Childrens insights and strategies in solving fractions problems

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Childrens insights and strategies in solving
fractions problems

• Terezinha Nunes
• Oxford Brookes University
• ESRC TEACHING AND LEARNING RESEARCH PROGRAMME

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With the collaboration of

• Peter Bryant
• Ursula Pretzlik
• Daniel Bell
• Deborah Evans
• Joanna Wade
• Support team
• Julia Carraher

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What are fractions?

• People tend to define fractions by the particular
  mathematical representations used ¼ or 0.25
• In order to understand and promote childrens
  reasoning, we need to consider the concept of
  fractions from other perspectives as well
• We need to think about the most important aspects
  of the logic of fractions and the situations in
  which fractions are used

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The logic of fractions

• Fractions, like natural numbers, involve the idea
  of equivalence (1/2 2/4 3/6 etc)
• They also involve the idea of order (1/2 gt 1/3 gt
  1/4 etc.)
• Why might these ideas be difficult?

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Difficulties in understanding equivalence

• Differentiating logic from perception
• Same fraction, identical wholes, different cuts
  is the quantity the same?
• The role of language
• Different number labels indicate the same
  quantity
• The same number may indicate different amounts

1/2 2/4
1/2 of 8 1/2 of 10
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Difficulties in understanding asymmetrical
relations (order)

• If the fractions have the same denominator, the
  larger the numerator, the larger the value
• If the fractions have the same numerator, the
  larger the denominator, the smaller the value
• This means that for natural numbers 1lt2lt3 etc but
  for fractions 1/2 gt 1/3 gt1/4 etc

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The importance of problem situations

• For whole numbers, the meaning of the number has
  an effect on how well the children can solve
  problems
• Mary had 7 fish 3 of them died how many does
  she have now?
• Mary has 7 fish Paul has 4 how many fewer does
  Paul have than Mary?

easy
difficult
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Fractions and the different situations

• A part-whole relation (cut a continuous quantity
  into 4 parts, take one part you have ¼) 1 and 4
  refer to parts

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Fractions and the different situations

• A quotient in a division (one pie shared by four
  children 1 divided by 4 is ¼) 1 is the number
  of pies and 4 is the number of children ¼ is
  also each girls share

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Fractions and the different situations

• An operator (¼ of 24 fractions of discrete
  quantities)divide into 4 groups, take 1 group

John lost ¼ of his marbles
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Fractions and the different situations

• An intensive quantity (¼ concentrate, ¾ water )

• An intensive quantity
• (¼ probability of drawing a white marble)

A relation the size of the whole does not matter
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• Our aim was to find out
• How do these situations affect childrens rate of
  success in solving problems?
• Which strategies do children use to solve
  problems?

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Quantitative results from two studies

• A survey of childrens performance in a fractions
  test (adapted from Brown et al. thankful for
  permission) carried out in Oxford
• An experiment comparing part-whole and quotient
  situations with stricter controls

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Survey methodology

• Schools covered a variety of socio-economic
  backgrounds
• Slides were projected with figures the children
  had booklets with the same figures the
  researcher gave instructions orally, no reading
  was required drawings contained the necessary
  information to avoid memory demands
• Survey was not designed for this purpose but
  contains questions that allow for comparisons

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Equivalence of fractions in part-whole situations
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Shade 2/3 of these shapes in b, c and d this
means shade the equivalent fraction
c
b
d
Y4 a 78 b 35 c 25 d 23 Y5 a 91 b56
c 48 d 44 a ns others p.05
Y5 perform at ceiling on 2/3 but only about half
of those succeeded in shading equivalent fractions
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correct Fraction unpainted 8-9 years 27 9-10
years 45
correct Fraction painted 8-9 years 32 9-10
years 51
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10
Peter cuts into 4, eats 2
Alan cuts into 8, eats 4
Peter eats more
correct 8-9 years 40 9-10 years 74
Alan eats more
Both will eat the same
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Equivalence of fractions in division situations
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11
Y4 64 Y5 96 p.001
Same Different
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3
Each girl eats as much as each boy
correct 8-9 years 76 9-10 years 87
Each girl eats more than each boy
Each boy eats more than each girl
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Proportion of correct responses by situation
Difference in performance in the two situations
is significant.
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An experimental study in Portugal (Mamede, Nunes
Bryant, 2005)

• The survey comparisons did not control for
  content and number used
• In this experiment, the problems used the same
  content (cutting chocolates, pizzas etc) and the
  same numerical values
• Children were randomly assigned either to
  part-whole or quotient situations
• Children solved equivalence and ordering problems
• At the end, they were taught how to represent
  fractions

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Participants

• 80 children (40 aged 6 and 40 aged 7 1st and
  2nd years at school)
• Half assigned to part-whole and half to quotient
  situations randomly
• Children had not received instruction on
  fractions yet

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10
Peter cuts into 4, eats 2
Alan cuts into 8, eats 4
Peter eats more
Alan eats more
Both will eat the same
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7
Each boy gets the same as each girl
Each boy gets more than each girl
Each girl gets more than each boy
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15
Each girl eats the same as each boy
Each girl eats the more than each boy
Each boy eats the more than each girl
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15
Armando cuts into 3, eats 1
Elsa cuts into 4, eats 1
Armando eats the same as Elsa
Armando eats the more than Elsa
Elsa eats the more than Armando
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Number of correct responses by task and situation
(maximum 6)
Differences are statistically significant
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Conclusion

• Children seem to have better insight into the
  equivalence of fractions in quotient problems
  than in part-whole problems
• They seem to be better able to understand the
  inverse relation between the denominator and the
  size of the fraction in quotient than in
  part-whole situations
• They also learn more easily how to name fractions
  in quotient than in part-whole problems
• Strategy analysis still being carried out

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Previous studies

• Analyses of childrens judgements of fractions
  have often concentrated on part-whole and
  numerical problems without reference to a meaning
• Empson used quotient situations but the children
  had learned fractions in part-whole situations
  and she did not observe different strategies

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Previous studies

• Strategies identified
• Partitioning children use concrete materials or
  drawings to represent the fractions and attempt
  to make judgements
• Children base their judgements on numbers
• Consider either the denominator or the numerator
  without considering the other value
• Consider both the numerator and the denominator

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Qualitative results from a micro-genetic study

• Teaching sessions used to investigate the
  childrens reactions to different cues
• Children in two or three age levels
• Year 3 mean age 7y6m
• Year 4 mean age 8y6m
• Year 5 mean age 9y6m
• Five different schools in Oxford

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Group discussion during teaching sessions

• Fractions as division using sharing
• Data from
• 3 small group sessions of about 45 minutes
• 3 different researchers worked with a total of 13
  children
• The children were always solving problems
  (actively engaged in reasoning)
• Children produced individual answers and
  discussed these in the groups

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The role of the researcher/teacher

• The researcher helped the children with the
  mathematical conventions
• The researcher gave hints to direct the children
  to use either a partitioning strategy or a
  sharing strategy (i.e., strategies that could be
  demonstrated with drawings) this allowed us to
  see how the strategies affected the childrens
  responses

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Introducing fractions through sharing situations
(problem taken from Streefland)
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Examples and comments

• The drawings showed that the children did not
  rely on perceptual comparisons they used
  correspondence strategies to define the divisions
• They also provided different written answers in
  one group two children wrote three quarters,
  one child wrote ¾ and a fourth child wrote ¼ ¼ ¼
• They all agreed that their answers meant the same
  thing

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• Illustrative dialogues (DE researcher)
• K I think its three quarters .
• DE You think its three quarters each, and
  youve written three over four, three quarters.
  How did you work that out ...
• K I split them into half first and that gave me
  six halves, and then I split one bar into
  quarters, on one of the bars.
• DE So you have two bars split in half and one
  bar split into quarters. Then what did you do?
  How did you work it out from there?
• K I took three of the quarters for one girl and
  that left a quarter (now indicating that the
  first chocolate could be divided into quarters
  but not actually drawing this), andI split them
  into quarters as well (but this is not shown in
  the drawing).

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An equivalence problem posed by one child
P writes ¾ and then estimates that this must be
the same as 2/3
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TN Researcher

• TN How did you do yours, P?
• P They each got two thirds so he gets
• TN Two thirds. How did you know it was two
  thirds?
• P Because two thirds is the same as a three
  quarters.
• TN Two thirds is the same as three quarters.
  Lets try it out.
• TN instructs the children to use a partitioning
  strategy to see whether the fractions are
  equivalent
• TN Here, do another bar here, divide it in four.
  Lets have a look. And do another one, just the
  same underneath, and divide it in three. Is it
  going to be the same, three quarters and two
  thirds?

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A
H tries several drawings but also gives up. They
cant provide an answer.
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• TN then directs the children to use a sharing
  strategy
• TN See how that would work out, if you were
  sharing the chocolates. So if you had three
  chocolates divided in thirds and you were sharing
  them for four people, would that work as two
  thirds for each?

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H One person would get more. TN Show me how.
Im not quite sure how. H Each one has one third
first and then they get two thirds and then
someone will get three thirds and then someone
else will get two thirds. If they all get equal
then there will be one left over.
P Its not the same. I thought one third was the
same as one and a half quarters. P goes back to
his work and crosses out 2/3
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Comments

• When using a partitioning strategy, the children
  rely on perceptual comparisons. The strategy is
  difficult to implement, as they cannot draw or
  cut with sufficient precision to trust their own
  demonstrations.
• When they use a sharing strategy, the use of the
  logic of correspondences makes the perceptual
  comparison irrelevant. The drawing supports
  logical reasoning rather than replaces it.

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6 children went to a pizzeria and ordered 2
pizzas.
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JW researcher

• JW What fraction of the first pizza do they
  receive?
• M One sixth.
• JW Why is that?
• M Because there are six children so they split
  the pizza in sixths no marks were made on the
  drawing.
• JW If they get one sixth from that one and one
  sixth from that one, how many sixths do they have
  altogether?
• St Two sixths.
• JW if the waiter brought the pizzas at the
  same time, how would they share them
  differently? What are the two ways that they can
  share it out?
• G They can share it in thirds.
• ST Those get a third from that one, and those
  three get a third from that one.

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Is one third the same as two sixths? Arguments
used by children in three different groups

• C (in TNs group) Because it wouldnt really
  matter when they shared it, because when they
  shared it in three, those three get it and that
  pizza is gone, and those three share this, and
  that pizza is gone. When they shared it at the
  same time, they share it fairly and the pizzas
  are gone.
• This argument seems to be based on the idea of
  exhaustive and fair division of the same whole
  done in different ways produce equivalent shares
  the child indicates the correspondences
  throughout without making drawings. Most children
  used this reasoning.

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Is one third the same as two sixths?

• H (in TNs group) Because one third is a third
  of three and two sixths is a third of six.
• This justification is based on numerical
  relations without reference to the context.
• C (in DEs group) I put it the pizza into
  thirds and I put the girls in half her gestures
  indicated that she separated half of the pizzas
  i.e., one pizza to each side and half of the
  girls i.e., 3 girls to each side if you apply
  the same operation to the numerator and the
  denominator, the fraction does not change.

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• P (in TNs group) I can explain in another way.
  Theres two sixes, add two sixes three times to
  make six sixths. With one third, you need to add
  one third three times to make three thirds. They
  are the same.
• This is a logical argument if you add the two
  fractions three times to make the same total,
  then the two fractions are the same if aaax
  and bbbx then ab. P is unsure about the
  conventions for denoting addition and uses two
  signs, then crosses out one. He writes 6 and 3
  but says six sixths and three thirds.

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Some pleasant surprises

• Some children went well beyond our expectations.
  We thought the problems might become repetitive
  but some children spontaneously used this
  repetition to extend their reasoning.

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The children proceeded to cut the sixths and see
what fractions they would get through reasoning,
without counting. They used the drawing as a
support (but not as the basis of comparison) up
to drawing 24 lines. After that they just
reasoned that 2x the number of parts meant each
got 2x more. They wrote 8/12, 16/24, 32/48, 64/96
and 128/192
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Conclusions

• A systematic investigation of how children
  understand equivalence of fractions in these two
  situations shows the different insights that
  children bring to understanding the equivalence
  of fractions
• Arguments in part-whole situations are based on
  perception in quotient situation situations,
  they are based on the logic of division
• Other situations should be investigated to extend
  our understanding (the use of fractions to
  represent intensive quantities fractions as
  operators)

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Assessing the impact of these interventions

• One study with small groups outside the
  classroom groups run by researcher
• Second study in the classroom
• Teacher presented the problems, children answered
  individually, agreed on group answers and then
  presented the work to the group
• Comparison group received regular instruction
  during the Numeracy Hour
• Problems included inverse questions 36 children
  went to a pizzeria each child received 1/3
  pizza how many pizzas did the teacher order?

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Results for the Whole Class Intervention
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Conclusions

• Quotient situations promote the use of strategies
  that do not seem to emerge in part-whole
  situations
• Pupils are able to engage in problem solving
  about fractions from the first day of instruction
• Learning promoted in this environment did not
  show decay
• It seems that teachers in Taipei use quotient
  situations what strategies do they observe and
  what strategies to they promote?

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