Development of concept of division – from intuitive models to division of fractions

$Page 1: Development of concept of division – from intuitive models to division of fractions$
42510011 0010 1010 1101 0001 0100 1011

Development of concept of division – from intuitive

models to division of fractions

Maja Cindrić, Department for Teacher Education, University of Zadar, Zadar, [email protected]

Irena Mišurac Zorica, Department for Teacher Education, University of Split, [email protected]

35th ANNUAL CONFERENCE OF THE ATEEBudapest, august 26th – 31st, 2010.

4251

0011 0010 1010 1101 0001 0100 1011

What does we want for our children to acquire by learning mathematics ?

• Kilpatrick and others (2001) :

• conceptual understanding

• procedural fluency • Strategic competence • Adaptive reasoning • Productive disposition

Figure1. : Interwined Strands of Proficiency, from Adding it up

4251

0011 0010 1010 1101 0001 0100 1011

• 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)


4251

0011 0010 1010 1101 0001 0100 1011

• 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


4251

0011 0010 1010 1101 0001 0100 1011

Contemporary mathematics curricula

• 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

4251

0011 0010 1010 1101 0001 0100 1011

We asked ourselves :

• what division means for children, students and mathematics teachers

• What means to develop conceptual knowledge of division

4251

0011 0010 1010 1101 0001 0100 1011

Children's intuitive knowledge of division

• 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

4251

0011 0010 1010 1101 0001 0100 1011


• 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?

4251

0011 0010 1010 1101 0001 0100 1011

• 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


4251

0011 0010 1010 1101 0001 0100 1011


• 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

• 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

4251

0011 0010 1010 1101 0001 0100 1011

• 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?


4251

0011 0010 1010 1101 0001 0100 1011

Definition of concept by Gerard Vergnaud

• 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.

4251

0011 0010 1010 1101 0001 0100 1011

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 heather uses 3,3 kW of electricity for 4,2 hours, how many kWh is that?

A heather uses 3,3 kW per hour. For how long can it be used on 13,9 kWh of electricity?

4251

0011 0010 1010 1101 0001 0100 1011

Problems situations for division of whole numbers invented by elementary school children

• for the study consisted of 135 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

4251

0011 0010 1010 1101 0001 0100 1011

True Fals Empty

True Fals

Partitive Measurment Partitive ∙ + -

12:365,9 28,9 5,2 96,6 3,4 41 0 5,1 41

45:359,3 33,3 7,4 86,5 1,1 46,6 4,4 2,2 31,1

72:12

50,4 34,1 15,5 95,6 1,5 39,1 2,2 2,2 43,3

Problems situations for division of whole numbers invented by elementary school children

4251

0011 0010 1010 1101 0001 0100 1011

Problems situations for division of fractions invented by elementary school children

• 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: .

2

1

True Fals Empty

True Fals


5,8 42,3 51,9 0 5,8 13,5 16,7 0 30,4

4251

0011 0010 1010 1101 0001 0100 1011

Problems situations for division of fractions invented by elementary school mathematics

teachers

• 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: , : .

2

1

2

1

4

3

4251

0011 0010 1010 1101 0001 0100 1011

True Fals Empty

True Fals


12:3 83,6 9 7,4 90,2 7,8 63,6 0 0 0

53,3 22,1 24,6 1,5 86,1 48,1 26 7,4 3,7

6,6 38,5 54,9 50 25 12,7 48,9 2,1 4,2

Problems situations for division of fractions invented by elementary school mathematics

teachers

4251

0011 0010 1010 1101 0001 0100 1011

Problems situations for Division of Fractions invented by students from teacher studies

• Subjects for this study consisted of 173 prospective teachers from University of Zadar and University of Split

True Fals Empty

True Fals

Partitive MeasurmentPartitiv

e ∙ + -

12:3 91,3 8,1 0 97,5 2,5 28,5 7 0 21

18,5 47,4 34,1 0 100 65,8 3,7 2,4 3,7

0,6 17,3 82 100 0 0 43,3 3,3 10

4251

0011 0010 1010 1101 0001 0100 1011

Measurment model Partitive model

Rectangular area

Equal groups

Equal measures

Multiplicative

conversi

on

Rate Total Equal groups

Equal measures

Multiplicativ

e conv

ersion

Rate Total

2nd - 4th grade

1 0,7 1,5 0 0 2,2 45,9 5,9 11,1 0 62,9 0,7

2 0,7 0 0 0 0,7 40 4,4 12,6 0 57 1,5

3 0,7 0 0 0 0,7 28,1 3,7 16,3 0 48,1 1,5

6th grade

1 2,9 2,9 0 0 5,8 0 0 0 0 0 0

students

1 1,2 0,6 0,6 0 2,4 86,1 2,3 0,6 0 89 0

2 17,9 0,6 0 0 18,5 0 0 0 0 0 0

3 0 0 0 0 0 0 0 0 0,6 0 0

teachers

1 0 6,6 0 0 6,6 67,2 5,7 2,5 0 75,4 1,6

2 32 13,9 0 0 45,9 0 0 0 0,8 0,8 6,6

3 0 0 0 1,6 1,6 0 0 2,5 0 2,5 1,6

4251

0011 0010 1010 1101 0001 0100 1011

Thank you !

Documents

Development of concept of division – from intuitive models to division of fractions