FUNCTIONAL OR PREDICATIVE? CHARACTERISING STUDENTS THINKING DURING PROBLEM SOLVING

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1 FUNCTIONAL OR PREDICATIVE? CHARACTERISING STUDENTS THINKING DURING PROBLEM SOLVING Adam Mickiewicz University, Poznań, Poland The article presents a part of a research, whose goal was to study and describe the ways of applying the graphic calculator by 14-year-old students and GeoGebra computer software by students of Department of Mathematics and Computer Science in Adam Mickiewicz University during solving a particular kind of tasks. This article attempts to answer the question: Can the recognition of students' ways of thinking and the discovery of their dominant cognitive structure cause the improvement of communication during Math lessons? Can we, by analyzing the students' output, discover and characterise the way of thinking they have applied? In the article we use Schwank s theory of predicative and functional thinking. INTRODUCTION Good communication between a teacher and a student during Maths lessons is one of the most important factors influencing the quality of Maths education. Effective communication in the area of both verbal and non-verbal behavior in educational situations is characterized by the contents of the school subject being presented by means of a clear language in a stimulating way. However, the teacher not only delivers the knowledge, but also formulates questions and tasks, as well as explains, explicates, answers the students' questions, analyses students' output, then comments on them, describes and evaluates (Sztejnberg, 2002). The teachers caring about good communication during their classes try to actively listen to each of their students. Uncommonly, nevertheless it happens, few students in the same class will tell us a totally different story. These may be ideas told in another mathematical language, maybe other ideas of solving the same problem, may be different work methods, and, first of all, these may be different ways of reasoning. I the book Fundamentals of Communication in Education (Sztejnberg, 2002) we can read: "In order to assure an effective performance of knowledge acquisition, students should be given such didactic tasks that are compatible with their sensory system. Such an approach will enable the students to learn with pleasure. They will discover the ins and outs of various school subjects with a great joy" (p.83).

2 Can the hypothesis referring to a student's dominant sensory system be applied to a student's dominant cognitive structure as well? Can the recognition of students' ways of thinking and the discovery of their dominant cognitive structure cause the growth of students' motivation for studying and the improvement of communication during Math lessons? Can we, by analyzing the students' output, discover and characterise their way of thinking? PREDICATIVE AND FUNCTIONAL THINKING Inge Schwank (1995, 1999, 2001) is the author of the theory of functional and predicative thinking. Schwank's study shows that in an every human being we can observe a relatively stable tendency to represent the way of thinking characteristic of one of the two cognitive structures: either the predicative or the functional one. However, it does not mean that there is any hierarchical dependence between them, or that any of these types of thinking is more advantageous for mathematical thinking. What is more, the inclination for a specified way of thinking does not exclude reasoning in the other way. Besides, one can think functionally while arguing predicatively. The predicative thinking structure is more effective for solving more complex cognitive problems (Schwank 1995, 1999, 2001). Predicative thinking is oriented to a structure analysis, looking for common features and similarities between objects, their systematization and creating structural connections. It is thinking in the categories of relations, which leads to generating the static mental representations of the considered ideas (Nowińska, 2008). Functional thinking involves a tendency to searching the differences and changes interpreted as the results of the ongoing process. It leads to the systematization of the observed objects according to the functional criterion and to the creation of dynamic mental representations of the considered ideas. It is thinking in categories of functions and processes (Nowińska, 2008). The figure below illustrates the differences between the ways of thinking described above.

3 Figure 1 Predicative versus functional cognitive organisation (Schwank, 1995) For diagnosing the way of thinking Schwank used the tasks from the Raven's Matrix Test 1. In each of the test tasks the ninth, missing element needs to be included. The analysis of the way to find the missing element from the test allowed Schwank to attempt to define the tested person's way of thinking. The research material consisted of the record of eyeball movements, which was obtained with an advanced research method using the EYEtracker. For the task presented below, the analysis of the material obtained from the device has shown that people with the predicative thinking structure had focused on looking for similarities - they had always analysed the same top in each column and line of the arrangement of drawings, looking for similarities in their 1 Raven's matrix test (progressive matrix test ) used to measure the index of general intelligence. Thanks to that test the information about a particular person's value of so called g factor, a general intelligence factor, is obtained.raven's matrix test in a standard version consists of 5 scales A, B, C, D, E, in each scale there are usually 12 tasks, which involve an examined person to grasp the relations among the elements of a matrix and point to the missing element from the matrix from the elements presented below the matrix. The level of difficulty is the smallest on the scale A and it increases in each following scale in such a way that the level of difficulty in the last scale is the biggest he test diagnoses so called non-verbal intelligence, independent of the examined person's experience, education, origin, etc. It checks the abilities of an individual to a logical induction, noticing the principles of continuity of patterns (scale A), observing analogies between pairs of figures (scale B), progressive changes of patterns (scale C), translocation of figures (scale D), taking figures apart (scale E).This tool is often used for the selection of employees, for instance in police forces, banks (in an advanced version). (

4 structure, while those with dominant functional thinking focused on the changes which had been the result of a certain process, happening in the structure of the drawings in the columns and lines of the arrangement, which has been shown in the picture below. Eye movement Eye movement dyring predicative analysis Eye movement during functional analysis Drawings come from the article I. Schwank On Predicative Versus Functional Cognitive Structures Figure 2 Eye movement METHODOLOGY OF THE STUDY The study of literature relating to predicative and functional structures of thinking provoked me to analysis accumulated research material, which were the effect of investigations the relating different ways of solving non-typical tasks by pupils and university students. I took the trial of the answer the question: Is the analysis of the students solution to the task sufficient to define their way of thinking, and discovery of their dominant cognitive structure - functional or predicative? I analyzed, so far, five of the junior high school students' work and five University student s work. And therefore, the presented in this article data are the result of small analysis the investigative test yet. The first grade junior high school students, whose work has been analyzed, took part in an examination referring to the methods of work on non-typical tasks with the use of a graphic calculator. The tasks were solved by four students, in their free time, that is after school classes, in the period covering their first grade. The extra classes took place once a week. During their work on the task the students used a graphic calculator, which was recording all the students activities step by step. The research material consisted of a film of the students work on the task with the use of a graphic calculator, the recording of a

5 discussion with the students which took place after they had finished their work on the task, and also the students work sheet. The students of Math and Computer Science Department of Poznań Adam Mickiewicz University who are specializing in teaching, have participated for many years in a research referring to the ways of working on non-typical tasks with the use of new technologies as well as in a classical, pencil and paper way. The research group consists of about 10 students every year, who during the academic year attempt to solve a dozen or so tasks. The first attempt to solve a problem happens without a IT, while the second one is performed with a chosen tool, that is with either a graphic calculator or a free computer software called GeoGebra. The research material collected during the performance of a certain work stage without using the IT, that is only paper sheets, has been analyzed. ANALYSIS OF STUDENTS WORK The analysis of the way in which the examined student had found the ninth element in Raven's test, either on the basis of a discussion with the examined student on the detailed strategy of work which they had applied or in the course of using advanced research tools such as EYEtracker, which enables to follow eyeball movements, very thoroughly allows us to define which of the two ways of thinking, the predicative or the functional one, is dominant for a particular person. Is the analysis of the students solution to the task sufficient to define their way of thinking? In order to answer that question I have analyzed the work of first grade junior high school students as well as the Math department students specializing in teaching. In both groups I have analyzed the solution to just one particular problem, different for each group. The remaining part of this article will present the essence of the tasks under the research, random solutions and the analysis of the research material, on the basis of which I have attempted to define the dominant thinking structure. Analysis of the junior high school students' work Task For what values of a the graphs of the function f(x) = ax are perpendicular to each other? A description of two students' work has been presented below. This characterization consists of a scheme including information about the students work stages, accompanied by a commentary on the student's activities.

6 Additionally, there is a detailed description of the part of the problem, which makes it possible to attempt to define the dominating way of thinking when working on that task. The final part of the characterization is devoted to the thinking structure with an attempt to support the choice with proper arguments. Student 1 Student (S1) procedure path Direction components have to be the opposite numbers He observes the graphs of functions y = x and y = -x He looks for the straight line perpendicular to the y=2x line, crossing the point with the coefficients ( -2,1) and the origin of the coordinates For which a values the graphs of the f(x)=ax + b, b 0, function will be perpendicular? Figure 3 The starting point to his reasoning was a graph of mutually perpendicular lines y = x and y =-x. The student noticed that the lines form four angles of the same measure, which means right angles. That enabled him to obtain the graph of one of the straight lines as the result of the rotation of another one by an angle of 90 regarding the origin of coordinates. The method, consequently, allowed him to find the image of a point belonging to the straight line on the line perpendicular to the first one. The student completed his reasoning by comparing the coordinates of the chosen point and its image, he looked for changes which appeared as an effect of transformation. This hypothesis of transformation in result which for given point of straight line the foundling his image on straight line to her the perpendicular he applies for concrete case. On straight line the y = 2x he chooses the point with the coordinates (1, 2), then he marks the point with the co-ordinates (-2, 1) (he tells that he follows on a previous case) and he looks for the formula of function, to its graph which second point would belong and this line would cross by beginning of origin of the coordinates. Student found the formula of the function after doing some calculation on the sheet of paper. The last stage of working it is finding the example of straight line crossing by this point. Realization of several trials with put rule assures the student in conviction that the coefficient a of the second function has to be an opposite number to an inverse number of the coefficient of the first function. The student observing the graphs of the pair of functions y = x and y = -x, that is such, which fulfilled the

7 conditions of task, he tried to find transformation in result which for given straight line foundling straight line perpendicular to its. S1 s work was characterized by their strong concentration on planning and performing the process, whose result would be several trials special cases of solution of task, and then their generalization. The student s thinking was dominated by looking for and analyzing the changes which had resulted from their applying of transformations within the frame of the thinking and acting process that had been planned by them. In S1 s work the functional thinking way is dominant. Student 2 Student (S2) procedure path Direction components have to be the opposite numbers An example y = 5 and y = -5 He observes the graphs of functions y = x and y = -x She looks for the straight line perpendicular to lines y = 2x For which a values the graphs of the f(x)=ax + b, b 0, function will be perpendicular? Figure 4 Student S2 draws in arrangement of co-ordinates the graphs of function, for these which look on perpendicular, it studies in what relationship are coefficients formulas of these functions. This hypothesis student checks on next examples, until to result. Working on that task student has made 22 attempts. The student in the course of his work looks for common features and similarities among the observed objects, that is the values of numbers and position of graphs in arrangement of co-ordinates, their way of thinking is performed in the relation categories. S2 s work on the task is characterized by their focusing on the discovery of the relation the looked for pair of numbers need to occur, and then generalization of features of these numbers. S2 s work is dominated by the predicative way of thinking. Analysis of University students' work The task content Find all the integer pairs (x, y) satisfying the equation: x 2 + x + 11 = y 2

8 The description of the students' work has been presented below. This characterization consists of a detailed description of the part of the problem, which makes it possible to attempt to define the dominating way of thinking when working on that task. The final part of the characterization is devoted to the thinking structure with an attempt to support the choice with proper arguments. University Student 1 Student 1 prior to the beginning work on the task had assumed that they have to transform the equation given in the content of the task in such a way to obtain such its form that would enable them to calculate all the integer pairs satisfying the equation. Each transformation is immediately interpreted with the reference to the instruction given in the task. As a result of the first transformation there are many fractions in the equation, which disturbs the student's thinking process. Figure 5 [bad thought because fractions appeared] The second transformation leads the student to an equivocal distribution of the task part, so he quit the solving process again and start looking for a new idea. Figure 6 [this way also is not good, because of lack of unique factorization] The third transformation, which again results in fractions in the equation, initially seems to be the wrong way once more, however, the student observes that as the result of subsequent two transformations and applying the binominal expansion formulas they would obtain the following equation form (2x+1-2y)(2x+1+2y)=-43, on the basis of which, immediately after solving four

9 systems of equations they will be able to come up with the right solution to the equation. The student finds all the solutions. Figure 7 [also fractions] Figure 8 although 43 it is a prime number, therefore the only possible distributions are: The first stage of the student's work on the task, during which S1 looks for an idea to solve the problem, is characterized by their searching for such a strategy that would bring about a predicted result. A detailed analysis of the course of that stage allows to notice the signs of the student's dominant cognitive structure. S1's work was characterized by their strong concentration on planning and performing the process, whose result would be such a form of the equation given in the content of the task that would enable the application of one of the familiar patterns for solving equations with two unknowns.

10 The student's thinking was dominated by looking for and analyzing the changes which had resulted from their applying of transformations within the frame of the thinking and acting process that had been planned by them. In S1's work the functional thinking way is dominant. University Student 2 Student S2 looking for an idea to solve the task for a given value of x, x being an integer, belonging to the interval [0,11], calculates the value of y. Figure 9 The student observes that for the 12 cases the values of y 2 always have the unit digit equal 1, 3 or 7. They conclude that integer radicals ending with 1,3 or 7 are for instance 121, 361, 441 Following this track the student obtains two out of four solutions to this equation (-11,11) and (10, 11). The student in the course of their work looks for similarities among the observed objects, that is the values of numbers, their way of thinking is performed in the relation categories.s2's work on the task is characterized by their focusing on the discovery of the relation the analyzed numbers need to occur, on the basis of which they formulate the solutions. S2's work is dominated by the predicative way of thinking. University Student 3 The student tries to transform the left and the right side of the equation to such a form that they become similar, or identical. Figure 10

11 They observe that for x= -11, x 2 = y 2,and then y = -11 or y = 11. Consequently they obtain two solutions. Assuming now that y 2 = 121, the result is that the equation is valid for x = 10 or x=-11. Figure 11 The student has obtained four pairs of solutions, all of which had been looked for. However, by using the empiric method of concluding, their reasoning should have resulted in the answer to the question whether those were all the possible solutions. S3's work, like S1's, was characterized by a strong focus on planning and performing the process, whose result should be such a form of the equation given in the content of the task, which would enable the application of some heuristic task-problem solving strategy. The student's thinking was dominated by searching for and analyzing the changes which were the result of the transformations they had applied in the course of their thinking and acting process. U1's work is dominated by the functional way of thinking. ADDITIONAL RESEARCH MATERIAL Unlike case of the junior high school students, where there was no premise referring to the dominating cognitive structure, the university students had solved few selected tasks from Raven's test before they participated in the research. Having finished those tasks they described on a sheet of paper the way in which they had been looking for the missing element, and then they also justified their strategy. On the basis of each of the examined students' way of thinking when solving the tasks from Raven's test, the conclusion is that the dominant thinking structure of S1 and S2 was the predicative structure, while S3's was a functional one. The above analysis of the task solution brings about a conclusion that S1 and S3 were thinking functionally, whereas S2 was thinking predicatively. So in one case the diagnosis of the thinking structure based on Raven's above task. That

12 confirms the fact, which had already been observed by Schwank, that the tendency to a specific way of thinking does not exclude thinking in other ways. CONCLUSIONS An attempt to define the way of thinking of the examined junior high school students and university students on the basis of their solutions has turned out to be not an easy task, particularly in case of the university students work. The working process of the junior high school students has been thoroughly recorded and described due to the possibility of recording their work step by step; this enabled a thorough description of the problem solving process as well as the description of the students way of thinking, and then identification the dominant cognitive structure. The documentation of the students work on the task (without a calendar or a computer) was only a worksheet, which was much poorer that the documentation of the junior high school students. This impossibility of observing every step of the students work narrows and restricts the way a teacher sees the thinking process engaged in the solution. Thus, a better documentation of the students work on the task increases the chances for a proper definition of the cognitive structure. The application of IT together with the tools recording the work on a calculator or a computer allows to observe the process of solving and thinking in a better way, and therefore to better define the students way of thinking. In this article are results of analysis of two junior high school students work and three students work. In total, so far, I have analyzed ten solutions. It is the small test yet, but the attempt to answer the question whether on the basis of a task solution we can conclude about the students dominant thinking structure, has completed successfully. However, the process itself was not equally difficult for each of the examined works. We have to remember, that All of them are tought in the same way to such an extent as the children s nature is one and unified, and the methods is suitable to the development laws and the subject. However, even here there are differences and departures, which take into consideration the children s properties. One child has an easy access to ideas, while the other one the viewing (Sztejnberg, 2002, p.87). I may add that one student prefers predicative thinking another functional thinking during solving specific mathematical problem. Not all can and should become the same and do the same. Thinking is never achieved in the vacuum. It is influenced by both the cognitive and emotional atmosphere, independently from whether you are conscious of it or not. If you have to think mathematically in an effective way, you need to have enough confidence in order to try out own ideas without any fear. Teachers must necessarily understand how important it is for the students to have that self-confidence. They should make an effort to create such conditions in which a student can be successful (Mason, 2005).

13 The understanding of the existence of the aforementioned differences and departures in the students ways of thinking and acting, the capability of diagnosing and defining them certainly may influence positively the quality of communication between a teacher and a student. It may cause a more thorough and deliberate selection of the material for the class work, which may enable an active functioning of students during lesson, therefore active communication. REFERENCES Cohors-Fresenborg e., Schwank I. Indyvidual Difrences in the Managerial Mental Representation of Business Processes, Juskowiak E. (2004). Sposoby wykorzystywania kalkulatora graficznego w procesie nauczania i uczenia się matematyki, praca doktorska, UAM, Poznań. Juskowiak E. (2010). Graphic calculator as a tool for provoking students creative mathematical activity, Motivation via Natural Differentiation in Mathematics, red. Maj B., Swoboda E., Tatsis K., Wydawnictwo Uniwersytetu Rzeszowskiego. Juskowiak E. (2012). Do the IT enable provoking and developing students mathematical activities?, Generalization in Mathematics at all Educational Levels, red. Maj-Tatsis B., Tatsis K., Wydawnictwo Uniwersytetu Rzeszowskiego. Kaune Ch., Nowińska E. (2012). Analiza dydaktyczna lekcji matematyki w oparciu o wybrane teorie ze szczególnym uwzględnieniem aktywności (meta)kognitywnych I dyskursywnych, Seria dydaktyczna Stowarzyszenia MATHESIS, nr 1, Pyzdry. Krygowska, Z. (1977). Zarys Dydaktyki Matematyki, części 1,2,3, WSiP, Warszawa. Mason, J., Burton L., Stacey K. (2005), Matematyczne myślenie,wsip, Warszawa. Nowińska, E. (2008). Myślenie algebraiczne. Współczesne Problemy Nauczania Matematyki, tom 1, Forum Dydaktyków Matematyki, Bielsko-Biała.

14 Polya, G. (2009). Jak to rozwiązać?wydanie trzecie, Wydawnictwo naukowe PWN, Warszawa. Schwank, I. (1999). European Research in Mathematics Education I, Osnabruck. Schwank, I. (1995). On predicative versus functional cognitive structures, European Research in Mathematics Education I, Osnabruck. Sztejnberg, A. (2002). Podstawy komunikacji społecznej w edukacji, Astrum, Wrocław. Schwank, I. (2001). Analysis of eye-movements during functional versus predicative problem solving,