as the importance of equivalence, estimation, unit fractions, and

part-whole relationship, students struggle to complete calculations with

fractions. As referenced in Kong (2008, p. 246), Huinker (1998), Niemi (1996b)

and Pitkethlyn & Hunting (1996) all confirm that “learners seldom

understand the procedural knowledge associated with fractional operations such

as addition and subtraction” and this is strongly connected to their lack of

foundational understanding of the meaning and ways of thinking about fractions.

Hasemann (1981) provide several possible

explanations for why children find simple fractions so challenging, including:

1.) fractions are not obviated in daily life, but instead are hidden in

contexts that children do not recognize as fractions situations; 2.) the

written notation of fractions is relatively complicated; and 3.) there are many

rules associated with the procedures of fractions, and these rules are more

complex than those of natural numbers. Moss & Case (1999) agree that

notation is a challenge for students, but they also suggest several other

pedagogical complications; to begin, when rational numbers are first introduced

to students they may not be sufficiently differentiated from whole numbers,

neglecting the importance of the relationship that a fraction names (Kieren,

1995).

When we add or subtract fractions, we have

to find a common denominator, but not when we multiply or divide. And once we

get a common denominator, we add or subtract the numerators, but not the

denominators, despite the fact that when we multiply, we multiply both the

numerators and denominators, and when we divide, we divide neither the

numerators nor the denominators (Siebert & Gaskin 2006, p. 394).

These “rules” might make sense to those who

already conceptually understand fractions operations, but they do not help to

support students who are just learning how to work with operations that include

fractions. Unfortunately, students are often presented with wordy rules for

procedures, such as the example above, that are difficult to understand and get

conflated with definitions of what it means to perform an operation. To further

complicate matters, Foundations to Learning and Teaching Fractions: Addition

and Subtraction Page 19 of 53 spontaneous or invented strategies for adding and

subtracting fractions are typically discouraged, inadvertently discouraging

students from sense making (see Confrey, 1994; Kieren, 1995; Mack, 1993;

Sophian & Wood, 1997).