8 Graphing formulas by hand to promote symbol sense

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Graphing formulas by hand to promote symbol sense

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For thirty years I have been involved in mathematics education in the Netherlands in several roles: as a teacher, mainly in upper secondary school (grades 10 to 12), participating in two curriculum pilots, as a constructor of national exams, as a teacher educator, as a board member of the Dutch Society of Math Teachers, and as a member of Committee innovation national math curriculum (Commissie Toekomst Wiskunde Onderwijs, cTWO). In all these different contexts algebra education was discussed: what algebra should students learn and how can they learn this? Below I sketch my personal ideas and experiences teaching algebra and my motivation for the research described in this dissertation.

At the start of this century, a radical innovation in upper secondary education was introduced in the Netherlands: at the same time, a new curriculum (Tweede Fase) and a new education concept (studiehuis) were introduced. Along these innovations, emphasis was placed on learning to learn and on (general) skills. In the mathematics curriculum and lessons the graphic calculator was introduced as a permanently available tool, and during exams a formula-card was introduced, containing information about rules for power and logarithm calculations, and a formula for a linear approximation of a function (tangent). Apart from the regular exam problems, students had to tackle larger mathematical tasks, using their problem solving skills.

The graphic calculator could be used e.g. to solve equations via graphs and/or a solver app which seemed to make by hand activities in algebra dispensable. The graphic calculator could be used as a black-box. As a consequence, structures of equations were hardly studied by the students, and the content of the algebra lessons changed.

Before long concerns were expressed by teachers and other stakeholders who realized that effective use of a graphic calculator also requires algebraic skills such as seeing through the structure of formulas and recognizing important characteristics. By then, universities and students had started to complain that students had not been adequately prepared at school and lacked algebraic skills. Universities introduced pen and paper basic skills tests (including solving simple equations and manipulations of algebraic expressions). Prospective students scored very poorly on these tests. To improve the students’ algebraic basic skills, many universities opted for explicit and exhaustive practice of basic skills by means of many back-to-back assignments, including simple fraction manipulation, expanding brackets, factorizing, etc. Increasingly more emphasis was put on these algebraic basic skills in secondary school math textbooks. Others, for example Wijers and Kemme (2000), stressed the importance of

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general algebraic skills like interpreting formulas, relating graph, table and formula, mathematical modeling, that are needed to solve more complex algebraic problems, and pointed out that algebra had to be meaningful for the students.

During this period, my students practiced by hand skills by working on large problems like the historic problem of l 'Hôpital (Drijvers, 1996) in which the minimum of the function 𝑦𝑦 = √0.42_{− (1 − 𝑥𝑥)}2_{+ 1 − √−0.84 + 2𝑥𝑥 had to be calculated, and calculating}

the minimum of 𝑦𝑦 = 𝑎𝑎𝑥𝑥 + 𝑏𝑏/𝑥𝑥, both with pen and paper, without the graphic calculator. To prepare students for the abovementioned university basic skills, I presented them the university pen and paper basic skill tests. They were able to solve the abovementioned large problems and performed well during the national math exams, but, to my surprise, had trouble with these basic skills tests.

From then on, I decided to pay more attention to by hand algebra activities and found that many students had difficulties learning effective and efficient methods to solve equations by hand, an important aspect of algebraic basic skills. Through ideas from cognitive psychology, I was encouraged to study expert behavior in solving equations by hand: what do they pay attention to when solving equations? Through introspection and interviews with my high-achieving students, who were proficient in this domain, we found that these expert-students use a limited number of categories of equations and could describe these categories (Drijvers & Kop, 2012). I used these categories successfully in my teaching algebra in secondary school and in algebraic courses for prospective university students.

Since the mid-10s, the need for a repertoire of algebraic basic skills which can be performed by hand is currently endorsed by everyone. The Dutch National exams require more algebraic skills by hand, students in the Netherlands have improved basic skills, and the university basic skill tests have silently disappeared. However, this shift towards basic skills had as a consequence that less attention was paid to students’ abilities to see structure in algebraic formulas and their reasoning in algebra (Turşucu, Spandaw & de Vries; 2018; Van Stiphout, Drijvers, & Gravemeijer, 2013).At the beginnings of 2010’s, the national

curriculum was changed again and I participated in the Committee cTWO that formulated new standards. The focus was on extra algebraic basic skills, and therefore analytical geometry was introduced. I was involved in writing the separate algebra section that was added to the curriculum in which algebraic skills were described. At the same time, the

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phrase “calculate exactly” was introduced to indicate that the graphic calculator could not be used in solving the problem.

The school textbooks have increasingly started earlier with algebraic basic skills and the algebra results of the National exams seemed to improve. However, students continue to have problems with algebra: it is very abstract for them and not very meaningful. Many students seem to use memorized tricks, and hardly learn to read through formulas. Since 2002, I have also been working as a math teacher educator at ICLON-Leiden University Graduate School of Teaching. In that role, I contributed several chapters on algebra to the Handboek Wiskundedidactiek (Handbook Mathematic Didactics), organized by Anne van Streun (Drijvers, Van Streun, & Zwaneveld, 2012). These writing experiences stimulated me to take the opportunity to start a PhD when it was offered by ICLON, which made it possible to investigate how students’ abilities to read through algebraic formulas and to give meaning to them might be promoted.

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