According adults continue to struggle when fractions are important

to Dr. Catherine Bruce, Diana Chang and Tara Flynn, Trent University Shelley
Yearly, Trillium Lakelands DSB, on assignment with Ontario Ministry of

     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

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     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

     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,

     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).