Learning and Teaching Mathematics  latest Issue
Volume 2016, Issue 20, 2016

From the Editors
Authors: Duncan Samson and Marcus BizonySource: Learning and Teaching Mathematics 2016 (2016)More LessWe hope you enjoy the wonderfully diverse array of articles in this issue, and remind you that we are always eager to receive submissions. Suggestions to authors, as well as a breakdown of the different types of article you could consider, can be found at the end of this journal.

Using arrays for conceptual understanding of multiplication and division
Author Debbie StottSource: Learning and Teaching Mathematics 2016, pp 3 –6 (2016)More LessResearch has shown that arrays are a useful tool for developing learners' conceptual understanding of multiplication and division across both the Foundation and Intermediate phases. Because arrays lend themselves to multiplicative understanding rather than additive understanding, they are a useful way of 'unitising'  i.e. seeing items in groups rather than as individual items. They are an important conceptual step between modelling multiplication with physical objects and using more algorithmic methods. Arrays also provide a way to make close connections between multiplication and division.

Creativity with areas  circles and squares
Author Ronit BassanCincinatusSource: Learning and Teaching Mathematics 2016, pp 7 –9 (2016)More LessFostering creativity in our pupils is essential if they are to successfully engage with the dynamic and rapidly changing world around us. Working with shape and space is one area of the curriculum where one can potentially encourage creative thinking through more openended activities and investigations. In junior grades, one of the goals of working with space and shape is the fostering of geometrical insight through the exploration of shapes, their properties and interrelations. In addition to fundamental properties such as area and perimeter, pupils should develop their visual and spatial perception skills in order to make the transition from basic to more complex shapes.

What is your philosophy of mathematics?
Author Laura De LangeSource: Learning and Teaching Mathematics 2016, pp 10 –13 (2016)More LessThe Curriculum and Assessment Policy Statement (CAPS) describes Mathematics as:
...a language that makes use of symbols and notations for describing numerical, geometric and graphical relationships. It is a human activity that involves observing, representing and investigating patterns and qualitative relationships in physical and social phenomena and between mathematical objects themselves. It helps to develop mental processes that enhance logical and critical thinking, accuracy and problem solving that will contribute in decisionmaking. (Department of Basic Education, 2011, p. 8)
The above description is multifaceted, incorporating notions of Mathematics as a language of symbols; a human activity involving processes such as investigation and generalisation; a means of describing the physical and social world in which we find ourselves; a deductive system of abstraction; and a means of improving broader decisionmaking processes. The above description incorporates ideas from a variety of different philosophies of mathematics. However, as a teacher, what is your own philosophy of mathematics?

Packing four spheres into a tetrahedron
Author Nicholas KroonSource: Learning and Teaching Mathematics 2016, pp 14 –15 (2016)More LessA few years ago, when learning about 3dimensional shapes, my Grade 9 Maths class was tasked with building containers to hold four ping pong balls of radius 2 cm so that the four balls fitted snugly inside the container. The simplest and most popular ways of doing this were to construct cuboids or cylinders with the correct dimensions. If you were ambitious you could try other shapes, such as a triangular prism, but whatever shape you chose it was important for the dimensions to be mathematically accurate. Calculating the dimensions of these shapes is well within the means of a competent Grade 9 learner.

Finding the maximum distance between two curves
Authors: Deepak Mavani and Beena MavaniSource: Learning and Teaching Mathematics 2016, pp 16 –17 (2016)More LessFinding the maximum distance between two curves

A multiple solution task : a South African Mathematics Olympiad problem
Author Michael De VilliersSource: Learning and Teaching Mathematics 2016, pp 18 –20 (2016)More LessExploring and engaging with multiple solution tasks (MSTs), where students are given rich mathematical tasks and encouraged to find multiple solutions (or proofs), has been an interesting and productive trend in problem solving research in recent years. A longitudinal comparative study by LevavWaynberg and Leikin (2012) suggests that an MST approach in the classroom provides a greater educational opportunity for potentially creative students when compared with a conventional learning environment.

Engaging with the unexpected  seek first to understand
Author Andrew MaffessantiSource: Learning and Teaching Mathematics 2016, pp 21 –22 (2016)More LessOne of the many joys of teaching is seeing how different pupils tackle problems in different, and often unexpected, ways. In formal assessments this of course makes the marking process a little more complex as one needs to be vigilant about properly engaging with unexpected responses  particularly if the solution provided by the pupil is different to the marking guidelines. When pupils approach questions in unexpected ways we have a duty to be openminded and to take the time necessary to properly explore and attempt to understand the reasoning behind the solution provided. In this article I share a recent episode that illustrates how engaging with the unexpected can lead to opportunities for deep learning.

Generalising the number of polygon diagonals
Author Duncan SamsonSource: Learning and Teaching Mathematics 2016, pp 23 –26 (2016)More LessGeneralising the number of polygon diagonals

Series summation  an extension of a 1972 Matric examination question
Author Alan ChristisonSource: Learning and Teaching Mathematics 2016, pp 27 –29 (2016)More LessSeries summation  an extension of a 1972 Matric examination question

Exploring rich diagrams
Author Marcus BizonySource: Learning and Teaching Mathematics 2016, pp 30 –34 (2016)More LessOne of the difficulties in finding suitable investigations for secondary school pupils is that interesting scenarios are likely to be too difficult, while those that are manageable are potentially uninteresting. More than that, in order to know whether a particular scenario is suitable, one must have worked through it oneself. Having done so, the temptation is then to scaffold the investigation process so as to lead students to the conclusions one found oneself  thereby eliminating the 'openended' nature of a proper investigative approach.

From counting line segments to counting rectangles
Author YiuKwong ManSource: Learning and Teaching Mathematics 2016, pp 35 –37 (2016)More LessThe problem of counting the total number of rectangles in a rectangular grid is periodically encountered as a problem solving activity in school mathematics and mathematical Olympiads.

Fractals in the mathematics classroom : the case of infinite geometric series
Authors: Atara Shriki and Liora NutovSource: Learning and Teaching Mathematics 2016, pp 38 –42 (2016)More LessThe world of mathematics is constantly evolving. However, the mathematics included in school curricula seldom reflects this evolving nature of the discipline. Appreciating the importance of exposing students to contemporary mathematics, we identified fractal geometry as a topic that could meaningfully be integrated into the regular curriculum. In this article we briefly introduce the idea of fractals and demonstrate how they can be integrated into the teaching of infinite geometric series.