Abstract: Merrotsy, Peter The product rule refers to the derivative of the product of two functions expressed in terms of the functions and their derivatives. This result first naturally appears in the subject Mathematical Methods in the senior secondary Australian Curriculum (Australian Curriculum, Assessment and Reporting Authority [ACARA], n.d.b). In the curriculum content, it is mentioned by name only (Unit 3, Topic 1, ACMMM104). Elsewhere (Glossary, p. 6), detail is given in the form.

Abstract: Turner, Paul The opinion of the mathematician Christian Goldbach, stated in correspondence with Euler in 1742, that every even number greater than 2 can be expressed as the sum of two primes, seems to be true in the sense that no one has ever found a counterexample (Boyer and Merzbach, 1989, p. 509). Yet, it has resisted all attempts to establish it as a theorem.

Abstract: Teia, Luis The grand architecture of nature can be seen at play in a tree: no two are alike. Nevertheless, there is an inescapable similarity that makes us identify a tree anywhere in the world. Just saying "tree" recalls words like green, root, leaves, still, strong, branches. The tree of primitive Pythagorean triples is no different. It has a root, or a beginning. It is rooted not on earth, but on the soil of our mind. It has branches that spring from that root as it grows with the action of nature and time. In this case, it is not the proverbial Mother Nature, but the human nature - a nature formed by the human interpretation of reality. The Pythagoras' tree presented by Berggren in 1934 has stood still and strong for almost a century, but probably it is even older. Its leaves are triples, and they grow throughout its branches. Ultimately, when one looks at the Pythagoras' tree, one looks at a 'tree'. The root is the triple (3, 4, 5). All branches and leafs emerge from, and are dependent, of this root. Like any tree, all it requires is a seed and soil, and all develops automatically. The architecture that defines the tree is present throughout the tree and is a reflection of the beginning - the root. In other words, the root (3, 4, 5) plus the same movement repeated over and over again creates the tree. In this paper, we will look at how this basic geometrical and mathematical movement governs the birth and growth of the Pythagoras' tree. Pythagoras is included in secondary education around the world including in Australian Curriculum (ACARA, n.d.), and hence this paper will be of interest to all.

Abstract: Fitzherbert, John Paolo Ruffini (1765-1822) may be something of an unknown in high school mathematics; however his contributions to the world of mathematics are a rich source of inspiration. Ruffini's rule (often known as synthetic division) is an efficient method of dividing a polynomial by a linear factor, with or without a remainder. Although not described by Ruffini, the process can be generalised to non-linear divisors. Ruffini's rule can be further generalised to evaluation of derivatives at a given point, does not require technology and, most importantly, it is not beyond the reach of high school mathematics students to prove why the rule works for polynomials of a specific degree.

Abstract: Bardell, Nicholas S The cubic polynomial with real coefficients y = ax3 + bx2 + cx + d in which a ≠ 0, has a rich and interesting history primarily associated with the endeavours of great mathematicians like del Ferro, Tartaglia, Cardano or Vieta who sought a solution for the roots (Katz, 1998; see Chapter 12.3: The Solution of the Cubic Equation). Suffice it to say that since the times of renaissance mathematics in Italy various techniques have been developed which yield the three roots of a general cubic equation. A 'cubic formula' does exist - much like the one for finding the two roots of a quadratic equation - but in the case of the cubic equation the formula is not easily memorised and the solution steps can get quite involved (Abramowitz and Stegun, 1970; see Chapter 3: Elementary Analytical Methods, 3.8.2 Solution of Cubic Equations). Hence it is not surprising that with the advent of the digital computer, numerical rootfinding algorithms such as those attributed to Newton-Raphson, Halley, and Householder have become the solution of choice (Weisstein, n.d.).

Abstract: Jourdan, Nicolas; Yevdokimov, Oleksiy Proof by contradiction is a very powerful mathematical technique. Indeed, remarkable results such as the fundamental theorem of arithmetic can be proved by contradiction (e.g., Grossman, 2009, p. 248). This method of proof is also one of the oldest types of proof early Greek mathematicians developed. More than two millennia ago two of the most famous results in mathematics: The irrationality of 2 (Heath, 1921, p. 155), and the infinitude of prime numbers (Euclid, Elements IX, 20) were obtained through reasoning by contradiction. This method of proof was so well established in Greek mathematics that many of Euclid's theorems and most of Archimedes' important results were proved by contradiction.

Abstract: Padula, Janice According to the latest news about declining standards in mathematics learning in Australia, boys, and girls, in particular, need to be more engaged in mathematics learning. Only 30% of mathematics students at university level in Australia are female (ABC News, 2014) although paradoxically it would seem, the majority of lawyers in Victoria are female, a profession which requires a good grasp of language, rhetoric and logic (Merritt, 2014). So why not engage girls - with one of their strengths in early childhood (and later) - language and its acquisition (Padula and Stacey, 1990) and at the same time assist boys in their teenage years when their language development has (usually) caught up with girls (Goodman, 2012; Gurian, Henley and Trueman, 2001)? One of the ways to do this would be to teach pure mathematics - in the form of proofs.

Abstract: Stupel, Moshe; Segal, Ruti; Oxman, Victor We present investigative tasks that concern loci, which integrate the use of dynamic geometry software (DGS) with mathematics for proving the obtained figures. Additional conditions were added to the loci: ellipse, parabola and circle, which result in the emergence of new loci, similar in form to the original loci. The mathematical relation between the parameters of the original and new loci was found by the learners. A mathematical explanation for the general case, using the 'surprising' results obtained in the investigative tasks, is presented. Integrating DGS in mathematics instruction fosters an improved teaching and learning process.

Abstract: Hassani, Mehdi; Kippen, Rebecca; Mills, Terence Life tables are mathematical tables that document probabilities of dying and life expectancies at different ages in a society. Thus, the life table contains some essential features of the health of a population. We will examine life tables from a mathematical point of view.

Abstract: Kissane, Barry Although the term is often used to denote electronic devices, the idea of a 'technology', with its origins in the Greek techne (art or skill), refers in its most general sense to a way of doing things. The development and availability of various technologies for computation over the past forty years or so have influenced what we regard as important in mathematics, and what we teach to students, given the inevitable time pressures on our curriculum. In this note, we compare and contrast current approaches to two important mathematical ideas, those of square roots and of integrals, and how these have changed (or not) over time.

Abstract: Galbraith, Peter Recognition that real world problem solving expertise is a major educational goal continues to be reinforced internationally, at least officially, through documents that set specific goals for the learning of mathematics - as in the following: Mathematical literacy is defined in PISA as the capacity to identify, understand and engage in mathematics, and to make well-founded judgements about the role that mathematics plays in an individual's current and future private life, occupational life, social life with peers and relatives, and life as a constructive, concerned and reflective citizen. (OECD, 2001, p. 22).

Abstract: Padula, Janice When hoping to initiate or sustain students' interest in mathematics teachers should always consider relevance, relevance to students' lives and in the middle and later years of instruction in high school and university, accessibility. A topic such as the mathematics behind networks science, more specifically scale-free graphs, is up-to-date (think of the recent award-winning film, The Social Network, Spacey et al., 2010), highly relevant to students' lives, and accessible. This article illustrates how mathematicians and scientists work together collaboratively when applying mathematics and developing new scientific theories, and describes ways of teaching mathematics in a highly meaningful, real-world context with a topic which is widely acknowledged as being important mathematically and scientifically.

Abstract: Tay, Eng Guan; Leong, Yew Hoong We look for teaching opportunities within the curriculum to "bring the practice of knowing mathematics in school closer to what it means to know within the discipline" (Lampert, 1990, p. 29). We should also be concerned about avoiding the scenario where instead of capitalising on such teaching opportunities, a wrong treatment of an unavoidable mathematical notion (such as a0) would leave an indelible impression on students that mathematics has strange inscrutable rules and devices or even more ironically mathematics is illogical.

Abstract: Bhattacharjee, Pramode Ranjan The theoretical study of Physical Science is based on many conventions. For a systematic study, one is to follow the normal conventions which have already earned international recognition. Now, what about those conventions which are not at all realistic and which have no resemblance with problems in real world' It is a high time to think of such conventions and to get rid of them with alternative replacements in compliance with real life situations or to deal with a problem where such a convention is used by alternative treatment so as to establish a bridge between theory and practice. Such attempts have been made in Bhattacharjee (2002, 2008, 2011, 2012). A means of getting rid of the most misleading sign convention of geometrical optics has been offered in Bhattacharjee (2002, 2012). Also considering the generalized equations of motion with the approach offered in Bhattacharjee (2008, 2011), one can easily do away with the need of using the ambiguous sign convention in regard to distance measurement which has been in regular use in solving typical problems of elementary mechanics for many years.

Abstract: Bardell, Nicholas S The roots of the general quadratic equation y = ax2 + bx + c (real a, b, c) are known to occur in the following sets: (i) real and distinct; (ii) real and coincident; and (iii) a complex conjugate pair. Case (iii), which provides the focus for this investigation, can only occur when the values of the real coefficients a, b, and c are such as to render the discriminant negative. In this case, a simple two-dimensional x-y plot of the quadratic equation does not reveal the location of the complex conjugate roots, and the interested student might well be forgiven for asking, "Where exactly are the roots located and why can't I see them'" In the author's experience, this sort of question is hardly ever raised - or answered satisfactorily - in school Years 11 or 12, or in undergraduate mathematics courses. The purpose of this paper therefore is to provide a clear answer to this question by revealing the whereabouts of the complex roots and explaining the significance of the conjugate pairing.

Abstract: Yevdokimov, Oleksiy As usual, the purpose of this section is to supply teachers and students with a selection of interesting problems. In this issue we invite readers to deal with determinants that remain a core topic of the first course on linear algebra at the undergraduate level.

Abstract: Stupel, Moshe The notion of periodicity stands for regular recurrence of phenomena in a particular order in nature or in the actions of man, machine, etc. Many examples can be given from daily life featuring periodicity: day and night, the weekdays, the months of the year, the circulation of blood in our body, the function of the heart, the operation of a clock, the natural circulation of water, crop rotation, and tree crop rotation. In astronomy there are many periodical phenomena: the revolution of planets around the sun, the Solar Cycle and the Lunar Cycle, the cycle of intercalation in a period of 19 years: "Every 19 years of which 7 are intercalary and 12 are regular, is called a Period" (Maimonides, Sanctification of the Month 6). Mathematically the meaning of periodicity is that some value recurs with a constant frequency.

Abstract: Dion, Peter; Ho, Anthony For at least 2000 years people have been trying to calculate the value of pi, the ratio of the circumference to the diameter of a circle. We know that pi is an irrational number; its decimal representation goes on forever. Early methods were geometric, involving the use of inscribed and circumscribed polygons of a circle. However, real accuracy did not come until the use of infinite series techniques, in which one can, by calculating more and more terms, obtain smaller and smaller corrections all leading to a precise value. Such series go on forever, so the limitation on accuracy is how much time one is willing to devote to the task and how fast the computer is, but mainly how quickly your series converges.

Abstract: Bernhart, Frank R; Price, HLee Mack and Czernezkyj (2010) have given an interesting account of primitive Pythagorean triples (PPTs) from a geometrical perspective. We wish here to enlarge on the role of the equicircles (incircle and three excircles), and show there is yet another family tree in Pythagoras' garden.

Abstract: Balasooriya, Uditha; Li, Jackie; Low, Chan Kee For any density function (or probability function), there always corresponds a cumulative distribution function (cdf). It is a well-known mathematical fact that the cdf is more general than the density function, in the sense that for a given distribution the former may exist without the existence of the latter. Nevertheless, while the density function curve is frequently adopted as a graphical device in depicting the main attributes of the distribution it represents, the cdf curve is usually ignored in such practical analysis.

Abstract: Watson, Jane; Chance, Beth Formal inference, which makes theoretical assumptions about distributions and applies hypothesis testing procedures with null and alternative hypotheses, is notoriously difficult for tertiary students to master. The debate about whether this content should appear in Years 11 and 12 of the Australian Curriculum: Mathematics has gone on for several years. If formal inference is not included in Years 11 and 12, what statistical content, if any, should there be' Should students continue learning more data handling skills, which are a feature of the F-10 curriculum (Australian Curriculum, Assessment and Reporting Authority [ACARA], 2011)' Perhaps the focus should be on procedural aspects, such as correlation and lines of best fit, employing principles from calculus. Or perhaps the curriculum should drop statistics and focus on the more complex theoretical aspects of probability.

Abstract: Yevdokimov, Oleksiy The purpose of the section is to supply teachers and students with a selection of interesting problems. In this issue we invite readers to look back as far as more than one hundred years ago and work on a selection of problems that had been proposed to students and teachers at that time. The tradition of publishing problems in periodicals was well established in Europe by the beginning of the nineteenth century. The first two problems appeared in The Mathematical Visitor (Martin, 1881) that was published in the seventies of the nineteenth century in the US. The next two problems are from the Russian periodical Vestnik (Newsletter) of the Experimental Physics and Elementary Mathematics (Tsimmerman, 1886-1917). The final problem comes from the Italian mathematical magazine Il Pitagora where it appeared in 1897.

Abstract: Valahas, Theodoros; Boukas, Andreas In Years 9 and 10 of secondary schooling students are typically introduced to quadratic expressions and functions and related modelling, algebra, and graphing. This includes work on the expansion and factorisation of quadratic expressions (typically with integer values of coefficients), graphing quadratic functions, finding the roots of quadratic equations and relating these to horizontal axis intercepts of corresponding graphs. For example, given the quadratic expression x2 - 5x + 6 = 0, students would attempt to factorise by seeking two integers m and n such that m + n = -5 and m x n = 6, and thus express x2 - 5x + 6 = 0 in the form (x + n) (x + n).

Abstract: Staples, Ed The Catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight... The word catenary is derived from the Latin word catena, which means "chain". Huygens first used the term catenaria in a letter to Leibniz in 1690 Hooke discovered that the catenary is the ideal curve for an arch of uniform density and thickness which supports only its own weight. (Wikipedia, catenary).

Abstract: Shriki, Atara A parabola is an interesting curve. What makes it interesting at the secondary school level is the fact that this curve is presented in both its contexts: algebraic and geometric. According to the intended curriculum in mathematics, in 9th grade, students should learn about quadratic functions, including simplification techniques. In the 10th grade, they are expected to solve a wide range of quadratic equations, construct graphs of parabolas, and connect algebraic and graphical representations of quadratic functions.

Abstract: Padula, Janice Have your students ever wondered what mathematics is, and exactly what it is that a mathematician does' In this paper different schools of thought are discussed and compared to encourage lively classroom discussion and interest in mathematics for high achieving Form 12 students and first (or higher) year university students enrolled in a mathematics degree program. (The topic also fits well under the rationale for Queensland Senior Mathematics B Syllabus, Queensland Studies Authority, 2008.) In particular the work and views of two mathematicians, Kurt G del (1931) and Ian Stewart (1996), mathematician and professor Reuben Hersh (1998) and university lecturer, researcher and writer Robyn Arianrhod (2003) are used to illustrate different views of mathematics. Two documentaries are suggested for viewing by students: Dangerous Knowledge, relating the work and place of Godel in the history and foundations of mathematics (Malone and Tanner, 2008), and How Kevin Bacon Cured Cancer (Jacques, 2008) which illustrates how mathematicians and scientists work together developing and applying mathematics.

Abstract: Fuentes, Sarah Quebec The promotion of proof as a process through which mathematics knowledge and understanding have been constructed will not necessarily motivate students, though, unless they believe that they are participating in meaningful mathematical discovery (Vincent, 2005, p. 94).

Abstract: Braiden, Doug The senior school Mathematics syllabus is often restricted to the study of single variable differential equations of the first order. Unfortunately most real life examples do not follow such types of relations. In addition, very few differential equations in real life have exact solutions that can be expressed in finite terms (Jordan and Smith, 2007, p. 2). Even if the solution can be found exactly it may be far too difficult to be clearly articulated such as those that form an infinite series. In either case, these real life problems are well beyond the scope of the secondary student to solve.

Abstract: Brown, Jill; Stillman, Gloria During July this year, the Australian Catholic University (ACU) hosted the Fifteenth Biennial Conference on the Teaching of Mathematical Modelling and Applications (ICTMA15). Teachers, mathematicians and mathematics educators from around the world descended on the Melbourne campus of ACU.

Abstract: Wongapiwatkul, Pimpalak; Laosinchai, Parames; Panijpan, Bhinyo An instruction method that uses Earth geometry and the great circle to enhance students' understanding of trigonometric ideas is described. Three visual aids to help students visualise the geometry of the Earth are presented, together with a method of calculating the shortest distance between two points on the surface of the Earth, that is, the great-circle distance.

Abstract: Sokolowski, Andrzej; Rackley, Robin An argument is presented that applying trigonometric functions to model harmonic motion offers a rich scientific context to exercise mathematical modelling through inductive inquiry in trigonometry classes as well. The paper describes an activity that uses a physics simulation called 'Wave on a String' created by the PhET Interactive Simulations Project at Colorado University at Boulder, and is available on the internet. The activity's cognitive learning objectives fit into the scope of the proposed Australian mathematics curriculum that highlights the development of the skills of mathematical modelling, data collection, and analysis.

Abstract: Padula, Janice The main elements of Kurt Godel's proof of the 'incompleteness' of a formal system such as Bertrand Russell and A.N. Whitehead's 'Principia Mathematica' are discussed together with ways to address potential difficulties for students. The article recommends the study of the logical-skeletal structure before students attempt the proof itself and describes how students may be introduced to the proof with a documentary highlighting its importance. In addition, the paper evaluates two books for the 'general reader', by E. Nagel and J.R. Newman and by Torkel Franzen, and a description of the proof's logical core written in clear English by Solomon Feferman.

Abstract: Griffiths, Martin A workshop for undergraduates and students in Years 11 and 12 on generating 'random' positive integers is described. The workshop explores a range of topics, from statistics to pure mathematics, including aspects of probability, random variables, and Fourier series. The teaching and learning that took place in the workshop is examined in the context of the 'Australian Senior Secondary Mathematics Curriculum'.

Abstract: Ghosh, Jonaki B A laboratory module on Fourier series and Gibbs phenomenon undertaken by Year 12 students using Mathematica is described. Paper and pencil methods were used to help students understand calculations while Mathematica added meaning to the calculations by providing graphical and numerical representations. Students were then able to focus on the behaviour of the graphs and the functions, which enabled them to visualize Gibbs phenomenon.

Abstract: Yevdokimov, Oleksiy Some interesting problems for teachers and students to solve are discussed. A few famous geometric inequalities for finding new proofs are highlighted.

Abstract: Mack, John; Czernezkyj, Vic The creation of a specific 'infinite ascent' from a single Primitive Pythagorean Triple (PPT) base and then a 'finite descent' from the PPT base is discussed. Some of the different challenges posed by Fermat as well as solutions for the same are highlighted.

Abstract: Hwang, Daniel The definition of 'twist', a transformation and its various applications are discussed. Some of the different classroom tasks to help illustrate the twist transformation are highlighted.

Abstract: Gough, John The need for proof to be and should be seen as a central component in school curriculum is discussed. Some of the various examples of proof and their application is highlighted.

Abstract: Galbraith, Peter; Lockwood, Terry The difficulty of a set shot in AFL football and how a set shot at goal varies with position on the field is discussed. A linking of the mathematics and modeling involved, to educational settings in terms of curriculum concerns and possible teaching approaches are highlighted.

Abstract: Kabael, Tangul Uygur The key aspects and features of the framework of 'three worlds of mathematics' that describes three levels of mathematics understanding are discussed. The teacher should design teaching activities that can help students relate prerequisite concepts with various chain rule applications.

Abstract: Farmer, Jim The key aspects and features of the Markov chain model used in an exercise to predict rainfall data for Darwin airport are discussed. The various factors that led to the failure of the model in such an exercise are highlighted.