ΤΤhe effects of different modes of he effects of different modes of representations in mathematical representations in mathematical problem solvingproblem solving

Athanasios GagatsisAthanasios GagatsisDepartment of Education-University of Cyprus

GRANDE FESTA DELLA MATEMATICAGRANDE FESTA DELLA MATEMATICA

Un’immagine vale più di mille parole…Ma che tipo di immagine risulta più

efficace nelle attività di problem solving?

What is the role of representations in the understanding and learning of mathematics?

Why do we use so many pictures in mathematics textbooks?

Can a picture attain the function of a text?

It is at the core of mathematical understanding. A concept in mathematics is accessible only

through its semiotic representations. A representation cannot describe fully a

mathematical concept. Each representation has different

advantages and capabilities (Duval, 2005).

The recognition and processing of a mathematical concept in a variety of representations and the conversion from one representation to another are considered as basic components of the understanding of the concept.

The importance of the diversity of The importance of the diversity of representations in the learning of representations in the learning of mathematicsmathematics

The focus of this presentationThe focus of this presentation

Number line Pictures

In elementary mathematics a representation that plays an important role in the teaching of basic whole number operations, and generally in arithmetic is the number line (Klein, Beishuisen & Treffers, 1998).

Despite the widespread use of the number line as an aid to whole number addition and subtraction, doubts about the appropriateness of using it have been raised (Hart, 1981).

It constitutes a geometrical model, which involves a continuous interchange between a geometrical and an arithmetic representation → Dual nature of number line

Geometric

dimensio

n

Arithmetic

dimension

The numbers depicted on the line correspond to vectors

Points on the line can be numbered

The simultaneous presence of these two conceptualizations may limit the effectiveness of number line and thus hinder the performance of students in arithmetical tasks (Gagatsis, Shiakalli, & Panaoura, 2003).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Test A

14+4= 13+ =16 19-4= 18- =16

8+7= 9+ =15 12-5= 13- =9

Write the appropriate number in the

Test B

14+3= 13+ =16 19-4= 18- =16

8+7= 9+ =15 12-5= 13- =9

Put the appropriate number in the If you want use the number line model.

Test C

12+4= 16+ =19 18-3= 17- =15

9+3= 7+ =11 13-4= 12- =9

Show the mathematical proposition on the number line and find the result.

Test D

Write the addition or the subtraction sentence that matches each number line model and find the result.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0

Figure 1: The model representing the abilities to conduct arithmetic operations by using or not the number line (NL) derived from Cypriot, Greek and Italian pupils’ responses.

T1

T2

T3

T4

T5

T6

T13

T14

T15

T16

T17

T18

T7

T8

T9

T10

T11

T12

T19

T20

T21

T22

T23

T24

T25

T26

T27

T28

Operations-no NL

Operations-optional use

of NL

Operations-required use

of NL

Recognition of operation

on a NL model

Ability to conduct

arithmetic operations

.88

.90

.93

.84

.83 .82 .80

.84 .83 .81

.68 .69 .70

.74 .72 .72

.79 .78 .79.72 .82 .83.75 .76 .78.74 .75 .75

.82 .80 .78

.81 .83 .80

.68 .67 .66

.74 .72 .82.79 .82 .79

.78 .78 .76.75 .73 .74.74 .79 .79

.83 .81 .81

.76 .77 .78

.78 .79 .79

.74 .72 .74

.79 .73 .76.74 .69 .70.77 .77 .77.76 .81 .81

.80 .79 .78

.79 .83 .82

.81 .81 .80

.82 .80 .81

Test A

Test B

Test C

Test D

The first, second and third coefficient of each factor stand

for the application of the model on the performance of pupils in

Cyprus, Greece and Italy, respectively.

The model raises a question of a great importance and interest for the mathematics educators of the three countries whether it is possible to teach or assess whole number operations in isolation from other skills and concepts such as those related to the use of number line diagrams.

The model that was empirically validated in this study provides evidence to the multidimensionality of pupils’ skill in conducting arithmetic operations and suggests that at least four different dimensions could be taken into account in the teaching and assessment of this skill.

Four different functions of pictures in Four different functions of pictures in problem solvingproblem solving

Informational pictures Organizational

pictures Decorative pictures Representational

pictures•do not give any actual information concerning the solution of the problem

•represent the whole or a part of the content of the problem

•provide directions for drawing or written work that support the solution process

•provide information that is essential for the solution of the problem

Organizational picture

Five children shared 15 bars of chocolate. How many bars of chocolate did each get? ……………. How many bars of chocolate remained? ……………

Join situation problem with the unknown in the transformation: The informational picture

?

The cakes I have had The cakes I have had ready since yesterday.ready since yesterday.

I made some I made some more cakes in more cakes in the morning.the morning.

These are the cakes I have These are the cakes I have now. How many cakes did I now. How many cakes did I make in the morning?make in the morning?

(IJb)

Verbal description(DeCorte & Verschaffel, 1987; Carpenter, 1985)

Representations used in additive problem Representations used in additive problem solvingsolving

Number line (Shiakalli & Gagatsis, 2005)

Schematic drawings,a triadic diagram of

relations (Willis & Fuson, 1988;

Vergnaud, 1982; Marshall, 1995)

Picture of a particular situation(Duval, 2005; Theodoulou,

Gagatsis & Theodoulou, 2004)

The effects of different modes of The effects of different modes of representation on additive problem solvingrepresentation on additive problem solving

To explore the effects of the informational picture, the number line and the verbal description (text) on the solution of one-step change problems of additive structures.

To investigate the possible interaction of the various representations with the mathematical structure and more specifically with the placement of the unknown, on students’ ability to provide a solution to additive change problems.

Purpose Purpose PhD Iliada Elia (2006)

The test

18 18 one-step change problems one-step change problems (measure-transformation-

measure)

9 9 join situationjoin situation ( (JJ)) 9 9 separate situationseparate situation ( (S)S)

VV= = verbalverbal, , PP= = informational pictureinformational picture, , LL = = number linenumber line

VV P P L L

start.amount(a) transf.(b) fin.amount(c)

Type of the relationType of the relation

The placement of The placement of the unknownthe unknown

RepresentationRepresentation

start.amount(a) transf.(b) fin.amount(c)

VV P P L L VV P P L L VV P P L L

VV P L P L VV P P L L

An example of the symbolization of the variables:

VJb= a verbal problem of a join situation having the unknown in the transformation

ccaabb

.71 .68 .69

.94 .90 .96

1.01 1.03 1.04

.93 .99 .93

1.02 1.00 .99

.73 .73 .66 .55 .53 .47 .59 .68 .57 .48 .57 .46 .63 .64 .63 .52 .57 .51

.67 .72 .66

.73 .74 .71

.68 .72 .65

.70 .67 .73

.72 .71 .73

.56 .56 .55

.73 .74 .71

.66 .70 .69

.78 .80 .76

.65 .65 .63

.74 .75 .69

VJa3

VSa6

VJb15

VSb12

ISa18

ISb8

LJb7

LJa11

LSa16

IJb17

IJa9

LSb5

VSc1

VJc10

IJc4

ISc13

Vab

Unk.c

Lab

Pab

LSc14

LJc2

AdPSA

Χ2(523)=1228.078, CFI=0.946, RMSEA=0.052

Figure 2: The model for the role of the positions of the unknown and of the representations on additive problem solving

Concluding remarks on the role of Concluding remarks on the role of representations in problem solvingrepresentations in problem solving

The findings revealed that pupils dealt flexibly and similarly with problems of a simple structure regardless of the mode of representation. However, when they confronted problems of a complex structure they activated distinct cognitive processes in their solutions with reference to the mode of representation.

Apart from the semantics of the problem, the different modes of representation do have an effect on additive problem solving.

There is an important interaction between the mathematical structure and the mode of representation in problem solving.

The connection among different representations in problem solving depended on the difficulty of the structure of the problem. Pupils carried out the transition from one representation to another more flexibly in solving problems with the unknown in the final amount compared to solving more complex problems, i.e. with the unknown in the other positions.

The assertion that the understanding and success in problem solving are closely linked with the ability to coordinate different representations of the same situation (Duval, 2002) is empirically validated in the case of the additive change problems.

The function of the informational picture in the second study

The function of informational pictures in problem solving was found to differ significantly from the use of the other forms of representation.

Pupils dealt in a unique and less flexible way with these problems.

The cognitive demands of the informational picture in the context of mathematical problems were different from the other forms of representations. It seemed to be incongruent with the mathematical

situation of the problems. It might require more complex mental processes

relative to the other representations as the understanding and the solution of the corresponding problem involves the coordination of three different modes of representations combined in the informational picture, i.e. pictorial, verbal and diagrammatic.

Both studies unraveled students’ differentiated ways of thinking in arithmetic or problem solving tasks of different representations. These findings provide a strong case for the role of different modes of representation on problem solving, as well as, on the understanding of mathematical concepts such as arithmetic operations.

It is important for mathematics educators to teach additive problem solving as well as arithmetic operations, by using the semantic approach integrated with the perspective which refers to the diversity of representations and transferring information from one representational mode to another.

Informational picture and number line need special attention within the context of mathematical problem solving and the learning of arithmetic operations respectively, during instruction.

Given that • a picture can be described through a sufficiently

rich language • not all verbal expressions can be represented

pictorially

•(Un’immagine vale più di mille parole…)•The well-known saying “a picture is worth a thousand words” refers to the economy of language and thought and not to the ability to represent. •“Representationally” speaking: “every word conveys a thousand pictures” but “a thousand pictures cannot represent some of the things we can represent using words”.