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 Volume 2005, Issue 2, 2005
Learning and Teaching Mathematics  Volume 2005, Issue 2, February 2005
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Volume 2005, Issue 2, February 2005

From the editors
Authors: Mellony Graven and Marcus BizonySource: Learning and Teaching Mathematics 2005 (2005)More LessWe are pleased to have included in this edition learners' work and learners' voices in several articles. We encourage members to continue to send us articles which illustrate exciting methods or innovative work that learners did (such as in the article by Michael De Villiers) or possible 'mis'conceptions which you wish to discuss and / or analyse further (such as in the article by M Faaiz Gierdien). In addition, some discussion or description of learners' experiences and feelings as they engage in a particular activity would be welcomed (such as the article by Bruce Tobias). We also welcome 'mathsart' of learners which we would love to use for our cover.

The curse of the calculator
Author Duncan SamsonSource: Learning and Teaching Mathematics 2005, pp 3 –4 (2005)More LessSo what goes wrong when a pocket calculator is brought into play? Are pupils relying so heavily on technology that mental arithmetic and critical thinking are starting to take something of a backseat? Or is the fundamental rigour of basic calculator usage simply lacking? The issue is no doubt a complex and multifaceted one.

Counting the trees  a statistics activity
Author Eric WoodSource: Learning and Teaching Mathematics 2005, pp 5 –6 (2005)More LessThis activity is designed to give students the opportunity to get some experience with ideas of random sampling and statistical inference in a context that is interesting and easy to understand. The worksheet (given on the following page) is selfexplanatory and can be completed within a typical 35minute block of class time, including some discussion of the results.

A geometrical introduction to the method of completing the square
Author Anesh MaharajSource: Learning and Teaching Mathematics 2005, pp 7 –9 (2005)More LessThe present Grade 11 Mathematics syllabus (Department of Education 2002a) and the new FET National Curriculum Statement (Department of Education 2002b) refer to three methods for solving a quadratic equation. In the FET National Curriculum Statement this is found under Learning Outcome 2: Patterns, Functions and Algebra, and the assessment standard on quadratic equations. Grade 10 learners cover the method of solving certain quadratic equations by the socalled factorisation technique. This implies that learners in Grade 11 have acquired the knowledge and abilities to solve quadratic equations by the factorisation technique. The new methods that they have to be taught are the method of completing the square and the use of the quadratic formula  and the latter itself is derived from the method of completing the square.

Help wanted! The journal's Question and Answer column
Author Alison KittoSource: Learning and Teaching Mathematics 2005, pp 10 –12 (2005)More LessThe following questions on the new FETC Mathematics Curriculum were sent to us by a concerned teacher. We asked Alison Kitto to respond.

A demonstration of convergence
Author Marcus BizonySource: Learning and Teaching Mathematics 2005 (2005)More LessAs a teacher one first encounters resistance to the idea of convergence when one discusses recurring decimals : very few learners will accept easily that 0,99999.... is the same thing as 1. They can accept that there comes a time when it might as well be, but both the notation and the fact that the recurring decimal never ends (so how can we know what its value actually is?), make the idea that it is actually equal to 1 a hard one to swallow. The argument below concerns a slightly less contentious case; it is an easy one to discuss with a class, and I find that the logical conclusion is quite comfortably accepted. Of course the argument is easily adapted to deal with other cases, such as 0,9999... = 1.

Investigating Leibnitz's Triangle in my classroom
Author Greisy Winicki LandmanSource: Learning and Teaching Mathematics 2005, pp 14 –20 (2005)More LessThis article describes a mathematics lesson I had the pleasure to teach to prospective elementary school teachers. I wanted to expose them to the need for algebra and to the different meanings the term "variable" can embrace.

What's the whole?
Author M. Faaiz GierdienSource: Learning and Teaching Mathematics 2005, pp 21 –23 (2005)More LessOn the facing page is the work of a fourth grade student from a class that I taught in a university town in the Midwest in the United States (US). At the time I was busy completing my doctoral studies and took advantage of an opportunity to teach at a local primary or elementary school. In the US the school year begins during late August and ends in June of the following year. The entry of the 8^{th} December date on the student's work shows that the school year had been going for more than three months. The class had not yet done operations using fractions.

Coming full circle
Author Kurt M. CoetzeeSource: Learning and Teaching Mathematics 2005, pp 24 –25 (2005)More LessMy story begins as a Wits drop out in 1989. It was a time when political upheaval was the order of the day. The Apartheid State was still the aggressor, fighting for its survival with its back against the wall. It was like watching a cornered animal putting up its fiercest fight in a futile, last attempt for freedom. As a young first year student at Wits failing to cope with the academic demands, I could too easily identify with the turmoil our country was experiencing.

Galileo and Newton
Author Tickey De JagerSource: Learning and Teaching Mathematics 2005, pp 26 –27 (2005)More LessWe are living in a world that has been completely revolutionized by science. As teachers of mathematics we should have some idea of the part mathematics played in that revolution.

Five Equations that Changed the World, Michael Guillen : book review
Author Tracey HowieSource: Learning and Teaching Mathematics 2005, pp 28 –29 (2005)More LessGuillen has written an easytoread and entertaining account of the lives of the people who developed equations that have had a fundamental impact on our lives today. He writes in the introduction : "I selected five equations from among dozens of serious contenders, solely for the degree to which they ultimately changed our world." (pg 4) He continues to argue, however, that what he has really done is to give a sense of the development of science in society from the 17th century until now. This book, however, is not a dry historical text. The writing centres around the lives of the five scientist / mathematicians who translated their findings into mathematical representations. What comes through clearly in these stories are the tribulations and joys of their lives as well as the sociopolitical contexts of the day. It was a time in history of great change and much of the way in which we view the world now has its roots in that epoch of scientific discovery.

Difficulties in teaching the beginnings of Maths
Author Tickey De JagerSource: Learning and Teaching Mathematics 2005 (2005)More LessMax Born : Flying to the moon is a magnificent triumph of the intellect and a total failure of reason.

Some applications of MS Excel in teaching Mathematics
Authors: M. ElGebeily and Barbara YushauSource: Learning and Teaching Mathematics 2005, pp 31 –33 (2005)More LessMS Excel was originally designed with business and financial applications in mind. However, the program has now grown into powerful software that can be used by virtually all branches of science and engineering. MS Excel has been used in mathematics education by many people, and this usage ranges from lower level to advanced level courses. We shall focus our attention on some of these applications of Excel in Mathematics which are not widely known. It is worth noting that all applications presented here can be implemented using other more specialized software, sometimes even more efficiently or effectively. The reasons for choosing Excel are that the program is widely available, and knowledge of programming is not a prerequisite for most of the tasks to be handled. Additionally, since users assume more responsibility in designing the application, they are in full control of the implementation and can achieve a certain level of creativity. We shall illustrate three different applications that can easily be used in the teaching and learning of mathematics.

Time to change the game
Author Gary FlewellingSource: Learning and Teaching Mathematics 2005, pp 34 –41 (2005)More LessThere are two fundamentally different games being played in our classrooms. I call the game that most of us have experienced as students, 'the knowledge game.' I call the other game 'the sensemaking game.' It is important that we should be able to recognize each game and understand their differing characteristics and purposes. I say this out of a conviction that only one of these games prepares students for life beyond the walls of their schools.

Factorisation : looking for patterns
Author Anesh MaharajSource: Learning and Teaching Mathematics 2005, pp 42 –43 (2005)More LessThe division of a polynomial by a binomial occurs in the context of applying the factor theorem when factorising third degree polynomials in one variable. After learners arrive at the linear factor they find it difficult to arrive at the corresponding quadratic factor. An investigation into the matric examiners' reports for 2001 (KwaZuluNatal Department of Education, 2002; Gauteng Department of Education, 2002) indicated that the technique of dividing a polynomial by a binomial (long division) was / is still taught to learners. The Interim Core syllabus for Ordinary Grade Mathematics 160407707 (Grade 9) had to be phased in by January 1998.

Word problems : friend or foe?
Author Bruce TobiasSource: Learning and Teaching Mathematics 2005, pp 44 –49 (2005)More LessIn my experience many students produce incorrect, and sometimes meaningless relationships to express a word problem mathematically. Sometimes in our efforts to make mathematics meaningful for our students we neglect to take their life experiences into account. Many traditional textbook word problems portray situations that are contrived (e.g. age problems), or that are socially, culturally or genderwise foreign (e.g. problems involving male dominated sports), or that are simply not applicable in real life (e.g. calculating area for wall papering an igloo). Why is it then that we continue to use these problems?

Ezit's rule : an explanation and proof
Author Michael De VilliersSource: Learning and Teaching Mathematics 2005, pp 50 –54 (2005)More LessAt the recent AMESA Congress in Potchefstroom during July 2004, Connie Skelton from Maskew Miller Longman told me about the following interesting discovery of a learner at Bridge House College in Cape Town, while she was still teaching mathematics there. Apparently the learners had been asked to explore some angle properties of a nonregular pentagonal star.