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 Volume 2006, Issue 3, 2006
Learning and Teaching Mathematics  Volume 2006, Issue 3, February 2006
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Volume 2006, Issue 3, February 2006

From the editors
Authors: Mellony Graven and Marcus BizonySource: Learning and Teaching Mathematics 2006 (2006)More LessWe are pleased to be publishing the third edition of Learning and Teaching Mathematics. This edition is full of exciting articles, covering both primary and secondary school interests, reviews, interviews and maths challenges. We have introduced a new column, which we hope with your input will be a regular feature, "Windows on a Child's Mind". Here you are invited to submit an interesting instance of a child's mathematical thinking. The instance discussed in this edition involves learners' (mis)conceptions of fractions. Hanlie Murray will then be asked to work with her team to provide comment, discussion and reflection on the instance. Our Question and Answer column in this edition is especially useful for everyone wanting to know more about the implementation of the new Mathematics and Mathematical Literacy curricula in the FET band.

Number patterns, cautionary tales and finite differences
Author Duncan SamsonSource: Learning and Teaching Mathematics 2006, pp 3 –8 (2006)More LessI recently included the following question in a scholarship examination for Grade 7 pupils :
Consider the following number pattern in which only the first two numbers have been given : 4 ; 8 ; __ ; __ ; __ ; . . .
Create three different number patterns (each starting with 4 ; 8) by writing out the next three numbers.
In each case explain the rule that controls your number pattern.

Ten times ten is twenty
Author Bertus Van EttenSource: Learning and Teaching Mathematics 2006, pp 9 –11 (2006)More LessFor the second time I had the opportunity to spend three months in a Senior Phase mathematics classroom in South Africa. "The learners can't do the basic calculations" is a recurring remark of the teachers. This plea is often made more explicit by saying : The learners don't know the tables. This is indeed often the case. Here are a few examples of my observations in class.

Help wanted! The journal's Question and Answer column
Author Aarnout BrombacherSource: Learning and Teaching Mathematics 2006, pp 12 –13 (2006)More LessDear Editors of LTM
I was hoping that you could find answers to the following pressing questions regarding the implementation of Mathematics and Mathematical Literacy in the FET band next year.
Regards, Erna Lampen

Adding tops and bottom of fractions
Author Mercy KazimaSource: Learning and Teaching Mathematics 2006, pp 14 –15 (2006)More LessI would like to share with fellow mathematics educators and teachers my experience with some primary school children on addition of fractions. Some months ago I worked with standard 6 and 7 children (about 1013 years old) in Malawi. One of the topics was addition of fractions. I observed that the most common error that the students made was to add the numerators and add the denominators.

HCF, LCM and the number of factors
Authors: Daniel Fresen, John Fresen and Johannes HeidemaSource: Learning and Teaching Mathematics 2006, pp 16 –17 (2006)More LessIn this paper we discuss two results both of which depend on the prime factorization of a natural number. The first result on the highest common factor and lowest common multiple is well known in textbooks on higher mathematics but we have not found it in a single South African high school text book. The second result, on the number of factors, we have not seen anywhere.

Interview with a prizewinning teacher
Author Mellony GravenSource: Learning and Teaching Mathematics 2006 (2006)More LessThere were two 2005 Aggrey Klaaste Maths Science & Technology Educator of the Year Awards, one for the GET phase, the other for more senior grades. The FET winner was Lazarus Lavengwa, from Phophi High School, and LTM's editor Mellony Graven interviewed him to find out what inspires him, and the secret of his success.

How far can we see?
Author Gerrit StolsSource: Learning and Teaching Mathematics 2006, pp 19 –21 (2006)More LessIn this paper an example is presented of how tangent properties of circles can be dealt with in context. The emphasis is on mathematical modelling and dealing with real life contexts. This paper wants to explore, use and discover some of these theorems in the context of a boy sitting in a tree on the lookout for cattle.

Locus  that mythical creature from Ancient Rome
Author Bruce TobiasSource: Learning and Teaching Mathematics 2006, pp 22 –26 (2006)More LessTeaching locus at the secondary school level can be both rewarding and at the same time frustrating. I find it rewarding to teach because for me loci represent a dynamic form of mathematics in that they pull together the more abstract algebra and the more visual graphs. However, very often at the upper secondary school level it appears that students are just not able to appreciate how the relationships between two variables that satisfy given conditions, produce such interesting results, despite the fact that these results can be represented graphically.

Two more simple proofs for the Theorem of Pythagoras
Author Daniel FresenSource: Learning and Teaching Mathematics 2006 (2006)More LessThis is a followup paper to "A Simple Proof for the Theorem of Pythagoras" published in the journal Pythagoras in 2000 by the same author.
The Theorem of Pythagoras is perhaps the most fundamental theorem in all of mathematics. It is so famous and has been proved in so many different ways that one can never be sure that a proof is new : the delight, however, remains.

Kids say the darndest things
Author Modisaemang MolusiSource: Learning and Teaching Mathematics 2006 (2006)More LessBelow I provide an example of an FET college student's work on adding simple fractions. This example exemplifies the way in which students (including college FET students) overgeneralize rules they learn in relation to specific fraction problems. I argue that the lesson to be learnt from such examples is that procedural (rulebased) teaching of fractions does not work! We have to get learners to understand the basic concept of fractions and to understand why certain procedures and rules work when working with specific operations on certain types of fraction.

Windows on a child's mind
Author Hanlie MurraySource: Learning and Teaching Mathematics 2006, pp 29 –30 (2006)More LessIn this feature, we look at specific instances of children's thinking, and reflect a little on the reasons for and implications of the episodes described. We invite all our readers to submit similar examples. Please give a brief description of the background, and if you want to, a discussion of the episode. From our side, we may add comments and discussion.

Making my unimportant mathematical discovery count
Author Craig PournaraSource: Learning and Teaching Mathematics 2006, pp 31 –35 (2006)More LessOne night I was sitting at my desk preparing the next day's lecture of a Functions and Algebra course for preservice maths teachers. We were dealing with the quadratic function at the time and in the next session I was planning to focus on the different forms of the quadratic equation [i.e. the general form y = ax^{2} + bx + c; the turning point form : y = a (x  p)^{2} + q; and the "root form" : y = a (x  α) ( x  ß) where α and ß are roots] and the different information that each equation gives us. Although this work is covered in Grade 11 mathematics, I knew from experience that students would benefit from revisiting aspects like completing the square and the origins of the quadratic formula.

The rule of 72  a handy one
Author Marcus BizonySource: Learning and Teaching Mathematics 2006 (2006)More LessRadioactive decay of certain isotopes means that within any time interval a certain proportion of the isotope converts. No matter how much of the isotope there is, the same percentage decays for any two equal time intervals. The halflife of the isotope is the time required for half of the isotope present at the start of the interval to decay; halflives can vary from a few minutes up to millions of years.

Teaching students logical reasoning
Authors: Sylvia Encheva and Sharil TuminSource: Learning and Teaching Mathematics 2006, pp 36 –39 (2006)More LessIn this paper we consider applications of logical connectives and the laws of logic in the process of mathematical education. One of the basic goals in education is that students obtain knowledge and mastery (Nesher and Kilpatrick, 1990). Another one, which is of no less importance, is to enhance students' logical reasoning (Dunkin and Barnes, 1986? Casanovas, 2003). Then a natural question arises : 'How can we teach them reasoning?'. Is it necessary to introduce a separate subject devoted to a study of human reasoning or can a student develop reasoning skills while studying different subjects?

Interview with a prizewinning teacher
Author Mellony GravenSource: Learning and Teaching Mathematics 2006 (2006)More LessThere were two 2005 Aggrey Klaaste Maths Science & Technology Educator of the Year Awards, one for the GET phase, the other for more senior grades. The GET winner was Rosy Ruiters, from Sanbersville Combination School in the Free State, and LTM's editor Mellony Graven interviewed her.

Marking matric scripts is a learning process
Author Vimolan MudalySource: Learning and Teaching Mathematics 2006, pp 41 –46 (2006)More LessThe marking of matric mathematics scripts at the end of each year is a long and deliberate process. Huge amounts of time and energy are invested in ensuring that the learners are accorded marks that are as correct and as fair as possible. In as much as it is a task for educators to mark the scripts, it also becomes a strong learning mechanism for educators to develop themselves as mathematics teachers. This paper then serves to look at the way the marking process impacts on the teaching and learning of mathematics. It has always been assumed that the lessons learned during the marking process will filter down to all mathematics educators so that useful adjustments can be made to their teaching style and the content they teach in the class.