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The challenge of capturing hearts and minds when considering the future or when engaging in statistical analysis is best tackled through dialogue so that pupils can consider and articulate their thinking and teachers can tap in to their reasoning. With this in mind, perhaps a better start point might be:
'In this area it may rain tomorrow or it may not, and yet the probability that it will rain tomorrow is generally not 50% - discuss and explain your thinking with evidence.'
The expectation that pupils support the value they choose for a probability with evidence gives them the opportunity to reveal their understanding. This kind of reasoning with evidence needs to be modelled through critical dialogue between the teacher and the pupils.
One way to help pupils develop the skills of reasoning and explanation is to work more frequently on the mental aspects of handling data including probability. Explicit links can be made between the handling data cycle and the way we work with probability. There are useful parallels with the cycle both in examples where we use an experiment to find the estimate for a probability or where we solve a problem using theoretical probabilities. In addition teachers need to plan for discussions which compare theoretical and experimental methods: their appropriateness, drawbacks and advantages in particular circumstances.
It is clear that skill in using fractions, decimals and percentages as part of a probability calculation need to be considered as a precursor to tackling probability problems. In the main, however, progress in probability depends largely on understanding ideas, rather than acquiring further skills. Finally, as probability is an evaluation of what might happen in future, it is important to carefully choose language so that the event described is placed in the future. For example, 'What is the probability that I will score 7 on two dice?' makes more sense to pupils than, 'What is the probability that I scored 7 when I rolled two dice?'
Representing: If pupils can represent data as part of a statistical enquiry then they are better positioned to become responsible citizens who can select and sift information thoughtfully and use mathematics with confidence to inform decision-making. Representation is a major focus of Probability, important in tying together the decisions pupils make at the different stages .
In a statistical enquiry, representing is part of almost all elements of the handling data cycle. It involves:
Suggesting a problem to consider using Probability methods, framing questions and raising conjectures
Deciding what data are relevant and identifying primary or secondary sources
Designing ways of capturing the required data, including minimising sources of bias
Creating representations of the data, including the use of ICT, for example, tabulation, grouping, arrays, diagrams and graphs.
Mathematical reasoning is required at all stages of finding the probability of an event
When specifying and planning by working logically, identifying constraints and considering available techniques; also by exploring conjectures and using knowledge of related problems
When collecting data by working systematically, exploring the effects of varying values in situations where there is random or systematic variation
when processing and representing data, making connections within mathematics and identifying patterns and relationships, and making use of feedback from different audiences
when interpreting and discussing results, explaining and justifying inferences drawn from the data, recognising the limitations of any constraints or assumptions made; using feedback to reassess initial conjectures and adjust aspects of the handling data cycle.
Using appropriate procedures involves manipulating data into suitable forms for accurate representation, calculation and communication. This will involve monitoring the accuracy of methods and solutions.
Appropriate procedures in a Probability enquiry are:
using systematic methods for collecting data from primary and secondary sources.
To construct table, diagrams , etc to present data in an organised form.
Calculating experimental and theoretical probabilities.
Interpreting and evaluating: Interpreting and evaluating results is fundamental to any statistical and probability enquiry. It includes:
interpreting probabilities when assessing the likelihood of a particular outcome
comparing distributions and making inferences
looking at data to find patterns and exceptions
considering the effects of changes to the data (e.g. removing outliers, adding items, making proportional changes)
appreciating why the interpretations placed on data have a degree of uncertainty and can be misleading
Appreciating convincing arguments, but knowing that these do not constitute proof.
Communicating and reflecting: Effective communication and reflection is of particular relevance in statistics. It includes:
preparing a brief report of a Probability enquiry, using tables, tree diagrams ,etc to summarise data and support interpretations and inferences drawn from the data
using precise language to summarise key features pertinent to the conjectures raised
presenting support for conclusions in a range of convincing forms
presenting a balanced conclusion where results are not convincing
Considering alternative approaches if results do not provide sufficient evidence
Range and content:
All my four chapters begin with use of an empty number line, and develop the concept of placing events on a scale along this line. I would like to note that there is a completely separate, but no less interesting, story surrounding the advantages of using a number line to help pupils form fundamental understanding of scale and an idea of place. In years 7 and 8 there is an emphasis on the language of probability (as mentioned earlier), and obviously there is a differing level of complexity to the questions covered in each book, but essentially we see a continuing metaphor and consistent type of question. The basic paradigms of picking cards from a pack and rolling dice are used in all three books, and we see a gradual shift towards problems with more than one variable. Until year 9 there is emphasis on the fact that a probability scale runs from 0 to 1, and work continues on manipulation of simple fractions, whilst the year 9 book assumes such knowledge and moves into considering relative frequency thus relating back to observable statistics. This seems vital to me, as we need to encourage the questioning of, and attempting to understand, results, and I would maybe have liked it to have been included earlier. Finally, in years 8 and 9 pupils are expected to make use of sample space diagrams, thus providing another graphical depiction of the probabilities of given outcomes.
Give pupils a selection of statements on cards and ask them to sequence on a probability continuum such as this
Sequencing events according to their probability can reinforce the usefulness of the probability line as well as stimulating discussion about the relative chance of different events.
The probability of getting at least one six when two dice are thrown
The probability of getting a multiple of 3 when one dice is thrown
The probability of getting a tail and two heads when three coins are flipped
Impossible Unlikely Likely Certain
The task gives practice in assessing an awareness of the outcomes which are possible in each context. Pupils may choose to calculate or may wish to illustrate some of the outcomes. Either will help to justify their ranking of the events relative to one another. We are sometimes expected to appreciate the chance of one event relative to the chance of another, quite different event, for example, 'You are more likely to die crossing the road thanâ€¦'
Matching Linking different circumstances to a given probability is an activity based around the number and colour of otherwise identical counters in a bag. This engages pupils in working out the possible number and range of colours of counters in a bag given a certain probability such as those shown below. Initially the work is in pairs moving to larger groups to share thinking.
P(Red) = Â½
P(Red) = 1/2 and P(Blue) = Â½
P(Red) = 1/2 and P(Blue) = 1/4
P(Blue) = P(Green)
P(Blue) = P(Red) and P(Green) = 1/2
P(Red or Green) = 2/5
P(Yellow) = 1/2 and there are 6 red counters
P(Red) = 3/7 and P(Green) = 1/3
P(Green) = 1/4 and there are at least 8 yellow counters
Together pupils should seek to find as many ways as they can of responding to the task, discussing results as a whole class with pupils taking on a critical role to discern similarities and differences between the solutions and to deduce the important features of the counters in the bag in order to satisfy the given probability. In other words, the joint thinking gives them the opportunity to generalise the solutions.
To simplify the task, the number of possible colours could be limited. To extend it, consider giving the probability of an event not occurring, for example P(not Red) = Â½
P(Pink) = 1/5 and there are 4 different colours
Which chair: trees to grouping branches.
This is a simple scenario which produces some unexpected results and so promotes further thinking about calculating combinations of outcomes.
One pupil sits on the middle chair of a row of seven:
an unbiased coin is flipped
a head means move one chair to the left
a tail means move one chair to the right.
Repeat the process twice more.
Pupils work in pairs to answer the question:
How many of the chairs is it possible to finish on after the three flips of the coin?
A 'tree diagram' could be used to build on the movement and visualisation to identify all possible sets of movement. It is interesting to discuss with pupils how the two forms of diagram both illustrate different aspects of the problem; see Resource sheet: Which chair? on page 67.
The ability to find and record all possible outcomes for successive events or a combination of two or more experiments is essential if pupils are to understand, find and use probabilities or estimates for probabilities in more complex situations
Using a probability fact
Two bags A and B contain identical coloured cubes. Each bag has the same number of cubes in it. An experiment consists of taking one cube from the bag.
The probability of taking a red cube from bag A is 0.5.
The probability of taking a red cube from bag B is 0.2.
All the cubes are put in an empty new bag.
What is the probability of taking a red cube out of the new bag?
Pupils should individually write down a 'gut' response and then compare their answers in small groups. The use of specific examples to answer the above will be useful but pupils need to share these and be encouraged to generalise.
What happens if the probability of picking a red cube is the same for both bags?
What happens if you change the probability of picking a red cube from each bag?
What happens if you change the number of red cubes in one bag? In both bags?
All stages of this problem demand that pupils identify the facts surrounding a situation. It has the potential to reveal misconceptions around probabilities of related events and offers the opportunity to generalise an outcome where the intuitive response is often incorrect.
Personal Learning and Thinking Skills (PLTS):
The Leading in learning programme has been developed as part of the National Strategies Secondary support for whole-school improvement. My scheme of work is deliberately structured so that pupils look beyond subject confines to thinking and learning more generally. There is a focus on specific thinking abilities and to encourage systematic development of thinking skills and transfer of learning across subjects and to other aspects of pupils' lives.
A fundamental understanding of probability makes it likely to understand everything from bowling averages in cricket to the weather report or your chances of being affected by snow! Probability is a significant area in mathematics because the probability of Particular events happening or not happening can be vital to us in the actual world.
Today the Probability theory used to make intelligent decisions in economics, Management, Operation Research, Sociology, Psychology, Astronomy, Physics, Engineering, and Genetics where risks and uncertainty are involved to draw a conclusion about the likelihood of events or values.
Here are given some examples of probability:-
What are the chances that England Cricket team will win the series?Â Â
What is the Probability that it will rain tomorrow?
What is the probability about stability in Gas prices in next month?
Planning for inclusion: Show how your scheme of work plans for inclusion
With Increased attention being paid to the results of national test and external examination statistics being published to assess the performance of schools, the potential value of assessment for pupil is often overlooked. All too often assessment is seen as an impersonal, formal process which is done to pupils. Their progress is measured, attributed a grade or score, and this is then reported to others the assessment process appears to have little value for the pupils themselves. However, if assessment is to enhance learning then its formative purposes must be emphasized. The pupils need to appreciate how the assessment may contribute to their learning and become involved in acting on the information which the assessment has provided.
My main concern in assessing my pupils' learning was the progress of my efficacy in teaching the topic. My assessment, therefore, needed to be effective and consistent with the expectations of pupil learning. Therefore i have chosen formative assessment as this would improve children's learning .
"The unique feature of formative assessment is that the assessment information is used by both teacher and pupils to amend their work in order to make it more efficient. There is little point in collecting information unless it can be acted upon, and since assessment information is sure to reveal heterogeneity in the learning needs of a class, the action needed must include some form of differentiated teaching."
(Professor Paul Black, 1995)
My formative assessment of my pupils' progress would include:
Assessment of descriptions and explanations given by pupils in both oral and written work. The medium for this assessment would include mental maths, questions in class, class exercises, homework and Plenary. Homework was set every Friday and collected in on Monday. As Tanner and Jones mention "Teachers assessment of students work is essentially an ongoing and informal activity consisting of asking questions, observing activities or evaluating progress. For such assessment to be formative there must be feedback into the learning process." Thus all the homework books were marked and given feedback on:
A grade, according to schools homework marking policy
A general comment(e.g. 'untidy work')
An instruction (e.g., 'show your workings')
A specific targets which indicates what needs to be done next in order to improve (e.g., 'revise your 8x table' ;)
Correction of errors (e.g., in calculation,spelling,method)
2. Assessments of individual's performance in pair/group work or whole class activities or discussion. This would be assessed according to:
a) Shared communication which reflects pupil's confidence with probability
b) Understanding of the problem which reflects on the level of the work (using traffic light signal)
c) Working on task - which may be subjective by the aptness of my activities
d) Communication - using language of probability
e) Attitudes - which may be influenced by the context of the problem
The framework for my formative assessment was based on assessment strategies adopted by the APU. I had considered only those strategies which I thought would transfer easily into the classroom for diagnosing or evaluating the achievement of individual pupils.