Content Listing

Summary

Learning takes place when students are asked to derive their own meanings and understandings from the experiences that are presented to them. The concept of discovery learning is challenging to many teachers, as it requires handing the content over to the students so they can independently investigate and understand the information. The teacher then becomes a facilitator of learning, and only mentors the students in their learning. Relinquishing the role of teacher as instructor can be a challenge for students as well as teachers, as it requires the students to be ready to take the lead and become more accountable in their own learning.

Like all popular terms in education, discovery learning has taken on a range of meanings, but it most commonly refers to a form of pedagogy in which students are exposed to particular questions and experiences that enable them to ‘discover’ for themselves the intended concepts (Hammer 1997). The learning is an inductive form, moving away from the teacher as the centre of learning to a student-centred approach.

Teachers and students have a set of stimuli, beyond which the students need to go to form concepts and relationships. The students can add to the stimulus material or rearrange it so it is more meaningful to them.

Summary

An experiment provides two ways to investigate information. The first is a formal procedure that is followed in order to learn about the effects of different things. The second involves experimenting to see how something works and how successful it is. The aim of an experiment is to validate a hypothesis.

Experiments give students the opportunity to take the leadership in the learning and to demonstrate their knowledge and understandings. They can utilise their skills to prove that they have an understanding of different concepts.

An experiment is asking, ‘What would happen if . . . ?’ and it is through the manipulation of variables that different results are obtained. It is the challenge of creating a situation, and varying or controlling the variables, that makes undertaking an experiment so rewarding for the students. In an experiment, the students are first-hand researchers of particular phenomena, and they undertake the task in order to gain the data to prove their work. An experiment is an alternative way to answer a question and to determine why certain phenomena happen in certain ways.

The role of the teacher is very important in organising experiments. The teacher becomes a facilitator by setting up stimuli to be used, and challenges the students to go beyond what they have been given to form concepts and relationships. Experiments combine a range of processes that are undertaken in learning, and may be part of any curriculum area.

Summary

A gambler may not know, off the top of his or her head, the exact probability of the winning cards in a hand of Poker or a doctor the exact likelihood of a particular diagnosis. They both may have a gut feeling for how likely or unlikely such an event is to happen. Calculating a probability provides, in this sense, a precise quantification for our intuition about the likelihood of an uncertain event.

In this chapter, we pursue a class of examples designed to stretch intuition and test beliefs about the idea of probability. Many of the examples presented here and in the exercises are commonly referred to as paradoxes. A paradox is a statement or problem leading to contradictory conclusions or solutions. True paradoxes in mathematics are rare and significant phenomena. For example, a simple paradox about “truth” known as the Liar Paradox (see Exercise 10.6) ultimately led to Gödel's remarkable Incompleteness Theorem in mathematical logic. In our own study, the Condorcet Paradox is the basis for Arrow's Impossibility Theorem (Theorem 18.19) and the central example in social choice theory. While we retain the term “paradox” in the names when this is common use, the question of which examples to follow are truly paradoxical is an interesting one and left to your interpretation. We begin with what is perhaps the most popular example of a probability puzzle.

Example 19.1 The Monte Hall Problem

Monte Hall was the flamboyant host of a television game show created in the 1960s called Let's Make a Deal. Contestants on the show were led by Hall through a series of guessing games and deals for cash and prizes. The scenario for our problem is the following: the prize is a new car, which is hidden behind one of three identical doors. Behind the other two doors are cows. The contestant must guess the correct door to win the car. Once the contestant has chosen a door, Monte Hall opens one of the remaining doors, and a cow walks through onto the stage.[…]

Summary

I was born and raised in Ballarat, a provincial city in Victoria with a history built on gold. My father’s people were ‘local’. They had participated in the gold rush of the 1850s and had made small fortunes as miners, mine owners and shopkeepers. These people had come from England. My mother was not a ‘local’. She was born on the banks of Orange Creek in the Northern Territory about 100 kilometres south of Alice Springs. Her people were the Pitjantjatjara people; they had come from the Tjukurrpa (or ‘The Dreaming’). As a small child, like small children everywhere, I was colourblind. I didn’t notice that my mother had brown skin and I didn’t know what people meant when they called her an Aborigine or a half-caste. I guess I thought that my family was just like everybody else in the suburban Australia in which I lived.

This myth was first shattered when I travelled to Central Australia for the first time. I was no more than four or fi ve years of age. The Stuart Highway was nothing more than a narrow, corrugated red ribbon that followed the old Ghan railway into the heart of the continent. The vast endlessness of the desert made a lasting impression on me. Alice Springs was still a frontier town of hard cattlemen. From here we travelled to Titjikala, the Aboriginal community where my mother had grown up. Established in the 1940s, the purpose of Titjikala was to provide rationed labour to the white people who ran cattle on Maryvale Station. I had arrived in the immediate aftermath of the equal pay decision, a decision that ended the participation of Aboriginal labour in the northern pastoral industry. This was another world, an Australia that I had never imagined existed. My grandmother and my other mother (aunt in whitefella terms) lived in sheds set on concrete slabs. Everyone at Titjikala lived in such houses.

Summary

Explanation is the development of a narration and the sharing of information with students, encouraging them to think logically, aesthetically and morally.

Increasing numbers of magazines and shows are offering instructions on how to make or create something. From creating a new look in your house to cooking tonight’s family dinner, new ways to exercise or techniques to take better photographs, we are amply informed on how to undertake tasks through visual, written and oral explanations.

Think of the radio commentary on a Rugby Union, football or cricket match. The commentators have the ability to create the visual images of the game for listeners. They explain the moves and the momentum of the sport, and provide a description of the intensity of the action. The success of the experience for the listener comes from the commentators’ in-depth knowledge of the game, the players and the expectations of the audience. Various nuances of the spoken word are required to lead to a good understanding by the listeners.

Although explanations are often considered to be one of the easiest learning strategies, they really require thorough preparation and planning. The examples of the novelist, the sports commentators and the journalists writing about cooking or home maintenance all demonstrate the importance of the knowledge of the field in which they work – so an explanation is always embedded in a particular context.

Summary

I was born in a small town in Queensland on the country of my maternal great-great grandmother; this country is the Gooreng Gooreng nation. I was raised and formally educated in another Queensland town on the country of my maternal great-great grandfather; this country is the Wakka Wakka nation. My interest in writing dates back to watching Nana reading old westerns and Mum reading historical romances. Like my three sisters, I was able to read and write before going to school, and my Year 3 teacher commented to Mum to the following effect: ‘Whatever Sandra does when she grows up, it should involve words.’ My lifelong interest in publishing had a humble origin in my place on my high school’s magazine committee, and continued through university as I served as a co-editor of the Aboriginal and Torres Strait Islander Student Handbook. After my fi rst university degree and some professional work I entered into editorial traineeships with Magabala Books in Broome then with University of Queensland Press in Brisbane, and later went on to become Manager of Aboriginal Studies Press in Canberra. I enjoy writing and as an academic I now have to write and be published in scholarly journals, but I do consider myself first and foremost an editor and a speaker. I have long contributed to public conversation about Indigenous writing and publishing, dating back to my fi rst conference presentation at the Aboriginal Publishers Conference in 1989 in Vancouver, Canada. I have also held leadership roles on arts organisations, including the Aboriginal and Torres Strait Islander Arts Board of the Australia Council and the Queensland Performing Arts Trust, and in 2012 I became a judge of the annual David Unaipon Award for Unpublished Indigenous Writers.

Summary

Discussions examine the pros and cons of a particular issue or idea through conversation or debate. We engage in discussions with friends and family every day, but in the classroom context they are a way to develop students’ knowledge on a topic or to assist them to learn more about what they already understand. Any discussion therefore needs to be carefully planned in advance, with consideration given to the roles of the teacher and students, as well as the content that needs to be covered.

The following topics will be explored in this chapter:

Summary

In this chapter, we delve deeper into the idea of an election as a game of strategy (Definition 2.9). The implications of this definition are somewhat unsettling. Is a popular election a game to be won? In a democracy, we expect social choice to be a faithful accounting of the true preferences of the people rather than an outcome arrived at by the strategic cleverness of certain of the voters. Unfortunately, the ideal of a strategy-proof, democratic election is an impossible dream. We state the Gibbard-Satterthwaite Theorem (Theorem 6.4), which asserts that with three or more candidates, every nondictatorial voting method is vulnerable to strategic manipulation. We verify the theorem for each of the democratic voting methods introduced in Chapter 5. We also show how strategic considerations in social choice can elect a third-party candidate in a popular election. The latter result is an example of the Paradox of the Chair. Finally, we give an example of the No-Show Paradox, a scenario in which voters in an election better serve their own interests by staying home as opposed to participating in the election.

The Condorcet Paradox

The developments in this chapter will all flow out of arguably the central example in social choice theory. We introduce this important example by means of the following simple puzzle:

Example 6.1 Choosing the Final

Twenty-five students in a game theory seminar are given the option of choosing the format for their final assignment. The options are to write a term paper (T), take an oral exam (O), or take a written exam (W). The professor proposes that the students vote on each of the options in head-to-head Simple Majority elections and thereby see which option is preferred to the other two. First, the professor pits term paper versus oral exam. The winner is term paper, by a vote of 18 to 7. Next, the students vote on writing a term paper versus taking a written exam. Here the written exam wins by a vote of 15 to 10.[…]

Summary

In this chapter, we explore the debate over voting methods that marked the birth of social choice theory as an academic field. Let us briefly recall the history and point of contention. On one side, we have the Chevalier Jean Charles de Borda, a mathematician and military scientist. Borda proposed his voting method in a lecture to the French Academy of Sciences urging the body to adopt his voting method for member elections. Borda's method, as we know, converts voter preferences into a single ranking, the social welfare. While ties do occur, they are not so common. In 1780, Borda published the first mathematical proof of fairness. He proved that the Borda Count can never elect a candidate ranked last by a majority of the voters (a Majority Loser). We proved this result ourselves in Theorem 12.15. In fact, Borda proved a stronger theorem. Answering the open question from Chapter 12, we state the following theorem:

Theorem 21.1 Borda's Theorem The Borda Count satisfies the Condorcet Loser Criterion.

On the other side of the debate is the Marquis Nicolas de Condorcet, a prominent intellectual of the Enlightenment period in France. Condorcet also had mathematical evidence supporting his political ideals for elections. Condorcet had a proof that the best chance for achieving the “correct” result in an election is to obey the majority opinion. Condorcet's Jury Theorem (Theorem 7.10) was stated in Chapter 7. For Condorcet, the majority opinion is expressed through head-to-head Simple Majority elections. As we have seen, the Borda Count does not always select the majority winner as the social choice and so violates a basic principle of fairness.

It was Condorcet himself who pointed out the principal obstacle to his own majoritarian ideals. When there are three or more candidates, the majority opinion, as expressed through head-to-head elections, can be fundamentally conflicted. We can have a Condorcet Paradox. Moving ahead almost 200 years to 1951, Kenneth Arrow used the Condorcet Paradox to prove his Impossibility Theorem, beginning the modern era of social choice theory.

Summary

The only two facts relayed to me about my natural family when I was growing up was that my mother was an Aboriginal woman named Mary Williams and that her father was a policeman in the Northern Territory. Consequently, at the tender age of six I resolved to do two things with my life: to fi nd my natural mother and to become a police officer, just like my grandfather.

Six months before I turned 16, I applied to join the NSW Police Cadet Corps. The year was 1967, the same year part of the Australian population voted overwhelmingly in a federal referendum for Aboriginal people to be included in the national census and for the Commonwealth to make laws for and on behalf of Aboriginal people. Although I hadn’t completed the mandatory four years of secondary schooling to qualify to sit the School Certifi cate examination, I was permitted to seek entry to the Cadet Corps and, subsequently joined the Cadet Corps one month before the end of that year. To my knowledge, I was the fi rst Indigenous person accepted into the Cadet Corps and one of only two Indigenous Police Cadets during the Corps’ 40-year history.

When I was 28 years old, I finally located my mother who was alive and living in Tennant Creek in the Northern Territory. At the time, I also found out that I was the eldest of nine brothers and sisters and one of 742 Western Arrernte family members. It was some time after meeting my mother that I also found out that the ‘policeman grandfather’ I was told about all those years ago wasn’t my mother’s father, but a whitefella who grew my mother up before she was removed from Alice Springs to Mulgoa Mission near Warragamba, New South Wales, at the eastern end of the Blue Mountains.

Summary

I was born in 1970 in Mackay, North Queensland. I am privileged to be Aboriginal and Australian South Sea Islander. My country is the gateway to the Great Barrier Reef - where subtropical rainforest meets mangroves and beaches. My family are saltwater people.

Our country has for many years (since colonisation and pastoralisation) been the sugar bowl of the world. The sugarcane industry was responsible for my South Sea Islander ancestors coming to Australia as blackbirded and indentured workers. This practice was a very small step above slavery. Mackay is a special place. Throughout Central and Far North Queensland, Aboriginal, Torres Strait Islander and South Sea Islander people have shared, married and lived in blended and distinct cultures. My greatest memories of growing up are those of my family - extended and joyful.

My involvement in human rights specifi cally relates to the rights of Indigenous Australians to education. This is a direct infl uence of my family. Education has always been extremely important to my parents. Both were sent out to work and were never formally educated beyond Year 4 in primary school. They wanted better for their children. After my sister was born (she was the youngest of the three Thomas children), my parents moved us from their sugar cane farm into town. My mother still lives in this home. My parents worked hard to ensure we were always able to fully participate in school and any extracurricular activities we were interested in. They attended parent-teacher interviews, school plays, concerts and sporting events throughout our school lives. This in itself took courage as they were often the only Indigenous Australian parents in attendance. The importance of education was accepted by my brother, sister and I as a given.

Summary

Every day, we use questions to gain different forms of information. Essentially, we are living and teaching with questioning in every aspect of our lives. Questioning is a natural part of our conversations and teaching. Yet in order for students to learn how to use questioning, they need to learn about questioning. This chapter explores questioning to assist in your own development and then the students’ understanding of questions. Finally, we will look at ways of teaching for questioning, covering how students can develop their questioning skills for research and investigation.

Many researchers and educators identify the different types of questions that will be covered in this chapter. The skill of questioning is paramount in teaching and learning, and this chapter aims to help you understand the impact it has on learning, as well as how you can use the strategy to its best advantage.

Activity 5.1 Asking Questions

Summary

I was born on my mother’s traditional lands and raised as a child between two households. My first home was my parents’, situated in the suburban sprawl of fibro ovens in treeless estates in western Sydney where we lived with a mix of Koori families, imports from war-torn Europe and the debris of Sydney’s post-colonial white society under the banner of ‘housos’, residents of the public housing estates of Bass Hill and Chester Hill. My second home was the coolness, calmness and stability of the loving arms and hugs of my mother’s mother and her Koori family in Harbord, Curl Curl and the Manly–Narrabeen Lakes areas, all traditional country of the Gai-mariagal. We swam, fi shed, prawned, regularly harvested swan and duck eggs, and ate possum and the odd goanna to supplement a diet rich in chayote, rhubarb, warrigal greens or anything else that was easy to grow or scavenge or that fell off the back of a truck.

I had an idyllic childhood, exploring Cowan Creek and the Hawkesbury River, camping with my uncles and cousins and having no knowledge of poverty or racism; then my cousins started to vanish. One by one they were plucked away, and the adults never spoke of them. We stopped visiting each other’s houses: card games that would last all night, the laughter and warmth of wider family, with Dad sharing long necks and Gran her blackberry or rhubarb pies – this all ceased abruptly. Then in ‘59 the camp at Narrabeen was bulldozed and the Elders trucked off, some to a mission place at Rooty Hill that I visited when my big sister took supplies to one of our Elders.

Summary

We represent the oldest culture in the world; we are also diverse and have managed to persevere despite the odds because of our adaptability and our survival skills and because we represent an evolving cultural spectrum inclusive of traditional and contemporary practices. At our best, we bring our traditional principles and practices - respect, generosity, collective benefi t and collective ownership - to our daily expression of our identity and culture in a contemporary context. When we are empowered to do this, and where systems facilitate this reclamation, protection and promotion, we are healthy, well and successful and our communities thrive (Dr Ngiare Brown, 2012 ). My name is Raelene Ward; I’m an Aboriginal woman originating from Cunnamulla in south-west Queensland. I am a descendant of the Kunja people on my grandfather’s side and the Kooma people on my grandmother’s side; both groups are in south-west Queensland. I was born and raised in Cunnamulla as well as a number of other smaller rural and remote communities in far south-west Queensland. I have worked in a number of mainstream hospitals and mental health institutions and community settings across the region. I have extensive knowledge of the Aboriginal community-controlled health sector, having worked in the sector for approximately eight years. From 2007 to 2010 I coordinated a number of suicide prevention projects across south-west Queensland, one of those being a three-year project to the value of $1.5 million. I have been a practising clinical nurse for the past 24 years, mainly in these contexts: suicide prevention, Aboriginal health, social and emotional wellbeing and mental health, all in the broader area of social determinants of health.

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Suggestions for Further Reading

Chapter 9 - Discovery learning

Summary

Chapter 8 - Experiments

Summary

19 - Paradoxes and Puzzles in Probability

Summary

Chapter 11 - Good sports: Representations of Aboriginal people in Australian sports

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Chapter 4 - Explanation

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Chapter 6 - Literature: Writing ourselves

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Chapter 3 - Discussion

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6 - Strategy and Choice

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Frontmatter

21 - Borda versus Condorcet

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Suggestions for Further Reading

Chapter 4 - Overkill, over time: The criminalisation of Indigenous society – or genocide by stealth

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Chapter 12 - The right to be human: Aboriginal and Torres Strait Islander peoples and human rights

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Chapter 5 - Questioning

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Contributors

Chapter 7 - Enterprise and entrepreneurial thinking: It’s a black thing!1

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Chapter 8 - Health and wellbeing

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Contents

Index

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