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FEATURES OF THIS CASE STUDY
Engaging a Wide Range of Students
Engaging a Wide Range of Students.
This is an engaging problem for pupils interested in sport. It also involves the notion of resolving ‘once and for all’ a controversy which has been raging for up to 40 years, and pupils are usually intrigued by the possibility of solving a problem or mystery which has not been achieved by others. A strong feature of this Case Study is its accessibility. Every pupil is able to put forward some sort of theory - some will judge the batsman ‘out’, others will disagree. These theories will be refined to varying degrees as pupils get further into the mathematics. The discussions that ensue as pupils justify their positions add to the richness of the task, especially if different models suggest different answers. The mathematics is accessible for those with lower levels of ability, yet challenging for more skilled and confident pupils. The pupils are taken through a sequence of activities that allow the mathematics to emerge gradually. The problem posed is open with no ‘correct’ solution, which allows pupils the freedom to investigate possibilities without being concerned that they may be ‘wrong’.
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Exploring Issues Associated with Mathematical Modelling
Exploring Issues Associated with Mathematical Modelling
Another key feature of this Case Study is the way in which the interconnections between the concepts of “estimate”, “measurement”, “error”, “assumption” and “conclusion” emerge and, indeed, are explicitly discussed. Pupils will have to make some assumptions in order to make their initial decision. Then, as the study progresses, some of those assumptions are quantified and then explored to see how they affect the likelihood of the batsman being judged either in or out, within reasonable limits. In this Case the relationships between the variables and outcomes are explored qualitatively before they are explored quantitatively. Research has shown that this reduces ‘cognitive overload’ and allows pupils to focus on the core thinking without being distracted by numbers, units and formulae.
You Have a Lot of Flexibility in the Detail of How you Might Run this Case
You Have a Lot of Flexibility in the Detail of How you Might Run this Case
This outline provides one suggested way that pupils could tackle the problem, but the Case Study provides flexibility for both the pupil and the teacher. It caters for those pupils who, no doubt, will think of 'other ways'. Indeed, a motivating factor for such pupils will be the opportunity to come up with a solution method of their own design. All aspects of the task allow flexibility: in the manner in which the teacher translates the concepts into actual classroom activities; in the balance the teacher strikes between control of the activity and pupil decision making; in deciding whether tasks should be done individually, in groups or as a whole class; in deciding the level of difficulty appropriate for the class in question; in allowing the pupils to explore ideas of interest in more detail; and in determining how the pupils’ work on the tasks should be assessed. Whichever approach is adopted, the nature of this problem is that it encourages pupils to explore a variety of creative solutions within groups and then to contrast them within the class.
What is the Maths in this Task?
What is the Maths in this Task?
The task is rich with mathematical possibilities and has strong links to the KS3 curriculum. As the lessons progress, the need to use and link together different mathematical skills will become apparent to pupils. Throughout the Case, important mathematics lies in Thinking how to make the critical decision of whether the batsman is in or not. This will necessitate pupils making decisions about what variables are important (such as the elapsed time since the bail was removed) and how they can collect the information they need. Subsequently they use Algebra to explore the variables in relevant formulae. Number and Measure are involved when pupils make estimates and then direct and indirect measurements from the photograph. Calculations of distances and times are compared with estimates, and conclusions justified. Another aspect of the mathematics that add to the value of the task is that pupils work with small decimal numbers (which tend not to feature in text book mathematics) and utilise skills in Handling Data when collecting the primary data (when determining the likely speed of a fully kitted-out test batsman). There is an important role for ICT in the development of spreadsheets to generate results and thus extend the mathematical analysis.
The maths in each lesson is summarized in the following table
MATHEMATICAL SKILLS | THINKING SKILLS |
Lesson 1: Sorting and classifying Researching data
| Lesson 1: Thinking, pairing and sharing Decision making Justification of responses. Information processing skills – using ICT or other resources Creative thinking :The thinking tool “Consider All Factors”
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Lesson 2 Estimation Considering factors in the design of a mathematical model The concepts of “variable” and “constant” Direct measurement : distance and units Indirect measurement : scale and ratio Decimal place value
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Lesson 3: Collecting primary measurement data Interpretation of data through discussion Calculation using scale factor Speed-distance-time relationship Substitution and evaluation using a formula
| Lesson 3: Enquiry skills: selection of factors suitable for investigation; breaking a complex problem into simpler steps Solving problems: design of data collection; interpretation and discussion of data Compare and contrast of estimates and findings in order to make a reasoned judgement Identifying assumptions Justifying conclusions, including any assumptions
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Lesson 4 | Lesson 4: |
Lesson 5: | Lesson 5: |
GENERAL ADVICE TO TEACHERS ON ADAPTING THIS CASE STUDY
The following suggestions are aimed at ensuring that the task can be approached confidently by teachers from all levels of experience and with differing mathematical skill levels:
- learn with the pupils – work the task in preparation for the lessons. Measure the relevant parts of the photo and form your own opinion. Create spreadsheets using Excel, and explore the assumptions yourself.
- allow the pupils the amount of freedom you are comfortable with. Discussions can be guided in a directed and ordered manner or in a less structured way with pupils exploring areas of interest on their own
- keep the goals of the Case Study (making a decision made on assumptions within reasonable limits) and the final task (justifying the decision-making process) in mind at all times, remind the pupils frequently to ensure that individual lessons have a focus on the task and the entire Case study has a clear and meaningful purpose.
- at all points where mathematical skills are required, quietly raise the pupils’ awareness of the fact that this is happening, but do not introduce the skill before the need is apparent and clear.
Prior Learning
Students should be familiar with the following concepts:
- Measurement of distance and time using simple instruments such as rulers and stop-watches;
- Scale and how it is used to convert a distance on a picture to a real distance. Pupils may have met this concept in their study of Geography;
- Small numbers, such as 0.1 and 0.23, being able to rank them from smallest to largest and their understanding of place value in this work;
- The substitution of numbers for pronumerals into a mathematical formula in order to calculate a result. Pupils may have met this work in formal algebra, or in their study of measurement (formulae for area and volume);
- The use of spreadsheets, especially in the way a mathematical formula is used to calculate a set of numbers in a column efficiently;
- The difference between a CONSTANT, a quantity whose value is FIXED such as the length of a cricket pitch, and a VARIABLE, a quantity whose value can CHANGE, such as the speed of a batsman.
ASSESSMENT
Assessment of pupils’ work should be based on their performance in the tasks themselves. It is suggested that criteria be established before the task begins. Four suggested criteria could be, accuracy (of measurements, calculations, etc); work habits (cooperation, responsibility for own tasks, etc); effectiveness of presentation (of graphs, summary charts, etc); and selection of appropriate strategies and mathematical and ICT skills for the analysis.
It is also strongly suggested that each student use a LOG BOOK in which to record all answers to questions, reflections, calculations and conclusions. This will assist teachers in their assessment of student work throughout this unit.