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The bar model as a visual aid for developing complementary/variation problems Eugenia Koleza To cite this version: Eugenia Koleza. The bar model as a visual aid for developing complementary/variation problems. Konrad Krainer; Naďa Vondrová. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Feb 2015, Prague, Czech Republic. pp.1940-1946, Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education. <hal-01288480> HAL Id: hal-01288480 https://hal.archives-ouvertes.fr/hal-01288480 Submitted on 15 Mar 2016 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. The bar model as a visual aid for developing complementary/variation problems Eugenia Koleza University of Patras, Department of Primary Education, Patras, Greece, [email protected] In this paper, we report the preliminary findings of a study that considered how third grade students represent multiplication and division problems using the bar model as a tool. Initial results indicate that students are able to create visuals, representing problem’s structure, and based on these visual representations they can formulate division problems given a multiplication one. Keywords: Multiplication, division, complementary problems, bar model. INRODUCTION Use of diagrams is considered an efficient strategy in teaching and learning mathematics and especially in mathematical problem solving. “From the most elementary class to the most advanced seminar, in both introductory textbooks and professional journals, diagrams are present, to introduce concepts, increase understanding, and prove results. They thus fulfill a variety of important roles in mathematical practice” (Mumma & Panza, 2012, p. 1). Diagrams make plain the quantities in the story context and the relationships that exist amongst them, limit abstraction and thereby aid in the problem-solving process (Bishop, 1989). Draw a diagram is a well known strategy for mathematical problem solving (e.g., Polya, 1957; Schoenfeld, 1985), grounding in the belief that generating a diagram enables deeper understanding of the situation described and facilitates the conceptualization of the problem structure (van Essen & Hamaker, 1990). Not all diagrams are beneficial or can easily be used by the students. Vosniadou (2010) distinguishes between external representations that are perceptually based (grounded on everyday observations) depictions and those that represent conceptual models (theory –based). Pictures used in mathematics and science textbooks, number lines and bar models that are CERME9 (2015) – TWG13 usually used on mathematical problem solving, are conceptual models. In our study, we used the bar model as a visual support for the resolution of simple (one operation) multiplication problems and the formulation of the two corresponding division problems. Children’s difficulty to understand the close relation between multiplication and division was a problem noticed very early in mathematics teaching literature. Nevertheless, in most western curricula, multiplication and division, as well as addition and subtraction are taught separately. Paraphrasing Herscovics (1989), it is as if students are taught the syntax of operations, without the semantics. In other words, students know the algorithms (the rules of the ‘grammar’) but they do not understand the meaning and their relations. This situation explains the difficulty that pupils often face when solving equations in algebra with an operational view of equality (Wagner & Parker, 1988). “The ‘one-thing-at-the-time’ design, provide fewer opportunities for ‘making connections’ compared to those adopted in eastern cultures” (Sun, 2013, p. 13). “In Chinese elementary schools, addition and subtraction are introduced simultaneously, and subtraction is introduced as the reverse operation of addition. Division is also introduced as a reverse operation of multiplication” (Cai, 2004, p. 110). As Cai refers (p. 112), “representing quantitative relationships in different ways will not only help students develop deeper understanding mathematics, but also will help them develop their flexibility of using equations to solve application problems”. Giving an example of a variation in a multiplication problem, Sun (2011, p. 104) explains that “within the problem set, there are two concepts of multiplication and division behind three similar problems made with 4, 6, and 24. Example problem: How many trees do 6 lines need so that each line can have 4 trees? Variation problem 1: 1940 The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza) How many trees will each line get if we plant 24 trees in 6 lines? Variation problem 2: How many lines do we plant if we plant 24 trees in order so that each line has 6 trees? Clearly, the intent of One Problem Multiple Changes is to enable students recapitulate the general relationship of multiplication and division, and the meaning of equal from the problem set 4 × 6 = 24, 24 ÷ 4 = 6, 24 ÷ 6 = 4 […]. The task draws students into a space of relations as opposed to directing attention to the object itself.” THEORETICAL FRAMEWORK meanings by the use of the bar model. In an initial phase the bar model may serve as a model of the mathematical structure of a word problem. Later, through a process of vertical mathematization, reflecting on the relationship between their actions upon a diagram and the effects of those actions, students may generalize and abstract those actions to successfully solve problems of the same semantic structure. In this way, bar model becomes, to those students, a model for the mathematical structure. Bar model is particularly useful for problems that involve comparisons, part-whole calculations, ratio and proportion. The ‘model method’, also known as graphical heuristic, In this paper, we will restrict to the use of the bar consists of the use of rectangular bars to represent model as a visual support in order to formulate mulnumbers rather than abstract letters to represent tiplication and division problems. unknowns in word problems. This method is often used in education systems of many countries under The Singapore education system’s approach concernvarious names: tape diagrams – in Japan (Murata, ing the use of diagrams, has close relation with the one 2008), strip diagrams – in US (Beckmann, 2004), or bar several soviet researchers (Bodanskii, Mikulina, in models – in Singapore (Hoven & Garelick, 2007). This Davydov, 1991/1969) has used in their studies concernspecial kind of diagrams “are clearly designed to help ing algebra word problems. For example in Figure children decide which operations to use. Instead of rely- 1 (on the left) is the diagram of the problem: “In the ing on superficial and unreliable clues like key words, kindergarten, there were 17 more hard chairs than soft the simple visual diagram can help children under- ones (labeled M). When 43 more hard chairs were addstand why the appropriate operations make sense” ed, there were 5 times more hard chairs than soft. How (Beckmann, 2004, p. 43). Cai and colleagues (2005) con- many hard and soft chairs were there?” as presented in sider the ‘model method’ as one of the big ideas related Bodanskii (1991/1969, p. 302), and on the right the bar to algebraic thinking in the Singaporean elementary model representation. curriculum. “Children solve word problems using the ‘model method’ to construct pictorial equations that In both cases, the common element is the concept of represent all the information in word problems as a “unit”. Units are not simply single discrete entities, but cohesive whole, rather than as distinct parts. To solve instead may be composed of one or more ‘shapes’ (in for the unknown, children undo each operation. This the diagram) of various types (Davydov et al., 2000). approach helps further enhance their knowledge of Taking as a “unit” the number of soft chairs, the sum the properties of the four operations”(p. 8). In other 17+43=60 is translated as 6 units, giving, thus, the anwords a basic property of the bar model is that it can swer of the problem. The problem solver is able to support an exploration and visualization of the ‘doing’ reason from the very diagram that was created as a and ‘undoing’ processes in mathematics. According model of the situation given. The algebraic equation to Hall and colleagues (1989) when the structure of that correspond to this visual solution- the model for a problem is recognized, a formal representation of this kind of relations- is just a step further. this relationship may be constructed. Departing from meaningful tasks, students may construct personal The research question behind our study was: Figure 1: Similar representations of the same problem 1941 The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza) researcher who hasn’t conducted a teaching experiment independently, but who wishes to do so, should engage in exploratory teaching first. It is important that one become thoroughly acquainted, at an experiMETHOD ential level, with students’ ways and means of operating in whatever domain of mathematical concepts and In this study, 19 third graders (aged 8) were involved, operations are of interest” (Steffe & Thompson, 2000, p. belonging in a class of a primary school in Patras. 274). In other words, we wanted first to explore the role During a period of eight 45-minutes sessions the this specific visual representation may have in helpstudents followed the regular lessons (teaching of ing students to understand how multiplication and the multiplication table) that were enriched by the division are connected, and their eventual difficulties, bar model as a way to solve and represent problems, in order to design our ‘teaching experiment’. Because and mainly as a way to connect multiplication and “incomplete understanding […] can result in inapprodivision, in a unified scheme. This connection was priately designed artifacts or artifacts that result in designed only for the needs of our experiment, given undesirable side effects.”(March & Smith, 1995, p. 254) that these two operations are taught separately in the Greek mathematics curriculum. Tasks given during DISCUSSION instruction were arithmetic and algebraic tasks (given a problem, the task asked for his complementary The first of the 8 lessons (the corresponding textbook ones-focus on relations) with three (group, restate and objective was the multiplication table of, 2, 5 and 10) vary) of the five semantic relations (the other two were began with a teachers’ question. change and compare) identified by Marshall (1995). Data was collected and analyzed from field notes, phoT: What is a problem? Who can make a probtos, pupils’ written work, and final tests. Learning lem? gains were assessed by means of a word-problem test S: I have 80 candies and I give 10 candies to just after the end of instruction and a second one a each one of 8 children month later. T: Is this a problem? What are you asking? In order to have a problem you must ask Hereafter, the problems given in the tests. for something. For example, 8 children bought 10 candies each one. How many Vary-problem in the short-term test: Grandma has in bought in total? Lets make a picture of the her sac 45 sweets in 5 bags. How many sweet are there problem. (The teacher designs the bar in each bag? model). In order to have a problem, we must ‘hide’ something. Vary-Problem in the final test: Grandpa gave to each one of his 5 grandkids 12 euros. How many euros he gave in total? Can the bar model support students in the resolution and formulation of multiplication and (complementary) division problems? For each problem was asked to the students: “Departing from the problem, with the same story and the same numbers make two other problems and ‘design’ them”. Teaching students for short periods of time could not serve as a basis for a solid understanding of their thinking and how it might be influenced by the use of a certain visual representation. Our ‘teaching experiment’ has not the characteristics described by Steffe and Thompson (2000), but it was an exploratory one which aimed only to the evaluation of the bar model as a visual aid for the resolution and formulation of complementary multiplicative-structure problems. “Any Figure 2 T: S: Can someone else make another problem, with the same story? We can ’hide’ the number of candies that each child will take. (The teacher changes the previous model) Figure 3 1942 The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza) T: S: T: T: S: T: Another problem? We can ‘hide’ the children. So, I will delete the squares. If I had 40 candies, and 10 children, how many candies could have each child? 4 Can you ‘design’ the problem? During the next 2 sessions students worked with the multiplication table (according to the official schedule), and solved multiplication problems following a ‘didactic contract’ of 4 precise steps that had emerged from the previous lesson: (1) we read first and we ‘design’ (make a representation of ) the problem, (2) in order to make the 2 ‘inverse’ (complementary) problems, we “hide” a number from the design, (3) we write (in words) the ‘inverse’ problems, and (4) we solve the problems. The complementary/inverse problems were introduced by the teacher as the two variations of an initial problem, and the bar model was presented as a tool of organizing problems’ data. Each complementary/ inverse problem was creating by ‘hiding’ a number on the bar. During this process, students faced two major difficulties: (1) Expressing in natural language the problem that they had already represented by the bar model, and (2) representing by the bar model the quotative/ measurement division problem-variation. For example, while the multiplication problem «If I save 8 euro in a week, how many euros I will have after five weeks?» and –‘hiding’ the 8- the corresponding partitive division problem “If I saved 40 euros in 5 weeks, how many euros I saved in a week” were easily represented by the bar model, it was not the same for the ‘quotative division’ problem. The rule of ‘hiding’ could not be applied: while in the “partitive division’ problem they had to ‘hide’ the numbers, in the ‘quotative division’ one, they had to ‘deconstruct’ the bar model. In the following pictures (Figure 4) we see (to the left) the incomplete diagram made by a student (reproduced in the middle), and the one suggested by the teacher (to the right). Teacher’s proposition in order to face the difficulty was to reformulate the problem as “How many 8s we want in order to make 40?” We present three different treatments of the “quotative division” problems representation during instruction a) An arithmetic treatment: no use of the bar model “30 ducks fly in groups of 5. How many groups of ducks are there?” Figure 5 b) The ‘quotative division’ as a subtraction: A “filling-in” use of the bar model. “I have 45 stamps in pockets of 9 stamps. How many pockets I have?” The thought behind the model was: “I take off each time 9 stamps from the 45 stamps” Figure 6 The ‘quotative division’ as the inverse of the multiplication: An ‘algebraic’ use of the bar model. Figure 7 Figure 4 1943 The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza) After instruction a final test was given. Hereafter, we comment students’ competency in using the bar model for the resolution of the ‘vary-problem’ given in this final test (see page 4). “Grand papa has 60 euros and he wants to distribute them to his 5 grandkids. How many euros will have each one?” From a total of 19 students, a) 13 students formulated correctly the two division-problems. The two main strategies observed during instruction, appeared also in the final test: ―― a mixed (‘filling-in’ and algebraic) strategy -MS(10 students), and ―― an ‘algebraic’ strategy -AS (3 students). Figure 10 c) The rest 3 students were not able to use the bar model at all. For example for the multiplication problem the diagram proposed was We give an example of each case. (MS) Fey Figure 11 or Figure 8 (AS) Joseph Figure 12 Figure 9 CONCLUSIONS The 3 students who used the ‘algebraic’ strategy were the higher achieving students of their class. This outcome confirms similar results by Booth and Koedinger (2007) according to which, higher-achieving middle school students do benefit from the diagrams while low-achieving students perform better on story problems that do not have accompanying diagrams. The results has shown, that the bar model is an effective model, but only for multiplication and partitive division problems. Though most students (13/19) after a relatively short term instruction (eight 45-minutes sessions) could formulate and represent by the bar model a multiplication and the corresponding division problems, further research is needed about the kind of representation that is more appropriate for the quantitative division problems. b) 3 students formulated only the partitive-division problem. 1944 The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza) The fact that a representation may be “clear” for a multiplication problem does not mean that it would be a useful tool for the formulation of the corresponding division problem. For example, lets take the array model. This model in a concrete (on the left) or in a more abstract (on the right) version, under certain conditions may be useful for students to understand the relation between a multiplication and the two complementary-division problems. It is useful, if all information is on the representation, but not if the students must construct the representation by themselves, especially in case of big numbers. The data collected do not permit us to know if the students who had used the algebraic strategy (AS) were acting in a pure algorithmic way, or ‘with understanding’. Eventually a confrontation with the strategies used by the same students in additive-structure problems may offer a more complete explanatory framework. A more accurate analysis of the relationship between the instrument and students’ meanings is required, and that is what is going to be done with the data analysis of the whole teaching experiment. REFERENCES Beckmann, S. (2004). Solving algebra and other story problems with simple diagrams: A method demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14, 42–46. Bishop, A. J. (1989). Review of research on visualization in mathematics education. 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De Corte, T. de Jong & J. Elen (Eds.), Use of representations in reasoning and problem solving (pp. 36–54). London, UK: Routledge. Wagner, S., & Parker, S. (1988). Advancing algebra. In B. Moses (Ed.), Algebraic thinking, grades K-12. Reston, VA: NCTM. 1946