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 Volume 2011, Issue 11, 2011
Learning and Teaching Mathematics  Volume 2011, Issue 11, August 2011
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Volume 2011, Issue 11, August 2011

From the editors
Authors: Duncan Samson, Marcus Bizony and Mellony GravenSource: Learning and Teaching Mathematics 2011 (2011)More LessIt was great meeting up with friends and colleagues from around the country at the recent AMESA National Congress held on the campus of the University of the Witwatersrand in Johannesburg. Forums such as the AMESA National Congress allow for wonderful interaction  the exchanging of ideas, discussions around policy and curriculum issues, comparisons of different teaching strategies, engagement with research findings, and so much more. In addition, forums of this kind serve as a constant reminder that we are not isolated practitioners, but are part of a broad community of practice  a community which is there to provide support and encouragement.

On numerical insight and the "magic" of square numbers
Author Dorit PatkinSource: Learning and Teaching Mathematics 2011, pp 3 –5 (2011)More LessA numerical insight is manifested by an intuitive view of mathematical constructs and their association with arithmetic operations; by a feeling that there is a connection between things; by the ability to mobilize knowledge and previous experience in order to develop various solution strategies; by comprehension of different solving methods; and by demonstrating openness for new ways. Different people solve the same task in different "individual" ways, evoking a feeling that mathematics is not just a rigid subject but is rather a subject with a wide perspective. Developing original solution methods allows control over the process and enhances selfconfidence. Numerical insight (like geometrical insight) relates to general mathematical orientation, not merely to a specific mathematical chapter.

The cure for mediocrity : Stand and Deliver
Author Liezel Du ToitSource: Learning and Teaching Mathematics 2011, pp 6 –9 (2011)More LessChaos... No, TOTAL CHAOS! Graffiti against the walls, learners doing whatever they wanted to do, a lot of noise inside as well as outside the classroom... After a while one of the girls (sitting on the lap of one of the boys) noticed that there was actually a teacher in the classroom. A new teacher  so here was a new opportunity! "Can we discuss sex today?" was her question. She did not expect the reply: "If we discuss sex, I have to give you sex for homework." Clearly this teacher was different!

Exploring online numeracy games for primary learners : sharing experiences of a Scifest Africa Workshop
Authors: Mellony Graven and Debbie StottSource: Learning and Teaching Mathematics 2011, pp 10 –15 (2011)More LessOn the 5^{th} of May we (Mellony Graven & Debbie Stott) ran a workshop aimed at Grade 35 learners, accompanied by their parents or teachers, on the use of free online numeracy game resources. This workshop formed part of the programme of Scifest Africa, the national science festival held annually in Grahamstown. We had such fantastic feedback from both learners and teachers that we thought we should share how we ran the workshop in the hope that schools with online computer facilities will run similar workshops not only with their own learners but with learners in nearby schools who do not have such facilities.

Staircase problem continued
Author Jim MetzSource: Learning and Teaching Mathematics 2011, pp 16 –17 (2011)More LessIn the May 2010 issue of Learning and Teaching Mathematics (No. 8) I reported on a variety of solutions offered by teachers to the "Staircase" problem ("Let Me Count The Ways", p. 56). The problem entails finding the sum 1 + 2 + 3 + 4 + 5 + ... + n by finding the area of a staircase figure that has a base of n squares as shown in Figure 1.

Factorization  variations on a theme
Authors: Duncan Samson, Cheriyaparambil K. Raghavan and Sonja Du ToitSource: Learning and Teaching Mathematics 2011, pp 18 –21 (2011)More LessFactorization remains a critical component of the South African school Mathematics curriculum. Factorization is used not only for simplifying algebraic expressions but also for determining the roots of equations and for determining the xintercepts of graphs. However, factorization continues to be an area which learners find particularly problematic.

Simply symmetric
Author Michael De VilliersSource: Learning and Teaching Mathematics 2011, pp 22 –26 (2011)More LessSymmetry is found in the visual arts, architecture and design of artefacts since the earliest time. Many natural objects, both organic and inorganic, display symmetry: from microscopic crystals and subatomic particles to macrocosmic galaxies. Today it features strongly in higher mathematics such as Linear and Abstract Algebra, Projective and Fractal Geometry, Algebraic Topology, Graph and Function Theory, etc., and in many other mathematical disciplines such as Quantum Physics, Relativity, String Theory, etc.

Equations with false solutions
Author Marcus BizonySource: Learning and Teaching Mathematics 2011, pp 27 –28 (2011)More LessWhen an equation involving algebraic fractions is solved, it is necessary to check apparent solutions against the original equation to weed out those that are inadmissible (because they require an original denominator to be zero). Sometimes all apparent solutions have to be rejected, sometimes there are some that remain valid.

Similarity of parabolas  a geometrical perspective
Authors: Atara Shriki and Hamutal  Technion DavidSource: Learning and Teaching Mathematics 2011, pp 29 –34 (2011)More LessThe issue of similarity of parabolas can be discussed from both algebraic and geometrical perspectives. Through an algebraic lens, Kumpel (1975) examined this issue and proved the surprising fact that any two parabolas are similar. In this paper we examine the similarity of parabolas by employing geometrical considerations and suggest an approach to determine the similarity ratio. Finally, we show that in an analogy to similarity of polygons, similarity of parabolas preserves ratios of distances, and that the ratio of corresponding areas is the square of this similarity ratio.

Error detection as mathematical catalyst
Author Deonarain BrijlallSource: Learning and Teaching Mathematics 2011, pp 35 –36 (2011)More LessThis paper stems from the detection of a mathematically unsound question. The question appeared in a provincial senior certificate preparatory examination paper in 2010. Dr Sudan Hansraj detected an error in the proposed solutions of the question, and this served as a catalyst for an interesting email discussion amongst a number of mathematics educators. This paper presents (1) the initial question, (2) the proposed solutions that were provided on the marking memorandum, both of which contained a serious flaw, and (3) a distillation of the email discussion including detections of counterexamples and existence conditionsfor such a problem to be true.

Using Microsoft Paint to illustrate a conceptual understanding of fraction addition and subtraction
Author Estella De Los SantosSource: Learning and Teaching Mathematics 2011, pp 37 –41 (2011)More LessThe National Council of Teachers of Mathematics has set, as one of its mathematics objectives, to "understand meanings of operations and how they relate to one another". In grades 68, students should "understand the meaning and effects of arithmetic operations with fractions". Instructional programs should enable all students to "select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods" (NCTM, 2000, pp. 148, 214).

Transformations in PowerPoint
Author Michael T. MuzheveSource: Learning and Teaching Mathematics 2011, pp 42 –43 (2011)More LessPowerPoint is a component of Microsoft Office, a program that comes with almost every school or home computer. This article describes how PowerPoint animations can be used to add a dynamic component to the study of transformations.

Encouraging participation in mathematics classes
Author Victor OdafeSource: Learning and Teaching Mathematics 2011, pp 44 –47 (2011)More LessNo matter what mathematics topic one is teaching, one should aim to involve every student productively with a goal of achieving conceptual understanding of the content covered. According to the National Council of Teachers of Mathematics (NCTM), "The need to understand and be able to use mathematics in everyday life and in the workplace has never been greater (NCTM 2000, p. 4). In the National Research Council document, EVERYBODY COUNTS: A Report to the Nation on the Future of Mathematics Education (National Academy Press, 1989, p. 60), it is stated that "Teachers must involve students in their own learning." In Beyond Crossroads (AMATYC, 2006), the Standards for Pedagogy call for instructional strategies that will promote students' active learning. The document goes further to state that "For today's students, learning is participatory  knowing depends on practice and participation" (p. 53). And the big question (and indeed the challenge) is: what do teachers need to do to get students to participate actively in mathematics classes?