My interest in this topic comes from somewhat lowbrow sources, as my friends were very good in card games and fairly adept at gambling (only amongst friends of course, and strictly no money changed handsâ€¦). They were always able to quite quickly evaluate whether their chances of winning was suitably large or not, and I always used to wonder how they make these decisions. I learnt more about probability in the school than college, thus enabling myself to learn more theoretically, but didn't learn to apply in real life. As Maier suggests that, individuals fail to utilise school learned procedures because they are not encouraged to relate school experiences to life outside school. "School children recognise that school mathematics is not a part of the world outside school, the world most important to most people". By learning Probability it helps pupil to make sense of the real world in situations involving risk, chance and uncertainty, thus it is one of my preferred topic to research and teach.
In today's world it is not hard to think that life around us is surrounded by games of chance and one have to be educated to predict the chance of winning and possibility of various outcomes."Learning about mathematical concept precedes the ability to apply these concepts". People who lack an accurate sense of probability are easily drawn in by false claims and pseudosciences are vulnerable to get rich-quick scheme. People expose themselves to dangers of taking risk at some point of their life, but they need to be able to estimate the probability of success in order to assess whether the risk is sensible. Thus understanding of Probability theory enables us to think about these decisions and also help us to take control of them. The national lottery is a perfect example of use of probability for estimating their success and allowing people to think that they would win one day. In professional life, more people use statistics and probability than any other branch of mathematics. Insurance company and Banks (lend money) have business plan that is entirely based on their understanding of statistical probabilities, assessing that their risk level is very low.
"Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities."
Today in this multi cultural society, topic such as probability need to be approached and addressed with care and consideration as these topics have links to many ethical and philosophical issues. Gambling as an aspect of topic itself is still forbidden for much people. In addition to this discussion of probability as a chance of outcomes comes directly in to conflicts with most religions and philosophies, which lecture to determinism and fate. Thus there is still a considerable debate as to whether probability should be taught and therefore the ways in which it is taught has to be discussed sensibly.
Finally ,the further motivation for teaching probability In this consumer-lead society, is important to develop people who are aware of the factors surrounding decision-making and will not be too easily swayed by attempts to persuade them to part with their hard-earned money. The concept of an informed consumer is crucial to the teaching of probability as we can help mould people who will not be taken advantage of too easily. One perspective of probability is that it addresses "evaluations of situations which are inherent in the subject's mind" (Borovcnik et al , 1991) and this has clear connotations for all of us in our everyday lives.
"The essence of probability lies in measuring and dealing with uncertainty. An understanding of this fundamental point can only come about by initially being involved in making judgements, whilst at the same time paying attention to the sources of uncertainty in decision making."
(Peter Gates, 1989)
Consequently, in preparing to teach probability, I had to explore ways of talking about uncertainty. Usually statements involving uncertainty are based on past experience and, as such, can often be subjective. These words and phrases may be called the language patterns of uncertainty or indeed probability.
In teaching probability to this class I hoped to put a clear emphasis on practical numeracy which I feel is particularly important for their progress in life. I would also like to emphasize the connections between branches of mathematics within the National Curriculum. As probability has strong links exist with AT2 Number, a course could aid the students' understanding of fractions, decimals and place value. Understanding and using relationships between numbers and developing methods of computation could also be enhanced by techniques developed for the calculation of probabilities.
Turning back to the idea of how these ideas should be taught, there is a considerable amount of experiment and investigation included in the texts, and this is always followed up with theoretical work of a similar nature. This brings me back to Crawford's paper on a particular scheme of work where he highlights the problematic aspect of experiments; they cause excitement. As he puts it;
"There was a great deal of activity and apparent enjoyment but little evidence of thought â€¦".
An Investigation of the Law of Thought
Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.
So, we need lessons which encourage participation and thought without sending pupils into a probability-fuelled frenzy. I feel that the Key Maths books make a pretty fair stab at this, and from my observation they work well when not the sole resource. Due to the nature of the topic, a balance needs to be struck between theory and practice, and this can be achieved in part through discussion and analysis of games. Games are often used because, as noted in the NCETM;
"Psychologically and socially, games are easily separated from real-life activity, so they provide a neutral, nonthreatening context in which fairness can be examined".
Again I should mention the fact that we must consider the issue of equal opportunities with regards to games. Different cultures have produced different games, and the use of a game unknown to some pupils in a class can be both problematic and rewarding. If knowledge of a game is assumed, pupils unfamiliar with it will feel alienated. Conversely, an unfamiliar game, if correctly explained and modelled, can open up new areas of cultural interest for pupils. Clearly, awareness of the issues within a class and school are crucial here, and it is with just such matters in mind that Shan & Bailey (1991) have devised probability activities which directly address ethnic and cultural issues to ensure there is no over-emphasis on Western viewpoints.
To further develop my subject knowledge in Probability, I have chosen to conduct the curriculum project with year 9; middle ability group (set 5) students. The Key Stage 3 SATs result showed that all of these students were able to achieve Level 5 with the exception of few who achieved a Level 4.
This year 9 class had four 50 minutes duration mathematics lessons per week, and I had responsibility for two of these lessons. Initially I was patronised working with this class because, this set was more challenging in both behaviour and learning. Most of the pupils from this class would be taking their GCSE -Foundation Level. The School's records indicated that 62.2% of the candidates who took GCSE achieved A* to C grades.
The math Teacher Mr X was head of math and his lesson is always functional and well controlled, and I was certain that I will not receive the same respect from the pupils. Before Christmas break my teaching experience with this class was centred mainly on issues of discipline. Most of the pupils learning were frequently disrupted by inappropriate classroom behaviour and absenteeism. They were also very difficult to encourage and motivate. Positive methods of discipline, detention and use of the School's behaviour policy and referral system gradually improved my relationship with the class and also made them to understand that I would always keep up to my word. This success strengthened my determination to improve the learning environment and introduce a sense of mathematical achievement within this class.
The Government, through the National Curriculum, has expressed their belief that probability is of great importance within the subject of mathematics. Understanding and using probability appears in Key Stage 2 and features strongly thereafter. Most of the School offer statistics at GCSE level and this trend reflects the importance of statistics to modern consumers in a society where the ability to make decisions, interpret data and communicate information is vital.
I planned to include this framework within my teaching methodology with careful planning. I intended to sharpen my skills particularly in relation to setting homework, marking and work with individual pupil mistake and misconception. As Boaler highlights "innovative ways teachers engage their students in learning and move math from a 'drill and kill' experience to one where students become mathematicians, not just rote memorizers "Thus in addition to this framework, I intended to teach the use of experimental activities with probability in lessons;
Bright and Hoeffner, 1993 states "Solutions to probability problems often seen counter intuitive even for teachers''. I, therefore, saw experimental activities playing an important role in linking the theoretical and empirical approaches to probability
Each of the pupils in my class had been issued with both a New National Framework Text Book and homework Book. I intended to use them as resources supplementing them with worksheets and other teaching resources drawn from internet.