BUILDING A VERSATILE UNDERSTANDING OF ALGEBRAIC VARIABLES WITH A GRAPHIC CALCULATOR

BUILDING A VERSATILE UNDERSTANDING OF ALGEBRAIC VARIABLES WITH A GRAPHIC CALCULATORALAN T. GRAHAM and MICHAEL O.J. THOMAS

INTRODUCTION: Algebra has been recognised for some time as a difcult topic for many secondary school students.

Understanding of the concept of variable is fundamental to further student progress in algebra.

In most areas of mathematics, conceptual entities present themselves with two distinct but complementary faces. They may be viewed as dynamic processes or as static objects, or in the words of Sfard (1991), as an operational and structural duality.

The experience of teachers, and a wide range of empirical research, inform us that children nd great difculty in understanding the algebra of generalised arithmetic (e.g. Kchemann, 1981; Wagner, Rachlin and Jensen, 1984). Researchers suggested that the use of literal symbols to generalise arithmetic relationships is too sophisticated to be useful as an introduction to the idea of variable.

Clearly, students nd difculty with this more sophisticated view of the role of symbols.

In other words, children need to see letters as capable of representing numbers before they can begin to use letters to generalise patterns of numbers.

Objectives: using a module of work based on a graphic calculator which provided an environment where students could experience some aspects of variables and hence begin to build an understanding of them. combine these advantages with the principles and techniques learned from research using the computer and apply them to the graphic calculator. describes their effective utilisation in the classroom to improve student understanding of variable. address the issue of student understanding of variable and describe an experiment to investigate the inuence of allowing students to experience variation and its symbolisation using graphic calculators.

Literature reviewsThere are a number of conceptual obstacles to progress in algebra (see e.g. Tall and Thomas, 1991; Linchevski and Herscovics, 1996; Stacey and MacGregor, 1997) and one of the most important of these is the failure to understand the concept of variable.

Since the concept of variable is more sophisticated than we often recognise and frequently turns out to be the concept that blocks students success in algebra (Leitzel, 1989, p. 29), it is important that students gain some understanding of it before other algebraic concepts are introduced if they are to progress beyond basic processes. In essence, then, one reason that algebra is hard is the wide variety of ways in which letters or literal symbols have been used in algebra and the sophisticated and multifaceted nature of the concept of variable (Schoenfeld and Arcavi, 1988; Wagner et al., 1984). The graphic calculator is now an available, portable and affordable alternative option to the computer for many schools. The value of the graphing capability of graphic calculators is well established and research has shown that it can help build relationships with other representations, such as symbolic forms, which can have a signicant inuence on attainment (Ruthven, 1990; Penglase and Arnold, 1996).

MethodTapping into Algebra was a classroom-based research project, using an experimental design to compare the teaching of variable in algebra with and without the use of the graphic calculator.

1. The algebra moduleThe basic premise of the overall module, which was designed to last about three weeks, was to use the graphic calculators lettered stores as a model of a variable. Each store is represented as a box in which changing values of the variable come and go, and above which sits its label. This model had been used successfully in the research using the computer (Tall and Thomas, 1991), and was fundamental to this study.almost no students would have had previous experience of using graphic calculators.

The rst section of the module comprised an introduction to using them. For example, students were shown how to perform simple calculations using the four operations, to edit key sequences and to store and retrieve numbers using the letter keys.

This sort of activity can help students begin to formulate theories about the consistency with which any given language handles the symbols, and to build an understanding of their purpose.One of the novel teaching aspects of the module was the use of screensnaps, where students were given a screen view and required to reproduce it on their calculator screen.

These screensnaps have the advantage of encouraging beginning algebra students to engage in reective thinking using variables.

Other topics covered included squares and square roots, sequences, formulas, random numbers and function tables of values.

In a pilot study in ve United Kingdom schools using the TI-80 graphic calculator a relative improvement of the experimental students was clearly seen, and on the basis of those results concluded that the students who used the graphic calculators could gain a better understanding of the use of letters as variables, without a detrimental effect on their basic ability to manipulate the symbols as encapsulated objects. The StudyTeachers from six New Zealand schools volunteered to take part in this controlled research project, agreeing to use the algebra module provided with one of their classes.Five of the schools used the TI-82 graphic calculator and the other (school C) used the Casio FX7700GH, giving an opportunity to explore the variable of calculator type.The control groups received corresponding algebra work to that of the experimental group, but were taught using their normal algebra teaching programme comprising primarily whole class, skills-based instruction and assessment, with their usual teacher and his/her presentation style. Hence the experimental and control groups had different teachers.Of the 147 experimental and 42 control students involved in the research project, 118 were from year 9 (age 13 years) and 71 from year 10 (age 14 years), and they covered all ability groups.

The module was taught during terms one and two of 1996 by the classroom teachers, and the researchers were not present in any of the classrooms while the students were learning.

It was considered important not to prescribe the learning micro world that the students worked in, but the teachers were encouraged to use their normal teaching approach. Each student was provided with a graphic calculator.

The classroom groups were given a pre-test and a post-test, based on Kchemanns (1981) study, and comprising 68 questions.

Students were not given their papers or any answers to the questions until after the post-test so that they were unable to obtain coaching or to memorise answers etc.Result and Discussion:As Table I shows, the results in the two schools with control groups suggest a signicantly better performance on the post-test for the students who had used the graphic calculators, compared with no such difference at the pre-test stage, conrming the pilot study results.

The use of the graphic calculators had signicantly improved understanding of the way symbolic literals are used in elementary algebra. In every case, understanding of the use of letters showed a strong improvement.

THE VIEW OF THE STUDENTS AND TEACHERSBoth the teachers and their students were asked to comment freely on their experiences with the graphic calculator teaching module.A formal qualitative analysis of the comments was not intended in this experimental study. Students CommentsTeachers commentsThe majority of students felt that the experience of using the graphics calculator was of benet in improving understanding and making the learning algebra more interesting.Generally, they also found the worksheets clear and easy enough to followThe majority of the participating pupils certainly enjoyed the experience, particularly their experience of the calculators

Overall the teachers comments were very positive and they felt that their pupils algebra did benet as a result of working on the calculator. All the teachers felt that the pupils enjoyed the work on the project, especially the screensnaps. Conclusion:In elementary algebra the use of letters has often been poorly understood. Yet an understanding of variable in mathematics seems to underpin all advanced work and so it is important that students gain condence at handling variables. The evidence presented from this study shows that students can obtain an improved understanding of the use of letters as specic unknown or generalised number from a module of work based on the graphic calculator. Questions such as those requiring students to say when L+M+N=L+P+N or A+B=B, are very difcult for the student who sees the letters as concrete objects, since they will probably think that two different objects can never be the same, or that adding an object necessarily increases something. However those who are thinking in a versatile way see the symbols as procepts, encapsulated objects representing a range of values. Hence it is possible for the values the letter represents to be sometimes the same. The graphic calculator helps to build such a versatile view of letters because the physical experience of the students is that of tapping keys to place different numbers into the calculator stores, changing the values in those stores and retrieving values from them.