35th ANNUAL CONFERENCE OF THE ATEE
Budapest, august 26th – 31st, 2010.
Development of concept of
division – from intuitive
models to division of fractions
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Maja Cindrić, Department for Teacher Education, University of Zadar, Zadar, Croatia
mcindric@unizd.hr
Irena Mišurac Zorica, Department for Teacher Education, University of Split, Croatia
irenavz@ffst.hr
What does we want for our children to
acquire by learning mathematics ?
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• Kilpatrick and others
(2001) :
• conceptual
understanding
• procedural fluency
• Strategic competence
• Adaptive reasoning
• Productive disposition
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Figure1. : Interwined Strands of Proficiency,
from Adding it up
What does we want for our children to
acquire by learning mathematics ?
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• Balance between conceptual understanding and
procedural skills
• ability to use a flexible application of knowledge
learned in appropriate situations
• combination of knowing the facts, knowledge of
procedures and conceptual understanding
• Students who memorized facts and procedures
without conceptual understanding often are not
sure when and how to use it so they know their
knowledge is very fragile (Bransford and others
1998)
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What does we want for our children to
acquire by learning mathematics ?
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• Well connected and conceptually grounded ideas
simply can be use in new situations (Skemp 1976)
• practice algorithms in mathematics, without
conceptual understanding are often quickly
forgotten or remembered incorrectly
• understanding of the concepts, without fluency in
the performance of algorithms, may present an
obstacle in solving problems
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Contemporary mathematics curricula
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• emphasizes the optimal balance between the
development of conceptual and procedural
knowledge
• many teachers are influenced by traditional
teaching, which emphasizes practicing algorithms
• teachers are aware of contemporary ideas, but do
not feel confident to change the way teaching …
• …or they don’t know how ?
• to be sure we conducted research on
understanding the concept of division
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We asked ourselves :
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• what division means for children, students
and mathematics teachers
• What means to develop conceptual
knowledge of division
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Children's intuitive knowledge of division
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• even young children, can solve many different
types of problem-solving tasks with direct
modeling of problem situations in the task
(Carpenter, Ansell, Franke, Fennema, Weisbeck,
1993)
• Children are insistent in using intuitive knowledge
despite the traditional methods explained by
teacher
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Children's intuitive knowledge of division
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• Mama sent Lucija to the
store and gave her 100
kuna, to buy two cakes,
and 3 packages of eggs.
Each cake cost 15 kunas,
a package of eggs, 11
kunas. Lucija wanted to
buy a chocolate egg,
which cost 3 kunas. Mom
told her that with the rest
money can buy what ever
she want. How many
chocolate eggs can Lucija
buy?
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Children's intuitive knowledge of
division
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• all basic arithmetic operations are associated with the
unconscious primitive intuitive model, which mediates in
search of arithmetic operations needed to solve a
mathematical problem (Fishbein 1985.)
• two intuitive models that children use when the situation
requires a division problem :
• partitive model and measurement (quotative) model
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Children's intuitive knowledge of division
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• On the table are 12 apples. I want to put apples into three
baskets, so that contains the same amount of apple. How
many apples are in each basket? "- Partitive division
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• On the table are 12 apples. I want to put apples in the
basket, so that in each basket contains three apples. How
many baskets will be filled with apples? "- Measurment
division
Children's intuitive knowledge of division
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• Fishbein and others argue that intuitive models can
impede, discourage or even prevent a child to
solve mathematical problem
• For 12 : 3 child is said : Grandfather has a 12
cookies and 3 grandchildren. How much cookies
will each grandchild get?
• For 3.21 : 0.75 child is said : Grandfather has a
3.21 cookies and 0.75 grandchildren. How much
cookies will each grandchild get?
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Definition of concept by Gerard Vergnaud
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• Concept is three-uple of three sets : C = (S,I,R)
– S: the set of situations that make the concept useful and
meaningful
– I: the set of operational invariants that can be used by
individuals to deal with these situations
– R: the set of symbolic representations, linguistic,
graphic or gestural that can be used to represent
invariants, situations and procedures.
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Class
Multiplication problem
Partitive division
Measurment division
Equal groups
3 children each have 4 oranges. How
many oranges do they have
altogether?
12 oranges are shared eqally among 3
children. How many does each
get?
If you have 12 oranges, how
many children can you give
4 oranges to?
Equals measures
3 children each have 4,2 liters of
oranges juice. How much
orange juice do they have
altogether?
12,6 liters of orange juice is shared
eqally among 3 children. How
much does each get?
If you have 12,6 liters of
orange juice, to how
many children can you
give 4,2 liters?
Rate
A boat moves at a steady speed of
4,2 m/s. How does it move in
3,3 secondes?
A boat moves 13,9 meters in 3,3
secondes. What is an average
speed in meters per second?
How long does it take a boat
to move 13,9 meters at
a speed of 4,2 m/s?
Measure conversion
An inch is about 2,54 cm. About
how long is 3,1 inches in
centimeters?
3,1 inches is about 7,84 cm. About
how many centimeters are
there in an inch?
An inch is about 2,54 cm.
About how long in inches
is 7,84 cm?
Multiplicative
conversion
Iron is 0,88 times as heavy as
copper. If a pice of copper
weights 4,2 kg, how much
does a piece of iron of the
same size weight?
Iron is 0,88 times as heavy as
copper. If a piece of iron
weights 3,7 kg, how much
does a piece of copper the
same size weight?
If equally sized piece of iron
and copper weight 3,7
kg
and
4,2
kg
respectively, how heavy
is
iron
relative
to
copper?
Part/whole
A colledge passed the top 3/5 of its
students in an exam. If 80
students did the exam, how
many passed ?
A colledge passed the top 3/5 of its
students in an exam. If 48
students passed, how many
students sat the exam?
A colledge passed the top 48
out of 80 students who
sat an exam. What
fraction of the students
passed?
Multiplicative change
A piece of elastic can be streched to
3,3 times its original lenght.
What is a lenght of a piece
4,2 meters long when is fully
streched?
A piece of elastic can be Stretched
to 3.3 times its original
length. When fully stretched
it Is 13.9 metres long. What
was its original length?
A piece of elastic 4.2 meters
long can be streched to
13.9 meters. By what
factor is it lengthened?
Cartesian product
If there are 3 routes from A to B,
and 4 routes from B to C how
many different ways are there
of going from A to C via B?
If there are 12 different routes from A to C via B, and 3 routes from A
to B, how many routes from B to C are there?
Rectangular area
What is a area of rectangle 3,3 m
long by 4,2 m wide?
If the area of rectangle is 13,9 m2 and the lenght is 3.3 m, what is the
width?
Product of measures
If a
A heather uses 3,3 kW per hour. For how long can it be used on 13,9
kWh of electricity?
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heather uses 3,3 kW of
electricity for 4,2 hours, how
many kWh is that?
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Problems situations for division of whole numbers
invented by elementary school children
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• for
the1101
study0001
consisted
of 135
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•
•
•
elementary school children
from 8 till 10 years old
to write three problem tasks for which solutions will
contain numerical expressions 12: 3, 45: 3 and 72: 12
How many children know to write a correct problematic
situation?
If the problem situation is the exact, a which classes of
situation students choose and which division model
If the child does not choose an adequate problematic
situation, where he/she make mistakes
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Problems situations for division of whole numbers
invented by elementary school children
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True
True
Fals
Fals
Empty
Partitive
Measurment
Partitive
∙
0
12:3
65,9
28,9
5,2
96,6
3,4
41
59,3
33,3
7,4
86,5
1,1
46,6
50,4
34,1
15,5
95,6
1,5
39,1
45:3
72:12
+
1
5,1
2
-
41
4
4,4
2,2
2,2
2,2
31,1
43,3
Problems situations for division of fractions invented
by elementary school children
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• subjects for the study consisted of 241 elementary school
children in 6th grade
• Students are asked to write one problem task for which
solutions will contain numerical expressions 12: 1.
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True
True
5,8
Fals
42,3
Empty
51,9
2
4
Fals
Partitive
Measurment
Partitive
∙
+
0
5,8
13,5
16,7
0
-
30,4
Problems situations for division of fractions
invented by elementary school mathematics
teachers
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• subjects for the study consisted of 122
elementary school mathematics teachers
• Teachers from different schools in the
southern Croatia (Split and Zadar County)
• teacher was required to write three problem
situations for three different division: 12: 3,
12: 12 , 12 : 34.
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Problems situations for division of fractions
invented by elementary school mathematics
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teachers
True
True
12:3
Fals
Fals
Empty
Partitive
Measurment
Partitive
∙
0
+
1
83,6
9
7,4
90,2
7,8
63,6
53,3
22,1
24,6
1,5
86,1
48,1
26
6,6
38,5
54,9
50
25
12,7
48,9
0
2
-
0
4
7,4
2,1
3,7
4,2
Problems situations for Division of Fractions
invented by students from teacher studies
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• Subjects for this study consisted of 173
prospective teachers from University of Zadar and
University of Split
True
True
12:3
Fals
1
Fals
Empty
Partitive
Measurment
Partitive
91,3
8,1
0
97,5
2,5
28,5
18,5
47,4
34,1
0
100
65,8
0,6
17,3
82
100
0
0
2
4
∙
+
-
7
0
21
3,7
2,4
3,7
43,3
3,3
10
Measurment model
Equal
gro
up
s
Equal
me
as
ure
s
Multipli
Rate
Partitive model
Total
c
a
t
i
v
e
c
o
n
v
e
r
s
i
o
n
Equal
gro
up
s
Equal
m
e
a
s
u
r
e
s
Multiplicati
ve
co
nv
ers
ion
Rate
Total
Rectang
ul
ar
ar
ea
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2nd
4
t
h
g
r
a
d
e
1
0,7
1,5
0
0
2,2
45,9
5,9
11,1
2
0,7
0
0
0
0,7
40
4,4
12,6
3
0,7
0
0
0
0,7
28,1
3,7
16,3
g
r
a
d
e
1
2,9
2,9
0
0
5,8
0
0
students
1
1,2
0,6
0,6
0
2,4
86,1
2
17,9
0,6
0
0
18,5
3
0
0
0
0
1
0
6,6
0
2
32
13,9
3
0
0
6th
teachers
0
62,9
1
0
57
0
48,1
0,7
2
4
1,5
1,5
0
0
0
0
2,3
0,6
0
89
0
0
0
0
0
0
0
0
0
0,6
0
0
0
6,6
67,2
5,7
2,5
0
75,4
1,6
0
0
45,9
0
0
0
0,8
0,8
6,6
0
1,6
1,6
0
0
2,5
0
2,5
1,6
0
0
Thank you !
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