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 Volume 2012, Issue 13, 2012
Learning and Teaching Mathematics  Volume 2012, Issue 13, January 2012
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Volume 2012, Issue 13, January 2012

From the Editors
Authors: Duncan Samson, Marcus Bizony and Lindiwe TshabalalaSource: Learning and Teaching Mathematics 2012 (2012)More LessIssue 13 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 Massachusetts), England, Israel, New Zealand and Hong Kong.

The role of physical manipulatives in teaching and learning measurement
Author Shakespear ChiphamboSource: Learning and Teaching Mathematics 2012, pp 3 –5 (2012)More LessMeasurement is a critical aspect of mathematics that affords opportunities for learning while applying and engaging with a host of other mathematical topics (Clements & Bright, 2003, p. xi). Although measurement is a theme that permeates all areas of mathematics as well as daytoday life, research has shown that many learners find it an aspect of mathematics that is difficult to grasp, with learners often "not understand[ing] the attribute being measured or the units that are used for measurement" (O'Keefe & Bobis, 2008, p. 391). Learners often find particular difficulty in determining the surface area and/or volume of a given object. Van de Walle (2004) argues that when learners are only taught the performance of the skills of a particular procedure at the expense of developing and engaging with the concept itself, they become reluctant to attach meaning to it. This problem poses many challenges for mathematics teachers.

Straight to the point
Author Colin FosterSource: Learning and Teaching Mathematics 2012, pp 6 –10 (2012)More LessIt is always nice when an idea for a task comes from a member of the class (Silver, 1994; Kilpatrick, 1987). I had asked Year 7 students to draw several graphs of the form y = mx + c, where m and c are constants, choosing for themselves what values to use for the constants. The idea was to look for the effect of m and c on the shape of the graph. Making the link between the form of an equation and the appearance of its graph is something that is reported as being difficult (Knuth, 2000a, 2000b).

A numerical method for finding the equation of any quadratic sequence
Author Ashley Ah GooSource: Learning and Teaching Mathematics 2012, pp 11 –13 (2012)More LessThere have been a number of articles in past issues of LTM that have focused on various methods for determining the general formula for a given quadratic sequence (see for example Samson, 2008; Bowie & du Plessis, 2009). This article adds to the growing discussion around quadratic sequences.

Geometry from the world around us
Authors: Dorit Patkin and Ilana LevenbergSource: Learning and Teaching Mathematics 2012, pp 14 –18 (2012)More LessGeometry forms an important component in both elementary and high school curricula. However, it is often perceived as being one of the most complex parts of the curriculum. Students frequently experience a sense of travelling to "an isolated island" where everything is structured in a "logical" or "unusual" way, without any relation to daily life.

Deriving the composite angle formulae for sine from Ptolemy
Author Michael De VilliersSource: Learning and Teaching Mathematics 2012 (2012)More LessPtolemy of Alexandria (approximately 100168 AD) not only developed his Planetary theory in his treatise Almagest, but also proved the following remarkable theorem, known as Ptolemy's theorem, which he used to compute his table of chords (trigonometric tables) that was in use for over 1000 years: "The sum of the two products of the opposite sides of a cyclic quadrilateral ABCD is equal to the product of the diagonals; e.g. AB.CD + AD.BC = AC.BD."

Searching for a neat cubic
Author Marcus BizonySource: Learning and Teaching Mathematics 2012, pp 20 –21 (2012)More LessIt's that time of year again: I have to set some calculus questions for school exams. Naturally I want to include a question about a cubic in which pupils would be expected to find the xintercepts as well as the turning points. It's easy if we allow one of the factors of the cubic to be repeated, so that the graph has one of its turning points on the xaxis, since that way we ensure that the turning points have rational coordinates and that the pupils have a moderately easy task.

Exploring language issues in multilingual classrooms
Author Lindiwe TshabalalaSource: Learning and Teaching Mathematics 2012, pp 22 –25 (2012)More LessMultilingualism is rapidly becoming a serious challenge for many schools in South Africa, perhaps most noticeably in the Gauteng province. Not only do many schools have learners with a variety of South African indigenous languages as their home language, but numerous schools also have learners from other African countries. As a result of this language complexity many schools have opted for English as the language of teaching and learning despite that fact that many teachers and learners are not fluent in English. As such, the teaching context in such schools is highly complex, and is heavily affected by multilingualism. During the course of learning Mathematics numerous language issues emerge which raise a number of critical questions. How does the fact that teachers teach mathematics in a language that is not necessarily the learners' home language affect the teaching and learning of mathematics? What impact does the teacher's home language have on the promotion of learners' conceptual understanding in the Mathematics classroom?

Using movement to teach geometry
Author Paula J. ArvedsonSource: Learning and Teaching Mathematics 2012, pp 26 –27 (2012)More LessLearners who are physically involved in their mathematics instruction delight in discovering and engaging with concepts as a result of the realworld connections which are forged through mind and body working together. They gain a depth of understanding far greater than any worksheet on its own can ever provide. Their mathematics vocabulary increases and they develop problemsolving skills that can be used in all subject areas. Sack and van Niekerk (2009) assert that "children should develop competence using physical and mental processes with visual representation modes in addition to verbal descriptions, regardless of the representation given in any particular problem" (p. 142). What follows is an example of a Grade 1 geometry unit on identifying and describing two and threedimensional shapes which incorporates body movement combined with verbal descriptions.

Viviani's theorem  a geometrical diversion
Author Duncan SamsonSource: Learning and Teaching Mathematics 2012, pp 28 –32 (2012)More LessViviani's theorem, named after the Italian mathematician and geometer Vincenzo Viviani (1622  1703), is a simple yet delightfully intriguing geometry theorem with a number of interesting extensions. Viviani's theorem states that for any point inside an equilateral triangle the sum of the perpendiculars from that point to the sides of the triangle is equal to the length of the triangle's altitude.

Teaching strategies and activities that enhance spatial visualization
Authors: William A. Jasper, Barbara E. Polnick and Sylvia R. TaubeSource: Learning and Teaching Mathematics 2012, pp 33 –37 (2012)More LessRussell and Nicole are working together on a project in their seventh grade mathematics class. Mrs. Jones, their teacher, asks them to count the number of squares that they can find on a standard 8 by 8 chessboard. Nicole has limited prior experiences with activities such as these, but she easily counts all the small squares and concludes that 64 is the total number of the squares on the 8 by 8 chessboard (or checkerboard). Russell, on the other hand, had played on his Nintendo and Game Boy for five years and was also a selfproclaimed expert at the game of Tetris. In addition to the 64 squares that Nicole found, Russell tells Mrs. Jones that there are additional squares formed by groups of the small squares. Nicole just did not "see" these other squares on her first count. Scenarios like these occur frequently in mathematics classrooms where students are developing the foundation required for higher levels of mathematics. It is important that teachers find ways to enhance students' spatial sense not only for future success in mathematics but also to enhance students' interests and talents in mathsrelated fields such as the physical sciences, architecture, computeraided design, geographic information systems, and graphic design (Allen, 2003; Lury & Massey, 1999; Mark & Egenhofer, 1994).

Generalising cubic sequences using a filtration method
Author Wandile HlaleleniSource: Learning and Teaching Mathematics 2012, pp 38 –39 (2012)More LessIn my experience, most learners have become quite proficient at determining the general expression for a given quadratic sequence. One can build on this proficiency to generalise cubic sequences of the form Tn = an^{3} + bn^{2} + cn + d by using a filtration technique. The filtration method provides a means of "filtering out" the cubic part of the general expression for a cubic sequence leaving behind a quadratic "residue", a general expression for which can be found using any number of conventional approaches.

Ladders and boxes
Author Peter WatsonSource: Learning and Teaching Mathematics 2012, pp 40 –45 (2012)More LessFinding problems that students actually enjoy doing can be a challenge. A good place to start might be to consider the problems that previously brought pleasure to you. For me a class of problems that I have enjoyed (and still do) are what Martin Gardiner (Gardiner, 1959, 1961a, 1961b, 1977) called mathematical puzzles, and I have used such puzzles in the classroom with great success. More often than not these kinds of puzzles are used by teachers for their recreational or 'fun' element, and it is often difficult to see how such puzzles might relate to the school curriculum. In recent times I have been interested in exploring puzzles that can not only be appropriately incorporated into a lesson, but which form an integral part of the lesson. This often requires a slight modification of the original puzzle in order to render it more suitable, a process which can be far from easy. In this article I present a particular type of puzzle based on Pythagorean triples and, through a process of investigation, show how similar puzzles could be constructed.

A study of a multiplestrategy approach to locating the centre of a circle
Author KinKeung PoonSource: Learning and Teaching Mathematics 2012, pp 46 –51 (2012)More LessProblemsolving plays a crucial role in mathematics education. One of the aims of teaching through problemsolving is to encourage students to refine and build their own processes over time as their experience allows them to discard some ideas and makes them aware of other possibilities (Carpenter, 1989). In addition to developing their knowledge, students also acquire an understanding of when it is appropriate to use particular strategies. Emphasis is placed on making students responsible for their own learning rather than letting them feel that the methods that they are using are the inventions of others. Considerable importance is placed on exploratory activities, observation and discovery, and trial and error. Students must develop their own ideas, test them, discard them if they are not consistent, and try something else (National Council of Teachers of Mathematics [NCTM], 1989). Students become more involved in problemsolving by formulating and solving their own problems, or by rewriting problems in their own words to aid their understanding. Crucially, students are encouraged to discuss the processes that they are using to improve their understanding, gain new insights into the problem, and communicate their ideas (Thompson, 1985; Stacey & Groves, 1985).

Considering terminology and notation of the decimal system
Author Richard KaufmanSource: Learning and Teaching Mathematics 2012 (2012)More LessHave you or any of your students noticed that terms on the left and right side of the decimal point can appear to be inconsistent? For example, the hundreds' digit is 3 places to the left of the decimal point, while the hundredths' digit is 2 places to the right of the decimal point. Students can sometimes find the terminology confusing since tens are not paired with tenths, hundreds are not paired with hundredths, and thousands are not paired with thousandths. This reflects an innate expectation for symmetric terminology.

Counting lattice points
Author Jim MetzSource: Learning and Teaching Mathematics 2012, pp 53 –54 (2012)More LessThe following problems challenge students to organize their work, recognize patterns, make generalizations, and apply a variety of algebraic concepts and formulas, all within the context of lattice points on a grid. This activity was inspired by the classic "painted cube" problem in which a painted cube is cut into smaller identical cubes and the problem is to determine the number of cubes with 0, 1, 2 and 3 faces painted. When students have completed the Counting Lattice Points activities, they should be encouraged to investigate the "painted cube" problem.

Problems with word problems in mathematics
Author Hanlie MurraySource: Learning and Teaching Mathematics 2012, pp 55 –58 (2012)More LessMany teachers complain that learners find word problems in mathematics more difficult than straight computation, and that many learners dislike and even fear word problems. Different reasons are given for this phenomenon, of which the most common is that learners cannot read with understanding and therefore do not know what is required of them. One way to resolve this dilemma is simply to sidestep it  teach the methods, tools, techniques, and formulas which society believes make up mathematical knowledge and then test and examine in such a way that this kind of knowledge will ensure a good pass rate. Although this was a popular and common choice in the past, extensive research and practical (and personal) experience show clearly that this type of mathematics education is no real education at all and does not equip learners to deal with mathematical problems or to cope with further education in mathematics.