Problem to be solved: Marie and Pavla each had some money but Marie had 10 CZK more than Pavla. Pavla managed to double the amount of money she had and Marie added 20 CZK more to her original amount. They now found that both of them had the same amount. How many crowns did each of them have at the beginning?
This record of the assignment did not allow him to find a suitable solving strategy. The experimenter recommends him to use the visualisation with the help of line segments, see (Novotná, Mnichov):
Experimenter (E): Try to record the situation at the beginning.
E: And after the change?
J starts to draw a new line segment.
E: Would not be better to record it in the same schema?
After a short discussion, J's graphical representation is
J: Aha * I do not need to construct an equation!
This is a small illustration of the power that graphical visualisation brings in the solving process of certain types of word problems.
Pictorial representation (diagrams) is one of the oldest and most used didactical tools for the solving of problems, see e.g. (Volkert, 1989). The importance of pictures, schemes, diagrams etc. grows with the expansion of new technologies such as audio-visual means, hypertext etc.
There are several studies, of which I will list four, analysing the role of figures as a didactical means for improving the learning process in different subject areas. Macek, 1984, Anschauliches Beweisen, 1989, Mares, 1995, Plass et al., 1998.
Mares's paper of 1995 is devoted to learning from pictorial materials. The child's understanding of pictographic materials depends on his/her cognitive structure development. It is connected not only with the age and spontaneous maturation of the child's intellect, but it also depends on the way the child's development is systematically influenced and on the level of thought-provocativeness of the child's environment. Mares characterises the differences between the pre-school and school periods from the point of view of verbal and non-verbal communication:
Mares (p. 319) refers to the term visual literacy which is used in literature in connection with understanding visual materials. It is considered either as an ability to understand ("read") and to use ("create") figures, to think and learn in the terms of figures, or as a set of skills that an individual has to his/her disposal in order to understand visually presented materials and to be able to use it for intentional communication with other individuals.
For Macek, 1984, the figure/diagram is characterised as a partly or completely constructed record. The author uses the term didactical figure for a visual two-dimensional or an audio-visual medium specifically constructed as a means of stimulating and regulating learning activities in the educational process.
The present article is connected with the role of figure in grasping word problem assignments. Some aspects in the specific domain of word problems dealing with the division of a whole into parts have already been presented e.g. in (Novotná, 1997b), (Novotná, 1998), (Novotná, 1999). The theory presented in these articles was supported by the results of experiments with 12-to15-year old Czech students.
We used the stages of word problem solving process presented in (Novotná, 1997c):
We will concentrate on the grasping of the assignment and consider only the case when the solver uses a figure.
The following terminology will be used (Novotná, 1999):
Coding of word problem assignment is the transformation of the word problem text into a suitable system (reference language) in which data, conditions and unknowns can be recorded in a more clearly organised and/or more economical form. The result of this process is a legend. The legend constructed in a pictorial form is called a graphic legend. The reference language contains basic symbols and rules for legend creation.
Note: There exist different reference languages for any one type of word problem. The solver's choice of one of them is influenced by several factors, e.g. by his/her previous experience, preferred information processing style, personal preferences.
Functions of graphical legends
The diagrams - graphical legends can fulfil different roles. We will specify those psycho-didactical functions given in (Mares, 1995), that are relevant for the situation of a graphical legend as a tool of getting insight in the word problem structure.
The functions of a graphical legend regarded from the point of view of a solver's emotionality are:
Specific features of graphic representations
Applying the criteria for the text reception studied in (Gavora, 1991), i.e. passive/active text reception, personal interpretation of the text, use of personal language, we conclude:
Pictographic communication differs basically from verbal communication (Macek, 1984) mainly in the following ways:
Information aspects of a graphic legend
The dominant role in the graphical legend creation is played by the teacher more often than when using verbal materials. Therefore the choice of a specific reference language is connected with the concrete group of people - a class, a family etc. In the affective plane, the relationships teacher-student and teacher-class are important. What is very successful in one class can be difficult to understand in another (icons, signs, their position etc. from one reference language can e.g. have completely different meaning in another). This feature of a graphical reference language can be the source of solvers' difficulties.
The following scheme is a modification of the general scheme of the pictographic communication scheme presented in (Macek, 1984) for the case of word problem assignment graphical coding:
Classification of graphical legends according to the impulse for their creation
In our research we identified three legend categories:
1. The solver forms his/her graphical legend spontaneously without (or with a minimal use of) a previously learned reference language (Novotná & Kubínová, 1999). The impulse for the legend creation goes out from the solver's internal need to visualise the problem structure.
For the graphical legends classification the following criteria were used:
a) Shape similarity
We use the classification presented in (Macek, 1984).
In our experiments (junior secondary level students) the pure iconic spontaneous legend was exceptional. Most students used the reference language consisting of both, iconic and symbolic elements.
Note: Most pupils representations are topologically correct rather than iconic.
b) Use of language and/or mathematical elements
In graphical legends reference language, the symbols used are not always pure pictographic ones. Words and/or mathematical symbols are attached. We distinguish two different roles of them:
c) Completeness of the record
The purpose why the solver creates a legend is in most cases his/her attempt to understand clearly and correctly the word problem structure. This grasping consists of several steps (Novotná, ERCME):
Not all these stages are necessarily recorded in the graphical legend. There are different reasons why the solver does not finish the detailed legend, the most usual are the following ones:
d) Procedural or conceptual graphical legends (Novotná & Kubínová, 1999)
We call a legend:
2. A model graphical legend for a family of problems is presented to solvers and they form graphical legends using the presented reference language.
The model legend should fulfil the following demands. It should
Creation of a model graphic legend for a family of word problems is always influenced by the author's past experience and personal preferences. It is not necessarily the best one for all other solvers. Therefore it is recommended to present the reference language of the model legend but not to insist on its use at any price. The individual differences usually touches upon the extent of the use of abstract symbols, the use of procedural/conceptual legend type, the composition of the legend and the richness and form of accompanying verbal and/or mathematical labels.
The educational climate and the relationships between the teacher and his/her students play an important role in the acceptance of a model graphical legend by the students. To increase students' creativity and ability to solve modified or non-standard problems, the teacher should not deliver the model legend as a completed algorithm, but construct it in co-operation with the students, let them discover the advantages (disadvantages) of its reference language themselves. In our experiments the usefulness of choosing well presented graphical legends was remarkable especially for weaker students.
3. The ability to solve a certain type of word problems can be increased by facilitating the positive transfer effect (the use of solver's past experience in solving similar problems). A simple figure can serve as a signal to activate the solving schemes stored in the solver's memory.
A typical example is the family of time - distance word problems. It is a difficult task for many students to solve these problems even if they are familiar with the model solutions to similar problems. Simple figures as
can activate the corresponding algorithm for the correct solving process and help the student to find the solution of the problem.
From the analysis of the graphic legends in solving word problems we find that the passive text reception has no place when creating a pictorial record of the assignment. The solver gives the assignment a new form, his/her personal presentation of the text. He/she adapts the text to his/her abilities and customs.
In the article the positive role of figures in the process of grasping word problem assignment was stressed. The questions of their possible negative influence were not analysed. There exist individual differences in dealing with pictographic materials from the point of view of solver's learning style, ability to draw pictures, influence of the solver's school and/or family background, age etc.
Gavora, P. (1992). Ziak a text (Student and text). Bratislava, SPN.
Macek, Z. (1984). Obraz jako didakticky prostredek (Figure as a didactical means). Pedagogika, 34, p. 453-467.
Mares, J. (1995). Uceni z obrazoveho materialu (Learning from pictorial material). Pedagogika, 45, p. 319-327.
Novotná, J. (1997a). Using Geometrical Models and Interviews as Diagnostic Tools to Determine Students Misunderstandings in Mathematics. In: Proceedings SEMT 97. Praha, Prometheus, p. 61-67.
Novotná, J. (1997b). Geometrical Models in Solving Word Problems That Include the Into Parts (Theory and Practice). In: Proceedings Interakcja teorii i praktyki nauczania matematyki w szkole podstawowej i sredniej. Ed. J. Tocki. VSP Rzeszów, p. 109-119.
Novotná, J. (1997c). Phenomena Discovered in the Process of Solving Word Problems. In: Proceedings ERCME 97. Praha, Prometheus, p. 98-102.
Novotná , J. (1998). Cognitive Mechanisms and Word Equations. In: Beiträge zum Mathematikunterricht 1998, Vorträge auf 32. Tagung für Didaktik der Mathematik vom 2. bis 6.3.1998 in München. Ed. M. Neubrand. Hildesheim, Berlin, Verlag Franzbecker p. 34-41.
Novotná, J. - Kubínová, M. (1999). Wie beeinflusst eine Visualisierung der Aufgabenstellung den Prozess der Lösung einer Textaufgabe. In: Beiträge zum Mathematikunterricht 1999, Vorträge auf 33. Tagung für Didaktik der Mathematik vom 1. bis 5.3.1999 in Bern. In print.
Novotná , J. (1999). Analyza reseni slovnich uloh (Analysis of word problem solving process). Praha, Karolinum. In print.
Plass, J.L. & al. (1998). Supporting Visual and Verbal Learning Preferences in a Second-language Multimedia learning Environment. Journal of Educational Psychology, Vol. 90, No. 1, p. 25-36.
Proceedings Anschauliches Beweisen. Eds. K. Kautschitsch - W. Metzler (1989). Schriftenreihe Didaktik der Mathematik, Band 18, Hölder - Pichler - Tempsky, Wien, B.G. Teubner, Stuttgart.
Volkert, K. (1989). Die Bedeutung der Anschauung für die Mathematik - historisch und systematisch betrachtet. In: Anschauliches Beweisen. Eds. K. Kautschitsch - W. Metzler. Schriftenreihe Didaktik der Mathematik, Band 18, Hölder - Pichler - Tempsky, Wien, B.G. Teubner, Stuttgart, p. 9-31.
Faculty of Education
116 39 Praha 1
The Czech Republic
Acknowledgement: The research was supported by the projects GACR No. 406/99/1696 and GAUK 306/1998/A.