Learning and Teaching Mathematics  latest Issue
Volume 2017 Number 22, May 2017

From the editor
Author Duncan SamsonSource: Learning and Teaching Mathematics 2017, pp 2 –2 (2017)More LessIn the first article of LTM 22, Marc North illustrates a variety of topics where bar models find applicability. In the second article in this issue, Debbie Barker discusses the design principles behind a polygon card activity that nurtures curiosity, creativity, resilience and collaboration. The third article, by Debbie Stott, describes a variety of activities aimed at developing number sense in Foundation and Intermediate Phase learners, while in the fourth article Lynn Bowie shares examples of where technology has proved particularly useful in an afterschool learning environment. Yvonne Sanders and Vasantha Moodley then tease out strategies and opportunities for teaching and learning in the context of the wellknown ‘Four 4s’ puzzle, while in the sixth article Shula Weissman explores a particular type of formative assessment aimed at providing insight into learners’ geometric understanding and reasoning.

Bars, bars and more bars
Author Marc NorthSource: Learning and Teaching Mathematics 2017, pp 3 –7 (2017)More LessTHE CHALLENGE WITH PIZZAS AND CAKES
It is fairly common practice for teachers to illustrate fractions by referring to slices of pizza or slices of cake. Using this model, each slice of pizza or cake is taken to represent a particular fraction of the whole. Although this method draws on the circle area model, it is questionable whether the students fully understand that one is referring to area in this scenario.

Calculating areas of polygons  conceptual engagement through card activities
Author Debbie BarkerSource: Learning and Teaching Mathematics 2017, pp 8 –14 (2017)More LessMathematics in Education and Industry (MEI) is a registered charity in the United Kingdom that works to support the teaching and learning of mathematics (mei.org.uk). In my work with MEI I aim to encourage the idea that mathematics is about curiosity, creativity, resilience and collaboration. In this article I discuss the design principles behind a set of polygon cards consisting of six triangles, nine quadrilaterals and a pentagon as an example of a teaching and learning activity that conveys these values.

Developing number sense in foundation and intermediate phase learners
Author Debbie StottSource: Learning and Teaching Mathematics 2017, pp 15 –19 (2017)More LessThe Foundation Phase CAPS document (DBE, 2011a) indicates that learners need to “exit the Foundation Phase with a secure number sense and operational fluency” (p. 8). The document states that number sense includes understanding the meaning of, and relationship between, different kinds of numbers; the relative size of different numbers; being able to represent numbers in various ways; 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. Estimation and the checking of solutions can also be included in the Intermediate Phase.

Technology as a support for mathematics learning teaching
Author Lynn BowieSource: Learning and Teaching Mathematics 2017, pp 20 –23 (2017)More LessIn the community centre in which I work, in Diepsloot, north of Johannesburg, we have been exploring the use of technology in supporting mathematics learning in an afterschool environment. In the process we have established a free online learning platform for grade 7 – 9 Mathematics. It has been an interesting journey with a great deal of learning along the way. A crucial insight gleaned is that no technology is a magic wand – it is a tool that, in the hands of a good teacher who thinks carefully about what learning aims it can fulfil, can be an effective aid for teaching. In this article I share a few examples of ways we have found technology to be particularly useful.

Exploring learners' geometric understanding
Author Shula WeissmanSource: Learning and Teaching Mathematics 2017, pp 26 –29 (2017)More LessGeometry plays an important role in the development of mathematical thinking. This article explores the use of a particular type of formative assessment that has the potential to provide insight into learners’ geometric understanding and reasoning, interrogating both concept image as well as concept definition. Examples of responses to these assessment items are provided and briefly analysed in terms of learners’ potential misconceptions. Suggestions are made as to how to move learners beyond such misconceptions.

Curious calculations that make you look twice
Authors: Duncan Samson, Benji Euvrard and Andrew MaffessantiSource: Learning and Teaching Mathematics 2017, pp 30 –34 (2017)More LessEvery so often one comes across a calculation where the structure of the answer, although entirely correct, causes one to doubt its veracity – where the answer seems to have been arrived at by an erroneous procedure and consequently looks decidedly dodgy! In this article we present four such cases and explore them in a way that we hope could be emulated in the form of a partially directed classroom investigation.

Mathematics competitions  problems based on cyclic patterns
Author Alan ChristisonSource: Learning and Teaching Mathematics 2017, pp 35 –38 (2017)More LessProblems based on cyclic patterns are often included in mathematics Competitions and Olympiads, and on many occasions are linked to the year of the competition. It is important to be able to recognise these types of questions and to understand the underlying pattern. Determining for example the exact value of 𝑎^{𝑏}, where 𝑎 and 𝑏 are positive integers, soon becomes impractical with a calculator when either 𝑎 or 𝑏 becomes large. However, irrespective of the values of 𝑎 and 𝑏, determining the last digit of 𝑎^{𝑏} becomes a simple matter once the cycling pattern in the final digit is recognised.

The parabola and the straight line : relating their equations
Author James MetzSource: Learning and Teaching Mathematics 2017, pp 40 –41 (2017)More LessThere is an interesting relationship between the equation of a parabola with turning point (𝑥_{1}; 𝑦_{1}) passing through another point (𝑥_{2}; 𝑦_{2}) and the equation of a straight line passing through the same two points.

A multiple solution task : another SA Mathematics Olympiad problem
Author Michael de VilliersSource: Learning and Teaching Mathematics 2017, pp 42 –46 (2017)More LessChallenging learners to produce multiple solutions (or proofs) for a particular problem has become an important research focus in problem solving and creativity in recent years. Research by LevavWaynberg and Leikin (2012), for example, has indicated that a Multiple Solution Task (MST) approach in the classroom can help to promote creativity among talented learners.

Assessment for learning
Authors: Ingrid Sapire, Yael Shalem and Yvonne ReedSource: Learning and Teaching Mathematics 2017, pp 47 –48 (2017)More LessThe Wits School of Education and the South African Institute for Distance Education (Saide) have developed materials for a 5unit short course based on the groundbreaking DataInformed Practice Improvement Project, a collaborative project with the Gauteng Department of Education that ran on the Wits Education campus under the guidance of Professors Yael Shalem and Karin Brodie.

Mathematical thinking in the lower secondary classroom, Christine Hopkins, Ingrid Mostert & Julia Anghileri (eds.)
Author Duncan SamsonSource: Learning and Teaching Mathematics 2017, pp 49 –50 (2017)More LessMathematical Thinking in the Lower Secondary Classroom is the joint output of a team of lecturers and teacher trainers from around the world who teach courses in South Africa under the auspices of AIMSSEC, the African Institute for Mathematical Sciences Schools Enrichment Centre. The ideas presented in the book were originally developed for AIMSSEC’s flagship Mathematical Thinking course – a 10 day residential course for educators who want to think more deeply about the way learners learn.