How does You Reckon? fit with the 2008 Programme of Study?
Appendix A shows the 2008 PoS; statements in bold are ones which the Plausible Estimation module addresses.
Who are PE tasks for?
PE tasks can be used with pupils (and adults) at any almost any age and stage.
Tasks should be chosen to suit the sophistication of the pupil.
Should calculators be used?
YES if you want students to use realistic, large numbers, and hand calculations would obscure the point of the task.
NO if students engage in busy work - notably aiming for spurious precision [the antithesis of what we are trying to teach!].
How much information should I include in the written prompt, or give out in class?
AS MUCH AS YOU HAVE - if this is an early task in your Plausible Estimation classes, and you want to emphasise key points about breaking tasks up, and making reasonable estimates.
NONE AT ALL - if you want students to do a full PE task, gathering all the information they need for themselves.
What should I do if pupils finish early?
Ask them to solve the problem in a different way.
Ask for a bounded estimate to check the sensitivity of their answer to the assumptions they have made.
Suggest they write a clear explanation of their answer.
Change some aspect of the task, and ask for a reworking.
What homework can I set?
Whole tasks can be given for homework.
Preparatory activities - notably searching the web for information - are appropriate, and are probably easier to do at home.
If the task can be solved easily using a spreadsheet, this might be easier to do at home, or in the library, than at school.
Can I change You Reckon? tasks?
Every You Reckon? task is a member of a family of tasks. After you have used some You Reckon? tasks in the way we suggest, you will see ways to change other tasks so as to match the needs of your class. Always think through the mathematical demands on pupils of the task variants. Here are some members of the Stop! Thief! Family.
A bank manager says that an armed youth stole a bag containing £5000 in £1 coins and ran away. Estimate the weight of the coins.
A bank manager says that an armed youth stole a bag containing £5000 in £1 coins and ran away. Could a thief run, carrying this bag?
The second task differs from the first in a number of important ways.
- pupils have to decide what they need to find out - they need to identify weight as one critical issue;
- there are more factors to consider in the second task - the dimensions of the bag should be considered - how big would it be? would it be unwieldy? Pupils also need to discover what weight they can run with;
- there are more stages (problem identification, calculations involving coins and volumes, estimates of what they can carry) in the second task.
Here is a third variant
A bank manager says that an armed youth stole a bag containing £5000 in £1 coins and ran away.
Write a report for the insurance company! Tell them if they should pay up, or prosecute the bank manager for attempted fraud. Explain your reasoning.
The third task requires the same mathematics as the second problem, but now pupils have to communicate their findings very clearly, with a sense of audience.
How do I see progression in pupil attainment?
PE tasks can be created that differ along a number of dimensions. We can see pupil progress in handling tasks that increase in sophistication on any of these dimensions.
Task openness: does the task prompt make it clear exactly what has to be done, or is planning how to do the task part of the challenge? Pupils will be able to tackle more open problems as they progress.
Exploration phase: how much effort is required to find important information? Telling pupils that a £1 coin weighs 9.5g is different to asking them to search the web or giving them coins and kitchen scales.
Abstractness of the activity: estimating the number of coins that can fit into a brief case, or the number of people who can stand in a 1 metre square, can be done via practical activities and with concrete materials to help the process, or without.
Number of stages: how many links are there in the chain of reasoning?
Number of factors: how many variables need to be taken into account?
Reasoning length: how long do pupils spend on task components? A task that takes 30 minutes, but comprises 20 explicit activities, has a much shorter reasoning length than a 10 minute 'open' task.
Technical load: tasks that benefit from powerful mathematical notation, or require careful conversion of units (e.g. US gallons to tonnes of biofuel) are harder than tasks that do not.
Sophistication in rounding: we want pupils to be able to do 'back of an envelope' calculations, so simplifying the numbers used (in the light of the accuracy required) to answer the problem is an important skill. Setting pi to 3, or rounding 1024 to 1000 in problems involving powers of 2 will often be appropriate.
Sensitivity analysis: an answer that talks specifically about the key quantities to be estimated, and the likely accuracy of the estimates made (9.5g for a £1 coin is a very accurate estimate; '4 people can stand in a metre square' has a large amount of uncertainty) is better than one that does not.
Accuracy of the answer given: spurious accuracy should be avoided (e.g. the CIA World Factbook gives the population of China as 1,321,851,888 (July 2007 est.).
Here is a short hierarchy of responses to the challenge 'how many times will your heart beat, before your 14th birthday?', from poor to good.
- Spurious accuracy - 'an estimated average rate of 72 beats per minute gives 72 x 60 x 24 x 365.25 x 14 = 530,167,680
- Answer rounded - the degree of rounding taking account of the likely accuracy of the numbers used in calculations, and is expressed in easily understood form; 'the number of heart beats will be around 500 million';
- Bounded estimates - an upper and lower bound for an estimate, based on low and high estimates of the numbers used in calculations, 'the number of heart beats lies between 400 and 700 million - corresponding to rates of 55 to 95 beats per minute'.
Knowing how confident you can be about the answer: in the Olympic Opening Ceremony task, students are asked if all the athletes can stand in the centre of the running track. The answer is very clear, and even if pi is set to 3, or the track is simply measured as a rectangle, the answer will be the same. If a PE estimate produces a result that is close to some critical value ('maybe they can just fit') then all rounding has to be examined very carefully.
Multiple methods: one can have more confidence in an answer that uses 2 or more different solution methods that produce very similar results.
Reporting phase: a simple written response requires different skills to a class presentation or a report to a defined audience for some purpose.
Quality of the explanation: solutions to PE tasks always need to be explained!