Algebraic thinking and problem solving using the spreadsheet

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1 Algebraic thinking and problem solving using the spreadsheet Sandra Nobre Doctoral student at Institute of Education, University of Lisbon Introduction Algebra and the development of algebraic thinking are currently deemed important to the mathematical training of all individuals, assuming a leading role for students who wish to pursue scientific and technological studies. However, most students show large difficulties in this area, which results in a lack of motivation and failure in Mathematics. One of the possibilities that make the learning of algebra more efficient, meaningful and attractive is the proposal of tasks involving problem solving and the use of the spreadsheet. The main purpose of this work is to study the role of the spreadsheet in problem solving and in the development of algebraic thinking. The aim is to develop research work in the classroom, in two stages. In the first phase, the work with students is being implemented in Tutored Study lessons, where there is greater freedom in the choice of problems to be proposed, as well as in the time to spend in their exploration. In the second phase, which will take place next year, similar tasks will be implemented in the Mathematics lessons. During both phases the solution of numerical and/or algebraic problems will be proposed, using the spreadsheet in two classes along 8 th and 9 th grades. The research methodology adopted fits within the qualitative/interpretive paradigm. For data collection the Excel files produced by the students will be saved, the dialogues from a few pairs of students will be audio-taped, the sequence of computer frames in Excel will be recorded and the participants observed, generating the researcher s field notes. The problem of the study The teaching of algebra has been a very strong subject in the research in mathematics education (Kieran, 2006). Knowledge in algebra is fundamental; the algebraic language is essential to further studies and to provide access to a wider range of future career choices (Ponte, 2006). Over time, several views on the teaching of this topic have emerged; however, there is still a small perspective on the study of algebra that considers it a mere symbolic manipulation. Usiskin (2004) states that algebra is rarely seen as a living language with a logical structure that can promote connections between various topics. On the other hand, it frequently appears as a dead language, where rules come from nowhere and whose applications appear as simple games. This makes its understanding complex and not always accessible to all students. Some researchers suggest that the difficulties faced by the students can be caused by a lack of preparation for the study of algebra. The traditional curriculum places great emphasis on arithmetic and the important aspects of preparation for the study of algebra are not addressed, making the poor performance in the first year of study in algebra one of the main causes of disengagement (Greenes & Rubenstein, 2008). In order to make algebra meaningful to all students, considering the facts previously mentioned, it is vital to change the way this is taught. The Principles and Standards of School Mathematics (NCTM, 2000) advocate Algebra for all". For example, students must understand the various meanings and uses of letters to represent variables in the context of problem solving. In Portugal, the curriculum for middle school mathematics released in 2007 states the teaching of algebra according to the following main purpose: To develop in students the language of algebra and algebraic thinking, as well as the ability to interpret, represent and solve problems using algebraic procedures and applying this knowledge 1

2 and capabilities in the exploration and modelling of situations in various contexts." (Ponte, J., Serrazina, L., Guimarães, H., Breda, A., Guimarães, F., Sousa, H., Menezes, L., Martins, M. & Oliveira, P., 2007, p.55). The methodological indications of this curriculum consider the use of algebraic expressions in representing conditions and data from problems, in a convenient way, using letters to denote unknowns or variables and introducing expressions with variables linked to a context. Thus, problem solving is an activity that enhances the potential of algebra (Canavarro, 2007; NCTM, 2000, Pereira & Saraiva, 2008; Ponte et al., 2007). On the other hand, current technological progress can help improve the teaching/learning of algebra. On a daily basis, there are many people who work with a spreadsheet and who inherently use algebra, even though not realising it, due to not associating this with school algebra. The spreadsheet can be used to help introduce the algebraic language. This does not mean that the paper and pencil algebra becomes obsolete; it is necessary to continue to learn how to solve simple equations and simple transformations by hand. However, school algebra needs to change to become relevant to the students and it is important to explore its relationship with technology, so that algebra becomes accessible to all (Usiskin, 2004). The spreadsheet is recognized by several authors (eg, Ainley et al., 2004; Dettori et al. 2001; Rojano, 2002) as a useful tool in problem solving and in particular in the development of algebraic thinking. The symbolic representation in the spreadsheet of the relations within a problem is initiated through the nomination of columns and the writing of formulas. This feature provides a stimulating working environment that favours a greater understanding of the relations of dependency between variables and encourages students to submit gradual algebraic solutions rather than arithmetic methods (Rojano, 2002). This tool also helps students to think algebraically, based on cell references to express their thinking, creating a bridge between natural language and algebraic notation (Ainley et al., 2004). The spreadsheet is hybrid and able to cohabit in a world of alternation/transition between arithmetic and algebra (Haspehkian, 2005); it is a good teaching solution to help students in the transition from arithmetic to algebra (Kieran, 1996; Rojano & Sutherland, 1997). Research focus and questions The main purpose of this project is to analyse the role of the spreadsheet in problem solving (numerical and/or algebraic) and the development of algebraic thinking in students, across 8 th and 9 th grades. Hence, the question guiding this study: What is the role of the spreadsheet in students work on solving numerical and/or algebraic problems and in the development of algebraic thinking? The following targets are set: - to analyse how students build representations in the spreadsheet in order to give meaning to problem solving and how do they justify these representations; - identify the type of use given to the spreadsheet; how and when to use that tool and how to combine it with pencil and paper; - to examine how students use algebraic language to solve problems; - describe the impact of the spreadsheet in learning and use of algebraic concepts (eg variables, expressions, equations); - consider the qualities and ownership of the spreadsheet presented by the students and the opportunities that they recognise in this tool. Research methodology 2

3 Methodological options Given that we want to understand how algebraic thinking is developed by students in solving problems with the use of the spreadsheet and to analyse the role of this tool, a qualitative methodology following the interpretative paradigm is the most suitable for this study. Participants The participants are researcher s students of two classes from a school located in the Algarve s coastal area. Concerning the students, it is important to say that problem solving was one of the strategies adopted by the teacher to improve students success in Mathematics in the 2008/2009 academic year, when the students were attending 7 th grade. Throughout this experience, the students had access to some technological tools in the classroom; however their knowledge of Excel was limited. The use of problem solving in a systematic and continued manner revealed a set of original thinking and strategies, and students creativity, which would not have been otherwise indentified. In general, they showed appreciation about problem solving; this strategy proved to be important to improve some of the students success in the subject. One of the strengths of the work on problem solving was revealed through the various typologies used (Amado, Nobre & Carreira, 2009). Content, ability and process problems were proposed (Palhares, 1997). In the content and ability problems, the difficulties caused by the need of unequivocal activation of mathematical school concepts were evident. Within the process problems, that involve mainly the search and implementation of a strategy that generates a model of the described situation, there was a wider variety in the strategies used and a greater involvement of the students. It was concluded that it is important to implement, in the classroom, various types of problems, for either a more robust learning of mathematical content, or to make the students feel that they are able to solve problems. In the early educational intervention, which is in the basis of this study, students attend 8 th grade. This choice was due to the fact that they already had contact with algebra, namely in the study of algebraic concepts as the notion of the unknown, 1 st degree equations and others. Although they already possess sufficient knowledge to solve some of the problems using algebraic language, in most cases, a large number of the students use another type of strategy. Data collection During data collection several tools will be used in order to obtain data that describes in detail the situations that occur. The first stage started in September 2009 and will run up until June It was implemented in the Tutored Study classes, due to being a privileged space in the development of mathematical competency. The work consists of the use of numerical and/or algebraic problem solving with the use of Excel. In these classes, the students have access to a computer and working in pairs has been favoured. Initially, during problem solving, the students were encouraged to use Excel and some of the activities were geared for the students to begin to engage with the functionalities of this tool. Recordings have been made: of the Excel files produced by the students, audio recordings of the dialogues between two pairs of students and the sequences of computer frames created in Excel. During key moments throughout the school year Diagnostic Tasks are being used; these consist of the presentation of a numerical/algebraic problem accompanied by a proposed resolution already created using the spreadsheet, but without having access to the formulas used. Several questions at a conceptual level are asked and finally, a question that will look for the students technical skills to find a resolution in Excel. These Diagnostic Tasks are used throughout the school year and will have an increasing degree of complexity, namely in the domain of the Excel capabilities. The second stage of data collection will begin in September 2010 and will run up until May

4 The intervention in this stage will be implemented in mathematics lessons. The work on 2 or 3 problems using the spreadsheet in 9 th grade will be proposed in each of the following curricular topics: equations systems, inverse proportionality, 2 nd degree equations and functions. As in the first phase, recordings of the screen sequences on the computer during the activity on problems, as well as the recording of the students dialogues during the exploration of the tasks, will be made. In this stage, it is also planned to interview the pairs of students selected during the study of each topic. During the implementation of the educational intervention, in the two academic years, the participant observation will also be essential as it allows a continued proximity to the phenomena under study and will always be accompanied by the writing of field notes. Data Analysis In this study, the data analysis involves essentially a content analysis. Once the audio recordings, the video of the frames sequence in Excel and interviews are transcribed, a deductive definition of the categories of analysis (Mayring, 2000) is foreseen -an approach that has been proven effective in an exploratory study already developed (Nobre, Amado & Carreira, 2009). 4

5 References Ainley, J., Bills, L., & Wilson, K. (2004). Construting meanings and utilities within algebraic tasks. In M. J. Høines & A. B. Fuglestad (Eds), Proceedings of the 28 th International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 1-8). Bergen, Norway. Bogdan, R. & Biklen, S. (1994). Investigação qualitativa em educação uma introdução à teoria e aos métodos. Porto: Porto Editora. Canavarro, A.P. (2007). O pensamento algébrico na aprendizagem da Matemática nos primeiros anos. Quadrante, 16(2), Dettori, G., Garuti, R. & Lemut, E. (2001) From arithmetic to algebraic thinking by using a spreadsheet. In R. Sutherland, T. Rojano, A. Bishop & R. Linz (Eds). Perspectives on school algebra (pp ). Dordrecht: Kluwer. Greenes, C. & Rubenstein, R. (Eds.) (2008) Algebra and algebraic thinking in school mathematics: seventieth yearbook. Reston, VA: NCTM. Haspekian, M. (2005). An instrumental approach to study the integration of a computer tool into mathematics teaching: The case of spreadsheets. International Journal of Computers for Mathematical Learning, 10(2), Kieran, C. (1996). The changing face of school algebra. In C. Alsina, J. M. Alvares, B. Hodgson, C. Laborde & A. Pérez (Eds.), International Congress on Mathematical Education 8: Selected Lectures (pp ). Seville: SAEM Thales. Mayring, P. (2000). Qualitative Content Analysis. Forum Qualitative Sozialforschung / Forum: Qualitative Social Research [On-line Journal], 1(2). [Retrieved 5 January, 2009, from NCTM (2000/2007). Princípios e Normas para a Matemática Escolar. Lisboa: APM. (Portuguese Translation of the original published in 2000). Nobre, S., Amado, N. & Carreira, S. (2009). Manifestações do pensamento algébrico na resolução de problemas. In J. Fernandes, M. Martinho & F. Viseu (Orgs). Actas do XX Seminário de Investigação em Educação Matemática. Viana do Castelo: Centro de Investigação em Educação da Universidade do Minho. (Published in CD-ROM, ISBN: ) Palhares, P. (1997). Histórias com problemas construídos por futuros professores de matemática. In D. Fernandes et. al. (Coords.), Resolução de problemas na formação inicial de professores de matemática, (pp ). Aveiro: GIRP. Pereira, M. & Saraiva, M. (2008). O sentido do símbolo na aprendizagem da álgebra em alunos do 7º ano de escolaridade. In R. González et al. (Eds). Investigation en education Matemática XII. Badajoz: Sociedade Española de Investigation en Education Matemática, SPCE e APM. Ponte, J. P. (2006). Números e Álgebra no currículo escolar. In I. Vale et. al. (Orgs.). Números e Álgebra na aprendizagem da matemática e na formação de professores. (pp.5-27). Lisboa: Sociedade Portuguesa de Ciências da Educação. Secção de Educação Matemática. 5

6 Ponte, J. P., Serrazina, L., Guimarães, H., Breda, A., Guimarães, F., Sousa, H., Menezes, L., Martins, M. G., & Oliveira, P. (2007). Programa de Matemática do ensino básico. [Retrieved from Rojano, T., & Sutherland, R. (1997). Pupils strategies and the Cartesian method for solving problems: the role of spreadsheets. Proceedings of the 21 st International Conference for the Psychology of Mathematics Education (Vol. 4, pp ). Lathi: University of Helsinki. Rojano, T. (2002). Mathematics learning in the junior secondary school: Students access to significant mathematical ideas. In L. English, M. B. Bussi, G. A. Jones, R. A. Lesh & D. Tirosh (Eds.), Handbook of international research in mathematics education (pp ). Mahwah, NJ: Lawrence Erlbaum. Usiskin, Z. (2004) A significant amount of algebra. (NAW 5/5 nr. 2 Junho de 2004) [Retrieved 8 April, 2009, from ] 6