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 Volume 2010, Issue 8, 2010
Learning and Teaching Mathematics  Volume 2010, Issue 8, January 2010
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Volume 2010, Issue 8, January 2010

Individualised teacher support intervention  a case study
Author Deonarain BrijlallSource: Learning and Teaching Mathematics 2010, pp 4 –13 (2010)More LessThe majority of secondary schools in the KwaZuluNatal Department of Education and Culture are underresourced in both material and human terms. A report published by EduSource in 1997 found that most mathematics and science educators were not qualified to teach the subjects. Although 85% of mathematics educators were professionally qualified as educators, only 50% had specialized in mathematics teaching (DoE, 2001). After a pilot study, we were aware of underqualified educators who were teaching mathematics at grades 10, 11 and 12. We found also that the learnereducator ratio was too high (in most cases about sixty to one). There was also an inadequate supply of textbooks at the two schools investigated. Other resources were also unavailable.

A new view of Pascal's triangle
Author Dolly NdudeSource: Learning and Teaching Mathematics 2010 (2010)More LessThe following problem was presented to educators at the Teachers Without Borders Workshop I attended at Trinset, Mthatha during the winter break. If you can move only right or down in this network of perpendicular roads, find the number of paths from A to each junction.

The influence of emotion, confidence, experience and practice on the learning process in mathematics
Source: Learning and Teaching Mathematics 2010, pp 15 –19 (2010)More LessA few years ago while I was teaching a grade 12 mathematics class I discovered that one of the girl learners, (Shahieda) had major difficulties in learning mathematics. While teaching the class I would do some examples of a topic on the board and then request that the learners attempt one or more similar problems on their own. While the learners were attempting the problems I would walk around in the class and invariably I found that Shahieda was struggling. I would then sit next to her and point out the mistakes in her solution and then proceed to correct her argument. Subsequently I would request the class to attempt another problem of the same kind. Shahieda would then provide the correct solution to this followup problem. Satisfied that she was now cognizant of the requirements of such problems I would proceed to the next topic.

Defensive teaching in mathematics
Author Betty McDonaldSource: Learning and Teaching Mathematics 2010, pp 20 –21 (2010)More LessFirsttime hikers know how important it is to have an experienced hiker lead the way. The experienced hiker is aware of the pitfalls, the snags and hazards along the path and knows how to avoid them. The novice hiker can safely follow the experienced hiker with the assurance that he would arrive at his destination safely. In a similar way, the defensive teacher operates like the experienced hiker in skillfully leading the student to his destination. Defensive education is not just about providing information to students. It anticipates possible misconceptions and errors and attempts to avoid them before they actually occur. Like any other teaching methodology it has its advantages and disadvantages. Some believe that it deprives students of the challenging and intense intellectual experience that comes with an informal relationship (Furedi, 2006).

A tessellating nosmoking sign
Author Michael De VilliersSource: Learning and Teaching Mathematics 2010, pp 22 –23 (2010)More LessDuring a recent visit to the Program Graduate Studies in Mathematics Education at the Pontifical Catholic University of São Paulo, I observed the interesting nosmoking sign shown below in buildings all over the city. It immediately struck me that one could easily use this picture for a little mathematical investigation by asking students whether they thought this figure would tessellate (tile) or not (and to explain why or why not). It might be a good activity for students to use cardboard cutouts and/or to model it with dynamic geometry software like Sketchpad, and then to use the builtin transformations to see if they could tile with it.

Grade 9 anecdotes
Author Andrew MaffessantiSource: Learning and Teaching Mathematics 2010, pp 24 –28 (2010)More LessGrade 9 anecdotes

Do we ever use parabolas in real life?
Author Marc NorthSource: Learning and Teaching Mathematics 2010, pp 29 –35 (2010)More LessEvery teacher's nightmare ... it's the last lesson on a Friday afternoon and I'm attempting (largely unsuccessfully) to teach parabolas. I'm trying desperately to get half of the class to understand the work and the other half to concentrate, and then some smartalec opens their mouth with the dreaded question : "C'mon Sir, where will we ever use this in life, huh, huh?"

What skills must a grade 7 have to succeed with high school maths?
Author Paul De WetSource: Learning and Teaching Mathematics 2010, pp 36 –40 (2010)More LessAbout eleven years back when I was on the Maths staff at Hilton College, each of our Maths teachers visited a different feeder school to get a sense of what they were doing in their Maths classrooms and to share some ideas about our expectations (mathematically) of the pupils they were sending on to us. This proved to be a very interesting and valuable exercise. There was considerable variation in the approach and in the scope of work being taught at the five schools visited. More recently, in October 2007, my friend and Headmaster of Felixton College, Ken Krige, asked me to talk to primary school teachers from a variety of Zululand prep schools on the skills required for High School Maths by a primary school leaver. My Head of Department here at Michaelhouse, Alan AdlingtonCorfield, and I decided to survey our Grade 8's and 9's at that time to get their take on the effectiveness of their primary school preparation for High School Maths. The results of this survey were also interesting and, together with our experience, they informed the talk that I subsequently gave at Felixton. Lastly, I recently read an excellent book entitled, "Where do I put the decimal point?" by Elizabeth Ruedy. It is a mustread for all Maths teachers as it explores Math Phobia and gives great tips for overcoming it. It too has been very useful in informing my thinking. I do not want this article to come across as prescriptive but rather hope that it will stimulate thought and discussion. If it adds any value to primary school teachers preparing pupils for High School Maths then it will have served its purpose well.

Pitfalls of personally constructed learning devices
Author Keith NabbSource: Learning and Teaching Mathematics 2010, pp 41 –45 (2010)More LessAs a mathematics teacher of several years, I have always taken an interest in what students write down on their homework and/or test papers. Especially intriguing are the scribblings that are foreign to me but seem to assist the student in problem solving. Here I do not speak of recurring slips in mathematical notation or even the occasional error in transcribing formulas from memory. Instead, I speak of the highly personal and admittedly invented expressions whose meanings continue to elude me. For example, I can understand "tan = sec^{2}" even if mildly insulted by its sloppiness  but things like DOKIDI and BBB slip right by me.

Pythagorean triples  a fractional approach
Author Duncan SamsonSource: Learning and Teaching Mathematics 2010, pp 46 –49 (2010)More LessA Pythagorean triple is a set of three positive integers which are able to form the sides of a rightangled triangle. The positive integers a , b and c form a Pythagorean triple if they satisfy the equation a^{2} + b^{2} = c^{2}. Such triples are commonly written in the format (a,b,c), and by the end of Grade 9 most pupils would have encountered wellknown examples such as (3,4,5) and (5,12,13). A primitive Pythagorean triple is one in which the three integers a, b and c are relatively prime, i.e. where they share no common factor other than 1.

How to make every graph a straight line (or not!)
Author Vera FrithSource: Learning and Teaching Mathematics 2010, pp 50 –55 (2010)More LessI have been teaching quantitative (mathematical) literacy to first year university students in various disciplines for the last ten years. Each year I observe a few students using an interesting technique for plotting graphs of functional relationships (I shall call it "the technique").

Let me count the ways
Author Jim MetzSource: Learning and Teaching Mathematics 2010, pp 56 –60 (2010)More LessAt the Teachers Without Borders Workshop held at Trinset, Mthatha during the winter break, educators were presented the problem of finding the sum, 1 + 2 + 3 + 4 + 5 +... n, by finding the area of a staircase figure that has a base of n squares.

Innovative pedagogical approaches based on solving a disjunctive probability problem
Authors: Sudi Balimuttajjo and Robert J. QuinnSource: Learning and Teaching Mathematics 2010, pp 61 –66 (2010)More LessProbability remains a pedagogically problematic topic in mathematics at both primary and secondary levels (Way and Ayres, 2002). Topics in statistics and probability require special attention, mainly because the underlying misconceptions students hold about probability are often not addressed by their teachers. Students find the abstract expressions, complex terms, and nested relationships contained in probability laws hard to understand and tend to rely on everyday reasoning skills that may be flawed (Tomlinson and Quinn, 1997). Developing sound reasoning skills is important in probabilistic thinking because many students and adults hold misconceptions about the likelihood of events based on their feelings (Stepans & Hutchison, 1998; Green, 1982). Additionally, students' understanding of probability does not improve naturally with age  teaching plays an important role (Li and PereiraMendoza, 2000). Furthermore, the difficulty of unlearning faulty beliefs lies in the fact that such beliefs are often grounded in the principles of probability but include subtle misconceptions that can be difficult to explain (Costello, 2008).

Sum to infinity  an openended investigation
Author Duncan SamsonSource: Learning and Teaching Mathematics 2010, pp 67 –71 (2010)More LessOpenended investigations are a wonderful way to access a diverse range of mathematical topics in a meaningful and engaging manner. Not only do such topics often arise unexpectedly or serendipitously, but openended investigations can provide an ideal context for nurturing such important dispositions as curiosity, creativity and selfconfidence, along with feelings of personal relevance and a desire to engage dynamically in a process of genuine mathematical discovery.

Tips from teachers of top performers
Source: Learning and Teaching Mathematics 2010 (2010)More LessLTM would like to encourage all our readers who have worked to help students gain exceptional mathematics results within various contexts such as national Olympiads and examinations or Sci Fest competitions to share their ideas of how to support students to achieve these results. Two LTM readers share their ideas in this issue.

Rectangles inscribed in a triangle (revisited) and an application of transformation geometry
Author Nic HeidemanSource: Learning and Teaching Mathematics 2010, pp 73 –74 (2010)More LessThe interesting problem solved by Poobhalan Pillay and Marcus Bizony (Learning and Teaching Mathematics, 7) was to find the largest rectangle that can be inscribed in an arbitrary triangle with the vertices of the rectangle on the sides of the triangle.