Abstract: Haggar, Fred; Krcic, Senida Selecting the 'better' solution to a problem between the first one that comes to mind and the alternative that may follow is not a 'fait-accompli'. After all, is it better if it is more economical' Or is it better if it is more elegant' Moreover, taking shortcuts does not always lead to a shorter solution. Consider the simple example of a rectangle modified so as to preserve its area.

Abstract: Gough, John Turner's opinion piece Relevance (2015) got me thinking. Do we really need to make the mathematics we teach relevant to our students' Can we do this' Or are there alternatives that circumvent this recurring nightmare catch-cry'

Abstract: Coupland, Mary Recently I attended a symposium for researchers in science education and those interested in such research. The topic was 'Imagining Futures', and as an exercise we were asked to consider two continuums/spectrums of aspects of education that were independent, and placed as intersecting axes. This created four quadrants as a frame for imagining possible futures for science education. For example, one axis could be 'purposes of education', varying from individual empowerment at one end to social good at the other. Another axis might be 'the nature of knowledge', varying from (say) a view that knowledge is fixed, to a view that knowledge is always contestable and changing. It was a very interesting day and I found myself thinking later on about the way that our opinions on educational issues are often framed by the use of continuums.

Abstract: Wilkie, Karina J Senior secondary mathematics students who develop conceptual understanding that moves them beyond "rules without reasons" to connections between related concepts (Skemp,1976, 2002, p. 2) are in a strong place to tackle the more difficult mathematics application problems. Current research is examining how the use of challenging tasks at different levels of schooling might help students develop conceptual knowledge and proficiencies in mathematics as promoted in the Australian curriculum - understanding, fluency, problem solving, and reasoning (ACARA, 2009). Challenging tasks require students to devise solutions to more complex problems that they have not been previously shown how to solve, and for which they might develop their own solution methods (Sullivan et al., 2014). Another key area of research is on formative assessment which has been found to be effective for increasing student motivation and achievement under certain conditions (for example, Brookhart, 2007; Karpinski and D'Agostino, 2013).

Abstract: Galligan, Linda A National Numeracy Report (COAG, 2008) and the Australian Curriculum (2014) have recognised the importance of language in mathematics. The general capabilities contained within the Australian Curriculum: Mathematics (2014) highlight literacy as an important tool in the teaching and learning of mathematics, from the interpretation of word problems to the discussion of mathematics in the classroom. The nationally commissioned National Numeracy Report (COAG, 2008), recommended that the language and literacies of mathematics be explicitly taught since language can be a significant barrier to understanding mathematics. As teachers routinely assess students' understanding of mathematics through literacy (often through reading and writing), students may struggle to understand the mathematics because they have specific language difficulties associated with assessment tasks set. Chapter 2 of the National Numeracy Review Report highlights the role of language in mathematics learning, and identifies a number of features of language that can have an impact on understanding mathematics.

Abstract: Lewis, Robert We have all heard the rhyme, and perhaps taught it ourselves to our students. When dividing one fraction by a second fraction, invert, that is, flip the second fraction, then multiply it by the first fraction. To multiply fractions, simply multiply across the denominators, and multiply across the numerators to get the resultant fraction. So by inverting the division of fractions we turn it into an easy multiplication of fractions problem.

Abstract: Cox, Teodora Mathematics teachers are frequently looking for real-life applications and meaningful integration of mathematics and other content areas. Many genuinely seek to reach out to students and help them make connections between the often abstract topics taught in school. In this article I share several ideas to help teachers foster student curiosity about mathematical ideas, by exploring children's literature and other fiction.

Abstract: Bentley, Brendan Have you ever looked at an object and found yourself thinking how visually attractive that object appears to the eye' Interpersonal considerations to one side, we might be referring to the intrinsic properties which appear visibly in physical objects or images. For instance, your mind might implicitly declare, "That painting is just fabulous", "What an amazing looking building", or "What an interesting shape". It may have been a piece of furniture, an item of clothing or even a flower in your garden that virtually demanded your attention.

Abstract: Coad, Lance The acronym STEM doubtless will conjure many meanings, and it will not serve my purpose to discuss or debate them all. I wish, rather, to reflect for a moment only on the function, role or place of mathematics when included as part of an integrated STEM activity, and this from the perspective of a teacher of mathematics. I will not attempt to expound upon what it is that constitutes an integrated STEM activity, save to suggest that it would be an educational activity designed to incorporate elements of some or all of science, technology, engineering and mathematics - and even the arts, in which case we would have a STEAM activity.

Abstract: Gough, John When we hear the words "strategy board game" we tend to think of classic counter-moving games such as chess, draughts (checkers), Reversi (Othello), Chinese checkers, or Stratego. Other board games, such as 'Ludo', and Backgammon use dice to direct the moves of counters.

Abstract: de Mestre, Neville Nichomachus was a Greek who lived around 100 AD near Jerusalem. He was the first to publish the link between number and geometry.

Abstract: Nivens, Ryan Andrew As many students can attest, cars are interesting. They come in various colours, they have interesting accessories, and they move us to where we want to go. Many years ago, the students of Pythagoras thought that numbers were interesting and had a saying that "All is number." As children are taught to embrace mathematics as a dynamic and useful subject, we can show them an interesting context of mathematics where cars and numbers intersect in licence plates.

Abstract: Espedido, Rosei Review(s) of: Learning to love math: Teaching strategies that change student attitudes and get results, by Judy Willis, M.D., Publisher ASCD Publications, ISBN 978-1-4166-1036-6.

Abstract: Moss, Diana L; Lamberg, Teruni This discussion-based lesson is designed to support Year 6 students in their initial understanding of using letters to represent numbers, expressions, and equations in algebra through making thinking explicit, exploring each other's solutions, and developing new mathematical insights.

Abstract: Prodromou, Theodosia New technologies have completely altered the ways that citizens can access data. Indeed, emerging online data sources give citizens access to an enormous amount of numerical information that provides new sorts of evidence used to influence public opinion. In this new environment, two trends have had a significant impact on our increasingly data-driven society: 1) the increasing use of large-scale databases within the open data movement, and 2) the growing use of big data.

Abstract: Ward, Lauren; Lyden, Sarah; Fitzallen, Noleine Context based learning (CBL) is a powerful tool that utilises areas of student interest framed in meaningful contexts to foster development of new skills and understanding. A rich context that students are familiar with can excite their desire to learn and motivate them to develop their knowledge in a wide range of disciplines simultaneously (Broman, Bernholt, and Parchmann, 2015).For middle school students, engineering activities that relate to real-world problems provide suitable CBL contexts for acquiring conceptual scientific and mathematical understanding.

Abstract: Patahuddin, Sitti Maesuri; Lowrie, Tom A critical incident is a situation or event that holds significance for learning, both for the students and teachers. It is "unplanned, unanticipated and uncontrolled" (Woods, 2012, p.1). Successfully using critical incidents in a classroom situation provides opportunities for rich analysis of classroom practices. The purpose of this article is to discuss how critical incidents can be harnessed for students' and teachers' development.

Abstract: Moule, Carol Review(s) of: Algebra tiles Australia: A concrete, visual, area-based model, by Lorraine Day, Publisher: A-Z Type ISBN 978-009807548-0-3; Maths with Mathomat, by Susie Groves and Peter Grover, Publisher W and G Australia Pty Ltd, 1999, reprinted 2008, ISBN 0-9780-958-610-308.

Abstract: de Mestre, Neville This Discovery article is based on a problem considered in this journal some 17 years ago by Professor Derek Holton of the University of Otago, New Zealand (Holton, 1997). The problem concerns the movements of frogs on lily-pads. We shall start with six frogs and seven lily-pads, although Derek Holton sought the solution for eight frogs and nine lily-pads.

Abstract: Gough, John It is almost impossible for a genuinely new strategy board game to be invented - although I live in hope. However, novel variations can be created that use interesting new playing rules, and these can stimulate fresh (mathematical) thinking. When the new game is easy to learn, and uses simple materials, and is immediately attractive or engaging, you have a winner! Here are some 'winners' I have found recently.

Abstract: Baroudi, Ziad Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities, together with sample work from students in the author's Year 7 classroom at a Catholic school in the South East of Melbourne.

Abstract: Graham, Pat; Chick, Helen In this article we look at a simple geometry problem that also involves some reasoning about number combinations, and show how it was used in a Year 7 classroom. The problem is accessible to students with a wide range of abilities, and provides scope for stimulating extensive discussion and reasoning in the classroom, as well as an opportunity for students to think about how to work systematically. Pat, the first author and a classroom teacher, used the problem with her students and we will present some of the strategies, solutions, and issues that they encountered and discussed. Helen, the second author who works with pre-service and inservice teachers, has used this problem with teachers and likes thinking about tasks that are good for fostering reasoning and problem solving.

Abstract: Fitzallen, Noleine Many fraction activities rely on the use of area models for developing partitioning skills. These models, however, are limited in their ability to assist students to visualise a fraction of an object when the whole changes. This article describes a fraction modelling activity that requires the transfer of water from one container to another. The activity provides the opportunity for students to explore the part-whole relationship when the whole changes and respond to and reason about the question: When does =1/3'

Abstract: Morony, Will A version of this paper was presented as one of several 'provocative papers' to the Connections and Continuity conference conducted in December 2014 by the AAMT working in partnership with the Australian Council of Deans of Science (ACDS). The focus of that conference was to explore the interface between school and university mathematics. Whilst the paper therefore takes as its starting point the Year 12 end-of-year examinations that have very high stakes for students' futures, the points it makes about the influence of assessment on what is valued as mathematics apply throughout schooling.

Abstract: Roddy, Mark; Behrend, Kat What do you do when you want to get your Stage 3 students authentically and enthusiastically engaged in the active construction of their understanding and fluency with measurement, data collection, representation and interpretation' How do you enable them to make choices about their learning, to measure with purpose, to record and organise the data they produce, to plot the points and to understand that the emerging line tells a story about something real, something changing. Here's one way to approach all of these objectives in an integrative and motivational context. Call in the Grow Beast!

Abstract: Gough, John Most readers would be familiar with the standard domino set which is played with rectangular domino tiles. The domino set, sometimes called a deck or pack, consists of 28 dominoes, colloquially nicknamed bones, cards, tiles, stones, or spinners. A domino set is a generic gaming device, similar to playing cards or dice, in that a variety of games can be played with a set.

Abstract: King, Alessandra During middle school years, students generally finalise their attitude towards mathematics and their perception of themselves as students of mathematics in terms of aptitude, motivation, interest, and competence (NCTM, 2000). Therefore, giving them varied opportunities to foster a positive and successful approach to the study of mathematics is critical, and can help them appreciate the relevance, usefulness, and creativity of the subject. Asking the students to create a website that focuses on mathematics can provide such an occasion; readily available, free internet tools that do not require any prior programming experience make this opportunity accessible to all students. In the school where I teach - an independent all-girls school in the suburbs of Washington DC - my Grade 8 Algebra 1 and Geometry classes have completed this activity successfully and with great interest. In our case, each student at the school is issued a laptop (and the school is internet-enabled), although all that is necessary is that students have access to a computer - whether in a lab, from a computer cart or in some other way.

Abstract: de Mestre, Neville See if your students can find the missing (') value. Perhaps you could give them a hint by saying that, first of all, they should try a linear combination. For example, if we label Input 1 'a' and Input 2 'b' then the problem is to seek the value [Output for Row 3] of 9a + 7b. The information in the first two rows can now be written as the pair of simultaneous equations, and labelled as equations (1) and (2) respectively.

Abstract: Seah, Rebecca Geometry belongs to branches of mathematics that develop students' visualisation, intuition, critical thinking, problem solving, deductive reasoning, logical argument and proof (Jones, 2002). It provides the basis for the development of spatial sense and plays an important role in acquiring advanced knowledge in science, technology, engineering, and mathematics. The Australian Curriculum: Mathematics (Australian Curriculum Assessment and Reporting Authority (ACARA), n.d) emphasises the need to help children develop an increasingly sophisticated understanding of geometric ideas, to be able to define, compare and construct figures and objects, and to develop geometric arguments. This article will look at some of the issues involved in the teaching and learning of two-dimensional shapes and illustrate how activities such as paper-folding tasks can be used to encourage visualisation and geometric reasoning.

Abstract: Carter, Pauline Review(s) of: Putting essential understanding of functions into practice 9-12, by Ronau, R. Meyer, D., Crites, T, Publisher National Council of Teachers of Mathematics Inc., 2014, ISBN: 978-0-87353-714-8.

Abstract: Gough, John What You Test Is What You Get (WYTIWYG for short) is a "provocative paper" (Morony, 2015). It raises fundamental questions about curriculum, classroom instruction and activities (pedagogy), and assessment practices. Why, for example, do we have relatively few secondary students progressing to the final years of school mathematics'

Abstract: Kissane, Barry Calculators have been around for about forty years or so now. There are many different species of them. Some are used in offices, some are built into devices, such as cash registers or rulers, some are on phones and some people even have them on their wrists in the form of a watch. There are online calculators, home loan calculators, superannuation calculators and the bank manager will probably pull out a financial calculator if you enquire about a loan or an investment. There is an almost uncountable number of calculator apps available for high-tech devices like tablets and smartphones.

Abstract: Clarkson, Philip In my first year of teaching in schools, there was an older staff member who only seemed to teach in the lower secondary years. I thought that was a bit strange; at that time, I assumed most people with a number of years experience would be teaching at least some senior mathematics. However, I found out that this colleague was a primary trained teacher and was therefore only able to teach up to Year 8. The Deputy Principal who did the staffing found him very useful as someone who could teach in any area of the curriculum, and he always had some mathematics to teach. Teaching mathematics for him was following the textbook and making sure the students could learn the number bonds and algorithms by rote.

Abstract: Lewis, Robert Maybe you, like me, have always had it instilled in you that you do not divide by zero or nasty things may happen' Some confusion still surrounds the number zero, particularly in using it in division.

Abstract: Richardson, Alice; Barker, Valerie; Ascione, Judith In many parts of the world Pi Day (www.piday.org) is celebrated on March 14 (3.14), but because of the day-month ordering of dates in Australia, and because March is very close to the start of the academic year, Australians prefer to celebrate Pi (Approximation) Day on 22 July (22/7).

Abstract: de Mestre, Neville I recently came upon an article containing a number of pentagon problems for senior high school students (Geretschlager, 2014). I decided to share the background and some of the easier problems with you and your students. Consider a regular pentagon which is convex with five equal sides. Ask your students to construct one by drawing a circle with a compass and marking five points on its circumference that are equally spaced apart. This means, of course, that they will have to use a protractor to ensure that the angle subtended at the centre of the circle is 360/5=72 degrees. The points are then joined as in Figure 1 to form the regular pentagon.

Abstract: Watson, Jane; English, Lyn By the time students reach the middle years they have experienced many chance activities based on dice. Common among these are rolling one die to explore the relationship of frequency and theoretical probability, and rolling two dice and summing the outcomes to consider their probabilities. Although dice may be considered overused by some, the advantage they offer is a familiar context within which to explore much more complex concepts. If the basic chance mechanism of the device is understood, it is possible to enter quickly into an arena of more complex concepts. This is what happened with a two hour activity engaged in by four classes of Grade 6 students in the same school.

Abstract: Prodromou, Theodosia The most recent Australian Curriculum: Mathematics document (ACARA, 2013) calls for students to develop the ability to use technology in their mathematical and statistical studies. Due to this increasing importance of integrating digital technologies in mathematics education, the national professional standards for teachers in Australia (AITSL, 2014) expect teacher education graduates to demonstrate technological, pedagogical and content knowledge (TPACK) in mathematics teaching and learning.

Abstract: Huber, Daniel; Jones, Leslie; Helminski, Christine The use of collaborative problem solving within mathematics education is imperative in this day and age of integrative science. The formation of interdisciplinary teams of mathematicians and scientists to investigate crucial problems is on the rise, as greater insight can be gained from an interdisciplinary perspective. Mathematical modelling, in particular, is increasingly recognised as a fundamental tool in understanding scientific phenomena, with models utilising mathematical disciplines ranging from statistics to differential equations (Giordano, Fox & Horton, 2014). Geometry is an effective tool in biomechanical modelling, and one that we have used to develop a series of lessons regarding the functional importance of mathematics in nature (see http://utweb.ut.edu/rwaggett/science-math-master.html). In this manuscript, we present a lesson in which geometry is used to model the suction feeding mechanism of the Goliath Grouper, and the consequences of geometric variability for organismal performance are explored.

Abstract: Collis, KF Contact with teachers, both formal and informal, over the last couple of years has revealed that, to the average teacher, the new mathematics curricula are causing mathematics to be regarded as more of a 'problem' subject than ever. This indictment of the subject is particularly prevalent among primary and lower secondary teachers, most of whom are not highly sophisticated in the subject matter area and find it difficult to re-orient their teaching and thinking to fit in with the new pattern demanded by what appears to be quite exotic subject matter. Let us examine briefly some of the basic problems involved.

Abstract: McQualter, JW Every mathematics teacher has differing views about how mathematics should be taught. These views overlap to form a body of accepted professional practice. Each mathematics teacher has his own way of selecting, organising and presenting mathematics to pupils. These have been called private theories about teaching mathematics (Bishop, 1971). The development of these private theories is the result of mathematics teachers feeling that public theories on mathematics teaching and learning cannot provide all the answers to the problems faced by individual teachers. Each teacher has developed his or her own set of beliefs about how to teach mathematics. Such beliefs can be grouped in four categories: beliefs about mathematics, beliefs about child development, beliefs about education psychology and beliefs about the relation between learning and teaching. (Rogers, 1979).

Abstract: White, Paul; Mitchelmore, Michael Some research into first year level tertiary students' understanding of calculus (White and Mitchelmore, 1992) revealed results as relevant to the junior years of high school as to the senior years. Basically, we found that many students were unable to use variables when they had to represent some actual, changing quantity and that their notions of functions were very hazy. As a result, even though they were all competent at factorizing,- expanding, solving and manipulating symbols in general, most could not cope with simple applied calculus problems.

Abstract: Moloney, Kevin; Stacey, Kaye Although students first learn about decimal notation in primary schools, it is well known that secondary students in many countries, including Australia, do not have an adequate knowledge of the concepts involved. Even some students who can calculate with decimals do not understand the comparative sizes of the numbers involved. Understanding decimal notation is an important part of basic numeracy. Our society makes widespread use of metric measurement for scientific and everyday purposes. Computers and calculators use decimal digital displays, so making sense of input and output decimal numbers is essential. In this article, we will demonstrate some of the ways in which students think about decimal notation and how this changes as students get older. Our testing at one school, which seems to us to be quite a typical Australian high school, showed that about a quarter of students still had important misconceptions in Year 10. We present a simple test that teachers can use to help them diagnose the mistakes that their students are making and we discuss some of the help that is appropriate. In this article 'decimals' will refer to 'decimal fractions' and 'fractions' will refer to 'common fractions'.

Abstract: Dossel, Steve Introduction teachers of mathematics have been faced for many years with the problems caused by the negative attitudes students have developed towards the subject. Such negative attitudes lead to avoidance strategies, disruptive behaviours, and maths anxiety. In this article, maths anxiety will be defined and its relationship to achievement in mathematics will be explored, before the factors leading to the creation of maths anxiety are discussed in some detail. Finally, methods of prevention or reduction of maths anxiety will be examined.

Abstract: Dawe, Lloyd In recent years there has been a growing body of research which is highly informative about the impact of language in the mathematics classroom. Thus studies on the reading of mathematical text, teacher speech, small group discourse and pupil's understanding of mathematical terms and symbolism appear in the literature. Studies of bilingual children learning mathematics in English as a second language have thrown a good deal of light on cultural forces which shape expectations about the way mathematics should be learned. A whole new perspective of pupils' use of language switching, the active role played by parents and the importance of first language competence has emerged. This paper is an attempt to draw together these different findings and 10 discuss their implications for classroom teachers.

Abstract: Perso, Thelma In discussing the widespread misunderstandings of Australian Aboriginal number systems, John Harris (1990, p.137) recently suggested that "Aboriginal people [want their children]...to learn the three R's [sic] and to grow up Aboriginal". In essence this means that most Aboriginal people recognise that for their children to be 'successful' in a Western society, they need to understand and be fluent or have facility with Western mathematics and standard Australian English, while at the same time embracing the Aboriginal culture and world view.

Abstract: Brinkworth, Peter The term 'numeracy' was coined by the writers of the Crowther Report (1959) in reference to the need for sixth formers in Britain to have an adequate background in scientific and quantitative thinking before proceeding to university or college. In effect, it was an elitist concept of numeracy aimed at providing the nation with numerate scientists, technologists, businessmen and industrialists, who could take advantage of new ways of solving problems using mathematical and statistical techniques.

Abstract: Watson, Jane Readers of E. T. Bell's Men of Mathematics might conclude that the only woman mathematician of any note in history was the Russian Sonya Kovalevsky. This is not so, and as the number of women in all fields of mathematics is on the increase, it is appropriate to fill in some of the gaps in our knowledge of the contributions made to mathematics by women. The first two discussions are followed by examples which are suitable for secondary schools and illustrate the opportunity for combining mathematical history and lore with the curriculum. The third portrait leads into a delightful story with a moral for teachers.

Abstract: Goos, Merrilyn Wondering whether we are really making a difference to young people's mathematics learning is a question that most teachers have probably wrestled with at some stage of their careers. However, evidence from a multitude of research studies shows that students' mathematics learning and their dispositions towards mathematics are indeed influenced - for better or for worse - by the teaching that they experience at school (see Mewborn, 2003, for a review of this research). In other words, teachers do matter. It is difficult for researchers to specify exactly how different types of teaching and teacher qualities affect student achievement because this would require untangling the complicated relationships that exist between teacher characteristics, teaching practices, and student learning. Nevertheless, the general trends in these relationships are clear. In this article I want to illustrate some of these trends by drawing on my experiences in working with pre-service and practising mathematics teachers and their students, and in doing so to propose three reasons why teachers matter. I will then give some examples of how teachers can matter to their students in a more practical sense.

Abstract: White, Paul; Sullivan, Peter; Warren, Elizabeth; Quinlan, Cyril Some teachers are concerned that a problem solving approach to teaching may reduce attention to the key concepts and procedures of mathematics. The polarisation of positions concerning problem solving and investigations versus the notion of a secondary mathematics teacher 'as an expositor and director of learning' (Allen, 1998, p.3) is illustrated by the debate raging in the US tagged the 'Math Wars'. A similar situation has arisen in New South Wales. The Stage 5 mathematics syllabus introduced in 1997 contained a whole strand on mathematical investigations. However, due to some strong opposition claiming such investigations take students away from content focused mathematics, this section was temporarily made optional and is currently under review.

Abstract: Truran, John Chance and data is becoming a more important part of the mathematics curriculum in both primary and secondary schools. It is natural to assume that a question like "What is the probability of ...'" is a good question. After all, it is often asked in school textbooks. But using this question can lead to very serious difficulties.

Abstract: Bouckley, Winifred Reference in the notes on the N.S.W. Higher School Certificate Third' Levell Mathematics Syllabus (1065) to Envelopes has focussed attention on the little-known, activity of Curve Stitching.

Abstract: Koop, Anthony J Hand-held calculators are rapidly becoming a part of the everyday environment of both adults and children. Hand-held calculators, hereafter called calculators, are inexpensive, readily available and can be used, even by young children, with relative ease. The 'Report of the Conference on Needed Research and Development on Handheld Calculators in School Mathematics' (National Institute of Education National Science Foundation, 1977) indicates that calculators "...have the potential for replacing the paper and pencil calculations that have been a major component of elementary school arithmetic."

Abstract: de Mestre, Neville If you missed the recent great article in the Australian Mathematics Teacher (2015) by Pat Graham and Helen Chick, I strongly recommend that you obtain a copy and put it into practice in your classroom. As outlined in their paper, an incredibly useful set of information about the mathematical ability of your students will be revealed. You can look at the way your students try to solve the 20 matchsticks problem, their method of recording, their errors, and also the ramifications of the questions that they ask. When your students have completed and discussed the 20 matchsticks problem, they should be ready to tackle the following extensions.

Abstract: Perkins, Karen The topics of decimals and polygons were taught to two classes by using challenging tasks, rather than the more conventional textbook approach. Students were given a pre-test and a post-test. A comparison between the two classes on the pre- and post-test was made. Prior to teaching through challenging tasks, students were surveyed about their mindset in regards to mathematics and how they think they learn best. They were surveyed again at the completion of the project to see if there were any changes.

Abstract: Espedido, Rosei Review(s) of: Maths handbook for teachers and parents; Explaining mathematical content and proficiencies, by Jack Bana, Linda Marshall and Paul Swan, Publisher RIC Publications Pty Ltd, ISBN 978-1-922116-79-6.

Abstract: Goos, Merrilyn At the special conference celebrating the 50th anniversary of the Australian Association of Mathematics Teachers (AAMT), I was asked to speak about challenges and opportunities in teaching mathematics as a stimulus for discussion of AAMT's "future directions". With an open invitation to be a little provocative I chose the five challenges depicted. I have framed these challenges as questions to draw you into a conversation with your colleagues and, vicariously, with me. I end my brief discussion of each challenge with another kind of question that I hope will point us towards opportunities and directions for the next 50 years.

Abstract: Prescott, Anne I stand in a stone-walled classroom in the middle of the hill country in Nepal, surrounded by the expectant faces of local school teachers. A space with no window glass, just shutters, a roof which leaks during the monsoon, and a floor which turns to mud during the same season.

Abstract: Chick, Helen The imagery from Google Maps can reveal some curiosities if you know where to look. Years ago someone discovered what looked like a jet plane sitting incongruously in an Adelaide park. Further analysis revealed the truth: the plane only appeared to be on the ground; it was, in fact, tracking into Adelaide airport and passing over the park when the photograph was taken.

Abstract: Beswick, Kim; Fraser, Sharon; Crowley, Suzanne According to McKenzie, Weldon, Rowley, Murphy and McMillan (2014, p. 67), in 2013, approximately 27% of Australian Year 7 - 10 mathematics teachers had received no teaching methodology education in mathematics and hence could be considered to be teaching out-of-field. The corresponding figure for science teachers was 20%. Furthermore, the likelihood of students being taught mathematics by an out-of-field teacher is greater in provincial or remote schools compared with metropolitan schools (Office of the Chief Scientist, 2012). Teachers in such rural or remote locations also tend to be less experienced and have limited access to professional learning and the support of expert colleagues compared with their metropolitan colleagues (Lyons, Cooksey, Panizzon, Parnell, and Pegg, 2006). Yet when an expert teacher is available, the task of mentoring out-of-field and less experienced colleagues is often undertaken with little acknowledgement or support. In this paper, we describe the initial stage of developing a framework designed to support out-of-field, less experienced or isolated mathematics and science teachers to make decisions about the use of resources in their teaching. The process highlighted the complexity and extent of the knowledge on which expert teachers draw in making such decisions and thus underscored the enormity of the task of teaching out-of-field. The eventual product, the Science, Technology, Engineering and Mathematics: Critical Appraisal for Teachers (STEMCrAfT) framework has proven useful not only for the target audience, but also as a tool for colleagues who take on a mentoring role. We begin with a brief description of teacher knowledge before describing the project and then presenting what we unearthed about expert teachers' thinking and knowledge.

Abstract: Thomson, Ian Using 'live editing' it is possible to write code that can be run a section at a time. This makes it easier to spot and correct errors. It can also be used to create an interactive mathematical story.

Abstract: Zembat, Ismail O Most students can follow this simple procedure for division of fractions: 'Ours is not to reason why, just invert and multiply.' But how many really understand what division of fractions means - especially fraction division with respect to the meaning of the remainder. Think about the 'bags problem' and its solution, using only diagrams.

Abstract: Miller, Geoffrey; Obara, Samuel A mathematical mnemonic is a visual cue or verbal strategy that is used to aid initial memorisation and recall of a mathematical concept or procedure. Used wisely, mathematical mnemonics can benefit students' performance and understanding. Explorations into how mathematical mnemonics work can also offer students opportunities to engage in proof and reasoning. This article will firstly illustrate how dissecting the so-called "butterfly method" for adding and subtracting fractions can deepen a student's comprehension of fraction arithmetic. Secondly, the article will exemplify how the same dissection process can be applied to other mathematical mnemonics.

Abstract: Quinnell, Lorna Compared to the use of the word "average" in the media and in everyday conversations, an average is used very precisely in mathematics. This is complicated by the fact that there are three different types of averages: means, medians, and modes. As with other concepts, scaffolding understanding of mathematical averages is important. Such understanding is one of the proficiency strands or key ideas in the 'Australian Curriculum: Mathematics' (ACARA, 2015a). The aim of a focus on understanding avoids superficial statements about averages, and of descriptions referring to methods of calculating averages with no connections to conceptual understanding and to real-life applications. Developing understanding of averages can be achieved through multiple scaffolding activities, including activities that focus on real data, and activities that extend to critical thinking about the use of the words average and median in the media.

Abstract: Roddy, Mark "Why do I need to learn this'" Every teacher has heard this question at one time or another. It is a particularly common refrain in mathematics, especially at higher grade-levels. Students struggle to learn concepts and to demonstrate their mastery of the skills we prescribe, and they want assurances that there is some point. We can, and should, provide students with examples of applications that come from both their everyday lives, and from the lives we see them coming to lead in the future. So, for example, it's not incorrect to say that you must know how to determine the area of a rectangle because then you will be able to estimate how much paint will be needed to cover a wall in your house. That's a good start but it's not enough.

Abstract: Wetherell, Chris This is an edited extract from the keynote address given by Dr Chris Wetherell at the 26th Biennial Conference of the Australian Association of Mathematics Teachers Inc. The author investigates the surprisingly rich structure that exists within a simple arrangement of numbers: the times tables.

Abstract: de Mestre, Neville Around 430 BCE it is reported that a typhoid epidemic carried off about a quarter of the population of Athens in ancient Greece. As usual the gods were blamed for this disaster, but could be approached for help through the high priests in the Temple of Apollo in Delos. When the gods were asked what could be done to halt this raging epidemic, they apparently replied that the altar in the temple would have to be doubled in size.

Abstract: Gough, John Div 'Blokus' is an abstract strategy board game for 2 to 4 players. It uses polyominoes which are plane geometric figures formed by joining one or more equal squares edge to edge (refer Figure 1). Polyominoes have been described as being "a polyform whose cells are squares", and they are classified according to how many cells they have. ie. number of cells. (Wikipedia, 2017 March 19). Polyominoes and other mathematically patterned families of shapes were invented by Solomon Golomb, and popularised by Martin Gardner.

Abstract: Gough, John We are familiar with metric units of measurement, such as metres for length; hectares for area (Figure 1); litres for volume; and grams, kilograms and tonnes for mass.

Abstract: Thomson, Ian When I was a child in a land far away, I scooped up snowflakes and stared at them in wonderment and awe. So meltingly ephemeral. Gone in an instant. And yet there, in the palm of my hand, I held all eternity. But how can that be'

Abstract: King, Alessandra During their middle and early high school years, students generally finalise their attitude towards mathematics and their perception of themselves as students of mathematics, in terms of aptitude, motivation, interest, and competence (NCTM, 2000). Therefore, giving them varied opportunities to foster a positive and successful approach to the study of mathematics is critical, and can help them appreciate the relevance, usefulness, and creativity of the subject. Most teachers are looking for innovative ways to capture, foster, and encourage their students' interest in mathematics, whilst at the same time conveying the required content. Furthermore, various educational organisations extol the power and usefulness of technological tools in the mathematics classroom. For example, the Common Core Standards (CCSSI, 2010) state that mathematically proficient students "are able to use technology to explore and deepen their understanding of concepts" (p. 7). NCTM's Principles to Actions states that "an excellent mathematics program integrates the use of mathematical tools and technology as essential resources to help students learn and make sense of mathematical ideas, reason mathematically, and communicate their mathematical thinking" (2014, p. 78). The Australian Curriculum: Mathematics (ACARA, 2014) includes the use of graphing software to foster the critical and creative thinking processes, and the ability to generate solutions.

Abstract: de Mestre, Neville Suppose that there is an inexhaustible supply of $3 and $5 vouchers from the local supermarket. They may only be exchanged for items that cost an exact number of dollars made up from any combination of the vouchers. What is the highest amount not able to be obtained'

Abstract: Robichaux-Davis, Rebecca R Progressing from additive to multiplicative thinking is critical for the development of middle school students' proportional reasoning abilities. Yet, many middle school mathematics teachers lack a thorough understanding of additive versus multiplicative situations. This article describes a sequence of instructional activities used to develop the proportional reasoning skills of middle school pre-service teachers. The activities could also be implemented in middle school classrooms.

Abstract: Neumann, Hanna Dr Peter Neumann, son of the late Professor Hanna Neumann, will be the keynote speaker at AAMT's biennial conference 'Capital Maths', which will be held in Canberra from 11-13 July 2017. Prof. Hanna Neumann gave the Presidential Address at AAMT's inaugural conference in 1966. The conference was held at Monash University and had the theme of 'mathematical unity'. Below is a transcript of Professor Hanna Neumann's address.

Abstract: Attard, Catherine Mathematics education features regularly in the media. The most recent international testing results highlight a decline in Australia's mathematics achievement when compared to other countries. So, who's responsible' Is it teachers, or should parents and the broader community share some of the blame' Typically, teachers are the first to be blamed because they work at the coal face, spending significant amounts of time with students, making them an easy target. But shouldn't the wider community, as a society that considers it acceptable to proudly claim "I'm not good at maths" take some portion of the blame'

Abstract: Gough, John A response to the opinion piece, The M in STEM: what is it really' John Gough provides a response to the opinion piece, written by Lance Coad (AMT, 72 (2)).

Abstract: Stohlmann, Micah Battleship is a game of guessing, strategy, and logical thought that has been around since the 1930s. In the game, players position ships of various sizes on a grid that their opponent cannot see. Players take turns guessing the location of the ships and the game continues until all of one player's ships are sunk. This game can be adapted to incorporate mathematics by using Desmos, a free online graphing calculator that runs in the window of any modern web browser.

Abstract: Espedido, Rosei; du Toit, Wilhelmina If, from a young age, students were taught that mathematics is not simply number-based content to be memorised but rather, in its purest form, is about acquiring and mastering a logical thought process, there would be less time dedicated to justifying why the learning of mathematics is important. In reality, students are unlikely to find themselves in a devastatingly life-changing situation should they not be able to demonstrate the congruent nature of two triangles. However the approach a student may choose to implement in dealing with the aforementioned situation is integral to the outcome. Learning experiences in mathematics foster the development of skills (including evaluation, reasoning and logical stepwise thinking) which all serve as invaluable tools for life beyond the classroom. So, why are mathematics teachers not more honest when addressing the questions posed by students? Why are they afraid of 'telling it like it is'? Why are we as teachers not making the relationship between mathematics and thinking processes clear enough that students also come to appreciate the content?

Abstract: Proffitt-White, Rob The Teachers First initiative is a grass-roots cluster-model approach for bringing together primary and secondary teachers and school principals: to analyse student performance data; design and practise activities and assessment tools; and promote teaching practices that address students' learning difficulties in mathematics. The balance of both top-down and bottom-up reform processes, seeded with the latest research evidence, allowed teachers to become both competent and confident in their effective teaching of mathematics. Its continued success is testament to our innovative school leaders and passionate teachers.

Abstract: de Mestre, Neville Consider N (> 1) people spaced regularly around the circumference of a circle. Now a circle has 360 degrees, and therefore there are many N for which the angle subtended at the centre of the circle by adjacent people is an integer. Ask your students to find them all. How can 19 19 = 361 help them?

Abstract: Kissane, Barry It seems that calculators continue to be misunderstood as devices solely for calculation, although the likely contributions to learning mathematics with modern calculators arise from other characteristics. A four-part model to understand the educational significance of calculators underpins this paper. Each of the four components (representation, calculation, exploration and affirmation) is highlighted and illustrated, mostly with relatively unsophisticated modern calculators such as those widely accessible to students in years 6-10, but also recognising some calculator features not available to younger Australian students. Intelligent use of calculators at these levels of schooling offers many opportunities for students to develop a solid understanding of key aspects of mathematics through their own actions, provided our apparentobsession with calculators as merely 'answering devices' is overcome.

Abstract: Dawe, Lloyd This paper addresses the continuing need for mathematics teachers to enrich their mathematical knowledge beyond the school curriculum, in order to effectively engage students in creative and imaginative thinking, particularly, but not exclusively, students who show exceptional promise. The author, a retired university professor, works staff and students in a girls' private school in Sydney for this purpose. The paper provides examples of imaginative problem solving gathered over a 5-year period, which has led to significant mathematical insight for both staff and students. It promotes the realisation of mathematical potential of students concurrently with the professional development of teachers. It is argued that this best happens in mathematics classrooms with experienced mathematics educators working alongside teachers.

Abstract: Espedido, Rosei Review(s) of: The smartest kids in the world and how they got that way, by Amanda Ripley, Publisher, Simon and Schuster Paperbacks, ISBN 978-1-4516-5442-4.