Jayasree Subramanian: Children's Reasoning Skills in Fractions
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Transcript of Jayasree Subramanian: Children's Reasoning Skills in Fractions
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Children's reasoning skills in Fractions A report based on a longitudinal study
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Subconstructs of Fraction
• ‘Fraction, ratio, proportion’ remains one of the most difficult area in school mathematics.
• There have been suggestion both in India and abroad that fractions be dropped from the primary school curriculum.
• However with Thomas Kieran’s work the notion of subconstruct of fraction and the need to bring in multiple subconstructs in designing curriculum has become clear.
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Part whole • ‘whole’ is divided into equal sized ‘parts’;
denominator counts the total number of parts and numerator counts the number of parts under consideration. Fraction is less than 1
3/5 3/5 • This is often the most commonly used
subconstruct to introduce fractions and also accounts for several difficulties- 3/2 or 3/5?
• How to represent 5/2? • 2/3+3/4=5/7
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Measure • Unit and sub-units of measurement. 1/n
denotes the size of the subunit and m/n is m pieces of size 1/n. Numerator can exceed the denominator.
• 1 • 1/2 • 1/3 • 1/4
• 1+1/4 or 5/4 • Particularly useful for introducing addition and
subtraction
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Quotient • equal share - numerator denotes the dividend
and denominator the divisor
6/2= 6!2 != 3!4
The work of Streefland and Nunes et al suggest that introducing fractions as share appeals to experiences, intuitive notions that children already bring to the classroom.
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Operator and Ratio Operator: Fraction operators as a function- 2/3 rd of something 1 ! of something
1/5 th of ". Operator interpretation of fraction is useful for introducing multiplication
Ratio: 3 spoons of sugar for every 4 cups of tea Distance travelled and time taken Equivalence of fraction for example can be best
explained by this subconstruct
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Complexity and Learning Trajectories Rational Number Reasoning is complex, and Yields to a Learning Trajectories strand analysis. • Permits us to respect complexity yet disentangle it; • Permits us to build from the cognitive resources
children • bring to school from informal settings; • Recognizes that the “logical structure of
mathematics” • and cognitive development in mathematics are not • identical; and • Permits us to view expertise as refinement of
approach • over time.
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Confrey-Learning Trajectories
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Our initial attempts • After initial attempts to introduce fractions
without appealing to subconstructs we realized that we need to fix the meaning of the fraction symbol m/n and we began by using the measure meaning and area model- rectangles
• This demanded precision in pictorial depictions – hard for both teachers and children
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Combining share and measure
• In collaboration with HBCSE we developed an alternative that introduced fractions as share and later brought in the measure meaning
• Short trials in 4 different locations • children in grade 5 and 6 • Comparison, equivalence and addition • Trials were successful • Can we begin from grade 3?
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Longitudinal study • Raipur girl’s primary school- Hoshangabad • Govt school, first gen learners, OBC, SC/ST, -
agricultural work
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Grade 3
• fraction as division and distribution • 12/3 was introduced as sharing 12 things
equally among 3 girls- divide, distribute and say how much each one got
• Often 12 rotis and 3 girls • children worked initially with material then
moved on to pictures, write the share • 5/2 was encountered- what to do with 1 extra? • How to write the share? Introduce the symbol
1/2
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Grade 3 • Fraction symbol m/n represented the share
situation ‘m rotis, n girls’ equal share • draw pictures • Name pieces – 1/3, " etc • write share, use = and + sign • #= "+1/4+1/4 or #= ! +1/4; • 7/3=2+1/3 • Were not sure if "+1/4+1/4= ! +1/4 • Could compare a few unit fractions
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Examples of children’s work
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Grade 4 • Used Jodogyan kit extensively • Introduced measurement activities- designed a
scale and subunits so that children had unit fractions from1/2 to 1/10 along with the unit scale
• They learnt how to figure out the name the subscale-
• Used it to measure and widea of iterationrite the length
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Grade 5 • Consolidate share and measure meanings • Learn to write a given fraction in many different
ways • 5+3/4=11 times 1/2+1/4 • =3+2+12/16 • = 5 times 1+6/8 • Learn to represent fractions on the number line • Equivalence • Comparison • Some addition
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Comparison tasks
S.N Nature of the task No. correct
resp out of 35 1 Comparing unit fractions Eg 1/11 and 1/7 34 (97.14%)
2 Comparing improper fractions (k+1/n , k+1/m) Eg: 10/3 and 13/4
29 (82.86%)
3 Comparing mixed fractions of the form Eg: 3+1/6 and 4+1/9
12 (34.29%)
4 Compare with ! Eg 5/7 and ! or 4/9 and 1/2 22 (62.26%)
5 m/n>1/2>p/q Eg: 7/13 and 2/7 12 (34.28%)
6 m/n and (m+l)/(n-r) Eg: 3/8 and 4/7 24.5 (70%)
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Nature of errors
• larger denominator means more girls to share with and hence smaller share.
• Many of the errors are due to focusing only on the denominator- i.e the size of the pieces without paying attention to the number of pieces
• 3/8= 1/8+1/8+1/8 and 4/7=1/2+1/14 more pieces in 3/8 so it is bigger
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Addition tasks
• Fewer N- distracter errors if for same denominators. O/w add numerators, add denominators.
• Use of recently learnt algorithm; one girl used equivalence
S.N Nature of the task No. correct resp out of 35
1 Unit fractions of the same denomination Eg: 1/7+1/7
20 (57.14%)
2 Non Unit fractions with same denominator Eg 2/7+3/7
19(54.28%)
3 Unit fractions with different denominators 12 (34.28%)
4 Any two fractions Eg:2/3+3/4 12 (34.28%)
5 Mixed fractions of the form k+1/n and l+1/n Eg: 2+1/5 and 3+1/5
7 (20.71%)
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Subtraction tasks
• Subtraction was not taught by us .
S.N Nature of task No. correct resp out of 35
1 Same denominator Eg: 9/11 – 5/11 9 (27.14%)
2 Integer – mixed fraction Eg 5- 2 ! 6 (16.42%)
3 !- unit fraction Eg: !- 1/3 3 (8.57%)
4 Any smaller fractions from a larger fraction Eg: 5/6 -1/3
1 (2.86%)
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Chhaya
More girls , so less share
4 3
7 7
7
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Poonam Rambarose
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Preeti Kamalkishore
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Ankita
"<1/2 1/8<1/7
Clearly More than 1/2
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Anjali Omprakash
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Nikita
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Anjali Vinod