Nurturing Fraction Sense
A paper from the
publishing team at 3P Learning
Nurturing Fraction Sense
Introduction
Reasoning and sense making are integral in developing conceptual understanding in all maths
topics, but none more so than the topic of fractions.
Teaching and learning fractions is a notoriously difficult undertaking, yet understanding fractions is
absolutely critical to overall success in mathematics. Before students move to fraction computations,
it is vital that they are presented with opportunities to develop a deep and flexible understanding of
fractions so that they do not resort to blindly following rules and procedures. In short, they need to
develop fraction sense.
Like number sense, fraction sense refers to the use of various strategies grounded in the conceptual
ability to reason and generalise as opposed to the reliance on procedural understanding. In order
to be able to develop and deepen their understanding about fractions, children need to explore
many representations and uses over a significant period of time. This post will look at some essential
understandings that underpin fraction sense.
Why are fractions so hard to learn?
Fractions are challenging because students have to work hard to adopt new rules that contradict
rules they have learned about whole number. For example, 1 is less than 1 because the more
3
2
pieces that something is divided into, the smaller the pieces, and therefore the smaller the fraction.
However, this generalisation only works for unit fractions – it is not so helpful when comparing 2
3
and 3 . Often students view fractions as whole numbers placed over another whole.
4
Pictures presented in most textbooks can reinforce this misconception.
For example:
Children will see 3 out of 4 circles coloured
rather than a relationship. Alternatively, students
will see that fractions are shapes because they
associate fractions with colouring and counting
parts of shapes.
Another reason that fractions are so difficult to
teach and to learn is because they describe a
relationship, rather than a fixed amount, which
also depends on the context.
Fractions can be used to describe part of a
whole, part of a collection of objects, part of
a number and a point on a number line. They
can also represent a measure or be used as a
ratio. Furthermore the representation of fractions
changes across contexts: fractions can be
represented as area models, number lines, set
models and ratios. Clearly to avoid frustration
and build conceptual knowledge, fraction sense
is vital and therefore needs to be developed
deliberately and gradually.
A common representation of 3 .
4
What fraction of the shape is
shaded?
A typical textbook question.
Nurturing Fraction Sense
A closer look at fraction sense
Students with fraction sense can access and apply mental images by using models, benchmarks,
and equivalent forms to judge the size of fractions. Fraction sense shows the student possesses a
deep and flexible understanding of fractions.
Some examples of fraction sense are the ability to:
reinterpret
3
as 3 lots of one half
2
represent fractions using words, a variety of models, diagrams and symbols and make
connections among various representations
reason that 1 is bigger than 1 only if the wholes are the same
2
3
understand that fractions are numbers where both the numerator and denominator need
to be considered in order to ascertain their true value
1
2
name a fraction between 3 and 3
sort fractions using benchmarks of 0,
1
and 1
2
16
15
16
1
14
and
by reasoning that
is more than
because
is
28
32
28
2
28
1
16
15
1
16
1
eactly and
is greater than that.
is less than because
is exactly and
2
28
32
2
32
2
15
16
15
is less than that. That means
is greather than
.
32
28
32
compare and order
Fraction sense means students will be better equipped to engage in rich tasks that stimulate and
challenge their thinking even further.
How to nurture fraction sense
Here are some rich experiences that lead to essential understandings that underpin fraction sense:
1. Provide lots of tasks that require partitioning and iterating
The most important and meaningful way for a teacher to
introduce and immerse students in the idea of a part-whole
relationship is to spend time partitioning and iterating.
Partitioning is taking a whole and dividing it into equal
pieces, while iterating is making copies of a unit and putting
them together to make a whole. The power of these two
actions is that they can build fraction sense even before
fraction notation is introduced and address misconceptions
before they set in. For example, through partitioning and
iterating students will realise that all pieces associated with
the same unit fraction must be the same size. Improper
numbers and mixed numbers are more readily understood
as they can be interpreted as multiples of a unit fraction.
Most crucially, images of partitioning and iterating provide
opportunities for students to work with changing wholes and
support their understanding that a fraction is not the name
of a part but a relationship between the part and the whole.
1
3
1
Nurturing Fraction Sense
2. Work with unevenly partitioned areas and number lines
Reasoning about unevenly partitioned shapes is another way to have students engage more deeply
with the relationship between the part and the whole and further bolster fraction sense. It can
obliterate the misconception that 1 can only be depicted by 4 congruent pieces.
4
3. Focus on equivalence and improper fractions early on
Doing this helps students to understand that there are multiple ways to name and represent one
quantity and that fractions are actually numbers. Providing a variety of interactive visual models to
work with helps make connections between different representations of fractions.
4. Use number lines often
Number lines make it easier for students to understand that between any two points on a number
line there are infinite fractions and decimals. They show students how fractions and decimals relate
and also equivalency between mixed numbers and improper fractions.
Furthermore, benchmarks can be used on number lines which helps students compare fraction
sizes.
5. Provide a variety of interactive visual models
This was mentioned in point 3 for making connections between different representations of
fractions. However, it is also important for comparing fractions and making sense of fractions as
operations.
6. Provide opportunities to apply their fraction sense in a variety of contexts
By encouraging students to share and discuss their ideas in new or non-routine contexts, they are
able to refine their thinking and gain new insights.
The topic of fractions is a complex area for both teachers and students. This post looked at why
fractions are so difficult and that in order for students to develop a deep and flexible understanding
of fractions, they need fraction sense.
Finally, some experiences and opportunities that should be used to nurture fraction sense were
outlined.
References
1. Siebert, D & Gaskin, N 2006 “Creating, naming and justifying fractions”, Teaching children mathematics, www.nctm.org, vol 12, issue 8, p 394.
2. Siemon, D “Partitioning - the missing link in building fraction knowledge and confidence”, July 2004, https://www.eduweb.vic.gov.au/edulibrary/
public/teachlearn/student/partitioning.pdf
3. Parrish, S 2011 “Number Talks Build Numerical Reasoning”, Teaching children mathematics, www.nctm.org, October 2011, vol 18, issue 3, p 198.
4. Assessment Resource Books, Fractional thinking concept map, http://arb.nzcer.org.nz/supportmaterials/maths/concept_map_fractions.
php#Partitioning viewed March 2014.
5. Fennel, F 2009 “Fraction Sense! Why? Fractions are Foundational!” National Council of Teachers of Mathematics presentation, 16 July 2009 Brown
Convention Center Houston, Texas, http://ffennell.com/presentations/FractionsCAMTJuly162009.pdf.
6. Clarke, D, Roche, A & Mitchell, A 2008 “Ten Practical Tips for Making Fractions Come Alive and Make Sense”, Mathematics teaching in the middle
school, www.nctm.org, vol 13, issue 7, p 372.
7. McNamara, J & Shaughnessy, M 2010 Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense, Grades 3–5, Scholastic Inc.
About the author
Nicola Herringer is a member of the award-winning educational publishing
team at 3P Learning and heads up Primary Year Publishing. With over 10 years
experience in education, Nicola is fascinated by how kids learn math(s)
–though bored by herbal teas…
Follow Nicola on Twitter @NicolaHerringer