 Home
 AZ Publications
 Learning and Teaching Mathematics
 Previous Issues
 Volume 2013, Issue 14, 2013
Learning and Teaching Mathematics  Volume 2013, Issue 14, January 2013
Volumes & Issues

Volume 2019 (2019)

Volume 2018 (2018)

Volume 2017 (2017)

Volume 2016 (2016)

Volume 2015 (2015)

Volume 2014 (2014)

Volume 2013 (2013)

Volume 2012 (2012)

Volume 2011 (2011)

Volume 01 (2011)

Volume 2010 (2010)

Volume 2009 (2009)

Volume 2008 (2008)

Volume 2007 (2007)

Volume 2006 (2006)

Volume 2005 (2005)

Volume 2004 (2004)
Volume 2013, Issue 14, January 2013

From the Editors
Authors: Duncan Samson, Marcus Bizony and Lindiwe TshabalalaSource: Learning and Teaching Mathematics 2013 (2013)More LessLTM14 begins with an interview with Rhodes University's Deputy Vice Chancellor, Dr Sizwe Mabizela. Dr Mabizela is a renowned South African mathematician who enjoys "fiddling" or "doodling" with mathematical ideas. The interview engages him in a discussion around his passion for mathematics and mathematics education. In the second article in this issue, Mark Rushby presents and discusses a number of warmup activities that he has found work well at a Grade 7 level and which have the potential to add to learners' enjoyment of mathematics. The third article explores conceptual and visual aspects of the process of "completing the square", while in the fourth article Mellony Graven and Hamsa Venkatakrishnan reflect on the Annual National Assessments (ANAs). Michael de Villiers then explores the idea that equality doesn't always represent the best solution to a modeling problem, while the sixth article reviews a useful website that allows one to tailormake number line images.

Interview with a Mathematics Doodler  Dr Sizwe Mabizela, Deputy Vice Chancellor, Rhodes University
Authors: Mellony Graven and Marc SchaferSource: Learning and Teaching Mathematics 2013, pp 3 –5 (2013)More LessIn November 2012 we interviewed Dr Sizwe Mabizela, our Deputy Vice Chancellor at Rhodes University, to find out about his love for mathematics and his enjoyment of "playing" or "doodling" with mathematics and mathematical ideas. Dr Mabizela is a renowned South African mathematician who has a deep concern about the teaching and learning of Mathematics. We were writing a paper (Graven & SchÃ¤fer, in press) for a book on mathematics knowledge for teaching entitled "Exploring Content Knowledge for Teaching Science and Mathematics" and wanted to make the point that often instilling a playful love of mathematics in learners is absent from the literature. We argue in the paper that a passion for mathematics and embracing a desire to explore and play with mathematics is as essential and important an ingredient for being an effective Mathematics teacher as content knowledge of the subject. We interviewed several of our favourite mathematics educators and asked them about how their love of mathematics played out in their life and how it all began. We share with you an edited excerpt from our extended interview with Dr Sizwe Mabizela and hope that it serves to inspire.

Warmup activities for Grade 7s
Author Mark RushbySource: Learning and Teaching Mathematics 2013, pp 6 –7 (2013)More LessAs teachers, we are usually keen to get into the day's Mathematics lesson for a particular class right from the outset, often forgetting that pupils have typically just been engaged with a different subject and as such may need a little time to make the mental transition to the Mathematics classroom. A warmup activity at the start of a Mathematics lesson is often a good way to help pupils make this transition. What follows are a number of activities that I have found work well with Grade 7 pupils.

"Completing the square"  a conceptual approach
Author Duncan SamsonSource: Learning and Teaching Mathematics 2013, pp 8 –11 (2013)More LessThe method of completing the square is a useful algebraic technique. Within the South African school context, "completing the square" is most often used for three specific purposes: (i) solving quadratic equations, (ii) writing parabola equations in turning point format, and (iii) writing circle equations in centreradius format. In my experience, learners who have good algebraic skills master the technique of completing the square fairly quickly. Learners who are less algebraically confident take a little longer to acquire the skill but are nonetheless able to master the technique with sufficient practice. However, for most learners the method of completing the square is little more than an arcane process of algebraic manipulation accomplished somewhat mechanically through the use of a guiding algorithm or mantra such as halve it, square it, add it to both sides. As such, most learners leave school without really understanding the conceptual basis of the technique. This is sad because the conceptual heart of the process is not only simple but beautifully elegant. A more conceptual approach to introducing learners to the technique of completing the square is likely not only to demystify the method for many learners, but is also likely to free learners from their reliance on rote algebraic manipulation. This article explores strategies for teaching the method of completing the square using a more conceptual approach.

ANAs : possibilities and constraints for mathematical learning
Authors: Mellony Graven and Hamsa VenkatakrishnanSource: Learning and Teaching Mathematics 2013, pp 12 –16 (2013)More LessThe introduction of the Annual National Assessments (ANAs) began in 2011. The ANA was explicitly focused on providing systemwide information on learner performance for both formative purposes, such as providing class teachers with information on what learners were able to do, as well as summative purposes, such as providing progress information to parents and allowing for comparisons between schools, districts and provinces (Department of Basic Education (DBE), 2011). The ANAs were written by all government school learners in Grades 16 as well as Grade 9 in September 2012. The ANAs focused on Literacy and Numeracy in the Foundation Phase, and Language and Mathematics in the Intermediate Phase. The 2012 national report of the ANAs (DBE, 2012) is available for downloading at http://www.education.gov.za.

Equality is not always 'best'!
Author Michael De VilliersSource: Learning and Teaching Mathematics 2013, pp 17 –21 (2013)More LessAs discussed in De Villiers (2007), the process of mathematical modeling essentially consists of three steps or stages as illustrated in Figure 1, namely (i) construction of the mathematical model, (ii) solution of the model, and (iii) interpretation and evaluation of the solution. Traditionally, much of schooling consisted of teaching learners how to solve given mathematical models such as linear, quadratic, trigonometric, exponential, and hyperbolic functions, hardly involving them at all in the construction and development of these models from real world contexts, and much less on evaluating the solutions and how well they fit 'reality'. In short, it consisted of teaching mathematics largely disconnected from the real world.

Number line image generator  a website review
Author Debbie StottSource: Learning and Teaching Mathematics 2013, pp 22 –25 (2013)More LessLast year, as part of the South African Numeracy Chair, I was involved in creating a supplement for the local Grahamstown newspaper (Grocott's Mail) called Fun with Maths. The aim of the supplement was to encourage parents and teachers to engage learners with various numeracy concepts in a fun way that differed from traditional teaching approaches. Supplements 4 and 5 of the series presented number line activities. Sizeable evidence shows that the number line is a powerful learning tool for children in primary school (see for example Beishuizen, 1997; Bobis & Bobis, 2005; Clarke, Downton & Roche, 2011). Researchers believe that regular use of number lines can develop learners' ability to form a mental number line, which in turn may assist learners in carrying out mental computation tasks. However, research has found that many learners are unsuccessful in using number lines effectively, a fact which may well be attributed to their lack of experience engaging with number lines. This review provides a brief background of various types of number lines, particularly structured number lines, and describes and reviews a webbased resource that could be used to produce a variety of closed number lines for classroom situations. The number lines incorporated in the Fun with Maths supplements were all created using this webbased tool.

Why increasing the number of compounding periods won't make you as rich as you might think
Authors: Duncan Samson and Craig PournaraSource: Learning and Teaching Mathematics 2013, pp 26 –30 (2013)More LessConsider the following scenario: R1000 is invested for a single year at 10% p.a. compounded monthly. At the end of the year the balance has accumulated to R1104.71. The total interest earned, i.e. R104.71, is equivalent to 10.47% of the amount invested. Since the power of compound interest lies in the iterative process whereby interest is earned on interest, it makes intuitive sense that if you compound interest more frequently, you will earn more money. Many school textbook tasks suggest that this is the case when they change the frequency of compounding from annual to monthly to weekly. But is this intuition correct?

An alternative trig formula for solving triangles
Author Letuku Moses MakobeSource: Learning and Teaching Mathematics 2013, pp 31 –33 (2013)More LessThis article presents an interesting trigonometry formula that can be used when solving triangles given two sides and an included angle. When the known dimensions are in SAS configuration one generally uses the cosine rule to first work out the length of the side opposite the given angle. This is then followed by a second step in which the sine rule is used to determine the size of the desired angle. The alternative formula presented here allows one to determine the size of the desired angle in one easy step. The formula is presented below in terms of angle B, the desired angle.

Reflecting on a 2^{nd} round 2013 SA Mathematics Olympiad Problem
Author Michael De VilliersSource: Learning and Teaching Mathematics 2013, pp 34 –35 (2013)More LessThe following problem was used as Question 13 in the 2^{nd} Round of the 2013 Senior South African Mathematics Olympiad (SAMO).

A quick tool for tracking procedural fluency progress in Grade 2, 3 and 4 learners
Author Debbie StottSource: Learning and Teaching Mathematics 2013, pp 36 –39 (2013)More LessThe South African Numeracy Chair (SANC) project works with fifteen schools in the broader Grahamstown area in the Eastern Cape. Among other things, the SANC project works toward improving numeracy proficiency among learners, basing its notion of numeracy proficiency on Kilpatrick, Swafford & Findell's (2001) definition of mathematical proficiency. This definition comprises five intertwined and interrelated strands: Conceptual Understanding, Procedural Fluency, Strategic Competence, Adaptive Reasoning and Productive Disposition. As part of the SANC project we run a number of regular afterschool maths clubs for learners, and in the club activities we strive to develop numeracy proficiency in the learner participants in each of these five strands.

Timelines for annuities  getting to grips with the conventions
Author Craig PournaraSource: Learning and Teaching Mathematics 2013, pp 40 –43 (2013)More LessTimelines provide a graphical representation that enables us to clarify the timing of transactions  whether payments or withdrawals. They are typically introduced at Grade 11 level, but sometimes the initial problems are so easy there seems little need to represent the information on a timeline. When annuities are learned in Grade 12, the use of timelines becomes more important, and then we need to be clear on the conventions. In this short article I discuss four important conventions of timelines, and I explain why these conventions are necessary. But first a brief comment on modelling the timing of payments.

Why still factorize algebraic expressions by hand?
Author Michael De VilliersSource: Learning and Teaching Mathematics 2013, pp 44 –46 (2013)More LessThis paper presents and briefly discusses an algebraic expression that came up in a proof that could not be factorized by current computer algebra systems, but had to be done by hand using high school techniques.

Nurturing curiosity and creativity through mathematical exploration
Author Duncan SamsonSource: Learning and Teaching Mathematics 2013, pp 47 –53 (2013)More LessInvestigations afford a wonderful opportunity for learners to explore a diverse array of mathematical ideas in a meaningful and engaging manner. Not only do investigations often lead to unexpected or serendipitous moments of mathematical discovery, but they also provide an opportunity to nurture curiosity and creativity, both of which are important components of a healthy mathematical disposition.
The investigation discussed in this article was presented to a group of secondary school teachers in the context of a teacher enrichment workshop. Teachers worked in groups of three or four and were tasked with generating as many different solutions as they could to the posed problem. The number and variety of solution strategies arising from what at first seemed like a simple starting point surprised many of the participants. In this article I synthesize the different solution methods that were generated in the workshop along with one or two additional solutions.

Deriving the composite angle formulae for sine from Ptolemy, Learning and Teaching Mathematics, Issue 13  Dec 2012 : pp. 19
Author Michael De VilliersSource: Learning and Teaching Mathematics 2013 (2013)More LessDeriving the composite angle formulae for sine from Ptolemy, Learning and Teaching Mathematics, Issue 13  Dec 2012 : pp. 19 : erratum