 Home
 AZ Publications
 Learning and Teaching Mathematics
 Previous Issues
 Volume 2012, Issue 12, 2012
Learning and Teaching Mathematics  Volume 2012, Issue 12, June 2012
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 2012, Issue 12, June 2012

From the editors
Authors: Duncan Samson and Marcus BizonySource: Learning and Teaching Mathematics 2012 (2012)More LessBy the time LTM12 reaches you the 2012 school year will already be in full swing. Issue 12 of LTM is once again filled with wonderful ideas that we hope you will enjoy trying out in your own classrooms. This edition contains a wide variety of submissions, both in terms of content, institutional affiliation, and country of origin. We have authors hailing from South Africa, the USA (Hawaii, Texas and Pennsylvania), Hong Kong and Israel.

Sorting mathematical representations : words, symbols, and graphs
Author Lynn ColumbaSource: Learning and Teaching Mathematics 2012, pp 3 –8 (2012)More LessMany students view the language of mathematics as a "foreign" language. Both require making connections to new vocabulary and using that vocabulary in context. Activities engaging students in communicating ideas about mathematics provide students with the opportunity to use mathematical language and make sense of what they are learning. Words, symbols, and graphs are powerful methods of communicating mathematical ideas and relationships. These tools allow students to express mathematical ideas to other people. Moving from one representation to another is an important way to enhance mathematical concepts. NCTM's Standards (2000) recommends that all students create and use representations to organize, record, and communicate mathematical ideas and consolidate their mathematical thinking through communication. In addition, NCTM's Curriculum Focal Points (2007) promotes a focused curriculum from both the concept and content perspectives in which students continually engage in and construct mathematical knowledge.

A simple visual proof of the Theorem of Pythagoras
Author Michael De VilliersSource: Learning and Teaching Mathematics 2012 (2012)More LessA wellknown circle geometry theorem is the socalled 'intersecting chords' theorem, which states that "if two chords of a circle, AB and CD, intersect at E, then AE.EB = CE.ED".

Probability and the ambiguity of language
Author Jim MetzSource: Learning and Teaching Mathematics 2012 (2012)More LessI have been travelling around various parts of South Africa doing voluntary work. Through my work with teachers I have noticed the issue of the ambiguity of language with regard to probability. As greater emphasis is placed on probability in the new curriculum, three important words must be clearly understood by learners and educators alike: "a", "one" and "or".

Foundation Phase Numeracy enriching encounters at the 2011 AMESA National Congress
Author Peter PausigereSource: Learning and Teaching Mathematics 2012, pp 10 –13 (2012)More LessThe 2011 AMESA Congress was hosted by the Wits School of Education from 11 to 15 July 2011. In this report I reflect on some of the enriching primary mathematics encounters I experienced as a congress delegate. With my research focusing on primary mathematics, I purposefully attended workshops, How I Teach sessions, formal paper presentations and Maths Market sessions that specifically targeted Foundation and Intermediate Phase educators. In this paper I report on a selection of these with a view to encouraging primary mathematics educators to attend future provincial and national AMESA conferences.

Multiply with your fingers
Author Jim MetzSource: Learning and Teaching Mathematics 2012, pp 14 –15 (2012)More LessIn his delightful book, Magic House of Numbers, Irving Adler shows how to use your fingers to get the multiplication table from 6 to 10. What follows is a brief summary of the method.

What is the function?
Author Michael De VilliersSource: Learning and Teaching Mathematics 2012, pp 16 –18 (2012)More LessCalculus as the study of the variation and behaviour of functions is an immensely powerful tool with which we can model and understand many real world phenomena. In fact, historically its origins in the 1600's and onwards were strongly rooted in the scientific desire to understand problems of speed, time, acceleration and forces. Unfortunately, calculus is often nowadays taught somewhat divorced from these historical roots. Such real world applications, however, could be a strong source for motivating students and gaining their interest.

A visual proof of the area of a trapezium revisited
Author YiuKwong ManSource: Learning and Teaching Mathematics 2012 (2012)More LessIn the July 2009 issue of Learning and Teaching Mathematics (see [1]), I described a proof without words of the area of a trapezium. Unfortunately, due to the given diagram being a special, symmetrical case rather than a general trapezium, some readers might have misinterpreted that the proof was only valid for an isosceles trapezium (see [2]). To clarify this point, we can refer to the visual proofs for two general trapeziums below.

Even oddness & odd evenness in figural pattern generalisation activities
Author Duncan SamsonSource: Learning and Teaching Mathematics 2012, pp 20 –23 (2012)More LessThe investigation of pictorial patterns has become a standard pedagogical practice in mathematics classrooms around the world. Such pictorial patterns are often used as an entry point to basic algebra and the concept of generalisation, but they can also be used to provide a meaningful visual/practical context for exploring algebraically equivalent expressions of generality. Typical pictorial patterns generally make use of either matchsticks or dots. A typical pictorial context is shown below.

Does the compound interest formula give us the "real" answer?
Author Craig PournaraSource: Learning and Teaching Mathematics 2012, pp 24 –26 (2012)More LessTake a look at the compound interest formula : A = P (1 + i)^{n}
I have always been astounded by its elegance. When you derive the formula, the terms collapse into common factors resulting in a neat, easytouse and easytoremember formula. But that's not what this article is about. Recently I started to wonder whether the answers we get from this formula are the same as the values the bank would get.

Error detection as mathematical catalyst : part 2
Author Deonarain BrijlallSource: Learning and Teaching Mathematics 2012, pp 27 –29 (2012)More LessFollowing the theme created in the paper by Brijlall (2011), we encountered another mathematically unsound question set in a provincial senior certificate preparatory examination paper in 2011. This question formed the basis of an email discussion within our mathematics education group. We decided to share our thoughts with a broader audience as two important issues arose from our interactions. In this article we present (1) the initial question, (2) the general strategies proposed by some text books, (3) an expected solution from the marking memorandum, (4) a learner solution that led to a contradiction to the expectation of the marking memorandum, (5) the reasons for such anomalies and (6) suggested pedagogical recommendations for teachers, writers of text books and educationists.

Seeing the average of averages
Author Jim MetzSource: Learning and Teaching Mathematics 2012, pp 30 –33 (2012)More LessA classic problem asks us to determine the average speed when the speeds over two equal distances are given.

LTM cover diagram, February 2006
Author Michael De VilliersSource: Learning and Teaching Mathematics 2012, pp 34 –35 (2012)More LessThe Learning & Teaching Mathematics journal, No. 3, February 2006, as shown in the first figure above, had an appealing geometric design on the cover that begs some further exploration. At first glance it may visually appear that the formed central octagon is regular, and one may thus be tempted to conjecture that it is a regular octagon (as many of my students did when asked).

The heuristic beauty of figural patterns
Author Duncan SamsonSource: Learning and Teaching Mathematics 2012, pp 36 –38 (2012)More LessI am generally left with a sense of unease whenever I encounter questions of the type shown below:
Write down the next three terms in the following sequence: 2 ; 6 ; 12 ; ...
My disquiet with such questions stems from the basic observation that any finite sequence of numbers can be justifiably continued in an infinite number of ways. Thus, any arbitrary choice of "the next three terms" should be given full credit. However, there seems to be a general acceptance that such questions are underscored by an unstated proviso that the three terms required should represent the "most obvious" or "most logical" sequence  whatever that's supposed to mean! Those questions that are prefaced, presumably with good intentions, with something along the lines of "Given that the following pattern continues in the same way..." are even more irksome, since there's an assumption that the "obvious" pattern is indeed that, i.e. "obvious", whereas I would argue that this notion is nonsensical. This can be understood by taking cognizance of the fact that a finite numeric sequence can be generated by an infinite number of functions, an idea that is readily supported by considering a finite number of points plotted in the Cartesian Plane where there would clearly be an infinite number of curves that could be drawn through the specified points. Thus, no finite sequence of numerical terms uniquely specifies the following term in a given sequence.

Sounding the numbers : an interdisciplinary teaching model
Authors: Dorit Patkin and Yifat SimpsonSource: Learning and Teaching Mathematics 2012, pp 39 –44 (2012)More LessThe article outlines an interdisciplinary project trialled at an Israeli teachertraining college. Devised by maths and music lecturers, it involved a class of firstyear students training to become earlyyears educators, and was incorporated into their weekly maths classes. As part of the first stage of the project, a teaching model, original teaching materials and a range of guiding pedagogical strategies were developed and tested at college level over one term. The article opens with a short theoretical examination of the advantages of an integrative approach to the teaching of music and mathematics in earlyyears education. It then goes on to present some of the teaching materials devised and tested as part of the project. The article will be relevant to colleagues across the spectrum of earlychildhood teachertraining and professional development programs.

Promoting studenttostudent discourse in small groups : findings of a practitioner action research project
Author Sarah Quebec FuentesSource: Learning and Teaching Mathematics 2012, pp 45 –50 (2012)More Less"Effective discourse happens when students articulate their own ideas and seriously consider their peers' mathematical perspectives as a way to construct mathematical understandings" (NCTM, 2010). Establishing a classroom community in which discourse is an integral part of teaching and learning can be challenging, especially if students are more accustomed to being passive listeners rather than active participants in the mathematics classroom. As a classroom teacher, I found myself in this situation. Although I incorporated both smallgroup and wholeclass discussions into my lessons, I was particularly interested in promoting discourse between students while they were working in small groups: How should I help students when they are working in groups? Do my interactions promote or hinder studenttostudent communication? How do I encourage students to listen to and evaluate each others' explanations?