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Summary

Despite the many years I have spent pondering and reading about the subject of race, I still feel a deeply rooted uncertainty about exactly what ‘it’ is and how to approach ‘it’. This, I think, is no bad thing and guards against the oversimplification of something that, rather than being a single object, is a mercurial, shape-shifting and slippery set of ideas and associated practices, which exist always in relation to other phenomena. On the one hand, I have long been dissatisfied with approaches that side-step an effective definition of race, such that any specificity is submerged into general ideas about inequality and difference. This seems to lose a grip on something that is characteristic of racial concepts, despite their diverse and changing manifestations, which makes them different from other forms of distinction. On the other hand, I have also been uncomfortable with approaches that are overly specific and define race in terms of ‘biology’, ‘colour’ or even broader terms such as ‘naturalisation’. These approaches tend to take for granted what all these terms mean and how their meanings change over time and space. My own reaction to this dilemma is outlined in the first chapter, in which I seek to pin race down, but in a way that acknowledges the precariousness and impermanence of the exercise, given that race is always on the move, is always embedded in a specific historical context and always exists in relation to other social connections and categories. These features undo an attempt to characterise race as a coherent object and point towards the existence of multiple processes of racialisation, in which very diverse social phenomena are freighted with racial meanings, which are still nevertheless recognisable as racial meanings, with their particular historical baggages and entailments.

My aim in this book is to place the concept of race in a very broad historical and geographical context, with a view to showing its affinities to other modes of social differentiation, which overlap with it, while also trying to grasp certain specificities and continuities.

Summary

Gambling games have inspired two major mathematical developments related to our work. In the seventeenth century, a writer named the Chevalier de Méré posed a question about probabilities with repeated rolls of the dice to the eminent mathematician and philosopher Blaise de Pascal. The problem led Pascal and another legend of mathematics, Pierre de Fermat, to develop the foundations for a modern theory of probability. In the twentieth century, the game of Poker with its complex mix of chance and strategy became a central problem for the development of game theory. Both Emilé Borel and John von Neumann published early work on the game in restricted settings. The psychology of betting and bluffing in Poker became a useful metaphor for the variables at play in economic and political interactions. While the complexity of Poker has proven too great to admit anything like a complete mathematical theory, working algorithms for playing the game can be extremely lucrative.

In this chapter, we discuss two popular gambling games: Craps and Poker. Our analysis of these games will motivate several new ideas. For the game of Craps, we explain the technique of “introducing a conditional,” which allows us to compute a probability in an infinite game. Straight Poker, in turn, presents the problem of counting card hands. We apply the Combinations Formula (Theorem 11.12) to compute the probability of Poker hands. We then return to the game Five-Card Stud and in particular address Question 1.2 from Chapter 1. Our resolution of this question will motivate Bayes' Theorem (Theorem 14.5).

Craps

The dice game Craps creates excitement from the simple of act of repeated rolling of a pair of dice. We introduce this casino game now.

Example 14.1 Craps

The game is played on a long velvet table with one dice roller (the shooter) and many side bettors and spectators. Craps can go on indefinitely, with the tension and side bets mounting with each throw. The rules are as follows: the shooter throws the dice.[…]

Summary

Teaching requires careful planning based on the needs of students, but it is influenced by many other components as well. At a whole-school level, curriculum planning determines the focus taken in particular subjects; consideration is also given to the number of hours allocated to teaching specific content. This is then translated into units of work, with each unit broken down further into lessons and activities, within which there are specific teaching and learning strategies that will be used to develop students’ knowledge and understanding of particular concepts, rules, facts or generalisations.

The planning for successful teaching and learning encompasses four major areas:

Summary

Graphic organisers are spatial representations designed to make efficient use of the information in a text. Visualisations are a technique to create still and moving images from given data. The ability to gather data and draw some meaning from it is the essence of learning. Through the use of graphic organisers and visualisations, students and teachers can demonstrate their understanding of discipline content and processes.

Graphic organisers

In all curriculum areas and topics, students are exposed to new and novel content. Graphic organisers can assist in the understanding and processing of this content. Each organiser relates to the processing of different information for a different purpose, so students need to understand the concept behind using graphic organisers in order to determine which one they will use in a given situation.

Graphic organisers may be used by individual students, a small group or the whole class to generate knowledge or to link information together. They may be used at the beginning of a lesson to investigate the level of knowledge the students have on a particular topic (pre-assessment) or added to as the students learn about an area, or presented as a demonstration of the learning from a series of lessons.

Summary

Arguably, the first principle of fairness for a voting method is that each vote should count equally in the final outcome. In this chapter, we study voting systems that violate this basic principle. These are voting systems in which differences between voters are purposely introduced to reflect differences in the rank of the player or, in the case of regions, the size of the population. Examples of such systems include the U.S. Electoral College, the U.N. Security Council, and countless corporate and legislative committees. As is customary, we study such voting systems under the simplifying assumption that there are only two possible outcomes of the election, “yes” and “no,” a yes-no voting system as in Definition 7.4. In this setting, the relative power of the individual voters is the central issue. We apply the counting techniques, notably the Combinations Formula (Theorem 11.12), developed in Chapter 11 to study the power of voters in weighted systems. We also consider the question of assigning weights to voters in a yes-no voting system. This problem leads us to the notion of a mathematical invariant. Finally, we show that Question 1.6 regarding the U.S. Senate introduced in Chapter 1 leads to a combinatorial identity called the recursive law for combinations.

Yes-No Voting Systems and Power

Much of the controversy over voting methods stems from issues arising from the presence of multiple candidates. Even the most democratic methods can appear “unfair” in the attempt to untangle social preferences when their are several competing alternatives. We remove this issue here and focus on elections with only two candidates.

Summary

Perhaps the most entertaining games to play are those involving both chance and strategy. This class includes popular board games such as Monopoly, Backgammon, and Scrabble, as well as a whole range of card games from Blackjack to Bridge. The roles of chance and strategy vary from game to game. For all these games, winning requires some combination of good fortune and strategic skill.

In this chapter, we analyze a simplified version of the game Poker (Example 8.1). Our example features the strategies of bluffing and calling that arise in the general game. We use this example to introduce two themes that we will return to in Part II of this text. First, we introduce the idea of treating an expected value as the payoff of a game. Second, we introduce the idea of mixed strategies. We conclude by considering a somewhat more complicated version of the original game (Example 8.8). This game includes the strategy of raising a bet in addition to the strategies of bluffing and calling.

Example 8.1 Betting on Diamonds

This is a two-player game with many possible variations. Each player antes $2. Player 1 chooses a random card from a deck and looks at it but does not show the card to Player 2. Player 1 may now either bet or fold. If Player 1 elects to bet, he or she bets $8. Player 2 may now either call or fold. If Player 2 calls and the card is a diamond, then Player 1 wins the pot; if the card is not a diamond, then Player 2 wins the pot. If either player folds, he or she loses the ante of $2 to the other player.

A first observation is that Player 1 should bet whenever he or she gets a diamond. There is no incentive to fold with the winning card.

Summary

My name is Jessa Rogers and I am an Aboriginal woman, born on Ngunnawal Country in Canberra. Through my mother I am a descendant of the Wiradjuri and Darkinjung peoples. It is important when introducing myself as an Aboriginal woman to give some information about my cultural background and family ties, so connections and relationships can be established (Martin, 2003). I provide this information gladly, as this chapter is also a personal story. My ancestors, many of whom shared my mother’s family name (Prince), lived at Karagi near the area known as The Entrance, in New South Wales. Today, my family members are dispersed along the East Coast of Australia. My responsibilities as a person are informed by my relationships. One of two children, I am a sister, a daughter, a granddaughter, a mother to two sons and an aunty. I am also a teacher. Elders continue to provide signifi cant guidance and support in all aspects of my life. In beginning this chapter, I pay my respects to Elders past, present and future. I recognise the Elders, parents, students, families and communities who have allowed me the honour of learning and teaching with them, and pay my respects to them also.

Summary

Narration is the art of telling a story through the use of words and images. Narrations may be written or spoken, but they need to draw the listener into the images created by the words. The presentation of stories requires careful preparation and delivery. The story assimilates information, and retelling it can be a powerful learning strategy; however, accuracy is needed to ensure that students understand the information that is being presented.

Types of narration

Heath and Heath (2009), in their discussion of explanations, highlight the value of stories. Storytelling can be a powerful way of explaining ideas or concepts because it gives both the teacher and the students a chance to be entertained but also instructed. They argue that ‘When we hear a story, our minds move from room to room. When we hear a story, we simulate it’ (2009, p. 210). The story is a valuable teaching tool ‘because it provides the context missing from abstract prose’ (2009, p. 214). By using one of the plot varieties discussed below, you can be sure of motivating students to listen and learn. With this in mind, we can choose the types of narration that we may want to use in teaching and learning.

Summary

In this chapter, we digress from our study of games and elections to focus more directly on the logic of mathematical proof. Our goal is to identify some proof methods that we have seen in the first part of the text and to introduce some that we will use in Parts II and III. We begin with the idea of a mathematical statement and the basic problem of constructing proofs and finding counterexamples. We then introduce proof methods for two types of mathematical statements: conditional statements and universal statements. We conclude with an introduction to the method of proof by induction. We apply this method to prove a version of Zermelo's Theorem (Theorem 4.17) for takeaway games.

Mathematical Statements

Like a coin flip, which offers two possible outcomes – a basic branching in a game of chance – the building blocks for a mathematical theory are statements with two possible values: true or false.

Definition 10.1

A mathematical statement is a statement that, by its wording, only admits two possible interpretations. The statement is true or the statement is false.

Mathematical statements arise frequently in ordinary language. For example,

“Today is Tuesday” or “It is raining”

Both are statements that can only be interpreted as true or false. Of course, there may be gray areas in such phrasings. Here are some examples of mathematical statements related to our topics of study:

Statement 1: “The Borda Count social choice always has a plurality of the first place votes.”

Statement 2: “A zero-sum game with a saddle point has a Nash equilibrium.”

Statement 3: “There is a guaranteed winning strategy for Player 1 in Chess.”

There is no gray area for these statements. Statements 1 is a false statement. We can quickly check, for example, that the Democratic candidate D is the Borda Count social choice for the Third-Party Candidate Election (Example 6.9), whereas the Republican candidate R has the most first-place votes.

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References

Index

Acknowledgements

List of figures

Frontmatter

Preface

Summary

Part II - Race in practice

Acknowledgements

Bibliography

14 - Gambling Games

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Introduction

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Chapter 11 - Graphic organisers and visualisations

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3 - Chance

Index

13 - Weighted Voting

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8 - Chance and Strategy

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Chapter 10 - Education

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Chapter 2 - Narration

Summary

10 - Proofs and Counterexamples

Summary

Part II - Basic Theory

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