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# Designing interactive representations for learning fractions

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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.

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### Designing interactive representations for learning fractions

1. 1. Designing interactive representations for learning fractions Alice Hansen, Eirini Geraniou & Manolis Mavrikis Institute of Education, London Knowledge Lab,London
2. 2. Slide 2 of 18
3. 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. 4. A lens on part-whole
5. 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. 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. 7. Introducing Fractions Lab
8. 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. 9. Fractions Lab tasks Make three fractions equivalent to ½
10. 10. 2/6 = 1/12 (Year 5)
11. 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. 12. Findings • Four aspects emerged as lenses for analysis: – Representation – Equivalence – Addition and subtraction – Fraction size
13. 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. 14. Representations Rectangle 50% PREFERRED REPRESENTATION Number lines 12% Liquid measures 35% No pref 3%
15. 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. 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. 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. 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. 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. 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. 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. 22. Reference • Cobb, P., diSessa, A., Lehrer, R., Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.