designing interactive representations for learning fractions
DESCRIPTION
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.TRANSCRIPT
- 1. Designing interactiverepresentations for learningfractionsAlice Hansen, Eirini Geraniou &Manolis MavrikisInstitute 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 mustbe of equal size the parts, taken together, must be equal to thewhole the more parts the whole is divided into, thesmaller the parts become the relationship between the parts and the wholeis conserved, regardless of the size, shape ororientation of the equivalent parts16124> 4. A lens on part-whole 5. Methodology Design-based research methodology (Cobb etal., 2003) so dual purpose: Trial and improve Fractions Lab Develop our understanding of how Fractions Labsupports students conceptual understanding offractions 6. Method 32 Year 5 (9-10 year old) and 35 Year 6 (10-11year 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 LabOperations areaCheck your work-in-progress using the addition,subtraction or equivalence boxes. FractionsLabwont give you the answer, but it will help you alongthe way. RepresentationstoolbarSelect therepresentation youwant to use andstart creatingfractions!SymbolsNumber linesRegionsSetsLiquid measuresHelpReceive support and guidance withbuilt-in helpInteract with thefractionsMany options allowyou to manipulatethe fractionrepresentations.These include:Adding: watch ananimation as twofractions are addedwith the join toolSubtracting: watchcompare or takeaway fractionsPartitioning: Seehow fractions canbecome equivalentthrough partitioning 9. Fractions Lab tasksMake three fractions equivalentto 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 fractionsdifferently? What is your preferred rep and why? (Y6 only) Pre- and post- test (Y6 only) Show 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 1/4100%90%80%70%60%50%40%30%20%10%0%No. students showingrepresentation priorto FLNo. students showingrepresentation afterFL 14. RepresentationsRectangle50%PREFERRED REPRESENTATIONNumber lines12%Liquid measures35%No pref3% 15. RepresentationsRectangle Im used torectangleswhen Im beingteached It helped meunderstandpartitioningbest It was biggerNumber line I found it mucheasier than theothers It really helpedme tounderstand Ive used thembeforeJugs You can put ajug on a jug tosee if it is equal I can do myworking outeasier thannormal It lays it outmoreunderstandably 16. Representations General statements about representationsFractions can be represented in different waysTotal statements: 42 Specific mention of one or morerepresentationsYou can show fractions with liquidTotal statements: 31 17. Equivalence General statements about finding equivalenceEquivalence. How to find them by going up inmultiplesTotal statements: 17 Equivalent fractions givenTotal statements: 7 PartitioningTo partition instead of times by 2Total statements: 10 18. Addition and subtraction General statements about +/-You can add them together; It has made memore confident at subtractionTotal statements: 22 Mention of denominatorsBefore you add two fractions together youneed to make sure the denominators are thesameTotal statements: 19 19. Size General statements about sizeI can see the fractions and so I know how big theycan be.Total statements: 14 Relationship between denominator and size ofpieceHow many parts are used to create a fractionTotal statements: 11 Quantity or amountHelped me to understand how much it wasTotal statements: 15 20. Conclusions Students tend to use intuitive ideas and thetools in Fractions Lab challenge and provokethem to think conceptually about new andfamiliar reps Students may benefit from a wider diet ofreps Use of virtual manipulatives enables studentsto witness dynamic changes to fractions andhas the potential to enhance conceptualunderstanding 21. Next steps Analyse video and voice data to triangulatepresent findings Introduce and trial sets representation How do the reps (esp. Liquid Measures)support students fraction understanding? How does FL support students to bridge gapfrom additive reasoning to multiplicativereasoning? 22. Reference Cobb, P., diSessa, A., Lehrer, R., Schauble, L.(2003). Design experiments in educationalresearch. Educational Researcher, 32(1), 913.