Nurturing Fraction Sense

A paper from the publishing team at 3P Learning

Nurturing Fraction Sense

Introduction

Why are fractions so hard to learn?

What fraction of the shape is shaded?

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.

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.

Fractions are challenging because students have to work hard to adopt new rules that contradict rules they have learned about whole number. For example, is less than because the more 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 and . Often students view fractions as whole numbers placed over another whole. Pictures presented in most textbooks can reinforce this misconception.

1 3

2 33

4

1 2

A common representation of .3 4

A typical textbook question.

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.

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Nurturing 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:

Fraction sense means students will be better equipped to engage in rich tasks that stimulate and challenge their thinking even further.

reinterpret as 3 lots of one half

represent fractions using words, a variety of models, diagrams and symbols and make connections among various representations

reason that is bigger than only if the wholes are the same

understand that fractions are numbers where both the numerator and denominator need to be considered in order to ascertain their true value

name a fraction between and

sort fractions using benchmarks of 0, and 1

compare and order and by reasoning that is more than because is

eactly and is greater than that. is less than because is exactly and

is less than that. That means is greather than .

3 2

1 2

1 2

1 2

1 2

1 2

1 2

1 3

16 28

16 28

16 28

16 28

14 28

15 32

15 32

15 32

15 32

16 32

2 3

1 3

A closer look at fraction sense

How to nurture fraction sense

3. Focus on equivalence and improper fractions early onDoing 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 oftenNumber 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 modelsThis 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 contextsBy 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.

2. Work with unevenly partitioned areas and number linesReasoning 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 can only be depicted by 4 congruent pieces.1 4

Nurturing Fraction Sense

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.

References1. 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.pdf3. 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 authorNicola 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