Discuss Various Theories of Management Essay
Portraits of the Case Studies The Case Study problems are teaching units, each of which supports 3 to 5 lessons on a topic from outside mathematics. They develop thinking, reasoning and problem solving skills and put substance into the Key Concepts and Processes of the new Programme of Study (PoS) for Key Stage 3 – aspects of the National Curriculum that are less familiar to many teachers. Between them, the Case Studies cover most of the PoS. The Case Studies are very different from each other and have been developed by a wide range of developers.
Each one is rich in mathematical possibilities and provides pupils and their teachers with interesting and exciting problems on topics that pupils find fun and engaging. The topics range from the real world to pure fantasy. The problems make extensive use of ‘open’ questions, to which there is often no one right answer. most require a number of steps and can be explored at various levels of depth by pupils of differing abilities. Each Case Study contains materials for pupils to use in the classroom as well as teachers’ notes and lesson plans.
Most of the Case Studies, but not all of them, require access to some form of ICT. Each Case Study took about 18 months to develop. In addition to fieldwork tests by its developer, each one was subjected to ‘action’ tests in two ‘cold trial’ schools by independent evaluators; the feedback from the trials was used for further development. Over 100 schools were involved in the development of the Case Studies. Choosing a case study The case studies are very diverse – over time it would be good to try them all.
To help teachers decide where to start this document provides the following summary information: Examples of mathematical activity provide a few lines that sketch the problem that each case study presents and the mathematics that it involves. Comparison charts summarise the case studies from various perspectives: • Type of problem: The case studies cover six broad types of problem – planning and organizing; designing and making; modelling and explaining; exploring and discovering relationships; interpreting and explaining; solving logic puzzles. A case study may have more than one of these aspects. Time required in terms of the number of (typically) 1 hour lessons suggested • Suitability for age and ability groups • Resources needed: particularly computers and interactive whiteboards – a few case studies require other resources • Links to the Key Stage 3 Programme of Study: This table indicates the potential contribution of each case study to developing: key concepts; key processes; content areas; curriculum opportunities, that comprise the KS3 Programme of Study Outline descriptions provide a short explanation of what each case study is about.
This document serves as a quick reference and comparison guide, but is no substitute for exploring the case studies themselves. Each case study itself includes a more extensive description of its content, pedagogical aims and practicalities. These descriptions can be accessed from the Case Studies section of the Bowland Player on the DVD or website. r4 © 2008 Bowland Charitable Trust 1 of 9 Examples of mathematical activities Alien invasion Locate spaceships using clues to estimate and calculate distances and directions. Interpret graphs and maps to plan an escape; crack a code to escape from a cell.
General problem solving. Control variables systematically (e. g. speeds, design of cars, barrier types). Make hypotheses and test them by observing the effects in crash test experiments. Present findings to the class. Plan a route in space, bearing in mind fuel, food reserves and distance. Trade between planets using fantasy units of currency. Use algebraic functions to decide where explosive charges should be placed to destroy asteroids. Propose the location of a by-pass, using data tables and graphs from the Highways Agency. Find ways to satisfy constraints (minimum radii of curvature, verge clearance, cambers etc).
Cost and present proposed solutions. Compare perceptions of the causes of death with actual statistics. Interpret very large and very small probabilities. Decide what these say about behaviour and attitudes. Explore random variation. Use photographic and slow motion evidence to decide qualitatively, then quantitatively, whether a batsman (in cricket) is ‘in’ or ‘out’. Select variables, make estimates of distances, times and speeds and use algebraic models. Choose packaging for a pizza. Measure temperatures as the pizza cools. Use data logging software.
Fit a graphical model to the cooling of a pizza. Calculate the longest reasonable travel time before a pizza becomes too cold to eat. Describe the characteristics of individual genres of music. Use the tempo of music and other variables to illustrate compound measures, eg beats per minute. Plan a 5-day trip to satisfy money/time constraints and to keep happy three sets of tourists with different requirements. Convert currencies, satisfy baggage allowances etc. Use coordinate clues to locate infected people. Mix ingredients in proportions to create an antidote.
Use resources optimally to design a vaccination programme. Solve number, spatial and logic puzzles to progress in an escape adventure game. Use number sequences to escape from a building. Use rotation and reflection to recreate a given pattern. Use codes and loci to escape from underground tunnels. Design a questionnaire and conduct market research for a new drink. Mix ingredients to obtain nutritional value and taste; design packaging for the drink. Explore a town’s accident database. Control variables to decide on the most effective allocation of a sum of money to provide safety measures.
Prepare a case and present it. Determine the age and species of a ‘Joey’ from tail and foot measurements and graphs of growth data. Devise an appropriate nutrition regime from tables of nutrient data. Present and justify this regime. Explore perceptions of randomness and relate this to the perceived effectiveness of speed cameras. Simulate the effects of different sitings Design, examine the maths and then construct a sundial, using symmetry, angles, nets, origami, graphs and charts. Analyse a complex decision faced by a water aid agency.
Devise and use a compound measure (eg per capita) to decide on a ‘fair’ distribution of resources. Break a problem into its component parts; combine everyday knowledge to create chains of reasoning that result in reasonable estimates of useful quantities. Crash test Explorers Highway link design How risky is life? In or out? Keeping the pizza hot My music Mystery tours Outbreak PointZero Product wars Reducing road accidents Save a baby kangaroo Speed cameras Sundials Water availability You reckon? r4 © 2008 Bowland Charitable Trust 2 of 9 Comparison charts:
Problem types & practicalities Problem type Time needed (1-hour lessons) Interpreting and estimating Age/Ability Resources needed Data projector or Interactive whiteboard Other materials/equipment: check details in Case Study 3 of 9 Discovering relationships Planning and organising Year 7 Year 8 Year 9 l m h l m h l m h Alien invasion Crash test Explorers Highway link design How risky is life? In or out? Keeping the pizza hot My music Mystery tours Outbreak PointZero Product wars Reducing road accidents Save a baby kangaroo Speed cameras Sundials Water availability You reckon? -6 3-5 3-5 4-5 3-5 5-7 4-6 3-6 3 3 3 3 4-5 5-7 4-6 3-10 2-4 2-5 Minor role Significant role Major role Note: These charts are intended to indicate the coverage of each Case Study and are not an assessment of relative “quality”. They reflect the diverse nature of the Case Studies: some focus on a specific topic or activity type while others cover a wider domain. Please check the individual case study documents for detailed ICT and other resource requirements well in advance of the lesson – the above is just a general guide. r4 2008 Bowland Charitable Trust Pupil PCs (usually 1 per pair or small group) Designing and making Solving logic puzzles Modelling Optional Some lessons Most/all lessons Comparison charts: Links to the Key Stage 3 Programme of Study Key concepts Key processes Content Areas Develop confidence in an increasing range of methods Curriculum opportunities Work on more challenging mixes of contexts and mathematics Work on open and closed tasks in real and abstract contexts Alien invasion Crash test Explorers Highway link design How risky is life? In or out?
Keeping the pizza hot My music Mystery tours Outbreak PointZero Product wars Reducing road accidents Save a baby kangaroo Speed cameras Sundials Water availability You reckon? Minor role Significant role Major role Note: These charts are intended to indicate the coverage of each Case Study and are not an assessment of relative “quality”. They reflect the diverse nature of the Case Studies: some focus on a specific topic or activity type while others cover a wider domain. r4 © 2008 Bowland Charitable Trust 4 of 9 Select from a range of resources, inc ICT
Tackle problems from other subjects and from outside school Work collaboratively and independently Link different concepts, processes and techniques Applications and implications of maths Communicating and reflecting Interpreting and evaluating Geometry and Measure Analysing (procedures) Critical understanding Analysing (reasoning) Number and Algebra Representing Competence Creativity Statistics Outline descriptions Alien invasion Alien Invasion is a set of interactive lessons about a full-scale alien attack that coincides with a class visit to Manford City.
To set the scene and support the lessons, live TV news bulletins, radio broadcasts and telephone messages help to develop the story line. The invasion leads to a series of non-routine problems for pupils to solve as the narrative unfolds. The problems are on the theme of mathematical communication and are intended to promote discussion, reasoning and creativity. Alien Invasion is an opportunity for pupils to apply and use skills that they have previously been taught and to see connections between mathematical topics. Alternatively, it can be used to introduce or extend skills.
Mathematical activities include estimating and calculating measures, interpreting graphs and maps, code breaking and problem solving. Alien Invasion includes lesson notes which teachers are able to adapt for their own classes, supported by a full range of video, audio and print resources. Crash test Pupils use computer software to explore the impact of car crashes under varying conditions. Pupils select a car, a crash point and a speed, then watch an animation of the crash and see the results as physical impact on a simulated dummy and as numerical data.
In Crash test, there are 3 stages to the software: researching, testing and presentation. In the research centre, pupils decide which data to collect, they collect and analyse it and test hypotheses. In the test laboratory, pupils focus on hypothesis testing; they are able to test up to 14 pre-defined hypotheses and choose 3 ‘test packages’ in each case. In the presentation suite, there are tools such as varieties of graph paper and also help to prepare a presentation of their results to share with others. Explorers In Explorers, pupils assume the role of various characters scratching a living at the very edge of our galaxy in 2084.
Pupils develop the skills of thinking, reasoning and problem solving as they undertake three activities in an era of reclaimed space vessels patched up beyond recognition and with barely competent crew members. Pupils have to find a safe route through a dangerous nebula, trade goods between distant planets and destroy asteroids blocking a space highway using sonic charges. The mathematics involved includes comparing probabilities, working with different currencies and working with linear and quadratic equations.
Explorers includes a series of graphically rich, engaging, exploratory applets, as well as complementary paper-based tasks, follow-ups and homework tasks. Pupils can work on these tasks in small collaborative groups or individually. Highway link design Highway link design is an opportunity for pupils to integrate school mathematics with wider social, economic, environmental and work-related concerns. Pupils work in groups to plan and cost a village by-pass. They engage in problem-solving to decide which solution best takes into account the variety of issues and viewpoints that different stakeholders have about such a development. 4 © 2008 Bowland Charitable Trust 5 of 9 The mathematical concepts include scale, speed and the measurement of curvature, which pupils need to balance against environmental and social factors; they then need to cost their chosen route. Pupils present their solutions and assess those of others. Specialised software is included, and extension activities create further opportunities for pupils to cost real by-passes, using Google Earth. How risky is life? In How Risky is Life? , pupils explore the risk of dying unexpectedly from various causes.
They start from fears they know and, by comparing them with real-life data, they recognise that their perception of risk is often driven by presentations in the media. Pupils learn how to calculate the risks involved for various activities and how these are related to the base risk of death for typical people of different ages and genders. The emphasis is on order-ofmagnitude comparisons, reflecting the variations in risk level between individuals and over time. Pupils work with real data; they deduce information about small probabilities and use measures of average and spread in real life.
They also work with orders of magnitude. They learn that mathematical thinking is essential for putting risks in perspective and that the media usually focus on stories rather than on information. In or out ? Pupils consider the evidence from a photograph about whether a batsman in cricket is ‘in’ or ‘out’. The original case arose from a controversial decision by an umpire in an ‘Ashes’ test match (between England and Australia) in the 1960s. Pupils use mathematics to examine the photograph to assess whether the batman was ‘in’ or ‘out’.
Initially, pupils construct a simple mathematical model of the situation by deciding what variables they need to measure and what assumptions they need to make. Using this evidence, they decide for themselves whether the batsman was ‘in’ or ‘out’. As the work develops, pupils explore these measurements and assumptions in detail, allowing them to refine their initial decisions and to understand that, sometimes, there is no single right answer! Pupils revisit their models, test their assumptions and apply their model to other situations.
The mathematical skills and thinking that are required emerge gradually during ‘In or Out? ’ Keeping the pizza hot Keeping the pizza hot involves building a mathematical model in the context of homedelivery pizza. Pizza home-delivery is dependent on being able to deliver pizzas quickly, in an edible condition. In Keeping the pizza hot, pupils explore ways to keep a pizza warmer for longer and the implications of doing so. Pupils are asked to help answer the questions: how long does it take a pizza to cool, how far can it travel in that time and what difference does the packaging make?
Keeping the pizza hot has a number of parts which include leading pupils to move from a practical problem of a cooling pizza to a mathematical representation of a cooling curve. This is a big step and is intended to induct pupils into the potential of mathematical applications. It demonstrates how mathematics can underpin scientific enquiry. The linking of the time to cool with possible distances of travel introduces further mathematics. Although not essential, this project would work well as a cross-curricular project with the science department. r4 © 2008 Bowland Charitable Trust of 9 My music My music uses the interest pupils have in music as an opportunity for mathematical investigations, using pupils’ own favourite music tracks as the raw data. Pupils work in small groups to listen to different tracks, take measurements and then interpret and present the results. They analyse the similarities and differences between types of track, looking first at tempo and then other variables such as track length, highest position or number of weeks in the charts, and album sales; they can also investigate trends in music over the years.
My music can work as an introduction to statistical work, including: the collection of numerical data, performing basic statistical calculations, forming and testing hypotheses, making inferences about a population, and identifying potential sources of error in data collection and calculations. Although not essential, this project would work well as a crosscurricular project with the music department. Mystery tours Mystery Tours is a cartoon-based role play. Pupils take the part of the Tour Manager of a struggling tour operator; they are asked to plan a fictitious three day trip around the UK using tools and data in the software.
They then lead a ‘simulation’ of the tour and write an evaluation report. There are three groups of tourists, categorised as ‘Nature Lovers’, ‘Thrill Seekers’ and ‘Culture Vultures’; data is available about the preferences of each group. Pupils work together in small groups, or individually, to create a successful trip. In the first stages of the exercise, the most important skills are working with data such as timetables and percentages. Other areas of mathematics are brought in when the trip begins.
The tourists are quite demanding, and it is up to the pupils to keep them happy by solving any problems that may arise, presented algebraically or geometrically. Outbreak Outbreak is centred on an outbreak of a fatal virus. Pupils play the role of a scientist trying to contain the spread of the disease. Pupils have to develop a strategy which will help find the infected people, create an antidote and plan a vaccination programme to minimise the further spread of the virus. Pupils work with different experts to help with the challenges.
Completing an activity in any one of the ‘bunker areas’, unlocks a code which can then be used in the Map Room to reflect the progress that individuals or groups have made. This provides the opportunity either for the whole class to work through different activities at the same time, or for independent progression. It also promotes group work discussion and real world interaction. PointZero PointZero is an adventure-driven puzzle game based around the central themes of survival, escape and the quest to uncover the truth.
Pupils assume the role of three lead characters who have awoken trapped in strange and varying locations in an unfamiliar urban environment, following an undisclosed event. They are encouraged to use their mathematical skills to overcome problems so that each character can gain access to the ‘PointZero’ Building. Examples of activities include exploring complicated number sequences to scale a high rise building, using loci to find the way out of a complex underground network and reproducing geometrical patterns to deactivate a museum security system.
PointZero encourages pupils to reflect on how numbers, algebra and geometry influence our daily lives, albeit in ways which may not be immediately apparent. r4 © 2008 Bowland Charitable Trust 7 of 9 Product wars Pupils are asked to create a new range of ‘smoothie’ drinks. They use proportional reasoning to analyse the nutritional value and geometry to design the packaging. In Product wars, pupils play the role of being part of a drinks company and work with other employees to research and design the ultimate range of ‘smoothie’ drinks. The Managing Director of the company, Brad King, asks pupils to carry out market research, develop mixes or some ‘smoothies’ and then design and create the packaging. Video is used at key points in the lessons to provide support and guidance. Activities include: using enquiry-based learning to collect and analyse information from peers in developing the product; using ratio and proportion, percentages and a spreadsheet to mix the ingredients in different quantities to obtain the right nutritional value and taste for the target sector; and identifying suitable packaging designs. Pupils receive feedback via texts from members of the product team and video messages from Brad King himself.
Reducing road accidents Pupils imagine that they live in a small town where, over the past year, there has been a large number of road accidents. The town council has set up an enquiry to see what could be done to improve the situation and has allocated ? 100,000 to spend on reducing the number of deaths and serious injuries. In Reducing road accidents, pupils choose from a wide range of possible initiatives, for example, to build new road crossings or roundabouts, to install traffic lights or to design publicity campaigns for specific groups of people.
Pupils work in small teams to plan the most effective way to allocate the money. To support this work, the police have provided data on all the road accidents. Pupils use a specially constructed computer program to analyse this data and build a convincing case for their proposal. Save a baby kangaroo Save a baby kangaroo is an authentic context in which the pupils find a young orphaned kangaroo just twelve centimetres long and weighing sixty grams. Different species of kangaroo have different nutrient needs at different stages of their growth.
Through video clips, photographs and data such as birth to adult weights, pupils become familiar with a range of data about the different species of kangaroo. They then use the data to identify which kangaroo they have found and develop a feeding programme to save the life of their own ‘Joey’ in a simulation. Finally they communicate what they have learned in order to help someone else save a Joey by making a poster for a Vet clinic. The mathematical content of Save a baby kangaroo includes creating alternative representations of data and communicating statistical information.
Speed cameras Speed cameras are a continuing source of controversy, and even the experts are divided on their effectiveness. This is partly because the random nature of accidents makes it difficult to draw valid conclusions, which opens up possibilities for accidental or deliberate misrepresentation of data. Speed cameras uses video and newspaper resources to motivate discussion with and among pupils; this is combined with the use of spreadsheets to model the random occurrence of accidents over a year.
Pupils realise that lower probabilities do not invariably lead to fewer accidents, and that the occurrence of more accidents in one year is not necessarily evidence of a higher probability. They learn that random variation can obscure r4 © 2008 Bowland Charitable Trust 8 of 9 underlying probabilities. These are difficult but fundamental concepts for pupils to understand, and the combination of ICT and continual referral to a real situation helps to bring them alive. The emphasis is on pupils interpreting and extrapolating from data and using data to support their arguments – and to examine the arguments of others.
Sundials Sundials introduces pupils to the idea of using the sun to tell the time, applying a range of mathematical skills to understand some of the theory – and to construct at least one sundial for themselves. A video about sundials provides the context, including footage explaining the history of sundials and how they work. An interview with Harriet James, a gnomonist (someone who makes sundials) shows how maths is essential to the construction of sundials. The classroom work is differentiated into three tiers.
Depending on the route followed, Sundials uses symmetry and the drawing of angles, nets, origami, circle work and comparing data. Each route includes reading information from graphs and calculating time. Sundials invites pupils to reach out to the clockwork of the heavens! Water availability Pupils take the role of administrators for an international aid agency charged with providing water resources to countries in the Middle East and North Africa. Pupils examine ways to compare the availability of water fairly between the countries and then determine which country is most in need.
In Water availability, pupils review documents that describe the importance of water in the region and assemble relevant data. Pupils come to recognise that a key aspect of data handling is to determine which data it is appropriate to use to answer a particular question. In Water availability, the analysis requires the creation of compound measures, such as per capita measures of water availability, which links to the maths of proportionality. Pupils realise that compound measures are important to enable fair comparisons to be made between countries of various sizes.
You reckon? The media (and political speeches) are full of claims about how long things will take, how much things will cost and how tricky problems can be solved. People need to be able to judge if such claims are reasonable. You reckon? develops pupils’ ability to make estimates about unusual quantities based on only limited information, by posing interesting questions such as “Is it possible to provide 20% of the diesel used for road transport in the UK by growing crops on ‘set aside’ land? . You reckon? develops mathematical thinking and requires pupils to communicate their solutions. Pupils see that the problems they are asked to solve are the same problems faced by aid agencies, governments, and salesmen! You reckon? helps pupils to recognise the power of even simple mathematics (together with smart thinking) when making decisions about important topics. r4 © 2008 Bowland Charitable Trust 9 of 9