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 Volume 2014, Issue 17, 2014
Learning and Teaching Mathematics  Volume 2014, Issue 17, December 2014
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Volume 2014, Issue 17, December 2014

From the Editors
Authors: Duncan Samson, Marcus Bizony and Lindiwe TshabalalaSource: Learning and Teaching Mathematics 2014 (2014)More LessLTM17 begins with an article by Debbie Stott who describes a card game she devised for developing learners' number sense and fluency. In the second article, Duncan Samson shares three varied ideas for classroom engagement at different levels. Graeme Evans then explores an interesting geometric scenario posed on the maths FET electronic discussion group, while in the fourth article Alan Christison investigates the average gradient between two points on a curve. Michael de Villiers and Nic Heideman then describe a mathematical exploration that highlights some of the main features of conjecturing, refutation and proving.

"I've got it!"  A card game for developing number sense and fluency
Author Debbie StottSource: Learning and Teaching Mathematics 2014, pp 3 –6 (2014)More LessThe Foundation Phase CAPS document indicates that learners exiting the Foundation Phase should do so with "a secure number sense and operational fluency" and that these learners should be "competent and confident with numbers and calculations" (Department of Basic Education, 2011a, p. 8). In the context of the Foundation Phase the development of number sense includes the meaning of different kinds of numbers, the relationship between different kinds of numbers, and the effect of operating with numbers. In the Intermediate Phase this development of number sense and operational fluency should continue, with the number range, kinds of numbers, and calculation techniques all being extended. However, from my experience in the South African context, many learners are still reliant on concrete onetoone methods of calculation such as finger counting or tally marks throughout the primary grades. This unfortunately does not match the expected level of fluency as reflected in the official CAPS documents (Department of Basic Education, 2011a, 2011b).

Reciprocals, equal areas and inverse functions
Author Duncan SamsonSource: Learning and Teaching Mathematics 2014, pp 7 –11 (2014)More LessOver years of teaching one gradually builds up a repertoire of interesting ideas for classroom engagement. In this article I share three of these ideas. The first relates to the ordering of fractions by using their reciprocals. The second centres on a simple technique for creating shapes with equal areas, while the third engages with the notion of inverse functions.

Anyone for tennis or volleyball?
Author Graeme EvansSource: Learning and Teaching Mathematics 2014, pp 12 –16 (2014)More LessThe following question was posed by Karen Ireland (St Henry's Marist College, Durban) on the maths_FET electronic discussion group.

Average gradient between two points on a curve
Author Alan ChristisonSource: Learning and Teaching Mathematics 2014, pp 17 –19 (2014)More LessThe notion of average gradient is analogous to that of average speed. In order to calculate the average speed of a vehicle between two points one would calculate the total distance travelled divided by the total time taken. This is of course a totally different matter to simply taking the average of the two speeds. Nonetheless, averaging the two instantaneous gradients to determine the average gradient between two points is a classic misconception and commonplace error at school.

Conjecturing, refuting and proving within the context of dynamic geometry
Authors: Michael De Villiers and Nic HeidemanSource: Learning and Teaching Mathematics 2014, pp 20 –26 (2014)More LessLockhart, a research mathematician, describes the present system of mathematics education at school as nightmarish, claiming that it destroys children's "natural curiosity and love of patternmaking" (Lockhart, 2002, p. 2). He goes further in his critique of school mathematics by suggesting that "there is no actual mathematics being done in our mathematics classes" (p. 14), and in the place of discovery and exploration there is only the mindless drill and exercise of given rules and algorithms.

The magic of "whole tens"
Authors: Dorit Patkin and Ronit BassanCincinatusSource: Learning and Teaching Mathematics 2014, pp 27 –31 (2014)More LessThis paper describes a series of activities that progressively explores the addition of two numbers whose sum is in whole tens. Amongst other things, the directed activities are designed to encourage the development of pupils' numerical insight through the decomposition and combining of numbers. The activities are suitable for pupils in Grade 1 to Grade 6.

Investigating cubic functions Using GeoGebra dynamic geometry software
Authors: Deepak Mavani and Beena MavaniSource: Learning and Teaching Mathematics 2014, pp 32 –35 (2014)More LessSketching and interpreting cubic functions is one of the main applications of differential calculus for Grade 12 learners. However, when drawing graphs of functions using paper and pencil, a great deal of time is spent in performing repeated, tedious computations. As such, learners often do not get sufficient time to explore the nature and properties of functions and their graphs. Dynamic geometry software such as GeoGebra, The Geometer's Sketchpad and Cabri have immense potential for learners to explore functions and their graphs without the tedium of repeated calculations in order to sketch the graphs. The dynamic nature of such software environments makes them ideal for mathematical exploration in which students can experience the processes of conjecturing and discovering.

The relationship between quadratic equations and their first difference equation
Author Ashley Ah GooSource: Learning and Teaching Mathematics 2014, pp 36 –37 (2014)More LessThe relationship between a quadratic equation and its linear first difference equation is perhaps not so obvious. After all, it is possible for different quadratic sequences to have the same first difference sequence.

Exploring mental computation strategies
Authors: Kwethemba Michael Moyo and Duncan SamsonSource: Learning and Teaching Mathematics 2014, pp 38 –41 (2014)More LessThe ability to perform mental computations both swiftly and flexibly remains an important mathematical skill despite the increasingly sophisticated technology that constantly surrounds us. Not only is mental computational agility considered a universally valued skill, mental computation skills provide opportunities to engage in mathematical thinking, thereby strengthening number sense and other general processes related to problem solving. Mental computation skills are a prerequisite for meaningful engagement with written algorithms. Furthermore, teaching mental computation strategies promotes creativity as learners are encouraged to think flexibly and independently.

Alternative quadratic formula
Author Moses L. MakobeSource: Learning and Teaching Mathematics 2014 (2014)More LessThis alternative formula is of course not quite as attractive as the standard equation (2) because its denominator is not rational.