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.

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.

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.

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.

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.

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.

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,

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:

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.

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.

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).

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).