Abstract: Kok, Xiao-Feng Kenan This paper outlines an approach to definitively find the general term in a number pattern, of either a linear or quadratic form, by using the general equation of a linear or quadratic function. This approach is governed by four principles:

- identifying the position of the term (input) and the term itself (output);

- recognising that each corresponding input and output form points on a straight line or quadratic curve;

- identifying the nature of the number pattern;

- using the general equation of a straight line (y = mx + c) or quadratic curve (y = ax2 + bx + c) to represent the general term.

Implications for teaching and learning are also offered.

Abstract: Mahmood, Munir; Al-Mirbati, Rudaina In recent years most text books utilise either the sign chart or graphing functions in order to solve a quadratic inequality of the form ax2 + bx + c < 0 This article demonstrates an algebraic approach to solve the above inequality.

Abstract: Haggar, Fred; Krcic, Senida Subdividing an equilateral triangle into four congruent triangles, then doing likewise to each of the three non-central triangles, and then again and again, leads to the Sierpinski gasket, from which the chaos game originated. An analogous procedure is hereforth applied to a circle, where a subdivision consists of two pairs of inscribed circles, with each circle tangential to the ones adjacent to it.

Abstract: Keane, Therese; Loch, Birgit While the flipped classroom has found much discussion recently, peer tutoring using screencast resources produced by students has not yet. In this paper, we describe student responses to the approach taken at a secondary Catholic school in Melbourne, Australia, where the mathematics teachers used the roll out of tablet PCs to all teachers and students as an opportunity to shift from teacher produced resources towards student produced resources to enhance student learning. Students were engaged in the production of short screencast videos explaining mathematical concepts. We analyse student feedback from a survey issued to all students, and focus our analysis in particular on: identifying where students go for assistance in mathematics; what benefits they see from creating these screencasts and also from watching screencasts produced by their peers; and who should create screencasts - students or teachers or both. While many students do recognise there are benefits for them, responses seem to indicate that they may not be ready to embrace student-centred education without significant guidance from their teachers and with a shift away from current perception that the teacher should do all teaching as this is what they are paid for.

Abstract: Vozzo, Enzo Ever since their serendipitous discovery by Italian mathematicians trying to solve cubic equations in the 16th century, imaginary and complex numbers have been difficult topics to understand. They were received with suspicion back then as it was difficult to visualise them, let alone make sense of them and apply them practically. It was not until around the turn of the 19th century that a graphical representation of imaginary number was realised first by the Norwegian surveyor Caspar Wessel (1799), then by the Swiss born French mathematician John-Robert Argand (1806), the French mathematician Abbee Adrien-Quentin Buee (1806) and the German mathematician Carl Friedrich Gauss (1831). They hit upon the idea to represent imaginary numbers as points on a plane similar to the Cartesian plane like the one devise by Rene Descartes in the 17th century. It was Descartes who first coined the term 'imaginary' to describe these numbers. The Swiss mathematician Leonhard Euler was the first to denote the imaginary unit as i =-1 and Gauss who first used the term 'complex'. The plane on which imaginary (and complex) numbers are plotted is called an Argand Diagram or the Complex Plane. On it, the horizontal axis is called the real axis and the perpendicular vertical axis is called the imaginary axis. (Subsequently, it was not until around the turn of the 20th century that they were used to understand a very practical problem: electricity in the form of alternating current, AC).

Abstract: Nillsen, Rodney Let us suppose a bank customer has $150 000 to invest. The bank says it can offer an account where the bank pays interest compounded daily at 3% per annum. As an alternative, the bank offers also another account, where the interest rate is 2.5% on the first $50 000 and 3.5% on any amount in excess of $50 000, where again the interest is compounded daily. The customer wishes to invest the $150 000 for, say, five years. Which account should the customer choose' If the customer were prepared to wait for 10 years instead of five, would this make a difference to the account the customer should choose' Is there much difference between the two choices' Does a small change in an interest rate lead to a possibly large change in the outcome' More generally, in what ways do the interest rates and the other variables affect the answers to such questions'

Abstract: Thomson, Ian This article is based on a problem-solving task for senior secondary school students. The skills and concepts involved in the task are related to the topic of linear programming. In the task, linear programming is applied to optimise the value of an investment portfolio. The context of the task centres around a scenario in which a large sum of money has been bequeathed in a will. As the tale unfolds, the instructions in the will become more and more complicated, and, in turn, this means that progressively more sophisticated techniques in linear programming need to be applied.

Abstract: Vincent, Jill; Pierce, Robyn; Bardini, Caroline Despite having passed the required level of secondary mathematics, many tertiary students struggle with first year tertiary mathematics. In recent years, university mathematics and statistics departments have been revising both curricula and pedagogy as well as providing extra support to help students through this transition (MacGillivray, 2008). In the study reported in this paper, we have analysed student work in order to track potential sources of students' difficulties. It seems that these difficulties may lie, not in new concepts, but in unsound foundations and in limited and inflexible structural understandings of mathematics established earlier in their experience of mathematics.

Abstract: Fitzherbert, John Jagadguru Shankaracharya Swami Bharati Krishna Tirtha (commonly abbreviated to Bharati Krishna) was a scholar who studied ancient Indian Veda texts and between 1911 and 1918 (vedicmaths.org, n.d.) and wrote a collection of 16 major rules and a number of minor rules which have collectively become known as the sutras of Vedic mathematics. The numbering of the sutras in this article has been adopted from Williams and Gaskell (2010) which matches the numbering from the online references.

Abstract: Turner, Paul; Thornton, Steve This article draws on some ideas explored during and after a writing workshop to develop classroom resources for the reSolve: Mathematics by Inquiry (www.resolve.edu.au) project. The project is well into its development phase, is funded by the Australian Government Department of Education and Training and conducted by the Australian Academy of Science in collaboration with the Australian Association of Mathematics Teachers. The project develops classroom and professional learning resources that will promote a spirit of inquiry in school mathematics from Foundation to year ten.

Abstract: Ferguson, Robert High school algebra and precalculus courses introduce students to the notion of a 'parent function' and a class of functions that can be derived from it through translation, compression, dilation, and reflection. The latter functions are referred to in this paper as 'generalised parent functions'. The parent function can be one of a broad variety of functions, e.g., trigonometric, exponential, logarithmic, polynomial, absolute value, etc.

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: 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: Gomes, Luis Teia Very much like today, the Old Babylonians (20th to 16th centuries BC) had the need to understand and use what is now called the Pythagoras' theorem x2 + y2 = z2. They applied it in very practical problems such as to determine how the height of a cane leaning against a wall changes with its inclination. This sounds trivial, but it was one of the most important problems studied at the time. A remarkable Old Babylonian clay tablet, commonly referred to as Plimpton 322 (Figure 1), was found to store combinations of three positive integers x, y, z that satisfy Pythagoras' theorem. Today we call them primitive Pythagorean triples where the term primitive implies that the side lengths share no common divisor.

Abstract: Merrotsy, Peter In the 'Australian Curriculum', the concept of mathematical induction is first met in the senior secondary subject Specialist Mathematics. This article details an example, the Tower of Hanoi problem, which provides an enactive introduction to the inductive process before moving to more abstract and cognitively demanding representations. Along the way, it is suggested that the Tower of Hanoi problem would also be suitable as an example for introducing mathematical inductive thinking to students in junior secondary school.

Abstract: Marshman, Margaret; Dunn, Peter K; McDougall, Robert; Wiegand, Aaron The new secondary Australian mathematics curricula have more statistics than the existing Queensland senior mathematics curricula. This paper discusses the attitudes to, and preparedness for, aspects of the implementation of the Australian Senior Mathematics Curricula within a group of Sunshine Coast (Queensland) mathematics educators. We found on the evidence presented that teachers value the importance of statistics, and see how technology can assist with teaching and learning statistics, but teachers are ambivalent towards statistics and feel less competent to teach statistics.

Abstract: Stacey, Kaye Review(s) of: Oxford figures: Eight centuries of the mathematical sciences (2nd ed.), Edited by John Fauvel, Raymond Flood and Robin Wilson Published 2013 by Oxford University Press.

Abstract: Pierce, Robyn; Bardini, Caroline Since the 1990s, computer algebra systems (CAS) have been available in Australia as hand-held devices designed for students with the expectation that they will be used in the mathematics classroom. Prior to the development of hand-held CAS (first the TI-92 in 1995), these programs (for example: Derive, Mathematica, Maple) were available for computers but had limited penetration in schools. In Victoria, CAS use at school and at home was permitted for senior secondary mathematics students completing externally set extended tasks known as 'Common Assessment Tasks' (Leigh-Lancaster & Rowe, 1999). However, in 2002 further institutional value was accorded to the use of CAS through the trial of a new senior secondary mathematics subject: Mathematics Methods (CAS). For this subject CAS use was also permitted in the final, high stakes, externally set examinations. After more than a decade with CAS readily available and its use encouraged for teaching, learning and assessment.

Abstract: Joarder, Anwar H An algorithm is presented for factorising a quadratic expression to facilitate instruction and learning. It appeals to elementary geometry which may provide better insights to some students or teachers.

Abstract: Bardell, Nicholas S An n-th root of unity, where n is a positive integer (i.e., n = 1,2,3,....), is a number z satisfying the equation Traditionally, z is assumed to be a complex number and the roots are usually determined by using de Moivre's theorem adapted for fractional indices. The roots are represented in the Argand plane by points that lie equally pitched around a circle of unit radius. The n-th roots of unity always include the real number 1, and also include the real number -1 if n is even. The non-real n-th roots of unity always form complex conjugate pairs. This topic is taught to students studying a mathematics specialism (ACARA, n.d., Unit 3, Topic 1: Complex Numbers) as an application of de Moivre's theorem with the understanding that the roots occur in the complex domain.

Abstract: Vincent, Jill; Bardini, Caroline; Pierce, Robyn; Pearn, Catherine In mathematics we frequently need to express equality between expressions. Robert Recorde, born in Wales in about 1510, is credited with inventing the equals sign that we use today. Up until this time, equality was expressed in words. Recorde's first use of the equals sign was in 1557 in The Whetstone of Witte. Translated into modern English, Recorde's explanation of his equals sign reads: "And to avoid the tedious repetition of these words 'is equal to' I will set as I do often in work, use a pair of parallel lines of one length, thus: =, because no 2 things can be more equal".

Abstract: Laine, AD There are many geometrical approaches to the solution of the quadratic equation with real coefficients. Here it is shown that the monic quadratic equation with complex coefficients can also be solved graphically, by the intersection of two hyperbolas; one hyperbola being derived from the real part of the quadratic equation and one from the imaginary part. Both hyperbolas are of relatively simple form. Special solutions correspond to one or both of the hyperbolas being degenerate.

Abstract: Grant, Ken Bernard Riemann (1826-66) was a student of Carl Friedrich Gauss (1777-1855) at the University of Gottingen in Lower Saxony. Since his youth, Gauss had conjectured that the distribution of the primes could approximately.

Abstract: Brown, Jill Review(s) of: Magical mathematics: The mathematical ideas that animate great magic tricks, by Persi Diaconis and Ron Graham, Published 2012 by Princeton University Press.

Abstract: Turner, Paul The idea of facilitating learning by providing ways for students to make connections between different parts of mathematics and between mathematics and the rest of reality is widely acknowledged to be a good one.

Abstract: Carley, Holly Usually a student learns to solve a system of linear equations in two ways: 'substitution' and 'elimination'. While the two methods will of course lead to the same answer they are considered different because the thinking process is different. In this paper we solve a system in these two ways to demonstrate the similarity in the computation. We then see that changing the point of view leads us to a 'simpler' way to solve a system of equations. This leads naturally to two other consequences, viz., what is known as Chio's pivotal condensation process for computing determinants and Cramer's Rule. While the condensation process for computing determinants is known, it is not widely known, and the manner of solving equations developed here has not been seen elsewhere. This should be of interest to anyone teaching solving systems of linear equations (especially by hand) and can be the basis for teaching the basics of solving systems of equations, or for use as a guided project. This material is particularly relevant for the topic of matrices in unit 2 of the Specialist Mathematics, the topic of Algebra and matrices in unit 1 of the General Mathematics curriculum, as well as anywhere where multivariate applications appear such as finding regression lines in the data collection topic in unit 3 of essential mathematics (perhaps as a special project) and the bivariate data analysis topic of unit 3 of the general mathematics curriculum.

Abstract: Lenard, Christopher; McCarthy, Sally; Terence, Mills There are many different aspects of statistics. Statistics involves mathematics, computing, and applications to almost every field of endeavour. Each aspect provides an opportunity to spark someone's interest in the subject. In this paper we discuss some ethical aspects of statistics, and describe how an introduction to ethics has been incorporated into teaching elementary statistical methods at La Trobe University, Bendigo.

Abstract: Farmer, Jim In which an argument with religious fundamentalists inspires an elegant expected number of trials exercise based on a generalisation of the matching problem.

Abstract: Bardell, Nicholas S This paper is a natural extension of the root visualisation techniques first presented by Bardell (2012) for quadratic equations with real coefficients. Consideration is now given to the familiar quadratic equation y = ax2 + bx + c in which the coefficients a, b, c are generally complex, as shown explicitly in Equation (1) with the usual notation.

Abstract: Bhattacharjee, Pramode Ranjan This paper being an extension of Bhattacharjee (2012) is very much relevant to Year 9 to Year 10A in the Australian Curriculum: Mathematics. It also falls within the purview of class IX to class XII curriculum of Mathematics in India (Revised NCERT curriculum) for students aged 14-17 years. In Bhattacharjee (2012), the discovery of flaw in the traditional definitions of trigonometric ratios, which make use of the most unrealistic concept of negative length or distance has been reported. With a view to getting rid of such unrealistic concept of negative length or distance, which has been in regular use in the sign convention of geometrical optics, in solving typical problems of elementary mechanics, efforts have already been made by the author earlier in Bhattacharjee (2002, 2011, 2012).

Abstract: Wares, Arsalan The purpose of this article is to describe a couple of challenging mathematical problems that involve paper folding. These problem solving tasks can be used to foster geometric and algebraic thinking among students. The context of paper folding makes some of the abstract mathematical ideas involved relatively concrete. When implemented appropriately these activities have the potential to address many of the mathematical proficiencies, as delineated by Australian Curriculum and Assessment Reporting Authority (ACARA, 2014).

Abstract: Salinas, Tracie McLemore; Lynch-Davis, Kathleen; Mawhinney, Katherine J; Crocker, Deborah A The differences between mathematical language and everyday language are explored in the mathematics education literature, demonstrating that mathematical definitions are negotiated and used differently, such as Landau's comparison of extracted definitions and stipulated definitions (as cited in Edwards and Ward, 2004) and how these differences influence students' thinking (Edwards and Ward, 2004; Zaslavsky and Shir, 2005). Not surprisingly, students' interactions with mathematical terminology may be more heavily influenced by their use of everyday language than by the role of definition within mathematics.

Abstract: Maltas, Dimitrios; Prescott, Anne Many people are discussing the issues surrounding mathematics at all levels of education. Politicians, parents, students, universities, education departments all have a view about what the problem is and all have ideas about what should happen. This paper represents a synthesis of the issues and implications of one of the problems evident in mathematics education in Australia: the reduction in students taking the higher mathematics subjects that involve calculus. It makes particular reference to NSW where it appears the problem has a critical edge as the number of students undertaking higher level mathematics subjects decreases and the number of students not taking mathematics increases.

Abstract: Kabael, Tangul The concept of limit is the foundation for many concepts such as the derivative and the integral in advanced mathematics (Parameswaran, 2007; Tall, 1992). According to the literature, there are two different kinds of notions of the limit, a dynamic limit notation and a formal limit notion. The dynamic limit notion is considered as and read, "as x approaches a, f(x) approaches L". The formal limit definition is given as follows: If the condition, is satisfied, the limit of the function of f at the point of a is L.

Abstract: Acosta, Daniel In Liping Ma's 1999 seminal work, Knowing and Teaching Elementary Mathematics, the author discusses a facet of teaching that relies on both content knowledge and attitudes toward mathematics, namely, exploring a student conjecture: as the perimeter of a closed figure increases, so too must the area. USA teacher responses to the conjecture were largely non-mathematical - blindly accepting the claim as true or deferring mathematical inquiry by expressing intentions to "look it up" or ask a more knowledgeable colleague. By contrast, Chinese teacher responses were largely mathematical, exemplifying what Ma termed four levels of understanding: (L1) disproving the claim, (L2) exploring possibilities, (L3) clarifying conditions, and (L4) explaining the conditions. These teachers had not been exposed to much content beyond elementary algebra and geometry, and yet provided arguments and justifications much like mathematicians would.

Abstract: Bardell, Nicholas S The roots of a quadratic equation with either real or complex coefficients can be found relatively easily from the 'standard' quadratic equation formula, A routine application of Equation will furnish the desired roots, but it gives no indication of the location of these roots if the result contains complex numbers. This paper describes how a simple application of de Moivre's theorem may be used to not only find the roots of a quadratic equation with real or generally complex coefficients but also to pinpoint their location in the Argand plane. This approach is much simpler than the comprehensive analysis presented by Bardell (2012, 2014), but it does not make the full visual connection between the Cartesian plane and the Argand plane that Bardell's three dimensional surfaces illustrated so well.

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 work on problems that have relationship to chess. All of them can be characterised as mathematical problems on a chessboard. Some problems are well-known and have historical flavour, while others are not so famous and rather forgotten. For some of the problems symmetry is in use. The first problem was considered by Leonhard Euler. All problems are from Wells (1992), though some were known long before his book was published. As usual despite the simple problem statements, finding a direct and certain way towards solution is often far from easy!

Abstract: Boudreaux, Grant; Beslin, Scott The purpose of this article is to examine one possible extension of greatest common divisor (or highest common factor) from elementary number properties. The article may be of interest to teachers and students of the Australian Curriculum: Mathematics, beginning with Years 7 and 8, as described in the content descriptions for Number and Algebra. The senior secondary curriculum makes no specific mention of greatest common divisor, but the article is nevertheless a good resource for revisiting with students at this level the concepts of greatest common divisor and lowest common multiple in greater depth, and with a view to critical thinking. Certain concepts and problems can be used even in post-secondary instruction. In particular, teachers may find it useful in designing projects for guided self-discovery or collaborative learning. The article is written as a hybrid: part guided discovery, and part exposition of interesting results and applications. Teachers who enjoy factorisation of positive integers and the concepts of divisor and multiple will hopefully find this content useful and meaningful in making connections of those concepts with fractional numbers. Sample problems and exercises are presented at the end of the article as self-tests and as vehicles for student investigations. Innovative teachers can also formulate their own conjectures and examples.

Abstract: Sokolowski, Andrzej This paper presents a study about using scientific simulations to enhance the process of mathematical modelling. The main component of the study is a lesson whose major objective is to have students mathematise a trajectory of a projected object and then apply the model to formulate other trajectories by using the properties of function transformations. It was hypothesised that situating the lesson in a modelling environment would enhance the meaning of transformations that are not often conceptualised in mathematics textbooks. The lesson is guided by inductive reasoning. As a medium of data gathering, a free simulation called Projectile Motion.

Abstract: Staples, Ed The Gregorian calendar in use today is not so old. The United Kingdom, its colonies, including the American colonies, adopted the reforms in September of 1752 during the reign of King George II. Before this, the realm was using the Julian calendar, and this had accounted fairly well for the Earth's last quarter turn in its annual orbit. However, after one and one half millennia, the more accurate fraction of 0.24219 was beginning to reveal itself in the form of a concerning week and a half retreat of the vernal equinox.

Abstract: Dayal, Hem Chand Mathematics teaching has undergone many reforms in recent decades. Much of the reforms have originated as a result of advancements in the fields of cognitive psychology, mathematics, and mathematics education (Begg, 2003; NCTM, 2000; Raizen, 1997). The changes have been particularly significant in the areas of mathematical curricula and instructional strategies, including the use of technology in teaching and learning mathematics. The above mentioned changes have led mathematics educators to re-look at the overall goals of mathematics teaching with particular emphasis to active student involvement in an enquiry-based learning as opposed to the expository style of teaching used traditionally. These innovations have led to newer and more challenging roles of the mathematics teacher.

Abstract: Stupel, Moshe; Ben-Chaim, David Based on Steiner's fascinating theorem for trapezium, a seven geometrical constructions using straightedge alone are described. These constructions provide an excellent base for teaching theorems and the properties of geometrical shapes, as well as challenging thought and inspiring deeper insight into the world of geometry. In particular, this article also mentions the orthic triangle and proves its special property, and shows some other interesting constructions, such as, for example, how to construct a circle's diameter using straightedge alone and having only a segment with its midpoint. In addition, it is enhanced by aspects of the historical background of geometric constructions, including reference to 'impossible constructions.' Application of the material presented in college or high school can enhance students' appreciation of the elegance, beauty, and fascination of mathematics. Through such 'adventures,' students will be encouraged to further pursue geometric problems and explore various methods of problem solving, especially those concerned with geometric constructions.

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 deal with points and lines problems. For many of them symmetry plays a key role. The first problem below is one of the most famous examples from this topic. It is attributed to Sir Isaac Newton. All other problems have historical flavour too and all problems are from Dudeney (1958). Despite the simple problem statements they have, finding a direct and certain way towards a solution is often far from simple!

Abstract: Dion, Peter; Ho, Anthony An interesting mathematical problem, 'The Dog Problem', is easily stated. Four dogs each at the corners A, B, C and D of a large square of side length L start walking directly towards the dog on his right (Dog A to B, B to C, C to D, and D to A) all at the same speed. They do not to anticipate the target dog's movement and attempt a short cut. The dogs continually change direction, and we know by symmetry that they will spiral around each other and to meet in the middle of the square. What is the path length walked by each dog'

Abstract: Zelenskiy, Alexander S In a recent issue of Australian Senior Mathematics Journal there has been published an interesting article by Galbraith and Lockwood (2010). In that article the problem of finding the most favorable points for a shot at goal in Australian football is considered from different points of view. A similar problem was considered by Galbraith and Stillman (2006) in the context of soccer.

Abstract: Stoessiger, Rex When we examine numbers in the newspaper or magazines we might expect that their first digits are just as likely to be 8 or 9 as a 1 or a 2 and we might assume that each of the nine digits (zero is not used as a first digit of course) will occur 1/9 of the time. Unexpectedly this turns out to be untrue in many situations.

Abstract: Obara, Samuel The National Council of Teachers of Mathematics (NCTM, 1989, 2000) and the new Australian Curriculum: Mathematics for senior secondary (ACARA, 2010) highlight the importance of teaching spatial reasoning as early as preschool when mathematics is introduced. Studies have shown that there is a relationship between spatial abilities and mathematical achievement (Burnett, Lane, and Dratt, 1979; Casey, Nuttall, Pezaris, and Benbow, 1995; Geary, Saults, Liu, and Hoard, 2000). Activities that enhance spatial reasoning skills are invaluable to, and should be encouraged in, classroom instruction. Casey, Andrews, Schindler, Kersh, Samper, and Copley (2008) define spatial skills as "the ability to think and reason through the comparison, manipulation, and transformation of mental pictures" (p. 270). In this article I present an activity (Aichele and Wolfe, 2007, p. 11) to pre-service high school teachers. Though the activity seems simple, it can be challenging to students who have not been exposed to spatial tasks. The activity goal is to create a square from a given polygon by making one straight cut so that two pieces, when put together without overlap or gaps.

Abstract: Kenney, Rachael; Kastberg, Signe The beautiful history of the development of logarithms (Smith and Confrey, 1994), coupled with the power of the logarithmic function to model various situations and solve practical problems, makes the continued effort to support students' understanding of logarithms as critical today as it was when slide rules and logarithmic tables were commonly used for computation. They continue to play an important role despite the fact that calculators are now used for many computations involving logarithms: logarithmic scales can increase the range over which numbers can be viewed in a meaningful way. As described in the senior secondary curriculum, logarithmic scales are used regularly in astronomy, chemistry, acoustics, seismology, and engineering and students should be able to "identify contexts suitable for modelling by logarithmic functions" and be able to "use logarithmic functions to solve practical problems (Australian Curriculum and Assessment Reporting Authority [ACARA], 2012)".

Abstract: Foster, Colin A post-16 mathematics student, Peter (not his real name), asked me, "Does Pythagoras' theorem generalise to three dimensions'" I said, "Yes." But it turned out that we had different things in mind.

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.