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38615 results in Cambridge Textbooks

Page 1475 of 1931

  • Chapter 6 - Demonstration

  • Diana Whitton, University of Western Sydney
  • Book:
    Teaching and Learning Strategies
    Published online:
    14 September 2019
    Print publication:
    29 June 2015, pp 76-88

  • Summary

    The learner’s role is to observe, to listen carefully and ask relevant questions and then to practise the task, always referring back to the demonstration that has been given.

    A demonstration is showing someone how to do something. From sports lessons to writing, mathematics, science or creating a blog, you will undertake demonstrations regularly in your teaching and learning. Your students will also demonstrate their skills and knowledge to you in many situations. Skills and knowledge about how to demonstrate are imperative for ensuring successful teaching and learning outcomes for both you and the students.

    Another thing to note is that demonstrations may be carried out by people other than the teacher, or by using media to assist with the students’ understanding of different concepts. The strategies of demonstration and application may be used in any curriculum area.

    Another thing to note is that demonstrations may be carried out by people other than the teacher, or by using media to assist with the students’ understanding of different concepts. The strategies of demonstration and application may be used in any curriculum area.

    The impact of a visual demonstration is very powerful and effective in learning. A teacher needs to be precise in the visual presentation and narration of the demonstrations as the demonstration will become the model the students emulate. If the demonstration is incorrectly or poorly executed, then that model will stay with the students.

  • 5 - Choice

  • from Part I - First Notions
  • Samuel Bruce Smith, St Joseph's University, Philadelphia
  • Book:
    Chance, Strategy, and Choice
    Published online:
    11 May 2018
    Print publication:
    29 June 2015, pp 57-76

  • Summary

    The shift of focus from games to elections represents a rather dramatic change in the objects of our study. We turn from probabilities, payoffs, and winning strategies to questions of the fairness, equity, and correctness of electoral outcomes. We have seen, however, that an election is not so different from a game at the most basic level. Recall in Definition 2.7, we defined an election as a game producing an outcome called the social choice. In this chapter, we study the mechanism for producing this outcome, the voting method.

    We begin with a discussion to motivate the idea of voter preferences. The summary table of voter preferences is called a preference table. We introduce this important device along with a number of voting methods that will be studied throughout the text. We conclude with a discussion of social preferences and social welfare.

    Preference Rankings

    In practice, most elections do not require voters to rank all the candidates. Usually a “vote” consists of simply an indication of first choice. As we have mentioned, the use of voter preferences is assumed to facilitate our mathematical analysis of elections. That said, we observe that there are elections that do explicitly involve preference rankings of the candidates. We give two “controversial” examples, both occurring in the year 1990.

    The world of sports provides a number of examples of elections in which voters submit preference rankings for the candidates. The Heisman Trophy in college football and the Most Valuable Player (MVP) awards and Hall of Fame elections for many professional sports are notable examples. Our example is from the National Basketball Association (NBA).

    Example 5.1 A Controversial MVP Award

    The 1989–1990 NBA season featured several future hall-of-fame players in their prime. Among the many strong performances, three players had standout seasons: Michael Jordan, Magic Johnson, and Charles Barkley. The MVP award was decided by the votes of 92 NBA experts (i.e., writers, broadcasters, etc.).[…]

  • 20 - Combinatorial Games

  • from Part III - Special Topics
  • Samuel Bruce Smith, St Joseph's University, Philadelphia
  • Book:
    Chance, Strategy, and Choice
    Published online:
    11 May 2018
    Print publication:
    29 June 2015, pp 299-314

  • Summary

    Combinatorial games offer an excellent example of the divide between theory and practice. On the one hand, we have Zermelo's Theorem (Theorem 4.17) asserting that one player can guarantee victory in any such game by following a particular series of moves. On the other hand, this theorem offers no practical help in executing the move-by-move play of a long and complicated game. While the theory of combinatorial games might be said to begin with Zermelo's Theorem, it certainly does not end with this result.

    Recall that we have defined a combinatorial game to be a sequential-move finite win-lose game of perfect information. We begin this chapter by extending our proof of Zermelo's Theorem for takeaway games (Theorem 10.9) to the general case of combinatorial games. We then introduce an example of a Nim-like game called Northcott's Game and apply Bouton's Theorem (Theorem 17.11) to determine the winner and winning strategy. Each takeaway game gives rise to a new combinatorial game called the Misère version. We analyze the Misère version of Two-Pile Nim using both the Binary and Ordinal Labeling Rules. Finally, we introduce the idea of strategy stealing for takeaway games and apply this argument to give weak solutions for the games Chomp and Hex.

    Zermelo's Theorem

    Mathematically, a combinatorial game is modeled by a game tree. We recall a game tree is a rooted tree with end nodes labeled with the winning player (Definition 2.3). For takeaway games, the end-node labels are unnecessary: the player who moves to an end node on his or her turn is the winner. In Chapter 10, we used the Binary Labeling Rule (Definition 4.16) and the Principle of Induction for Trees (10.2) to prove Theorem 10.9. The idea was simply to label the game tree and then read off the winner and the winning strategy from the labels.

    Here we extend this argument to prove Zermelo's Theorem for combinatorial games (Theorem 4.17).

  • 2 - Games and Elections

  • from Part I - First Notions
  • Samuel Bruce Smith, St Joseph's University, Philadelphia
  • Book:
    Chance, Strategy, and Choice
    Published online:
    11 May 2018
    Print publication:
    29 June 2015, pp 11-22

  • Summary

    In this chapter we formulate mathematical definitions of a game and an election and give examples. The examples of games will indicate some of the categories we will study and will also serve to introduce the two main devices we will use to represent games: the game tree and the payoff matrix. For elections, we introduce the notions of true preferences and insincere voting. We consider the famous example of the 2000 presidential election in the state of Florida as an illustration of the connection between games and elections.

    We begin by thinking a bit about how to define the notion of a game. The term has broad meanings. What properties should we use to limit the scope? We can probably agree that a game should involve an act of competition with roles for skill, ingenuity, and maybe luck. Should a game be fun? Should it be fair? Fun is, of course, a matter of taste. Fairness also seems difficult to ensure. A grandmaster can play a game of Chess against a novice, but surely this this is not a “fair” game. However, allowing a weaker player to have a handicap destroys the intrinsic fairness of a game but may be the only way to create a good match.

    We define a game in a manner that is quite inclusive, making no commitment either way as to the questions of fun, fairness, or other such qualities. We focus instead on what a game produces. This is the outcome, by which we mean some particular, well-defined final status such as “win,” “loss,” or “tie.” Whatever the method or means of play, when the dust settles and the game is over, we expect to have a well-defined final result.

    Further, we often want to know the score of the game or the winnings for each player. We will refer to these as the payoffs. Precisely, the payoffs are numeric quantities assigned to each player for each particular outcome. We can think of the payoff to a player as either points or a dollar amount.

  • Chapter 2 - Reconciliation

  • Kaye Price, University of Southern Queensland
  • Book:
    Knowledge of Life
    Published online:
    07 August 2018
    Print publication:
    29 June 2015, pp 26-38

  • Summary

    I was born in 1961 in central-west New South Wales on my father’s traditional country of Wiradjuri. I would later learn in adult life that this was in the period of the Australian government’s ‘assimilation’ policy, in force from the mid-1930s to the 1960s. In 1961 our family lived on what was called the Murie Aboriginal Reserve, with several other Aboriginal families, a few miles just outside the township of Condobolin. Our lives were typical of those of many Aboriginal families across Australia at the time - thousands lived on government- or mainly church-controlled or run missions or reserves, somewhat segregated from the mainstream white or non-Aboriginal population. As a young Aboriginal child growing up in the 1960s in rural Australia at the time, I did not realise that there was a difference, ‘a black and white issue’ - us young ones were just busy growing up rich in culture, with strong family kinship, language and spiritual connection to traditional country, and not realising the struggle for human rights and social justice Aboriginal people were going through at the time.

    My father was a shearer, so our family moved regularly for his work, which saw us living on a number of Aboriginal reserves in small rural towns in country New South Wales, such as Gulargambone and Quambone in the north-west. At this time little did I know that my rich Aboriginal family and cultural story world, as I then knew it, was about to dramatically change my whole life experience and destiny, directly impacted by the government’s assimilation policy. Legislative provisions under that policy allowed the welfare authorities, with the assistance of the police, to forcibly remove Aboriginal children from their parents, families and communities, largely on the assumptions by the authorities at the time that Aboriginal children were neglected, and should be disconnected from the family, culture and traditional spiritual homelands or country for assimilation in white Anglo society.

  • Chapter 5 - Art: Connecting cultures

  • Kaye Price, University of Southern Queensland
  • Book:
    Knowledge of Life
    Published online:
    07 August 2018
    Print publication:
    29 June 2015, pp 77-97

  • Summary

    ‘I paint stories that we grew up with.’

    Ray Blacklock

    In 1995 I reconnected with my Uncle Mooka (Michael Brown) when he was released from prison after a very long sentence. A very talented artist, he had painted and taught art classes in prison; he could make a canvas out of a sheet and board: prison know-how and Aboriginal improvisation taught him to work with limited resources. Along with my dad, Ray Blacklock, he taught me about my culture: we shared stories about land, growing up in the bush and hunting for animals. They taught me colour, movement and artistic techniques. My continuing love of art was firmly cemented. Many in my family are artists, including my dad, and two of my other uncles, Nudge Blacklock and Keith Brown, who are both award-winning artists. I also paint, as do my two sons, aged eight and fi ve; none of us has any formal art training. Such is the case with the majority of Aboriginal artists today.

    I completed my Bachelor of Arts (Communication) at the University of Technology in Sydney, the fi rst of my family ever to get to university. My fi rst gig was curating the ‘Mum Shirl Tribute’ exhibition for Boomalli Gallery in Sydney, which showed art by Aboriginal and non-Aboriginal artists. An amazing Wiradjuri woman, Mum Shirl tirelessly fostered children, represented people in court, and visited prisoners and their families; she is still a legend in the Aboriginal community. Even though I never had the privilege of meeting her, I was passionate about this exhibition because Mum Shirl was a voice for the voiceless, particularly prisoners.

  • Chapter 13 - Checklists

  • Diana Whitton, University of Western Sydney
  • Book:
    Teaching and Learning Strategies
    Published online:
    14 September 2019
    Print publication:
    29 June 2015, pp 172-182

  • Summary

    A checklist is an information aid used in many different types of tasks to ensure all facets of a task are completed accurately and resources are organised.

    Academic, social/emotional and organisational skills are important in any classroom, for both the teacher and students, and using different types of checklists is one way to utilise both academic skills and organisational skills.

    Have you ever created a shopping list or a to-do list to guide you in what you must remember and achieve over a given time? Checklists are not just the shopping list or a reminder list - they are far more comprehensive, and have real value in many workplaces. They can be used to remind workers of what to do and how to do it, but they have also been shown to have a positive impact in many professions, including medicine and construction.

    This chapter explores how you can use checklists as an effective and meaningful teaching strategy in the classroom. It looks at:

  • types of checklists

  • why we use checklists

  • how to create a checklist, and

  • the advantages of checklists.

    • Types of checklists

    Three broad types of checklists may be considered relevant in the classroom:

  • organisational - to assist in the planning of an activity

  • implementation - the generation of a list of how to undertake an activity

  • review - what needs to be checked at the end of an activity, a form of review or evaluation.

  • Web Resources

  • from Part I - First Notions
  • Samuel Bruce Smith, St Joseph's University, Philadelphia
  • Book:
    Chance, Strategy, and Choice
    Published online:
    11 May 2018
    Print publication:
    29 June 2015, pp 135-136

  • Summary

    The Web has an enormous number of free sites with games, blogs, and articles relevant to our study. We mention some sites featuring future topics.

    Apps

    Games of chance including Poker, Blackjack, and Craps can be played at the site wizardofodds.com.

    We will introduce the Iterated Prisoner's Dilemma in Chapter 16. You can play the game at the website s3.boskent.com/prisoners-dilemma/fixed.html. This app implements state-of-the-art strategies for the game discovered by William Press and Freeman Dyson. There is a link to the article as well.

    Nim can be played at www.kongregate.com/games/ricardorix/nim.

    We will introduce a number of Nim-like games in Part II. You can try some of these now. The site www.cut-the-knot.org has, in particular, versions of Northcott's Game (Example 20.3) and Turning Turtles (Exercise 20.5).

    We discuss the combinatorial games Chomp and Hex in Chapter 20. These games are available online as well. Chomp can be played www.math.ucla.edu/∼tom/Games/chomp.html. Hex is available at www.mazeworks.com/hex7/.

    The site egwald.ca/operationsresearch/cooperative.php has a number of partial- conflict and zero-sum games.

    The site www.270towin.com has interactive electoral maps.

    Content

    The Stanford Encyclopedia of Philosophy at plato.stanford.edu has extensive, accessible articles on game theory and social choice theory with references.

    The organization FairVote.org is devoted to promoting fair elections. It advocates for the use of Instant Runoff Voting (the Hare Method).

    The site rangevoting.org advocates for Range Voting. The site contains helpful information about other voting methods as well.

  • Preface

  • Samuel Bruce Smith, St Joseph's University, Philadelphia
  • Book:
    Chance, Strategy, and Choice
    Published online:
    11 May 2018
    Print publication:
    29 June 2015, pp ix-xii

  • Summary

    This book was written to teach an accessible and authentic mathematics course to a general audience. Games and elections are ideal topics for such a course. The motivating examples are familiar and engaging: from the 2000 U.S. presidential election, to economic pricing wars, to gambling games such as Poker and Blackjack. The elementary theory of these topics involves manageable computations and emphasizes analytic reasoning skills. Most important, the study of games and elections has produced a wealth of beautiful theorems, many within the last century. Games and elections offer a wide selection of elegant results that can be approached without extensive technical background. A course on these topics provides a view of mathematics as a powerful and thriving modern discipline.

    There are excellent texts covering aspects of the theory of games and elections, such as those of Straffin [48] and Taylor and Pacelli [49]. The topics are also frequently covered as chapters in liberal arts mathematics texts. This text is pitched at a level between these two types of treatments. The book is designed to be more mathematically ambitious than the general textbook but to allow for a less demanding course than one offered using a more specialized text.

    A novelty of this book is the integrative approach taken to the three subjects: probability theory, game theory, and social choice theory. This approach highlights the mix of ideas occurring in seminal results on games and elections such as the Jury Theorem, the Minimax Theorem, and the Gibbard-Satterthwaite Theorem. On a practical level, the integrative approach allows for a more gradual development of material. Rather than taking a steep descent in one area, the chapters follow a spiraling path from chance to strategy to social choice and back again, exploring examples and developing techniques and intuitions in all areas while delving deeper into the theory of games and elections.

    Structure of the Book

    This book is divided into three parts. Part I introduces the main devices used in the text.

  • 17 - Takeaway Games

  • from Part II - Basic Theory
  • Samuel Bruce Smith, St Joseph's University, Philadelphia
  • Book:
    Chance, Strategy, and Choice
    Published online:
    11 May 2018
    Print publication:
    29 June 2015, pp 251-262

  • Summary

    We have seen that gambling games and zero-sum games are described by a real number, the expected value E. For a gambling game, E is the average payoff for a round of the game. For a zero-sum game, E is the expected value at the mixed-strategy equilibrium (Definition 15.4).

    In this chapter, we consider a class of strategic games described by an ordinal number, by which we mean a nonnegative integer n = 0,1,2, …. Recall that a takeaway game is a finite two-player sequential-move strategic game with perfect information for which the last player to move wins. In Chapter 10, we proved Zermelo's Theorem for this class of games (Theorem 10.9), establishing that one player has a strategy that guarantees victory. Our proof of Theorem 10.9 depended on the Binary Labeling Rule (Defintion 4.16) and the Principle of Induction for Trees. In this chapter, we introduce a refinement of the Binary Labeling Rule we call the Ordinal Labeling Rule. This new rule assigns an ordinal number called the Sprague–Grundy number to a takeaway game. Like the Binary Labeling Rule, the Ordinal Labeling Rule determines the winner and winning strategy for a takeaway game.

    We then revisit the prototypical takeaway game: Nim. With many piles and many chips, the game Nim is too large to solve directly by labeling rules. Charles Bouton produced an elegant solution to the general game in 1901 involving binary numbers. We introduce binary numbers here and use them to prove Bouton's Theorem. We apply Bouton's Theorem, in turn, to address Question 1.4 from Chapter 1.

    Sprague–Grundy Numbers

    We introduce a new labeling system for game trees of takeaway games. Given a finite set of ordinal numbers S, the number Mex (S) is defined to be the smallest ordinal not in S. The term Mex(S)stands for the minimum excluded number of S.

Page 1475 of 1931