Prospect Research Project
Research Project
Historical Aspects of Classroom Mathematics
Changing Perceptions
of
Probability Theory
through
History
June 17, 2010
Zoebaida Fiamingo & Evelien Kerkmeijer
Table of Contents
Motivation 3
Implementation 8
Students’ material 9
Teachers’ material 14
Field tests 17
Conclusions and Recommendations 18
References 19
Motivation
The target group of our project is high school HAVO 3 students. We set up a 2-hour lesson for these students about the history of probability theory. The reason for choosing this subject is that students who choose Math A in their profile seem to have difficulties with probability theory: what is it and why do we need it and why is this math?
Our main goal is that they will have a better understanding of what probability theory is and why it has been developed if they know more about the history. Furthermore we hope to achieve a ‘Hey, math is fun’ attitude towards the end of the year after a year full of mainly Math B topics. Math B choosers tend to look down on Math A because they think it is easy, and hopefully they will come to appreciate it more as well.
Manual for teacher
The game on probability theory in the next section has been developed for HAVO 3 students, who are just starting with this topic in class. An extensive study on the integration of history in classroom mathematics [1] showed that there are many benefits for both students as well as teachers. It shows that a deductively-structured mathematical theoretical process does not only define mathematical knowledge, but that for a good understanding of the theory the procedure that originally led to it is equally important. From a didactical point of view the process of actually doing mathematics plays a significant role.
In the game, the students will do exercises based on probability theory. It starts very basic, just like how it started in history. The exercises follow the chronological development of probability theory, they build up in difficulty, taking into account the students’ level of knowledge and capabilities.
In order for you as a teacher to assist the students as best as you can, it is highly recommended that you first read chapter 7 of the ICMI study [1]. It helps you to see the benefits and the goals. Furthermore reading up on the history leading to the probability theory uses nowadays enables you to understand the problems that could come up in class and how you can respond to that. If time is an issue, we have included a short overview of the history which is relevant to our game.
Historical overview
People enjoy games of chance, form very early times on. From the prehistorically period and Roman Empire animal bones have been found that served as some sort of dies. These were biased dies.
In the fourteenth century card games/decks of cards were appearing. That was also the time that mathematicians were discovering gambling-theories.
Several concepts had been developed, but it took until mid seventeenth century before a proper probability theory was developed and it was accepted as a branch of mathematics [2].
During the fifteenth century several probability works emerged and calculations of probability became more noticeable, although mathematicians in France and Italy remained quite unfamiliar with these methods [3].
Fra Luca Paccioli wrote the first printed work on probability in 1494: Summa de arithmetica, geometria, proportioni e proportionalita [3]. In 1550, Geronimo Cardano inspired by the Summa wrote a book about games of chance called Liber de Ludo Aleae (‘Book on Dice Games’) [3]. This book did not only cover dice games but card games as well. It had promising ideas on probability, but his notes remained unpublished until 1663 and then made little impact [2].
In 1654, a wealthy French nobleman called Chevalier de Méré with a passion for gambling, proposed a gaming problem to the mathematician Blaise Pascal. The problem was how to distribute the stakes in an unfinished game of chance [4]. This is also known as ‘the Problem of Points’. The problem was far from new at that time [3], but up until then a correct solution had not been given.
This problem proposed by De Méré is said to be the start of a famous correspondence between Pascal and his friend, Pierre de Fermat, also a famous mathematician. They exchanged their thoughts on mathematical principles and problems through a series of letters. Over a period of four months the two of them worked out a clear way of thinking about the odds in dice games. Historians think that the first letters written were associated with the above problem and other problems dealing with probability theory. Therefore, Pascal and Fermat are the mathematicians credited with the founding of probability theory [3].
One of the other problems was a more simple question used in the game we developed.
In 1657, the Dutch scientist Christiaan Huygens published a book ‘On Reasoning in Games of Dice’. He pulled together the ideas of Pascal and Fermat and extended the gaming theory to three and more players [2],[4]. The probability theory slowly evolved into a widely accepted branch of mathematics. One of the reasons for acceptance being the need for reliable calculations to support risks on payments (economics, life-insurances). Huygens made the link between insurance and probability theory.
Council pensionary of Holland Johan de Witt was the one to apply Huygens’ theories in insurance. This was initiated by a 53 year old aunt of De Witt who asked his advise in 1656 on what the best option to be paid was for her: 6000 Dutch guilders at once, or a yearly payment of 800 Dutch guilders.
Huygens expanded his theories applied on life insurances with the help of mortality tables derived from the works of the English merchant John Graunt (1662).
One of the best known early books on probability theory was Jakob Bernouilli’s “Aars conjectandi (published 1713). He introduced the notation of chance as a number between
0 and 1. One of his other discoveries was the binomial probability distribution.
For further reading we recommend you to take a look at the books and documents mentioned in the bibliography at the end of this paper.
Follow up on manual
In order to achieve our goal, we developed a 2-hour lesson. It is a game, during which the students will do different math exercises in groups, based on the history of probability theory. It consists of 4 parts (rounds).
In the first part, introduction to probability theory, we make sure every student is at the same level, understands what the basis is of throwing a die, and knows what ‘fair’ means in chance.
In the second round the students are introduced to Pascal and Fermat, the founders of the probability theory. It shows them how and why the probability theory was developed. In this time of fast communication they are returning to the old days, when communication was done by letters, which could take some time. By this way of communication they have to solve one of the problems posed by De Méré to Pascal. Furthermore they are introduced to Bernoulli’s discovery / definition of probability as a number between 0 and 1 followed by a related exercise.
The third round focuses on applications of probability theory. First by doing an exercise on basic life-insurance, using historical data, followed by discussing with the whole group suitable applications (e.g. weather forecasts).
During the last round, the students have to come up with their own game of chance with a deck of cards. The teacher chooses 2 or more (depending on how much time is left) to play with the whole class and gives prizes to the winners, or the groups can exchange their games and try to solve it with their own group.
We also described in detail how our work fits into the framework of the ICMI study. This will show you how and why the games’ ingredients will lead to a better understanding of the subject.
Relation to ICMI study
The ICMI study focuses on the benefits of integrating the history of mathematics in mathematics education. Section 7.2 of this study deals with reasons why the history of mathematics may be relevant to the educational process. It is divided into five main areas.
As to how our game is related to this section can be explained like this. Instead of providing the students with ready to use approach to solve the problems of chance it is better to let them discover on their own what is chance. And from there try to develop their own theory on chance: how to calculate, compare and express probability. This is done by using the history. First struggling with what is chance and then trying to make it their own by rediscovering the problems posed to the great mathematicians of history and the concepts and structures that followed from that. In short, work on the basis of understanding rather than just applying rules to solve the problem.
The history serves as a resource. It provides relevant questions and problems that will help motivate and interest students to this topic. They’ll come to understand why it is necessary to have a probability theory. It makes them aware of the wide spread applications of this theory, not only in mathematics but, most importantly, in everyday life. And that it is therefore necessary that research, development and optimization are still being done and the theory is still evolving
As to the didactical background of teachers, this basic reconstruction of the historical development of probability theory helps them come aware of the difficulties they may encounter in the classroom (what is chance, how to compare or notate). Also to what extent should you go into the theory for students of this age. It helps the teachers to get involved into the creative process of doing mathematics, rather than just providing and listening, a one way process. This also tackles the different types of learning that work best for the different types of students.
The teacher is also in the position to show the students that it is not wrong to make mistakes, but that mistake helps you in getting a better understanding of your problem and can lead to beautiful theories…isn’t that how some of the most beautiful mathematics came into being?!
It is good to see that their isn’t usually one good approach to a problem, but a variety of ways.
Section 7.3 of the study deals with how the history of mathematics may be integrated in mathematics education. This is characterized by:
a) Learning history by provision of direct historical information.
b) Leaning mathematical topics by following a teaching and learning approach inspired by history.
c) Developing deeper awareness, both of mathematics itself and of the social and cultural contexts in which mathematics has been done.
We aimed that the students both learned history as well as mathematical topics.
Point a) we have covered by using examples from history, with dates/names/works/data/famous problems and questions. Furthermore there is the short summary of evolution of probability theory through history in the next part of this section addressed to the teachers. With this summary we hope to enrich the knowledge of the teachers concerning this topic.
Point b) is matches completely. The topics/problems we handle in the game are in chronological order. We took crucial steps for the probability theory from history, in a way that it became didactically appropriate for classroom use. The problems have increasing level of difficulty, each one building on the knowledge and experience from the previous one. We specifically looked at the level of mathematics of our target group. And that the teachers who will use this don’t have to be historians in order to grasp the historical evolution.
By providing the historical overview the teacher is hopefully encouraged and motivated to add any (historical) examples, problems or theory they think is relevant. In this way it is possible to teach the subject according to the specific needs of the class.
Point c): are aspects related to intrinsic and extrinsic nature of mathematical activity included?
The intrinsic aspects are included by letting the students discover the evolving nature of probability theory. For example by how to notate and what the terminology is. Also by showing the difficulties and intuitions used in order to produce a new part/domain of mathematics.
With showing the way probability theory was developed and all the examples we put into the game we also included extrinsic aspects. It shows that this subject of mathematics has relations to other sciences and problems in everyday life. That social and cultural milieu stood at the basis of chance and had a big influence it it’s development. The history is used as primary source material as well as secondary and tertiary as stated on page 212 of the ICMI study.
Section 7.4 of the ICMI study deals with ideas and examples for classroom implementation. The way our game matches the possible ways mentioned in this section 7.4 is mainly based on point 4: worksheets. It is meant for students to work in groups and contains a collection of exercises in order to master a topic which was introduced briefly in classroom. The students solve and tackle the problem by discussing with each other interacting with the teacher. For the teacher it is a fun way to communicate with the students, let the questions and progress be dictated by the students themselves.