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 Volume 2011, Issue 10, 2011
Learning and Teaching Mathematics  Volume 2011, Issue 10, May 2011
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Volume 2011, Issue 10, May 2011

From the editors
Authors: Duncan Samson, Marcus Bizony, Mellony Graven and Lindiwe TshabalalaSource: Learning and Teaching Mathematics 2011 (2011)More LessThis issue of Learning and Teaching Mathematics marks the 10^{th} publication of LTM since the inception of the journal in 2004. Paging through previous issues one is struck by the wonderful variety of topics, issues and themes that have found voice over the last few years. It is also encouraging to see how widely read the journal has become  in this issue for example we have contributors hailing not only from South Africa but also from as far afield as The Netherlands, Canada, Ghana, Hong Kong, and the USA (Hawaii, Texas, Florida, Indiana and South Carolina). We are of course excited by the international participation in our journal, but equally want to encourage more submissions from South African teachers.

Mathematical literacy for higher education
Author Vera FrithSource: Learning and Teaching Mathematics 2011, pp 3 –7 (2011)More LessMathematical (or quantitative) literacy plays an important role in the university curriculum for many disciplines, and the object of this article is to enrich teachers' understanding of what may be expected in terms of learners' mathematical literacy in higher education. Although I am writing mainly about mathematical literacy, what I say is also relevant to Mathematics teachers, because learners in a Mathematics class should ideally be developing their mathematical literacy at the same time. However, it is easy for Mathematics teachers to lose sight of the applications of the mathematics concepts in real contexts, especially as the examinations do not emphasize these.

The ratio table as learning tool in Grade 7
Authors: Bertus Van Etten and Stanley A. AdendorffSource: Learning and Teaching Mathematics 2011, pp 8 –13 (2011)More LessRatio tables have been found to be an effective teaching and learning tool to solve problems related to ratio, scale and percentages. One context where this is evident deals with conversion from one unit of measurement to another, for instance from centimetres to metres and vice versa.

Laphum'ikhwezi Ngede Mathematics Club : an interview with Wandile Hlaleleni
Source: Learning and Teaching Mathematics 2011 (2011)More LessThe Laphum'ikhwezi Ngede Mathematics Club was established on 24^{th} April 2010. This is the first maths club to be established in the Centane district of the Eastern Cape. LTM interviewed the club director, Mr Wandile Hlaleleni, to find out more about this exciting initiative.

Capitalising on mathematical moments
Author Wandile HlaleleniSource: Learning and Teaching Mathematics 2011 (2011)More LessImagine that Sange is a learner in your Grade 9 Mathematics class. One day Sange points out that he has just turned 15 and as a result is a quarter of his grandfather's age. How could you as a teacher capitalise on this simple observation and explore its potential richness? Perhaps a starting point would be to set up a table of ages in order to compare Sange's age with his grandfather's at different times in both the past and future. A partial comparative table is shown below which includes simplified ratios of Sange's age to his grandfather's age. The shaded column shows their present ages.

Building students' mathematical proficiency : connecting mathematical ideas using the Tangram
Author Mourat TchoshanovSource: Learning and Teaching Mathematics 2011, pp 16 –23 (2011)More LessThis article describes a classroomtested approach that brings together different subdomains of secondary school mathematics using the Tangram: Number Sense and Algebra (unpacking the idea of a square root and irrational numbers, understanding a reverse relationship between area and side length of a square); Geometry and Measurement (spatial reasoning, understanding properties of geometric shapes, measuring area and length).

Proof without words : parahexagonparallelogram area ratio
Author Michael De VilliersSource: Learning and Teaching Mathematics 2011 (2011)More LessGiven a convex hexagon ABCDEF with opposite sides equal and parallel, with G, H, I and J being the respective midpoints of sides AB, BC, DE and EF, prove that area ABCDEF = 2 x area GHIJ.

Running a Maths Bonanza
Author Zonia JoosteSource: Learning and Teaching Mathematics 2011, pp 24 –26 (2011)More LessWe recently organized a Maths Bonanza focused on Grade 2 to 4 learners with the aim of encouraging learners to work collaboratively to explore mathematics and to have some fun while learning. The Maths Bonanza was organized as a collaboration between AMESA, the FRF Numeracy Education Chair and the Albany Museum in Grahamstown. We advertised locally for schools to book a one hour slot for their classes of learners. Ten classes from six schools participated in the Maths Bonanza week and feedback from teachers, learners and staff at the museum was very positive. I have had numerous requests to do this again and we are considering taking the Maths Bonanza on a road show to other schools in the neighbouring towns. In this paper I want to share with you four activities that we used that you can easily duplicate. For all of the activities learners were arranged in groups around a large table.

Cubic curiosities
Author Duncan SamsonSource: Learning and Teaching Mathematics 2011, pp 27 –31 (2011)More LessCubic graphs form an important part of the South African school Mathematics curriculum in terms of curve sketching, graph interpretation and contextualised maxima and minima problems. The purpose of this article is to highlight a number of interesting features of cubic graphs that can be used to illustrate interconnections between different aspects of school mathematics.

Did you know? Euclid's partition definitions
Author Michael De VilliersSource: Learning and Teaching Mathematics 2011 (2011)More LessDid you know that Euclid defined isosceles triangles so as to exclude equilateral triangles, and similarly rectangles (oblongs), rhombuses and parallelograms (rhomboids), so as to exclude special cases?

Using spreadsheets to "get inside" annuities
Author Craig PournaraSource: Learning and Teaching Mathematics 2011, pp 33 –35 (2011)More LessOver the past few years I have spent a great deal of time working on the use of spreadsheets for learning financial mathematics. My goal in doing this has been to "get inside" the formulae for simple and compound interest and for annuities. In doing so, I wanted to understand more deeply how the money was growing each month, and to help teachers and learners to do the same. The formulae that we use in financial maths at school provide efficient ways to calculate answers but they don't really help us to understand what is happening monthbymonth, particularly with annuities. In this article I wish to share four spreadsheets that have become an integral part of my teaching of financial maths with preservice secondary maths teachers. All four spreadsheets focus on annuities. The first two spreadsheets focus on future value of an ordinary annuity while the third and fourth spreadsheet focus on present value of an annuity.

Basic geometric configurations and teaching Euclidean geometry
Author Margo KondratievaSource: Learning and Teaching Mathematics 2011, pp 37 –43 (2011)More LessMathematicians have always recognized geometry as an important source of meaning in mathematics (Hilbert & CohnVossen, 1952). Mathematics educators also regard geometry as "a natural area of mathematics for the development of students' reasoning and justification skill" (NCTM, 2000). The visual aspect of geometric problems is their distinctive and invaluable property since geometric questions appeal to solvers' natural spatial intuition. While solving them, students need to become aware of geometric properties and their relations, drawing conclusions not immediately evident in the diagram (Henderson & Taimina, 2005), or learning from resolving apparent contradictions (Kondratieva, 2009).

Factorising quadratic trinomials : an alternative approach
Authors: James Dogbey, John Gyening and Gladis KersaintSource: Learning and Teaching Mathematics 2011, pp 44 –45 (2011)More LessPolynomial functions are important concepts in school mathematics due to their ability to model numerous realworld problems. Quadratic functions, which are secondorder polynomial functions of the form f(x) = ax^{2} + bx + c where a, b and c are real number constants with a ≠ 0, are probably the most frequently used polynomials to apply school mathematics to other disciplines, and to realworld situations. Quadratic equations result, for example, when solving equilibrium problems in chemistry, when computing trajectories in projectile motion, and when dealing with optimization situations in management sciences  analyzing cost, revenue, and profit. The ability to solve quadratic equations efficiently is therefore an essential skill in many applications in school mathematics, in the physical sciences and in commerce.

A correct visual proof of area of trapezium
Author Michael De VilliersSource: Learning and Teaching Mathematics 2011 (2011)More LessThe July 2009 issue of Learning and Teaching Mathematics (no. 7, p. 28) contains the following "proof without words" for the area of a trapezium as submitted by YiuKwong Man... Unfortunately the proof is only valid for an isosceles trapezium, and not for a general trapezium.

The development of numeric and geometric patterns
Authors: Tiritoga Takawira and Zonia JoosteSource: Learning and Teaching Mathematics 2011, pp 47 –52 (2011)More LessThe study of pattern affords opportunities to observe, hypothesize, experiment, discover and create (NCTM, 1989). Patterns are not simply about describing what comes next in a particular sequence, but rather about finding relationships. It is important for young learners to recognize that patterns are predictable, and it is the duty of the teacher to help learners understand and appreciate this predictability. The most important aspect is to generate rules about a pattern and use known information to predict unknown information. Learners need to move from "What comes next?" to analyzing the structure of the pattern (Economopoulos, 1998:296298). We need to focus on how to develop "functional sense" without simply asking for the "next number" (NCTM, 1998:100101).

Thinking mathematically  the connection between two games of "Nim"
Authors: KinKeung Poon and TakWah WongSource: Learning and Teaching Mathematics 2011, pp 53 –57 (2011)More LessAlthough many people dismiss games as purely recreational in nature, evidence from psychological studies shows that games offer significant opportunities for students to learn and develop their problemsolving ability (Bright et al., 1985; Peters, 1998). Studies in the extant literature have found that games provide a problemsolving context that supports and enhances student involvement in lessons, synthesizes their ideas and knowledge, and arouses and develops their curiosity and creativity (Carlson 1969). The use of games and puzzles in the teaching of mathematics, especially in the area of problem solving, has increased dramatically. The use of games in teaching and learning has also been emphasized in mathematics curricula, for example in Hong Kong, particularly in primary schools (HKG, 2000, 2007).

Probability revisited
Author James MetzSource: Learning and Teaching Mathematics 2011 (2011)More LessIssue 8 of Learning and Teaching Mathematics (May 2010) included an informative article by Sudi Balimuttajjo and Robert J. Quinn entitled "Innovative Pedagogical Approaches Based on Solving a Disjunctive Probability Problem" (pp. 6166).

Search for meaning : a guided discovery teaching method
Author KyongHee M. LeeSource: Learning and Teaching Mathematics 2011, pp 58 –62 (2011)More LessMost teaching methods currently used in teaching mathematics to college students involve direct lecturing. Teachers tend to prepare all the teaching materials to present in class while students mostly spend their efforts in regurgitating or mimicking what was taught. Students memorize definitions, mathematical concepts, and how to solve some problems, only to forget them after they are done with their exams. However, society and industry require more selfdirected learners (Fink, 2003). How then should students be taught? Weimer (2003) places emphasis on the importance of producing independent and autonomous learners who assume responsibility for their own learning. Bean (2001) observed that students tended to be poor readers and were often overwhelmed by the density of their textbooks. He found that using a cooperative type of learning in small groups could be a powerful and effective form of active learning. DeLong and Winter (2002) believe that the textbook should be a vital part in learning. Halsey (1977) reported that a guided discovery teaching method in an elementary Algebra course achieved a high degree of success for pupils in their subsequent mathematics courses. Students in group discovery classes are actively involved in their own learning in and outside of the classroom (Flahive and Lee, 2007).