Children's reasoning skills in Fractions A report based on a longitudinal study

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.

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

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

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.

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

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.

Confrey-Learning Trajectories

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

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?

Longitudinal study • Raipur girl’s primary school- Hoshangabad • Govt school, first gen learners, OBC, SC/ST, -

agricultural work

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

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

Examples of children’s work

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

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

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%)

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

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%)

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%)

Chhaya

More girls , so less share

4 3

7 7

7

Poonam Rambarose

Preeti Kamalkishore

Ankita

"<1/2 1/8<1/7

Clearly More than 1/2

Anjali Omprakash

Nikita

Anjali Vinod