HOW CAN WE IDENTIFY AND MOTIVATE MATHEMATICAL TALENTS? CASE OF WORD PROBLEMS Jarmila Novotná Charles University, Faculty of Education M.D. Rettigové 4, 116 39 Praha 1, Czech Republic, jarmila.novotna@pedf.cuni.cz ABSTRACT In this paper, one type of didactical unit suitable for the identification and motivation of mathematical talents is presented. It is developed for the case of solving word problems but can be easily modified for other mathematical domains. As an example, a situation intended for students before starting algebra, namely word problems dealing with the division of whole into unequal parts, is used. 1. INTRODUCTION 1 In September 2003 the Socrates project MATHEU: Identification, motivation and support of mathematical talents in European schools was launched within the Socrates Comenius programme. The three-year project is coordinated by G. Makrides (Intercollege, Nicosia, Cyprus) with participation of institutions from eight European countries. One of the aims of this project is to design pedagogical methods and educational tools to identify and motivate potentially talented students in mathematics in both primary and secondary education levels (talent is seen here as an ability to face and solve problematic situation and to appreciate the role of theoretical thought). There are several activities proposed to achieve the project goal including: Identifying mathematical talent through a range of measures that go beyond traditional standardized tests. Presenting interesting tasks that engage students and encourage them to develop their mathematical talents. 1 The ideas presented in the Introduction are based on (MATHEU, 2003).
Improving opportunities for mathematics learning through challenging activities which are needed to help students develop mathematical talents. Students must be challenged to create questions, to explore, and to develop mathematics that is new to them. They need outlets where they can share their discoveries with others. Improving the ways in which students learn mathematics. Teachers must become facilitators of learning to encourage students to construct new, complex mathematical concepts. Students must be challenged to reach for everincreasing levels of mathematical understanding. The study described in the paper is a contribution to the above mentioned aims. It is a part of the author s longitudinal research focused on students cognitive processes when solving word problems 2. The function of word problems in developing students mathematical abilities and in identifying misunderstandings in their understanding of mathematical concepts and procedures is generally accepted by mathematics educators; see e.g. (Novotná, 2003). In this paper, the perspective of the theory of didactical situations (Brousseau, 1997) is used. Basic concepts of the theory are presented in par. 3. 2. TALENTED STUDENTS IN MATHEMATICS Students may show their special talents in mathematics in various ways. There exist several lists of characteristics of talented students. Let us mention here those which are common in most of the lists. Students talented in mathematics are likely to learn and understand mathematical ideas quickly, work systematically and accurately, see mathematical relationships, make connections between the concepts and procedures they have learned, apply their knowledge to new or unfamiliar situations, communicate their reasoning and justify their methods, take a creative approach to solving mathematical problems, persist in completing tasks, construct and handle high levels of abstraction, have strong critical thinking skills and are self-critical, can produce original and imaginative work. 3 This variety in characteristics induces the difficulties in the identification of students talented in mathematics. No student demonstrates all characteristics, but he/she shows a significant number of them. The identification is not, and cannot be, perfect. 2 The most typical word problems in school mathematics are represented by a brief text description of a situation with not all quantities being explicitly given; the solver is required to give a numerical answer to a question using the information given in the text (Verschaffel, Greer & De Corte, 2000). Moving from the given text to its mathematical model is a process that can involve en route several direct and indirect transformations. No teacher would be surprised by the amount of difficulties that most students face when solving word problems. 3 See e.g. www.eddept.wa.edu.au/gifttal/giftiche.htm.
The term problem solving is a broadly used but vague notion, a kind of umbrella under which different theoretical approaches take place. Mathematicians agree that problem solving occurs in cases where there is no clear algorithm to be performed. The use of problem solving raises certain questions 4, some of them related directly to the identification and motivation of talents in mathematics: Which are the properties of problems that help to develop solvers creativity? How to recognise that a taught mathematical knowledge was incorporated into students structure of mathematical knowledge? This question is crucial because learning mathematics should not be restricted to the repetition of taught algorithms (Sarrazy, 2002). Which difficulties that students face when solving a problem can be overcome without an external help and which need teacher s intervention? 3. PREPARATION OF A DIDACTICAL SITUATION 5 The conditions for the particular use of a piece of mathematical knowledge are considered to form a system called a "situation". A situation is characterized in an institution by a set of relations and reciprocal roles of one or more subjects (pupil, teacher, etc.) with a milieu, aimed at transforming that milieu according to a project. The milieu consists of objects (physical, social or human) with which the subject interacts in a situation. The subject determines a certain evolution amongst the possible, authorized states of this milieu which he judges to conform to his project. A didactical situation (see Fig. 1) is a situation in which an actor (e.g. a teacher), organizes a plan of action with the aim to modify or cause the creation of some knowledge in another actor (e.g. a student). In a non-didactical situation, the evolution of the actor is not submitted to any didactical intervention whatever. Modeling effective teaching leads to the combining of the two: certain didactical situations make available to the subject of learning situations which are partially liberated from direct interventions: a-didactical situations. The Theory of didactical situations classifies situations according to their structure: Situation of action (the actor decides and acts on the milieu, it is of no importance whether the actor can or cannot identify, make explicit or explain the necessary knowledge) 4 For the case of word problems see e.g. (Novotná, 2003). 5 The characterization of a didactical situation was put together using (Brousseau, 1997) and (Brousseau, Sarrazy, 2002). Only items related to the focus of the paper are mentioned.
Situation of formulation (at least two actors are put into relationship with the milieu; their common success requires the formulation of the knowledge in question). Situation of validation (a situation whose solution requires that the actors establish together the validity of the characteristic knowledge of this situation; its effective realization thus depends on the capacity of the protagonists to establish this validity explicitly together). Situation of institutionalization (a situation which reveals itself by the passage of a piece of knowledge from its role as a means of resolving a situation of action, formulation or validation to the new role of reference for future personal or collective uses). Devolution is the process by which the teacher manages in a didactical situation to put the student in the position of being a simple actor in an a-didactical situation (of a non-didactical model). The student feels responsible for obtaining the proposed result and accepts the idea that the solution depends only on the exercise of knowledge which he/she already has. Devolution Situation of action formulation validation Institutionalisation A-didactical situation Fig. 1. Scheme of a didactical situation For each problem there exists knowledge that enables to solve it. Students do not have all of them at their disposal. Students learning can be characterized as widening their repertory of available solving means. The teacher s task is to create a suitable didactical situation where such widening occurs. Brousseau (1996) stresses out the importance of student s encounter with the situation.
4. DIDACTICAL SITUATION PROPOSED FOR IDENTIFICATION OF STUDENTS TALENTED IN MATHEMATICS The three types of a didactical situation presented above offer opportunities to identify characteristics of students talented in mathematics: Situation of action: Learning and understanding mathematical ideas quickly, working systematically and accurately, seeing mathematical relationships, making connections between already learned the concepts and procedures, applying knowledge to new or unfamiliar situations, taking a creative approach to solving mathematical problems, persisting in completing tasks, having strong critical thinking skills, being self-critical, producing original and imaginative work. Situation of formulation: Communicating reasoning, justifying used methods Situation of validation: Working systematically and accurately, seeing mathematical relationships, making connections between already learned the concepts and procedures, applying knowledge to new or unfamiliar situations, having strong critical thinking skills, producing original and imaginative work. Situation of institutionalisation: Working systematically and accurately, seeing mathematical relationships, making connections between already learned the concepts and procedures, applying knowledge to new or unfamiliar situations, persisting in completing tasks, having strong critical thinking skills,. In the following text, a didactical situation will be analyzed from the perspective of identification and motivation of students talented in mathematics. The situation is intended for students before starting algebra who were not taught to solve the problems of the presented type. The possibility to the independent discovery of a successful solving strategy is offered to them. As the example, word problems dealing with the division of whole into unequal parts 6 were chosen. 4.1. THE TYPE OF WORD PROBLEM The basic problem for the situation is the problem dealing with the division of a whole into two unequal parts, where the relationship between the parts is multiplicative and the whole is given. The structure of the word problem dealt with is schematically presented at the fig. 2. For the assignments see the attachment. 6 Word problems dealing with the division of a whole into unequal parts are usually used in school mathematics as means for constructing and solving linear equations and their systems. Most often in the traditional teaching is based on the application of algorithms presented to the student usually by the teacher. The student practices their use when solving (individually or in a group) analogous problems.
Whole A k times B Fig. 2 The variations used in the experiment included similar problems with more than two parts. The following arithmetical solving strategies were anticipated: a) Division of the whole by the number of unit parts: As B = k.a, in the whole is A included (k + 1)-times, therefore A = W : (k + 1). b) Systematic trial c) Approximation d) False position: If the smaller part is increased by 1, the biggest by k and the whole by k + 1; the whole equals W, therefore A = W : (k + 1). 7 The following difficulties were expected (besides numerical mistakes): Division of the whole into equal parts Incorrect mathematisation of the relationship B is k-times more than A Forgetting the part A (division of W by k) 4.2. PROPOSED DIDACTICAL SITUATION The mathematical problem was put into various contexts: dividing an amount of money between two people, putting marbles into two boxes, pouring water into two glasses, and dividing a group of children into two swimming pools, number of kilometres done in two days of excursion. (See the attachment.) The goal of the context variation was to see the influence of discrete and continuous environment on the used solving strategies. The situation is composed of three phases: Phase 1: The goal of this phase is to let students tackle the problem and try to create a successful solving strategy. (Situation of action.) Students are working in pairs or threes. Each group receives the assignment of one problem. The teacher expresses clearly that they are allowed to use any solving strategy they want. Besides this, no hint is given by the teacher. In case that a group finishes 7 The final formula for A is the same in the case a) and d). However, the solving strategy differs significantly.
the solution faster than others they may ask to be given another problem (of the same type). The situation finishes when each group finds a solution (correct or incorrect) of at least one problem. The work in pairs and trees enables students to start formulating their ideas already in Phase 1. Phase 2. Speakers of all groups present to the whole class their solutions of the first solved problem. In order to be understood, groups have to find a clear way of describing their solving strategies. They might use any means to express clear ideas, including any type of models (e.g. graphical or symbolic representations or concrete objects). At the same time, they are obliged to compare their strategies with these already presented by other groups in order to recognise the differences or congruities in them. In case that an incorrect solving strategy is presented, the task of the class is to persuade the presenting group in a suitable form about the mistakes in their solution. (Situation of action and validation.) Phase 3. Students work again groups. They are solving similar problems with three and four parts where all parts are related to the smallest part. The change of the number of parts represents for them a new quality. The teacher does not give any hint e.g. by expressing the similarity in the structure of the previously solved problems or a possible successful solving strategy. The generalisation of the solving strategy for the problem with two parts represents a form of institutionalisation of the knowledge gained in phases 1 and 2. 4.3. FIRST EXPERIENCES WITH THE PROPOSED SITUATION The above described didactical unit was experimentally used in one Prague school in the June 2004 (Pelantová, Novotná, 2004). The whole teaching sequence lasted 45 minutes (one lesson). In the experiment 28 students from the 6 th class participated. The class was taught in the traditional way with a negligible attempt for using elements from the constructive approach to teaching mathematics; several students in the class are evaluated by the teacher of mathematics as naturally curious, usually working with a high interest. In each pair or three there were both, students with excellent results in school mathematics and with poorer ones. The problems with the division of a whole into parts represented a new topic for the students. The experimental lesson was taught by a mathematics teacher (T) who was not the teacher of mathematics in the class. The whole unit was video recorded by two cameras a fixed one recording the situation in the whole class during the whole lesson and a portable one recording selected actions in individual groups (operated by the observer O). The analysis following the lesson was done by T and O. The aim of the analysis was to identify characteristics of talents in mathematics as mentioned above in the phases of the unit and determine potential talents in mathematics using them. Not all students classified as talents in mathematics (shortly talents) showed all characteristics. A
student was classified as talent in case of the presence at least of the following characteristics: working systematically and accurately, seeing mathematical relationships, applying knowledge to new or unfamiliar situations, taking a creative approach to solving mathematical problems, persisting in completing tasks, being self-critical, justifying used methods. Using these characteristics, three students were identified as talents. Further findings based on the lesson observations: From the social perspective, there was not a unified situation by all talents. Some of them were able to work collectively, discuss their ideas, share the work. Others established themselves as leaders who offered the other members of the group the role of observers with a very little possibility to become involved in searching the strategy in a creative way (in some cases, one of the members served as the recorder of ideas and solving procedure). Surprisingly, all groups used the same successful strategy strategy a). No other of the a priori specified strategies was applied; no difference occurred for different contexts of the problem. Talents showed a variability of ways of visualisation of the problem structure. Talents were able to generalise the strategy a) to the case of multiplicative relationships among three and more parts see fig. 2 c), d). Talents, having discovered the strategy a), asked for other problems to be solved. They were offered problems with two parts and the additive structure between them (see the attachment). Strategy a) was not suitable for solving them. Talents started by an attempt to do it. But they showed a high degree of critical thinking and creativity and proposed a new strategy for this type of problems see fig. 2 b). a) Problem E b) Additive analogy of Problem B c) Problem S d) Problem Z Fig. 3. Solutions of Adéla identified as a talent
5. CONCLUDING REMARKS The list of talents identified by T and O (in the following text labelled as List 1) was compared with the evaluation of individual students by the class mathematics teacher (List 2). The two lists were not identical. One of students from the List 1 was missing on the List 2. In the experiment, his solutions were correct, he was able to formulate and justify his strategy, generalize it, but his written record was incomprehensible for the others. His graphical display is the most probable reason of the mathematics teacher s underevaluation of his mathematics abilities. The case study described here confirmed the usefulness of the proposed structure of the didactical situation for the identification, motivation and development students potentially talented for mathematics. In phase 1 they had a broad field of activity mainly in proposing methods for solving the problem, defending the proposed activities, evaluating the correctness of the result, applying the chosen strategy to problems with different contexts. In phase 2, their contribution to formulating clearly and precisely the ideas was important. In phase 3 they were again the leading persons in finding the analogies and generalisation of the previously used strategy. The didactical situation of the presented type is very flexible. It can be easily transformed for other types of problems. REFERENCES Brousseau G. Fondements et méthodes de la didactique des mathématiques. In: J. Brun (Ed.). Didactique des mathématiques. Lausanne: Delachaux et Niestlé, (1996). Brousseau G. Theory of Didactical Situations in Mathematics. [Edited and translated by Balacheff, M. Cooper, R. Sutherland, V. Warfield]. Dordrecht/Boston/London: Kluwer Academic Publishers, (1997). (French version: Théorie des situations didactiques. [Textes rassemblés et préparés par N. Balacheff, M. Cooper, R. Sutherland, V. Warfield]. Grenoble: La pensée sauvage, 1998.) Brousseau, G., Sarrazy, B. Glossary of Terms Used in Didactique. DAEST, Université Boerdeaux 2, (2002). Translated into English by G. Warfield.) Makrides G. et al. MATHEU: Identification, motivation and support of mathematical talents in European schools. Final application, (2003). Novotná, J. Etude de la résolution des «problèmes verbaux» dans l enseignement des mathématiques. De l analyse atomique à l analyse des situations. Bordeaux: Université Victor Segalen Bordeaux 2, (2003). Pelantová A., Novotná, J. Nepodceňujeme naše žáky? Objeví žáci samostatně strategie řešení slovních úloh? [Do we not underestimate our students? Do
students discover word problem solving strategies independently?] In: Proceedings 9. setkání učitelů matematiky. Plzeň: JČMF, (2004). (In Czech.) Sarrazy B. Effects of variability on responsiveness to the didactic contract in problem-solving among pupils of 9-10 years. European Journal of Psychology of education. XVII. 4, 321-241, (2002). Verschaffel L., Greer B., De Corte E. Making Sense of Word Problems. Sweets & Zeitlinger Publ., (2000). Appendix: Assignments of word problems used in the experiment A. Divide 185 Czech crowns between two people in such a way that one of them receives four times more crowns than the other. B. Put 174 marbles in two boxes in such a way that in one of them there will be five times more marbles than in the other. C. Pour 204 ml of water in two glasses in such a way that in one of them there will be three times more water than in the other. D. In the health center, there are two swimming pools, small and big one Split 90 children in the swimming pools in such a way that in the big one there are four times more children than in the small one. E. During the two-day excursion, children made 85 km. In the second day they made four times more kilometers than in the first one. How many kilometers did children make in each day? S. 96 students have joined three sport clubs. All the clubs have their meetings at the same time. Two times more students join the basketball club than the gymnastics club, in the swimming club there are three times as many students as in the gymnastics club. How many students have enrolled in each sport club? Z. Lenka, Cilka, Slávek and Aleš play marbles. They have 108 marbles together. Aleš has two times more marbles than Cilka, Slávek four times more than Cilka and Lenka five times more than Cilka. How many marbles has each of them got? Problems with two parts and additive relationships between them were analogous to problems A to E (the same contexts, relationships expressed using more by ). Acknowledgment: The research was supported by the Socrates Comenius project No. 112212-CP-1-2003-1-CY-COMENIUS-C21 MATHEU: Identification, motivation and support of mathematical talents in European schools and by the Research project MSM 114100004 Cultivation of mathematical thinking and education in European culture.