PROBLEM SOLVING IN AND BEYOND THE CLASSROOM: PERSPECTIVES AND PRODUCTS FROM PARTICIPANTS IN A WEB-BASED MATHEMATICAL COMPETITION 1

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1 12 th International Congress on Mathematical Education Program Name XX-YY-zz (pp. abcde-fghij) 8 July 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not change this part.) PROBLEM SOLVING IN AND BEYOND THE CLASSROOM: PERSPECTIVES AND PRODUCTS FROM PARTICIPANTS IN A WEB-BASED MATHEMATICAL COMPETITION 1 Hélia Jacinto Research Unit of the Institute of Education of the University of Lisbon helia_jacinto@hotmail.com Susana Carreira FCT, University of Algarve & Research Unit, Institute of Education, Univ. of Lisbon scarrei@ualg.pt Being aware that school is just one of many places where youngsters learn, the research community has been stressing the need to deepen knowledge about the role and relevance of beyond school mathematical learning contexts. In this paper we are focusing on the mathematical problem solving activity that occurs at Sub14 - a beyond school web-based competition. Our purpose is to describe and understand how do participants engage in solving mathematical problems within the competition, and what are their perspectives regarding the differences and the similarities of the problem solving activity in the mathematics classroom and in the competition Sub14. Participants perspectives show an overlap between mathematical activity developed in those two contexts. The analysis of participants productions illustrate how they merge knowledge about problem solving acquired in this beyond school context with mathematics content knowledge, learnt at school. Keywords: mathematical problem solving; beyond school learning; web-based competition. INTRODUCTION Portuguese schools provide a large number of mathematical competitions for students to enroll in, according to their personal preferences and adding up to all class oriented activities. In this paper we discuss how a particular mathematical competition is influencing the participants problem solving skills in the south of Portugal. Sub14 is a web-based mathematical problem solving competition, addressed to years old students, and it is organized by the Mathematics Department of the Faculty of Sciences and Technology of the University of Algarve ( Sub14 comprises two stages. The Qualifying consists of ten problems, each one posted online every two weeks. The Organizing Committee posts every new problem and provides updated information on the website, which also allows participants to send their answers using a simplified text editor and attach a file of any format, containing their productions on solving the problems. The participants may solve the problems working alone or in small teams, using their preferred methods and tools, but they must send their solution and a complete explanation of their reasoning. The Committee assesses every answer and replies to each participant with some constructive feedback about the given answer. Participants are welcome to complete or improve their answers, according to the feedback received. Participants must have answered correctly to, at least, eight problems, to attend to the Final. abcde

2 In this paper we will focus on part of a broad research study, anchored in the Sub14 s mathematical problem solving activity, and underline some implications of the results obtained which are triggering new paths for research. Particularly, we will present the perspectives of the participants regarding the differences and the similarities of the problem solving activity, which they engage in at the mathematics classroom and at Sub14. There are four focuses in the theoretical approach that we used: (a) looking at mathematics as a human activity, (b) taking problem solving as an environment to develop mathematical thinking and reasoning, (c) exploring the concept of being mathematically and technologically competent and finally (d) considering the role of home technologies in beyond school mathematics learning. In this paper, we will focus on mathematical knowledge and problem solving skills, fostered both in the classroom and in the online mathematics competition Sub14. THEORETICAL BACKGROUND Mathematical knowledge and problem solving As Stanic and Kilpatrick (1989) stated, problems have been a constant in the mathematics curricula, but only in recent decades the debate has arisen regarding problem solving and its importance for the learning of mathematics (NCSM, 1978). The work that has most inspired the development of problem solving theories is undoubtedly that of George Polya. Ever since, problem solving has inspired a multitude of studies, with just as many purposes, and a common thought: the notorious importance of improving the ability to solve problems to the development of mathematical skills (APM, 1988/2009; APM, 2007; ME, 2007). Polya (1978) described problem solving as the art of discovery. He considered that a student is doing mathematics, when solving a problem that challenges his curiosity and evokes his creative abilities (p. V). Polya s widely known problem solving model emphasizes the importance of spotting and exploring a brilliant idea, through a questioning stand that allows looking at the problem from different perspectives. Such model comprises four stages: (i) understanding the problem the solver should read and analyse the problem carefully, interpret and identify some relevant aspects such as the data or the conditions; (ii) devising a plan the solver must decide which is the best strategy for that specific problem; (iii) carrying out the plan the solver must be persistent in executing the devised plan; (iv) retrospective analysis the solver should reconsider and review the path taken and check the solution. In addition to Polya s model, we considered three other problem solving models, due to its relevance: the three-step strategy proposed by Schoenfeld (1985); the model proposed by Lester, Garofalo and Kroll (1989, quoted by De Corte, 2000), which establishes the cognitive processes involved in problem solving; and the five stages model of self-regulated learning created by Verschaffel (1999). A brief analysis, focusing on the processes included in these four models, reveals that the different phases almost match each other. The main differences between these models lies in the theoretical assumptions and motivations that support them but, at the level of the structure and organization of processes, the distinction lies in the extent and refinement of each phase of each model. The model presented by Polya seems to be the one that best captures the consistency of the ideas proposed by the other authors. Abcde+3 ICME-12, 2012

3 Problem solving as a major activity for the learning of mathematics Jacinto, Carreira Learning to solve problems is the main reason to study mathematics, as claimed by the NCSM in In fact, problems have taken a prominent place in school mathematics (Stanic & Kilpatrick, 1989) for such a long time that, at the heart of the western society, endures the idea that a good student is necessarily a good problem solver. In Portugal, in line with international trends, problem solving is seen in curricular documents as a kind of major activity in mathematics (APM, 1988/2009, p. 42), for students to strengthen, broaden and deepen their mathematical knowledge (ME, 2007, p. 6). This view on mathematical problem solving entails a conception of mathematical knowledge that is not reducible to proficiency on facts, rules, techniques, and computational skills, theorems or structures. It moves towards broader constructs that entails the notion of mathematical competence (Perrenoud, 1999; ME, 2001) and problem solving as a source of mathematical knowledge. In solving a problem there are several cognitive processes that have to be triggered, either separately or jointly, in pursuing a particular goal: to understand, to analyse, to represent, to solve, to reflect or to communicate. One of the purposes of mathematical problems should be to introduce and foster mathematical thinking (Schoenfeld, 1992), adopting a mathematical point of view, which impels the solver to mathematize: to model, to symbolize, to abstract, to represent and to use mathematical language and tools. Many authors suggested several approaches to teaching problem solving, thus, it can be seen in various forms and its teaching can assume several functions in the mathematics classroom. Hatfield (1978) presented three perspectives: teaching for problem solving which focuses on the acquisition of mathematical tools (algorithms, techniques and procedures) that are obviously useful to solving problems; teaching about problem solving focused on the teacher and how to guide students in processes that lead to the solution, often using Polya s model; and teaching through problem solving that focuses on the discovery of new content or mathematical techniques, resulting from the problem solving activity, i. e., mathematical concepts are introduced with a problem (Schroeder & Lester, 1989). This latter perspective reflects the kind of teaching that Polya and other researchers emphasize in their work. Sub14 s mathematical problem solving activity Since Sub14 is a competitive activity, the Organizing Committee recognizes its importance in facilitating mathematical problem solving and in complementing the curricular tasks. Therefore, the Committee encourages creativity at several levels namely in the reasoning, finding a strategy, or communicating those ideas. However, every participant must present a complete and detailed explanation of the reasoning developed during the solving process, and such mathematical thinking should be as clear as possible. Another Sub14 s feature is the nature of the feedback sent to the participants by the Committee aimed at appreciating their ideas and stimulating self-correction (Jacinto, Amado & Carreira, 2009). Such feedback may complement the work that teachers provide in their classrooms regarding problem solving. Some teachers neglect the creativity involved in finding a path and, sometimes, student s self-confidence is shattered to the point that he is not fully able to appreciate mathematics and its nature. ICME-12, 2012 abcde+2

4 The problems posed by the Committee are word problems, in the sense that Borasi (1986) states: the context is fully expressed in the statement, the formulation of the problem is unique and explicit; the desired solution is almost exclusively unique and precise, and the strategic approach involves combining several algorithms or techniques known (p. 134). Although there is not a strong concern in following the syllabi and its contents, the Committee considers the mathematical knowledge needed to solve each problem, since Sub14 is addressed at students from two different school levels. METHODOLOGY This paper reports part of a larger study focused on the mathematical competition described previously, and that had as major objective the understanding of the participants perceptions regarding (i) the mathematical activity, (ii) the competences involved in taking part in the competition and (iii) the role of the technological tools they have used. In particular, this study aimed at understanding how the students engaged in solving mathematical problems within the competition, and what were their views on problem solving when comparing school to the competition environment. The methods we used in that larger study were essentially qualitative, since we were looking for the participants perspectives regarding their activity while involved in Sub14 and to their particular approaches to the problems proposed. The data collecting involved the gathering of documental data, such as the electronic files containing solutions from participants of several editions of the competition and the feedback sent to the students, by . We then administered a questionnaire to 86 participants who attended the Final, aiming at a broader identification of their personal views regarding their habits of using ICT, on their attitudes towards mathematics, problem solving and their participation in the Sub14. We also applied semi-structured interviews to eleven of the Finalists. We selected those interviewees intentionally in order to obtain diversity of perspectives, since they showed different problem solving and technology skills along the qualifying stage of the competition. The interview aimed at bringing out the competitors perceptions regarding mathematics, problem solving and the use of ICT in two different settings: the classroom and the Sub14. This paper presents data from these three sources since we seek to combine the quantitative results from the questionnaire, with short excerpts from interviews to elucidate a particular point of view and the analysis of the solution sent by a team of three participants to a numerical/algebraic problem. Combining such information, we hope to describe the problem solving activity that participants experience at the Sub14 and at their classrooms. DATA ANALYSIS AND MAIN RESULTS Participants perceptions on problem solving in and beyond the classroom According to the participants, the problems that teachers occasionally propose in their mathematics classes differ from those that they solve for the Sub14. Besides being scarce, problem solving activity in the classroom is often associated with a specific mathematical content or procedure that the teacher aims to develop. Therefore, the challenging aspect of the problem diminishes considerably since the students already know what procedure or rule they Abcde+3 ICME-12, 2012

5 must use in order to reach the solution. This type of problems, described as specific problems by an interviewee, usually appears after the teaching of a particular content. Some participants also state that more challenging problems are usually proposed as homework, which, in a way, gives problem solving a secondary place in the class mathematical activity. In the classes we do things related to the contents we are studying. [Ana] Yes, related to the contents the teacher is teaching. [Gonçalo] From time to time, our teacher gives us a mathematical challenge to solve at home. In the class, he does not. [Filipe] Although some school problems may not involve a direct application of a formula, the path to the solution does not require an in-depth reasoning or a creative solving process. In fact, the contestants stress that their teachers do not usually recommend looking for a different and innovative method when solving a problem. Roberto was one of the participants that showed to be most disappointed with such attitude: Often, my math teacher tells me that I must solve the problem using a certain method... and I always find another [Roberto] On the contrary, throughout the competition, the participants are encouraged to generate and develop their own problem solving methods. This kind of mathematizing seems to be almost absent from the classroom activities, since most part of the interviewees do not recognize, immediately, the Sub14 problems as standard mathematical problems. It s not entirely different, but there are some differences. When I start solving, sometimes I don t even think of mathematics, I think I have to solve it, I must find a way! [Lucia] I had to find other ways to get the solution. And that had little mathematics. [Andre] Oh! It is very different! You don t just solve and use techniques, you have to think and look for where the problem is ( ) It s not just about using mathematics and numbers. [Isabel] However, we identified an overlap between the Sub14 and the classroom mathematical activity, despite the fact that the latter arises in close connection with the curriculum content. The most persistent contestants seem to be able to appreciate the intrinsic worth of each environment. In fact, 80% of the finalists who replied to the questionnaire stated that the participation in the Sub14 contributed to the improvement of their results in Mathematics at different levels. They recognize their personal development of skills in interpreting and understanding of the problems posed or in selecting relevant information. Their reasoning has also improved as well as their focusing abilities. Other participants mentioned that they have discovered new techniques and methods, unusual in the classroom, stressing that the Sub14 problems have made them think differently from what they were used to. The problems posed by the Organizing Committee are different from the ones teachers pose at the mathematics classroom, according to these respondents views. The interviewees acknowledge an improvement in their reasoning abilities and in explaining what they thought during the problem solving process. In fact, they consider these developments so important, that they take them into the classroom. ICME-12, 2012 abcde+2

6 I improved my methods to get the solution. The way I explain how I got there has also improved. Now I can explain myself better. [Gonçalo] I pay more attention to little things ( ), and that has an impact on the tests and on the problems that my teacher presents, that may have some tricks. [Lucia] Participant s production on a competition problem Alexandra, Filipe and Angela have known each other for as long as they can remember, so they like working together and they always help each other. The group revealed an increasing engagement in the elaboration of their answers to the problems, working hard to be as clear as possible regarding the process developed to solve each problem. The following figure shows the statement of one of the problems posed by Sub14. Most part of the contestants solved this problem using a trial and improvement approach. Cages and Parakeets Carla has several parakeets and birdcages. If she puts a parakeet in each cage, one bird will be left out. If she puts two parakeets in each cage, then a cage will be empty. How many parakeets and cages does Carla have? Figure 1 Problem #9 of the qualifying stage This group of participants produced and sent, via , a text document explaining their activity during the problem solving process, as shown in figure 2. Assuming that we have 5 cages, which we call "G", and 6 parakeets, which we call "P", for the first hypothesis of the statement, we have: That is, using an equation, for the first hypothesis we have that the number of cages plus 1, equals the number of birds, i.e., G +1 = P For the second hypothesis, let us say we had 4 cages and 6 birds. Then we would get: That is, forming groups of 2 parakeets we would have three groups of parakeets and four cages. The number of birds over 2 plus 1 equals the number of cages. So we use this information to build the equation: Using these equations in a system with 2 variables, we have: 1 st Hypothesis: 2 nd Hypothesis: Answer to the problem: Carla has 4 parakeets and 3 cages. Figure 2 Alexandra, Filipe and Angela s solution to problem #9 Abcde+3 ICME-12, 2012

7 These participants use a typically formal mathematical language that is clear in their vocabulary:... then we have to ; assuming ; i.e., 1 st hypothesis. They also define variables, G and P, in order to translate the problem into two equations in two variables. They find each equation by interpreting, manipulating and mathematizing an example. As Alexandra stated in the interview, they had to learn how to solve those two simultaneous equations. It seems that they face this new learning as inevitable, mostly because they were not able to solve the problem using any other strategy. They asked their mathematics teacher to help them, and he taught them to solve a system of two equations in two variables, a specific content they would be learning only on the following grade. Looking at their solution, we can find the different phases of Polya s problem solving model, namely, understanding and interpreting the conditions they use particular cases and manipulate images to translate the statement into formal mathematical language; devising a plan they use the previous knowledge to build a system of two equations in two variables; executing the plan they solve the system of equations; looking back they check if the solution that they found verifies the initial conditions, that they call 1 st hypothesis and 2 nd hypothesis. In these students productions, we found evidences of their mathematical competences. They seem to be able to formulate valid arguments using visualization; they have a predisposition to find patterns and regularities, and to generalize; they are skilled in analysing numerical relations and in explaining them using different types of languages that include the use of symbols; they are able to use equations as representing a problematic situation and they can solve those equations (ME, 2001). They also show traces of digital fluency since they use the text editor as the main support for presenting their solution, but it comprises much more than plain text. Besides using an equation editor, the participants customized the initial birdcage image that only contains one bird. By editing the image, they managed to create a cage with no birds and a cage with two birds, using those new images and the initial one throughout the solution, in order to translate the formal statements into iconic representations. The file sent by the participants describes, in a powerful manner, the processes used to solve the problem posed. It reveals a clear concern with intertwining common sense knowledge, that characterizes the typical first approach of these participants, to formal mathematical contents, that characterizes their common approach to problem solving in the classroom. We used trial and error, but also used equations. We prefer to do it mathematically since this is a mathematics competition; we like to see the calculations on paper. [Alexandra] Despite explaining why they combined these two levels of mathematizing, their reasons are strictly related to a personal perspective on the nature of the mathematical activity, which involves numbers, calculations, algebraic relations, and formalization. CONCLUSIONS The formative aims of the problems posed at Sub14 are essentially in line with the perspective of giving students the experience of mathematical thinking and the opportunity to bring forth mathematical models and particular kinds of reasoning, which many authors claim to be ICME-12, 2012 abcde+2

8 essential in the mathematics classroom learning (Lesh & Doerr, 2003; Lesh & Zawojewski, 2007; ME, 2001; Schoenfeld, 1992). However, from the interviewee s standpoints, those objectives are not so obvious in their mathematics classroom. From the data analysis, we have evidences that point to the main differences between problem solving activity in the classroom and that of Sub14. This is particularly interesting because these contestants have an inside knowledge of these two environments, namely on how they work and which are their specific purposes. The type of problems presented in each of these environments differs since the problems that teachers pose at the classroom are targeting the application of a concept or a specific mathematical procedure. The Sub14 problems demand the drawing up of a personal approach, which involves combining facts and mathematical procedures but also the use of common sense knowledge or technology. The problem solving that these youngsters seem to find in their classes fits the perspective of teaching for problem solving (Hatfield, 1987), in which the teacher starts by teaching a specific topic of the syllabus and then asks students to solve a specific problem. Another distinctive aspect is the communication of the strategy. Whilst in the classroom the focus is the product that results from solving the problem, that is, communicating the solution, at the Sub14 it is crucial to allow full access to the process elaborated, that is, communicating not only the solution but also the solving process. At Sub14, parents or teachers are welcome to assist in the describing processes of the contestant s reasoning, so that it becomes clear and accurate. The tools that participants use to communicate their strategies also vary from those normally allowed in the classroom. The solving processes also differ greatly between the two environments. Traditionally, students have a limited time to solve a problem in the classroom, which may affect the nature and complexity of the problems posed, they only have the help of colleagues and the teacher, and may use a very limited number of tools. At Sub14, participants have 15 days to plan, create and adjust their strategy; they can look for the help of colleagues, family or teachers, and are free to use any technological tool that allows them to find the solution of the problem. We identify, therefore, a back and forth of learning experiences and knowledge between the Sub14 s problem solving activity and the classroom s problem solving activity (Figure 3). Figure 3 Mathematical knowledge exchange between school and Sub14 There is no doubt that problem solving involves the mobilization of a set of content knowledge, most likely learnt in the classroom context. However, it is not just a recap of contents, as it involves the selection and organization of knowledge in the establishment of a Abcde+3 ICME-12, 2012

9 strategy and its implementation. Once the solution is found, the participant must explain the process and reasoning followed, which fosters the development of various forms of representation that must be expressed as clearly and accurately as possible. The approach used by these contestants to the Cages and Parakeets problem allows us to understand how they engage in solving mathematical problems within Sub14, using digital tools, and how they merge their perspectives on school problem solving and beyond school problem solving. Polya s model is useful when analysing such engagement, since we were able to identify and describe two distinctive features in their activity: (i) experimentation and testing particular cases, supported by technology, characterized by their initial approach to the problem, which contrasted with (ii) the influence of school mathematics in the generation of a formal mathematical technique to sustain the solution. Such contrast meets their perspectives regarding mathematical problem solving in each of the two environments focused. At Sub14, contestants are free to develop a deep understanding of the problem, they are welcome to use all digital tools needed, may use a variety of forms of representation, they value and customize the appearance of the solution. At school, there is a focus on formalizing, on using mathematical symbols or specific techniques for specific problems. Thus, this solution brings together competences that competitors develop beyond the classroom as the understanding phase of Polya s model, with those developed in the classroom such as the implementation phase of Polya s model. In addition, as Sub14 is an inclusive competition, every participant has the same opportunity to get to the Final, despite being extraordinarily mathematically competent. Therefore, we think that this particular kind of accomplishment, which may result from the combination of common sense and formal learning, suggests that these participants are problem-solving competent, in the Sub14 context, and may not be extremely good achievers in school mathematics. Further research will focus on seeking a clear understanding of the mathematical problem solving activity that youngsters engage in when participating at Sub14, considering the role of the digital tools that they use on a daily basis, and what implications might those technological experiences have in the learning of school Mathematics. 1 This work was partially supported by project PTDC/CPE-CED/101635/2008 entitled Mathematical Problem Solving: Views on an interactive web-based competition - Sub12 & Sub14, and the PhD student grant SFRH/BD/73363/2010, both from Fundação para a Ciência e Tecnologia. REFERENCES APM (1988/2009). Renovação do currículo de matemática. Lisboa: APM. APM (2007). Princípios e Normas para a Matemática Escolar (NCTM, transl.). Lisboa:APM. Borasi, R. (1986). On the nature of problems. Educational Studies in Mathematics, 17. De Corte, E., Verschaffel, L., & Op t Eynde, P. (2000). Self-regulation: A characteristic and a goal of Mathematics Education. In M. Boekaerts, P. Pintrich, & M. Zeidner (Eds.), ICME-12, 2012 abcde+2

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