The Influence of Symbolic Algebraic Descriptions in Word Problem Assignments on Grasping Processes and on Solving Strategies Jarmila Novotná Marie Kubínová Charles University in Prague <jarmila.novotna@pedf.cuni.cz> <marie.kubinova@pedf.cuni.cz> In our research presented in the paper we concentrate on the influence of elements of symbolic algebraic description in the word problem assignments on grasping processes and mathematisation of problem structures. We also monitor the different interpretations of letters in student s solving processes - the variable and the unknown. Prior students experiences of solving purely arithmetical problems are confronted with their experience of the use of letters in mathematics at an intuitive level during two school periods: the very beginning of algebra and one year later. The stages of working with word problem assignments containing algebraic elements are identified and described. 1. Introduction Many researchers have been concerned about the difficulties which pupils have with the transition from arithmetic to algebra (e.g. Bednarz, Filloy, Goodson-Epsy, Hejný, Herscovics, Janvier, Kieran, Linchevski, Littler, Novotná, Royano, Sfard). In (Filloy & Rojano, 1989) it is stated that Recent researches have pointed to certain conceptual and/or symbolic changes which mark a difference between arithmetical and algebraic thought in the individual. Tools for identifying them differ from analysing cognitive processes connected with purely algebraic problems (e.g. Cifarelli, 1988) to diagnosing the difficulties connected with the transition from arithmetical to algebraic problems (e.g. Bednarz & Janvier, 1994). In the Czech Republic, problems caused by the formalistic approach to algebra survive. At present large changes in the educational system have been prepared that accentuate among others, these problems. Our research represents a contribution to the discussion about the changes. 1.1. Pre-algebra and Algebra in Czech Schools The results presented in our paper are closely connected with the way children meet and use algebraic elements at the pre-algebraic and algebraic levels in Czech school mathematics. To enable the reader to understand this background we briefly describe the main characteristics of teaching pre-algebra and algebra in the Czech Republic. Traditionally, letters (or other symbols, e.g. *, ) are commonly used in school mathematics already from first grade of elementary school. The impulse for introducing letters comes from the teacher and is significantly supported by textbooks and students working materials (working books, working sheets etc.) in other subjects (pre-science, later physics, chemistry, science). When solving some types of problems (word problems, calculations of perimeters, areas, volumes, etc., simple constructive tasks in geometry, dependencies), students are asked to follow rules (given by the teacher, textbook etc.) for describing the solving process using letters. This is demanded even in cases when it is possible to solve the problem easily, e.g. by insight, and when a strict use of letters makes the solution more difficult.
While in mathematics teaching, letters always represent a variable, an unknown, a constant or a parameter (depending on the context) and are manipulated correspondingly, in science teaching (especially in physics) letters are used only as labelled numbers 1. Because operating with letters is introduced later in mathematics than in other school subjects there is a lack of interdisciplinary links. In the mathematics curricula of all three Czech educational programmes, letters are used: In the first six grades at the intuitive level, operations with expressions containing letters are not developed; there is a strong support of acquiring algorithms for calculating with numbers. Students start to operate with letters in a systematic way from the seventh s grade. In most cases acquisition is based on very formalistic knowledge and most often it is taught in an instructive nature. The visualisation is seldom used. The commonly used letters for labelling unknowns and variables are x and y (in the connection of mathematics as a scientific discipline) or first letters of names of objects being considered as unknowns or variables (the attempt to put school nearer to the life). 1.2. Cifarelli s Levels of Reflective Abstraction In his unpublished doctoral dissertation 2, Cifarelli studied the activities and structures developed by college students as they solved a set of algebra word problems. Reflective abstraction was introduces by Von Glasersfeld (1991) as one of three types of Piaget s abstraction (reflective abstraction, reflected abstraction and pseudo-empirical abstraction). It is an important mechanism for explaining how students construct conceptual knowledge. Cifarelli introduced the following four levels of reflective abstraction: 1. Recognition: The level where a student is able to recognize characteristics of a previously solved problem in a new situation and believes that one can do again what one did before. Solvers operating at this level would not be able to anticipate sources of difficulty and would be surprised by complications that might occur as they attempted their solution. 2. Re-presentation: The level where a student becomes able to run through a problem mentally and is able to anticipate potential sources of difficulty and promise. Solvers who operate at this level are more flexible in their thinking and are not only able to recognize similarities between problems, they are also able to notice the differences that might cause them difficulty if they tried to repeat a previously used method of solution. 3. Structural abstraction: This level occurs when the student evaluates solution prospects based on mental run-throughs of potential methods as well methods that have been used previously. The student is able to discern the characteristics that are necessary to solve the problem and is able to evaluate the merits of a solution method based on these characteristics. This level evidences considerable flexibility of thought. 4. Structural awareness: The level where a student is able to anticipate the results of potential activity without having to complete a mental run-through of the solution activity. The problem structure created by the solver has become an object of reflection. The student is able to consider such structures as objects and is able to make judgments about them without resorting to physically or mentally representing methods of solution. 1 The student uses a letter but works with it as though it was a concrete number that he/she is using as a cryptogram. 2 The levels are cited from (Goodson-Epsy, 1998).
1.3. Hejný & Littler s Stages of Student s Coming to a Knowledge of Algebra In the as yet unpublished paper Protoalgebra, major stages which the student must pass through during the evolutionary process before the student comes to a knowledge of algebra. 1. The student is able to translate a concrete /spoken form of a problem into a mathematical symbolic form and understands the meaning of the symbols. 2. The student can solve spoken problems which involve the use of words to represent unknowns. 3. The student can interpret abstract arithmetical symbolism and undertake the necessary operations correctly and with confidence. 4. Statements involving any of the arithmetical operations can be interpreted by the students in whatever form they appear. 5. The students look for patterns. 6. The students are able to verbalise relationships from the recognition of pattern. 7. The student is able to recognise a relationship by means of a pattern and to express this relationship as an algebraic expression or equation. 2. Our Research 2.1. Instruments In our research, we studied differences between arithmetical and algebraic thought in the individual in the situation of one word problem which was modified in three ways: Problem 1: A packing case full of ceramic vases was delivered to a shop. In the case there were 8 boxes, each of the boxes contained 6 smaller boxes with 5 presentation packs in each of the smaller boxes, each presentation pack contained 4 parcels and in each parcel there were v vases. How many vases were there altogether in the packing case? Problem 2: A packing case full of ceramic vases was delivered to a shop. In the case there were 8 boxes, each of the boxes contained k smaller boxes with 5 presentation packs in each of the smaller boxes, each presentation pack contained 4 parcels and in each parcel there were v vases. How many vases were there altogether in the packing case? Problem 3: A packing case full of ceramic vases was delivered to a shop. In the case there were b boxes, each of the boxes contained k smaller boxes with r presentation packs in each of the smaller boxes, each presentation pack contained s parcels and in each parcel there were v vases. How many vases were there altogether in the packing case? 2.2. Methodology We have used the following research methods: didactical analysis of textbooks, analysis of written tests. These methods provide a rich source of data from which we selected elements for the classification of levels of understanding of a word problem structure in case of assignments containing algebraic elements. 2.3. Analysis and Results In the following analyses we will recognise three types of letter roles: variables, unknowns (pre-algebraic and/or algebraic use) and letters as signals labelling here is something to be calculated (closely connected with students prior experiences and teachers demands). 3 3 Theoretically this category could also be indicating an unknown, but in all the solutions analysed above, only those indications mentioned were true.
2.3.1. Problem 1 Problem 1 (one letter used in the assignment) was given to 28 students from grade 7 (age 13) at the beginning of the introduction of algebra. In (Novotná, 2000), Problem 1 was used for the classification of stages of understanding of assignments containing algebraic elements. The following four stages of dealing with the assignment where one element of the language of algebra (number of vases v) was presented. Stage 1: The solver ignores data which are not assigned as concrete numbers. At this level, the solver does not see the letters as representing amounts that need to be taken into account. His/her previous experience of work with letters is forgotten in the environment of the word problem. He/she has only superficial knowledge based on the key words or the layout of the problem. The ability to work with algebraic representations is not developed. Example: 4 parcels; presentation packs: 5x4=20; smaller boxes: 6x20=120; boxes: 120x8=960 Stage 2: The solver is aware of the fact that he/she is asked to work with letters but he/she is not able to understand the meaning of the symbols in the given context. When working with letters, he/she tries to use his/her previous experience in a mechanical way. In most school mathematics situations, letters are only used as labels for something that is to be found by calculations. The amount v is taken as an unknown. Example: v=8x6x5x4 Sometimes more than one experience is recalled and the solver combines several. Stage 3: The solver is aware of the nature of data assigned as letters but the symbolic algebraic description of the situation is not yet fixed in his/her knowledge structure. He/she substitutes a concrete number for v and thus changes the problem into a pure arithmetical one. Example: v=6; 8x6x5x4x60=57 600 Stage 4: The solver is able to work successfully with data assigned in both arithmetical and algebraic languages. The understanding of the problem structure does not depend on the nature of the assigned data. The conditions for the successful use of algebraic methods have already been created. At this level, an abstract lift occurred in the solver's knowledge structure. Example: 8x6=48 smaller boxes; 48x5=240 presentations packs; 240x4=960 parcels; in 1 parcel v vases. 960v vases A more detailed analysis of students written solutions, when taking into account letter types, showed that this four-stage classification is too coarse. Each of the stages can be divided into substages according to the nature of the letters used in the written solution: a) No other letters than those given in the assignment are used in the written solution. b) The solver uses letters (different from v given in the assignment) in his/her written solution. The letter acts as a signal indicating that a calculation is performed there. The influences of the student s previous experience and the teacher s demands are strong. We identified two different forms: Student uses the same letter for calculating different amounts. Student uses another letter for each individual calculation. 2.3.2. Problem 2 Problem 2 (two letters used in the assignment) was given to 23 students from grade 8 (age 14) after one year experience with algebra in school. There are two letters, k and v in the assignment. Their position differs: k occurs as the first letter and is surrounded by numbers from both sides, v is placed at the end of the assigned
information. Solvers always do not take them as letters bringing information of the same quality. Theoretically, the combinations of all for stages mentioned in Problem 1 analysis for k and v can take place. In the following text (i, j) describes the solution in which the stage for k equals i and the stage for v equals j. Theoretically, all (i, j)-situations could occur. Solvers (our longitudinal experience shows that regardless of their age) prefer the variable that occurs as the first. In the case of Problem 2 this tendency is reinforced by the position of k being surrounded by numbers from both sides. Therefore we can expect that there will be more solutions described by the (i, j)-situation with i > j (the solver grasps the first letter at a stage with a higher number than the second one) than those with i < j. The (i, i)-situation means that the solver grasps letters at the same stage regardless of their place in the assignment. In the Table 1, all possible (i, j)-situations are recorded. The sign + represents the combination that occurred in the solution in our study, the sign - represents the combination that did not occur. Our results are in accordance with the above mentioned expectations. k v Stage 1 Stage 2 Stage 3 Stage 4 Stage 1 + - + - Stage 2 + - - - Stage 3 + + + - Stage 4 + + + + Table 1. (i, j)-situations Examples: (1, 1)-situation: 8x5x4 = 160 (2, 1)-situation: k = 8x5x4 = 160 (3, 1)-situation: k = 3; 8x3x5x4 = 480 (4, 1)-situation: 8xkx5x4 = 160k (3, 2)-situation: k = 6; v = 8x6x5x4 = 960 (4, 2)-situation: v = 8xkx5x4 = 160k (1, 3)-situation: v = 2; 8x5x4x2 = 320 (3, 3)-situation: k = 3, v = 2; 8x3x5x4x2 = 960 (4, 3)-situation: v = 6; 8xkx5x4x6 = 960k (4, 4)-situation: 8xkx5x4xv = 160kv Note: There were solutions that cannot be uniquely put into one slot in the table. We put such solution in one group that we call overhang. These solutions need to be analysed individually. As an example we can mention here the solution containing the following results (written one after the other): v = 160k, k = 160 v, which does not fit in any of (i, j) positions. This situation usually means that the solver does not understand properly the problem and tries to offer to the teacher several reasonable results hoping that one of them could be correct and could satisfy the teacher s demands. 2.3.3. Problem 3 Problem 3 (only letters used in the assignment) was given to 16 students from grade 9 (age 15) with two year experience with algebra.
In this case students had already at least two-year experience with algebra. The assignment contained only letters. There were mostly correct solutions, in the others total misperception of the assigned information occurred. 3. Concluding Remarks 3.1. Our Results and Cifarelli s Levels Our solvers did not reach the fourth level, Structural awareness. This is not surprising because Cifarelli worked with students at the college level with a more developed knowledge of algebra. Our students had either none or much smaller experience with algebra. Most of them worked on the level Recognition or Re-presentation. The level Structural abstraction can be detected by those who worked at our stage 4 with all letters. 3.2. Our Results and Hejný & Littler s Stages Students worked at stages 1 to 5 described by Hejný & Littler. Our stages are in accordance with what Hejný & Littler say: It can be argued that students come to a knowledge of algebra through an evolutionary process beginning with the interpretation of abstract symbolism when they are introduced to arithmetical forms and continuing through to the manipulation of algebraic expressions and equations. 3.3. Students Changing Stages during the Solving Process Students when grasping the information given in the assignment and trying to understand and use the assigned information do not always keep to the stage at which they work. They come to another stage when they feel the necessity to do it (either intentionally or unintentionally). In case of Problem 1, in our sample we detected only transitions from a lower to a higher stage. It shows that the student during the solving process improved his/her understanding of the nature of the assigned information and grasped the information hidden in the assigned letter in a more correct way. A different situation occurred in case of Problem 2. Changes occurred in both directions in both variables. The switch to a lower stage indicates that the student was not able to work at a higher stage and tries to reach the stage where he/she is more able to deal correctly with the assigned information. From the above explanations one can deduce that the passage from (i, j) to (i, k) with k < j is more frequent than the passage from (i, j) to (p, j) with p < i. Note: In case of Problem 1 some students operating at stage 4 when solving it moved even to stage 1 when solving the problem with a more complicated structure but containing the same numbers and letter in the same order. It indicates that their recognition of the algebraic nature of assigned letters - variables is only superficial and needs to be further developed. The abstract lift has not been completed, the cognitive gap (Linchevski & Herscovics, 1996) remains. References Bednarz, N. & Janvier, B. (1994). The Emergence and Development of Algebra in a Problem Solving Context: An Analysis of problems. In Proceedings Int. Group for the Psychology in Mathematics Education. Lisbon. Cirafelli, V.V. (1988). The role of abstraction as a learning process in mathematical problem solving. Unpublished doctoral dissertation, Purdue University, Indiana. Filloy, E. & Rojano, T. (1989). Solving equations: the transition from arithmetic to algebra. For the Learning of Mathematics 9, 2, 19-25.
Goodson-Epsy, T. 1998. The role of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra. Educational Studies in Mathematics, 36, 219-245. Hejný, M. & Littler G.H. Protoalgebra. To be published. Linchevski, L. & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30, 39-65. Novotná, J. (2000). Students levels of understanding of word problems. Regular lecture, ICME 9, Tokyo/Makuhari, Japan. Abstract in H. Fujita (Ed.) ICME 9. Abstracts of Plenary lectures and Regular Lectures, pp. 96-97. Tokyo/Makuhari, Japan. Von Glasersfeld, E. (1991). Abstraction, re-presentation, and reflection. In Epistemological foundations of mathematical experience, pp. 45-67. Acknowledgments: The research was supported by the projects GAČR No. 406/99/1696 and Research Project Cultivation of Mathematical Thinking and Education in European Culture.