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 Volume 2009, Issue 7, 2009
Learning and Teaching Mathematics  Volume 2009, Issue 7, January 2009
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Volume 2009, Issue 7, January 2009

Turning Points!
Author Jogy AlexSource: Learning and Teaching Mathematics 2009 (2009)More LessMy passion for mathematics started when I was in standard 6. In the first term Mathematics examination I got only 27 out of 50. My elder sister then told me that I must practise my mathematics by writing it many times. I started practising like that and for my second term Exam, I got 49,5 out of 50!! Since then I was crazy about Maths and always got high marks in all my Maths Exams.

Illegal operations with fractions that work
Author Marcus BizonySource: Learning and Teaching Mathematics 2009, pp 4 –5 (2009)More LessWe know that when you want to multiply two fractions you just multiply the tops and multiply the bottoms. How many of us have seen learners try to do the same when they add fractions, i.e. add the tops and add the bottoms?

Do infinite things always behave like finite ones?
Authors: Jurie Conradie, John Frith and Lynn BowieSource: Learning and Teaching Mathematics 2009, pp 6 –12 (2009)More LessIn a calculus class for preservice teachers the lecturer asked the prospective teachers to explore whether 0.999... is equal to 1. Unsurprisingly the majority of the class's initial response was that 0.999... is less than 1. After some research and through discussion with the lecturer some groups of students came up with arguments that show that 0.999... = 1.

Rectangles inscribed in a triangle
Authors: Poobhalan Pillay and Marcus BizonySource: Learning and Teaching Mathematics 2009, pp 13 –15 (2009)More LessGiven an arbitrary triangle, we wish to form a rectangle with all four vertices on sides of the triangle, and one side on one side of the triangle, so that the rectangle shall be as large as possible. The suggestion is that this is achieved by having two of the vertices at midpoints of sides of the triangle.

Exemplifying the fundamental principles of counting
Authors: Bonnie H. Litwiller and David R. DuncanSource: Learning and Teaching Mathematics 2009, pp 16 –17 (2009)More LessSuppose that a traveler wishes to cross two rivers consecutively by car. The first river has three bridges (A, B, C) while the second river has two bridges (D, E). In how many ways can the traveler cross these two bridges?

Patterns with a constant second difference
Authors: Lynn Bowie and Jacques Du PlessisSource: Learning and Teaching Mathematics 2009, pp 18 –24 (2009)More LessIn a course on algebra with practising teachers we were reading a rather nice paper by Alan Bell entitled "Purpose in School Algebra" (Bell, 1995). In this article he offers and discusses a variety of tasks that he claims support the aims of school algebra. One particular task that provoked interesting discussion within our group and then further exploration was the following one (Bell, 1995, p68).

Mathematical proficiency and practices in Grade 7
Author Rasheed SanniSource: Learning and Teaching Mathematics 2009, pp 25 –28 (2009)More LessThe evolvement of a mathematically literate society is one of the goals of mathematics education. One way of achieving this is by promotion of mathematical reasoning. RAND Mathematics Study Panel's (2002); Mathematics practices and Kilpatrick et al.'s (2001) Strands of Mathematical Proficiency provide frameworks for developing mathematical reasoning in our learners. Practices and proficiency can be achieved by exposing of learners to appropriate experiences in the classroom.
In this paper I attempt to investigate the extent to which the assessment standards in respect of different learning outcomes of the South African school mathematics reflect the tenets of mathematics practices and mathematical proficiency, with particular emphasis on Grade 7.
Each of the verbs that were used in describing assessment standards was classified according to the strand(s) of mathematical proficiency and / or practices that it promotes. The results show that conceptual understanding and adaptive reasoning are mostly promoted by the assessment standards with 37.3% and 24.1% occurrences respectively. On the other hand, justification 2.4%; generalization 1.2%; and productive disposition 0.0% are least promoted. These findings are discussed at the end of the paper.

Financial maths  moving beyond formulae
Authors: Susana Da Silva, Craig Pournara and Nontando MafuyaSource: Learning and Teaching Mathematics 2009, pp 29 –33 (2009)More LessFinancial mathematics in high school has tended to be dominated by substituting values into formulae. We believe that learning financial maths could be much more than this. So Susana and Nontando set out to investigate what happens when teachers try to move beyond formulae in financial maths. This investigation was the focus of their research projects for their honours programme. In this article we share the tasks that we gave to Nontando's Grade 12 mathematics class. We discuss some of the learners' responses to the tasks and make some suggestions for the teaching and learning of financial maths.

A simple approach deriving the quadratic formula
Author YiuKwong ManSource: Learning and Teaching Mathematics 2009 (2009)More LessMost teachers use the method of completing the square to accomplish this task. However, from the teaching point of view, it is not an easy approach for the students to master, especially when they are learning this topic at an early stage.

Visualising series
Author Duncan SamsonSource: Learning and Teaching Mathematics 2009, pp 35 –39 (2009)More LessVisualisation is recognised as being a central component in mathematical activity (Cunningham, 1991; Hershkowitz et al., 2001; Arcavi, 2003). Furthermore, visualisation as both the product and process of creating, interpreting and reflecting upon images, is gaining increased focus in the fields of both mathematics and mathematics education (Zimmermann and Cunningham, 1991; Arcavi, 2003). It has even been suggested that visual thinking may well become "...the primary way of thinking in the future" (Hershkowitz and Markovits, 1992:38).

Generalizing the golden ratio and Fibonacci
Author Michael De VilliersSource: Learning and Teaching Mathematics 2009, pp 39 –41 (2009)More LessIn a golden rectangle, the rectangle obtained by removing a square from one end is similar to the original rectangle (see figure). The ratio of the length to the width of such a rectangle is called the golden ratio and is often denoted by the symbol Φ.

The plight of the petrol pump attendants
Author Marc NorthSource: Learning and Teaching Mathematics 2009, pp 42 –46 (2009)More LessEarlier this year I drove into a petrol station to fill up with diesel and was confronted by the most unusual sight : two very unhappylooking petrol attendants and a broken diesel pump.

'Reasoning and reflecting' in mathematical literacy
Authors: Hamsa Venkat, Mellony Graven, Erna Lampen, Patricia Nalube and Nancy ChiteraSource: Learning and Teaching Mathematics 2009, pp 47 –53 (2009)More LessIn December 2008 we at the Marang Centre, Wits University decided to hold a two day workshop focused on a review of the recently completed Mathematical and Sciences matric examinations. Since this was the first set of matric examinations for the new National Senior Certificate for the Further Education and Training (FET) band, this seemed an important activity for academics, lecturers and researchers involved in Mathematical Sciences education research and development.

Reflecting on "Talking Geometry"
Author Michael De VilliersSource: Learning and Teaching Mathematics 2009, pp 54 –56 (2009)More LessI greatly enjoyed the chatty, heuristic style of Erna Lampen's article "Talking Geometry : a Classroom Episode" in Learning and Teaching Mathematics 6, pp. 38, describing how she creatively and interactively guided students in a large class to think geometrically with Sketchpad, stimulated by herself and their own thinking. However, I wish to raise some important mathematical issues about the end of the article.

LTM Cover Problem, June 2008
Author Michael De VilliersSource: Learning and Teaching Mathematics 2009 (2009)More LessThe June 2008 issue of Teaching and Learning Mathematics depicted the following lovely, geometric result, though with no proof given anywhere in the journal : "A cyclic quadrilateral defines four arcs. The lines joining the midpoints of opposite arcs are perpendicular."