In this presentation given at the BERA 2014 Conference in London we share how we have designed an exploratory learning environment (Fractions Lab) that allows students to interact with various fractions representations, add or subtract them and check their equivalence. Associated exploratory tasks challenge students to solve problems while the ‘epistemic affordances’ of Fractions Lab take advantage of their intuitive ideas but also challenge them to reflect on the feedback provided. The main objective is to challenge pre-conceived ideas of how fractions are represented and how Fractions Lab can create an environment for students to develop 'situated' abstractions about fractions. In the presentation we will identify some key design decisions (e.g. introducing explicit tools to encourage students to understand that there are underlying structures common to all representations) and discuss how they evolved from the literature and during design experiments, as well as how they impacted upon students' conceptual change. We conclude that students' interaction with Fractions Lab provokes them to think conceptually about fractions and to capitalise on their intuition, discouraging them from simply procedurally calculating an answer.

1. Designing interactive
representations for learning
fractions
Alice Hansen, Eirini Geraniou &
Manolis Mavrikis
Institute of Education,
London Knowledge Lab,London

2. Slide 2 of 18

3. A lens on part-whole
• For part-whole students need to understand:
– the parts into which the whole is partitioned must
be of equal size
– the parts, taken together, must be equal to the
whole
– the more parts the whole is divided into, the
smaller the parts become
– the relationship between the parts and the whole
is conserved, regardless of the size, shape or
orientation of the equivalent parts
1
6
1
24
>

4. A lens on part-whole

5. Methodology
• Design-based research methodology (Cobb et
al., 2003) so dual purpose:
– Trial and improve Fractions Lab
– Develop our understanding of how Fractions Lab
supports students’ conceptual understanding of
fractions

6. Method
• 32 Year 5 (9-10 year old) and 35 Year 6 (10-11
year old) students
• Visited each cohort once in June and July
• Each student used FL 15-30 mins
• ‘Reflection on my learning’ (All students)
• Pre-test / post test (Y6)

7. Introducing Fractions Lab

8. Introducing Fractions Lab
Operations area
Check your work-in-progress using the addition,
subtraction or equivalence boxes. FractionsLab
won’t give you the answer, but it will help you along
the way. Representations
toolbar
Select the
representation you
want to use and
start creating
fractions!
Symbols
Number lines
Regions
Sets
Liquid measures
Help
Receive support and guidance with
built-in help
Interact with the
fractions
Many options allow
you to manipulate
the fraction
representations.
These include:
Adding: watch an
animation as two
fractions are added
with the ‘join’ tool
Subtracting: watch
‘compare’ or ‘take
away’ fractions
Partitioning: See
how fractions can
become equivalent
through partitioning

9. Fractions Lab tasks
Make three fractions equivalent
to ½

10. 2/6 = 1/12 (Year 5)

11. Students’ written statements
• Reflection on learning (Y5 & Y6)
– Fractions Lab helped me to learn …
– How has FL helped you to think about fractions
differently?
– What is your preferred rep and why? (Y6 only)
• Pre- and post- ‘test’ (Y6 only)
– Show ¼ in as many ways as you can
– What do you know about fractions?

12. Findings
• Four aspects emerged as lenses for analysis:
– Representation
– Equivalence
– Addition and subtraction
– Fraction size

13. Representations: Show 1/4
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
No. students showing
representation prior
to FL
No. students showing
representation after
FL

14. Representations
Rectangle
50%
PREFERRED REPRESENTATION
Number lines
12%
Liquid measures
35%
No pref
3%

15. Representations
Rectangle
• I’m used to
rectangles
when I’m being
teached
• It helped me
understand
partitioning
best
• It was bigger
Number line
• I found it much
easier than the
others
• It really helped
me to
understand
• I’ve used them
before
Jugs
• You can put a
jug on a jug to
see if it is equal
• I can do my
working out
easier than
normal
• It lays it out
more
understandably

16. Representations
• General statements about representations
Fractions can be represented in different ways
Total statements: 42
• Specific mention of one or more
representations
You can show fractions with liquid
Total statements: 31

17. Equivalence
• General statements about finding equivalence
Equivalence. How to find them by going up in
multiples
Total statements: 17
• Equivalent fractions given
Total statements: 7
• Partitioning
To partition instead of times by 2
Total statements: 10

18. Addition and subtraction
• General statements about +/-
You can add them together; It has made me
more confident at subtraction
Total statements: 22
• Mention of denominators
Before you add two fractions together you
need to make sure the denominators are the
same
Total statements: 19

19. Size
• General statements about size
I can see the fractions and so I know how big they
can be.
Total statements: 14
• Relationship between denominator and size of
piece
How many parts are used to create a fraction
Total statements: 11
• Quantity or amount
Helped me to understand how much it was
Total statements: 15

20. Conclusions
• Students tend to use intuitive ideas and the
tools in Fractions Lab challenge and provoke
them to think conceptually about new and
familiar reps
• Students may benefit from a wider diet of
reps
• Use of virtual manipulatives enables students
to witness dynamic changes to fractions and
has the potential to enhance conceptual
understanding

21. Next steps
• Analyse video and voice data to triangulate
present findings
• Introduce and trial ‘sets’ representation
• How do the reps (esp. Liquid Measures)
support students’ fraction understanding?
• How does FL support students to bridge gap
from additive reasoning to multiplicative
reasoning?

22. Reference
• Cobb, P., diSessa, A., Lehrer, R., Schauble, L.
(2003). Design experiments in educational
research. Educational Researcher, 32(1), 9–13.