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Creative Education, 2010, 2, 115-120 doi:10.4236/ce.2010.12017 Published Online September 2010 (http://www.SciRP.org/journal/ce) Copyright © 2010 SciRes. CE 115 Using an Interactive Computer System to Support the Task of Building the Notions of Ratio and Proportion Elena Fabiola Ruiz Ledesma Escuela Superior de Cómputo del Instituto Politécnico Nacional, Departamento de Posgrado, México. Email: elen_fruiz@yahoo.com.mx Received February 27th, 2010; revised July 18th, 2010; accepted July 25th, 2010. ABSTRACT This article describes the design and general outcome of app lying a computer system tha t includes interactive a ctivities to the topics of ratio and proportion. The work was undertaken with Mexican students attending primary school grade six (11 year olds). In designing the activities, our work was based on the studies of researchers who have focused on such topics, as well as on the work of researchers in the field of computer scien ces and education techno logy. Support of the activities designed is found in psycho-pedagogy, in knowledge of mathematics and in the fields of computer sci- ences and education technology. Keywords: Ratio, Proportion, Interactive Activities 1. Introduction There is little doubt of just how important it is that stu- dents attending b asic education classes develop cogn itive structures in the area of mathematics, structures that pro- vide them with the basis needed to successfully meet the academic demands of the next school system level. Hence development of software for basic education in mathematics becomes just as significant in the current Mexican context because incursion of computers into that educational level raises the need to analyze which pedagogical strategies recommended by Mexican re- searchers to teach the concepts of ratio and proportion can be incorporated into an educational software system, and precisely how that should be done. Consequently the authors of this article have focused their attention on supporting the task of building the no- tions of ratio and proportion. And in doing so, they have developed a computer system that uses technology to carry out interactive activities. In this article, the authors introduce a teaching pro- posal developed in a computer system with interactive activities that make us e of technology. The design of the proposal activities was based on a pr ior study undertak en by Ruiz, [1] with students who course six grade of pri- mary school and that deals with the topics of ratio and proportion. These topics were selected in view of the fact that their instruction is begun in primary school and that they serve as the basis for subsequent concepts, such as those of direct proportional va ri at i on or li near function. 2. Portrayal of the Problem To identify whether the ratio and proportion teaching activities designed through a computer system enable the 11 year-old student to build notions or ratio and propor- tion, developing said student’s qualitative proportional thought and aiding him/her to recover the sense of his qualitative proportional thought. Qualitative proportional thought is supported by lin- guistic recognition, creating comparison categories such as large or small. Qualitative thought also in clud es things intuitive that are based on experience and that are em- pirical, and things perceptual that are supported by the senses. Qualitative proportional thought refers to activities that enable students to count measure and employ quan- tities in procedures. 3. Theoretical Basis 3.1. On Ratio and Proportion According to Piaget [2] one can see in 11 and 12 year old subjects the presence of the notion of proportion in dif- ferent areas, notions such as: spatial proportions (like Using an Interactive Computer System to Support the Task of Building the Notions of Ratio and Proportion Copyright © 2010 SciRes. CE 116 figures), the relation between the weights and length of arms on a scale, probabilities, etc. [3], also states, based on his experiments, that children acquire their qualita- tive identity before their quantitative conservation, and further distinguishes between qualitative comparisons and true quantification. In fact for Piaget the notion of proportion always be- gins qualitatively and logically prior to being structured from a quantitative standpoint. He stresses that in order for a student to develop his / her qualitativ e proportional thought that student must necessarily begin with the no- tions of enla rgement and reduction, followin g the idea of a photocopier or scale drawing, assuming that at a very early age the student is able to recognize what is propor- tional. According to Piaget and Inhelder [3] after the student develops his / her perceptual ability, an ordering takes place by way of comparisons, which can be seen when students use phrases the likes of “larger than…” and “smaller than…” and which are known as verba l catego- ries. In this sense Piaget says that during the transition from things qualitative to things quantitative, the idea of order appears without a quantity having yet emerged, an event that Piaget calls intensive quantifications. Subsequently when students use measurements to make comparisons they first confront parts of the object and superimpose one figure on top of another and after- ward use a measuring instrument, be that a conventional instrument or not. Freudenthal [4] states that compari- sons make it possible to measure and measurement is shown by way of two modes: direct and indirect. The direct mode of comparing is when an object is superim- posed onto another object, while the indirect mode is when there are two objects (A and B) and a third element to compare (C). With respect to the emphasis that should be placed on early education of ratio and proportion, Streefland [5-7], states that the point of departure should be qualitative levels of ratio and proportion recognition and that use should be made of didactic resources that foster devel- opment of perceptual patterns in support of the corre- sponding quantificatio n pr ocesses. The didactics of mathematics are referred to as an essential activity in teaching ratio and proportion, as is the importance of didactic tools developed by the de- signer. Freudenthal’s Didactic Phenomenology [4] is particularly mentioned, together with other background considered in order to attain a realistic building of ma- thematics. Freudenthal indicates that comprehension of ratio can be guided and deepened by way of visualizations. He moreover states that such visualizatio ns can be illustrated by way of detailed constructions in which the drawings are differentiated and in which the drawings depict which points of the original and image actua lly coincide. An example of the foregoing is in two contig uous figures one of which is an enlargement or reduction of the othe r, and in which the same linear ratio can be established in each segment of the figure. Freudenthal suggests that when working with the ratio of longitud es, flat figures is used as a means of representation because of their global expressiveness, in the sense that a student’s qualitative and quantitative comprehension is facilitated by visual perception. Ruiz and Valdemoros [8] found that the students she worked with presented differen t difficulties, specified as: a) The students’ qualitative though t dealing with propor- tionality has not been explo ited to the utmost, which was observed when they demonstrated that they were cen- tered on one of the dimensions of the figures they had been asked to reduce or expand; b) In some students, things qualitative are barely raised as an introduction to things quantitative, given that in the linguistic categories detected among them, one found the following: “it is larger than…”, “it is smaller than…”, which reflects a certain understanding of proportion; yet among these same students one did not find other categories through which they showed a greater understanding of the idea of proportion; c) They appeared confused when establishing relations between quantities, which is why is became necessary to emphasize that in order to reach the notion of ratio. 3.2. Technological Elements Lim [9], states that technology makes it possible for teachers to be more flexible thus enabling them to ad- dress the different needs of students with varying levels of capabilities – all of whom may be sharing one single classroom- by using software that can be adapted to the teaching and to the particular conditions of each student or group. Technology makes it possible for teachers to divide their groups of students into teams and to work with each one at their own pace. What is more, technolog y can make it possible for the students to determine the pace at which they feel comfortable working. Some research has been undertaken for the purpose of finding out if usage of a Web-based environment bene- fits learning. The study carried out by Galbraith and Haines [10] shows that students who use a computer in their daily mathematics learning enjoy mathematics. They like the flexibility provided by the computer and spend a great deal of time at the computer to complete a task, and enjoy trying out new ideas. The researchers also concluded that Web-based applications increase levels of confidence, motivation and interaction. Nguyen and Kulm [11], Combs [12], Gourash [13] and Using an Interactive Computer System to Support the Task of Building the Notions of Ratio and Proportion Copyright © 2010 SciRes. CE 117 Engelbrecht and Harding [14,15], point out that usage of computers in educational terms enables students to find the meaning of what they are doing in that their ability to discover is developed and they are able to delve deeper. Working with the points mentioned by the researcher in the previous paragraphs, Table 1 depicts a specifica- tion of the indicators that refer to ratio and proportion, as well as to the didactic actions associated to those indica- tors. 4. Methodology Under this heading the authors of this article have in- cluded the subjects with whom work was undertak en, the elements of the computer system used to carry out didactic activities and the didactic activities themselves. Subjects The research was carried out with a sample of 29 stu- dents all in the same grade six class of a primary school in Mexico City, to whom the activities designed were applied. Sequence of activities and their presentation in the computer system Taken into account in the task of designing the se- quence of activities are the didactic actions specified in the theoretical framework, the computer activities, as well as the theoretical elements entailed in the concepts of ratio and proportion. Activity 1. Choose the reduced or enlarged figure of the figure provided, by way of visualization. This activity is based on ideas of “reduction” and “enlargement” supported by models of the scale and photocopier drawings-type experiment, using things perceptual and observation. This was used as the point of departure in view of that indicated by Piaget, [12], Streefland, [5-7] and Ruiz y Valdemoros [8], regarding the point that early instruction on ratio and proportion needs to begin with qualitative levels of that learning, which is why activities that do not require use of quantities for solution of the activities are first used. Table 1. Objectives, indicators and didactic actions of the concepts of ratio and proportion. Concepts Objectives Indicator Didactic Actions Intuitively establish ratios Compare Superimpose figures Use verbal categories such as “one side fits twice in the other side” or “one side is one third of the other side” Ratio: Relation between two magnitudes expressed through a quo- tient Explicitly establish ratios Express the ratio in the form of a fraction Count the sides of a square (in a grid) Use a table Count the sides of a square and wri te out the r a ti o as a frac tion . Qualitative proportional thought -Visualize -Use linguistic expressions Choose reduced or expanded fig ures. Enlarge and reduce figures. Use verbal categories, such as “larger than” or “smaller than” Transition from qualitative to quantitative propor- tional thought -Compare -Measure indirectly Superimpose figures Count sides in the squares of a grid Proportion: Re- lation of ratios or the equivalence of two or more ratios Quantitative proportional thought -Measure directly -Use the rule of three or of the excluded third Measure using a conventional instrument Use the table Carry out number operations Table 2. Contains the relation existing between Table 1 activities and the actions to be undertaken on the computer. Didactic Actions Actions on the Computer Superimpose figures Drag the mouse Use verbal categories Use voice Count the sides of a square (in a grid) Use a pencil, like in Paint Choose reduced or enlarged figures Click on the option Expand or reduce figures Use a pencil, like in Paint Measure using a conventional instrument Use ruler Use the table Table to the filled in by the student Carry out number opera ti ons Calculator Consideration has furthermore been given to Freuden- thal’s points that deal with guiding and deepening com- prehension of ratio through visualizations. By the same token the figures designed in the com- puter system relate to the knowledge possessed by 11 year old children, to whom the figures are aimed. Exam- ples of the foregoing could be a boat, a bus, a star, a dog, albeit all drawn in straight line seg ments. How the concepts appear in the computer system Shown are four similar figures, all of which have small differences. One of the four figures is also depicted in a double or triple linear amplification or reduced by half or to a third of the linear size. The user is asked to choose among the four figures and find the figure that is a reduced version of the original. (See Figure 1). Using an Interactive Computer System to Support the Task of Building the Notions of Ratio and Proportion Copyright © 2010 SciRes. CE 118 Figure 1. Source: Created by the authors. After having chosen one of the figures, the choice is then analyzed and an answer is immediately provided bearing the result of the analysis. The user is asked if he / she wants to try again, the answer is received and, de- pending on that answe r, the exer cise is either restarted or work is continued with the next exercise. Activity 2. Choose the figure that is a reduction of amplification of the figure provided, by way of com- parison. Case 1 Superimpose one figure over another Superimposition of one figure on top of another en- ables subjects to recognize relations of similarity among the figures in intuitive terms. The action of comparing figures is the beginning of measurement, yet without using a conventional instru- ment since it is achieved by superimposing the figures just as specified by Freudenthal, 1983. How the activ i t y appears in the comp ut er system? The student has the option of using his / her mouse to drag any of the four figures in order to place the figure on top of the original and review the result, by way of visualization, to see if the figure is an enlargement or reduction on all sides by the same amount. See Figure 2. Case 2 Use of a grid The transition from qualitative to quantitative propor- tional thought is achieved by using the grid [1] Counting is used and the measurement unit is one side of a square in the grid in which the figures are portra yed. The result of the process of counting the sides of the figures is used in order to establish quotient relations with the results obtained, th at is to say ratios. This is also done for the purpose of establishing relations of equiva- lence between t wo rat i os or proportions. How the activity appears in the computer s ystem The student is presented with a figure on a grid, as well as another empty grid in which he / she can draw and use the grid itself as a means of support. The student Figure 2. Source: Created by the authors. is asked to draw the figure at double or half or one third, etc., of its original size (See Figure 2). The drawing in the first grid is a means of support, used to compare with the drawing done by the student. The student is told whether his / her answer is correct and asked if he / she wants to repeat the exercise, and the student decides if he / she wants to repeat the exercise or go on to the next one. Activity 3. Use of the table The table was used as a means of representation in or- der to determine internal and external ratios Freudenthal [4]. The students work with proportional variation prob- lems and obtainment of the quantities is not only achieved through use of the operator, but also by estab- lishing relations among ratios. Finally, work was done on equivalence relations as a proportionality relation. Outcomes of the didactic activities Activity 1. Choose the figure using visualization Eleven of 29 students used observation to choose the reduced figure. Whereas the remaining 18 decided to compare the figures by superimposing one figure over the other s. They used the mo use to drag the figu re to see if it was, from the side or width, double, half or one third of the original. All of this was commented upon during the session; and furthermore coincides with Freudenthal [4]. If the figure in question was a circumference, the students compared the radii or the diameters. Conse- quently all of the students were ab le to determine that the answer was the reduced figure, and each student decided for him / herself whether to drag the figure and superim- pose it over the other figure in order to compare them. This student reaction is similar to comments made by Combs [12]. Activity 2. Grid Of the entire group of 29 students, 23 were able to draw correctly on the grid figures that were similar to Copyright © 2010 SciRes. CE 119 those presented to them. The remaining 6 students made two to three attempts before they were successful with the task. The computer system was extremely useful in that it presents the student with the option of working with different figures, thus making for work that is not mechanical. What is more, students are given the oppor- tunity to discover what is taking place. The work reached the level of recognition of ratios as a comparison by quotient of two quantities. The group worked on notation of ratios in the form of a / b fractions, in which b is a different from 0. Activity 3. Use of the table Once again of the entire group of 29, 20 students were able to fill in the table correctly. They accomplished this by deciphering the unit value, in the ev ent that it was not provided, wh ile others established ratios by reading them from the table and writing them down in the form of fractions. All students used the calculator, first deter- mining the operations that had to be undertaken; for in- stance, in Figure 3, reference is made to the case of Luis. When faced with the activity, this is what Luis did: In the case of Luis, one can see that he establishes rela- tions between two variables, the milk and the chocolate bars. In Freudenthal terms, these are external ratios. Manuel is another case: What Manual expressed when solving the activity can be seen in Figure 4. Manuel was asked to provide a conclu sion, as follows: “The conclusion you’re asking me for is that… well like, to get the [number of] chocolate bars you have to use the 6 times table. So, first you need 6 bars for 3 liters of milk, then 12 bars for 6 liters of milk, then 18 bars for 9 liters. So if I compare liters to chocolate bars I can see that it’s double the number of liters in chocolate bars.” There is indeed food for thought in Manuel’s work. Before asking him to provide a conclusion, Manuel had already established relations among the data in one of the columns, the liters of milk column. Freudenthal refers to this as internal ratios. Usage of the table enabled the students to establish relations between magnitudes of figures, to the point of establishing both internal and external ratios (as per th eir definition by Freudenthal). Lastly the students were in- “I saw that 3 fits 2 times into 6, so for 3 liters of milk you need 6 bars of choco- late. In other words, you need double the amount of chocolate bars, so one liter needs two bars, two liters four bars, three liters six bars. I’ll just use the calculator to multiply by two to fill in the data I’m asked f o r” Figure 3. Response provided by Luis in the activity of filling in the table. “I saw how the quantities in the choc- olate bars column changed, because they gave us more numbers in that column. I saw that it began with 6, then there was a blank that I had to fill in, then came the 18 and right after that the 24. I used the calculator to divide 18 by 6, and it gave me a result of 3; then I divided 24 by 6 and it gave me 4. So I took the 6 and multiplied it by 2 and got 12, and that’s the value of the second blank, and the last blank in the column is the result I got from 6 times 5, which is 30.” Figure 4. Manuel fills in the table, first filling in the liters of milk column. Copyright © 2010 SciRes. CE 120 deed able to use the different representation registers – drawing, table and things numerical- when they were solving ratio and proportion problems. One is therefore led to point out that work with th ese interactive activities enabled the students to build the concepts of ratio and proportion, investing them with meaning and sense, ra- ther than just using the algorithm to work on the topics. This is indeed similar to the finding s obtained in the stu- dies by Galbraith, P. and Haines, C. [10], Nguyen and Kulm [11], Combs [12] and (Engelbrecht and Harding [15]). 5. Conclusions According to the findings obtained we are in a position to state that the activities de signed to support students in the task of consolidating the concepts of ratio and pro- portion were adequate given that the students demon- strated a great deal of interest when working with the computer system and worked independently to solve the problems. Their teacher did not have to instruct the stu- dents to compare one figure with another in order to de- termine which one was the reduction, rather the students developed that ability than ks to the activities propo sed in the system. The foregoing leads one to infer that the stu- dents were able to develop both their qualitative and quantitative proportional thought; they showed a good deal of freedom in dragging the mouse, using the grid and filling in the table, activities for which there was no need for their teacher to give them instructions. Working in this manner moreover enabled the students to develop visually and perceptually, in other words they were able to further develop asp ects that constitute their qualitative proportion a l thought. Use of the table made it possible for the students to establish relations among figure magnitudes, establishing internal and external ratios (as defined by Freudenthal). Finally the students were able to use the different regis- ters of representation (the drawing, the table and things numerical) when solving ratio and proportion problems, which enab les the authors of this article to point o ut that working with interactive activities made it possible for the students to build th e concepts of ratio and prop ortion, and imbue those concepts with meaning and sense rather than simply using the algorithm to work on the topics. This is quite similar to the findings obtained in the stud- ies undertaken by Galbraith, P. and Haines, C., Nguyen and Kulm, (Combs, and (Engelbrecht and Harding, The computer system made it possible for the students using it to develop skills that were not developed by all of the other students in their group using pencil and paper and the blackboard. REFERENCES [1] E. F. Ruiz, “Study Of Solving Strategies and Proposal for the Teaching of Ratio and Proportion,” Proceedings of the 22nd Annual Meeting North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 2, 2000, pp. 395-396. [2] J. Piaget, “Psicología del Niño,” Madrid: Ediciones Morata, 1978, pp. 131-150. [3] J. E. Piaget and B. Inhelder, “Las Operaciones Intelec- tuales y su Desarrollo, Lecturas en Psicología del niño, I. Madrid, 1978, pp. 70-119. [4] H. Freudenthal, “Didactical Phenomenology of Mathe- matical Structures,” D. Reidel Publishing Company, Holland, 1983, pp. 178-209. [5] L. Streefland, “Search for the Roots of Ratio: Some thought on the Long Term Learning Process,” Educa- tional Studies in Mathematics, Vol. 15, No. 3. 1984, pp. 327-348. [6] L. Streefland, “Fractions in Realistic Mathematics Educa- tion,” Tesis Doctoral Publicada Por la Kluwer Academic Publishers, Netherlands, 1991. [7] L. Streefland, “The Design of a Mathematics Course a Theoretical Reflection,” Educational Studies in Mathe- matics, Vol. 25, No. 1/2, 1993, pp. 109-135. [8] E. F. Ruiz and M. Valdemoros, “Concepts of Ratio and Proportion in Basic Level Students: Case Study,” Pro- ceedings of the 24th Annual Meeting of the North Ameri- can Chapter of the International Group for the Psychol- ogy of Mathematics Education, Vol. 4, 2002, pp. 1651- 1657. [9] C. K. Lim, “Computer Self-Efficacy, “Academic Self- Concept and Other Predictors of Satisfaction and Future Participation of Adult Distance Learners,” American Journal of Distance Education, Vol. 15, No. 2, 2001, pp, 41-51. [10] P. Galbraith and C. Haines, “Disentangling the Nexus: Attitudes to Mathematics and Technology in a Computer Learning Environment,” Educational Studies in Mathe- matics, Vol. 36, 1998, pp. 275-290. [11] D. M. Nguyen and G. Kulm, “Usi ng Web-Based Practice to Enhance Mathematics Learning and Achievement,” Journal of Interactive Online Learning, Vol. 3, No. 3, 2005, pp. 1-16. |