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 Volume 2007, Issue 5, 2007
Learning and Teaching Mathematics  Volume 2007, Issue 5, November 2007
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Volume 2007, Issue 5, November 2007

From the Editors
Authors: Mellony Graven and Marcus BizonySource: Learning and Teaching Mathematics 2007 (2007)More LessWe are pleased to be publishing the fifth edition of Learning and Teaching Mathematics. This edition is full of exciting articles written by a range of South African and international authors. Our journal is also now available online through the Sabinet SA ePublications. This means that if your school or institution subscribe to the journal you will have portal IP access to it.

Some novel cyclic quadrilateral proofs
Author Michael De VilliersSource: Learning and Teaching Mathematics 2007 (2007)More LessRecently in the geometry courses I am teaching at Kennesaw State University, a circle geometry result not so wellknown in the USA was posed as a problem, namely, to prove that the opposite angles of a (convex) cyclic quadrilateral are supplementary. One of the students, Matt Hickman, came up with the following proof I had not yet seen before, which readers may find interesting.

Patterns of visualisation
Author Duncan SamsonSource: Learning and Teaching Mathematics 2007, pp 4 –9 (2007)More LessMathematicians have always been fascinated by the art and science of patterns (Joseph, 2000). In a parallel with the visual arts, to meaningfully engage with a pattern requires a necessary discernment of the principle on which its elements are ordered. Pattern itself does not lie in the individual elements, but rather the rule which governs their mutual relationship (Taylor, 1964:6970).

Which agenda drives you in your teaching of Mathematical Literacy?
Authors: Mellony Graven and Hamsa VenkatSource: Learning and Teaching Mathematics 2007, pp 10 –11 (2007)More LessIn our work with Mathematical Literacy teachers in schools and through post graduate courses here at the university we have identified four agendas that teachers tend to be working with in their mathematical literacy teaching. Feedback from teachers at various workshops including workshops at the 2007 AMESA conference in White River indicated that reflecting on the spectrum of agendas was a useful activity for current and prospective mathematical literacy teachers. Teachers noted that having to think about which agenda they mainly work towards in their teaching helped them reflect on the assumptions (sometimes unconscious) they have about mathematical literacy and also opened up other possible agendas which they had not considered.

An 8^{th} grade geometry problem from Japan and American teachers' solutions
Authors: Darryl Corey, Hasan Unal and Elizabeth JakubowskiSource: Learning and Teaching Mathematics 2007, pp 12 –16 (2007)More LessIn the TIMSS Videotape Classroom Study, tasks on which students were working during seatwork were coded into three categories (Stigler et al., 1999, p102). The results are shown in Table 1, and one of the major distinct differences in Japanese instruction compared to US is spending more time on inventing new solutions or thinking about mathematical problems. According to Shimuzu (2000) the frequent exposure of students to alternative solutions methods to a problem and discussing multiple solutions to a problem in a whole class mode is a common style for teaching mathematics in Japanese elementary and lower secondary schools.

Writing for television : Kato's conjecture, ketchup and earthquakes
Author John WebbSource: Learning and Teaching Mathematics 2007, pp 17 –18 (2007)More LessFilm crews on the scenic UCT campus are a familiar sight. They use the classical architecture of Jameson Hall in scenes set in ancient Greece or Rome, or the views of Devil's Peak and across the Cape Flats as background for a fashion shoot. One might think that mathematics would be far removed from the world of grips, gaffers and best boys. However, in November last year I found myself in the middle of cries of Lights! Action!! Camera!!! when I was asked to provide professional mathematical advice and assistance in making one of a series of television commercials.

Problemsolving and proving via generalization
Authors: Michael De Villiers and Mary GarnerSource: Learning and Teaching Mathematics 2007, pp 19 –25 (2007)More LessA very useful problem solving strategy often emphasised at school and regularly tested in Mathematical Olympiads and Challenges is to consider special cases of a problem. Not only are the special cases usually easier to solve, but they often allow one to identify a pattern or give some clue towards a general solution or proof. Less well known (or utilised) appears to be the opposite strategy, namely, to consider a more general case than the given problem. Contrary to what one might expect, the general case is sometimes much easier to solve than the special case.

Using messiness to appreciate the nature of Mathematics as a discipline
Author Paul BettsSource: Learning and Teaching Mathematics 2007, pp 26 –30 (2007)More LessI am concerned with the nature of mathematics because, according to Hardy (1992), school maths is not mathematics. Burton (2001) believes that school maths can and should reflect the discipline of mathematics, and that children can emulate what mathematicians do as they try to make sense of mathematics. I believe that looking closely at the nature of mathematics does matter if school maths is to reflect the discipline of mathematics.

Mathematical Literacy  a lifeline for many
Authors: Esme Buytenhuys, Mellony Graven and Hamsa VenkatakrishnanSource: Learning and Teaching Mathematics 2007, pp 31 –35 (2007)More LessMathematical Literacy was introduced in the FET band at the start of 2006. As expected the new and unfamiliar nature of this learning area has caused a great deal of insecurity for teachers, learners and parents. Following a Marang Centre annual workshop on Mathematical Literacy by Aarnout Brombacher some teachers suggested that Hamsa and I (Mellony) start a mathematical literacy support group where we could engage together on an ongoing basis with some of the issues arising from the implementation. This group has had several meetings where we have discussed the successes and tensions of mathematical literacy. Esmé is one of the teachers we met through this support group and have subsequently been working with and learning from.

Connecting roots, function extrema and inflection points
Authors: David R. Duncan and Bonnie H. LitwillerSource: Learning and Teaching Mathematics 2007 (2007)More LessThe relationship between the roots of a quadratic polynomial and the maximum or minimum point on its graph is easy to see. Suppose that the two quadratic roots are x = α and x = β. The quadratic function can then be written as f(x) = κ(x  α)(x  β) = κ[x^{2}  (α + β)x + α β]. To find the maximum or minimum of this function, its first derivative must be found, and set equal to 0.

Is it the Maths teacher's job to teach Physics?
Author Serkan HekimogluSource: Learning and Teaching Mathematics 2007, pp 37 –40 (2007)More LessThroughout the course of history, the influence of ideas from physics has remained at the forefront of mathematical developments. The demands of physics led to the invention of calculus by Newton and Leibniz in the seventeenth century. Hilbert, Maxwell, and Lagrange were physicists as much as they were mathematicians. Much of the impetus for new mathematical development continues to emerge from the field of physics. In May 2000, the Clay Mathematics Institute announced seven $1 million prizes for the solutions to each of seven unsolved problems of mathematics, two of which came directly from physics, YangMills Theory and the Mass Gap Hypothesis and the NavierStokes Equations (see Devlin, 2002).

Discovering Pythagoras' Theorem through guided reinvention
Authors: Bertus Van Etten and Stanley A. AdendorffSource: Learning and Teaching Mathematics 2007, pp 41 –47 (2007)More LessThe current Senior Phase mathematics curriculum guidelines compel educators to afford learners the opportunity to explore mathematical aspects on their own. In a way that enhances learning, they should be allowed to experience how mathematical findings, theorems, axioms, etc were arrived at by past mathematicians. This, however, seldom happens, either because educators feel learners may not be able to adequately deal with it on their own, or educators themselves do not know how to go about it, or because this kind of activity would slow the pace considering the loaded curriculum. In this regard educators need content that lends itself to be used in this way, and they need guidance as to how to facilitate situations in which this teaching methodology is used.

Designing spot tests to facilitate Maths teaching
Author Aneshkumar MaharajSource: Learning and Teaching Mathematics 2007, pp 49 –50 (2007)More LessMathematics educators should have some way of determining whether previous knowledge, skills and abilities that they assume are in place, really are. This could be done by oral questioning or designing a spot test which learners should attempt. Spot tests could also be used to determine whether learners are able to outline the key steps in the proof of a theorem, based on the context of a given diagram. For such activities in spot tests, a time limit should be set. Also, such a test should be followed by a class discussion.

The Midmar Mile, mixing concrete, and other anomalies in my Maths Literacy classroom
Author Marc NorthSource: Learning and Teaching Mathematics 2007, pp 51 –57 (2007)More LessHave you ever experienced one of those "Light Bulb" moments in your teaching career, when something happens in class that opens your eyes to the wonder and complexity of teaching Mathematics? In 2006 while teaching Mathematical Literacy, I experienced several of these moments, each showing in a different way the complexities involved in using mathematics to solve problems in the physical world.