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Functional Mathematics and Teaching Modelling
FUNCTIONAL MATHEMATICS AND TEACHING MODELLING Hugh Burkhardt Shell Centre for Mathematical Education University of Nottingham, UK and Michigan State University USA Abstract-Mathematical literacy (ML) is the capacity to make mathematics functional in everyday life - that is, to make effective use of it in better understanding the world and in meeting its challenges. The current focus on ML provides a unique opportunity to make modelling a reality in schools, at last. This paper will j r s t review evidence that much of school mathematics is currently non-functional for many people at all levels of achievement. It will then look at the missing ingredients in current curriculae that are needed to develop ML. notably the explicit teaching of modelling skills. Using existing examples of proven effectiveness, it will outline what is needed to make ML an outcome of school mathematics.
1. FUNCTIONAL MATHEMATICS - A DESCRIPTION WITH EXAMPLES Mathematical literacy (ML) under various more-or-less equivalent names, including functional mathematics, quantitative literacy, and numeracy, is now becoming a major concern of mathematics curriculum improvement programmes in many countries. This movement is epitomised by the emergence of PISA (OECD 2003), which aims to assess ML, as a prime international comparison of standards of performance in mathematics. PISA is both a symptom of, and a support for, this new emphasis on mathematical literacy for all citizens as a curriculum responsibility. Functional mathematics (FM) is mathematics that nonspecialist adults, if they are taught how, will benefit from using in their everyday lives better to understand the world they live in, and to make better decisions. This reflects the PISA definition but may be less susceptible to misinterpretation as "business as usual". FM is distinct from specialist mathematics that prepares people for certain professions like physics, engineering, economics or accounting. It is also much more than applicable mathematics - that is, mathematics that is potentially useful in applications and modelling (as most mathematics is). Current curricula do address both these important needs; however, they lack key elements so that, for most people, secondary school mathematics is almost entirely nonfunctional. If you doubt this, perhaps surprising, assertion, ask any adult with a job that doesn't involve mathematics when they last solved a quadratic equation, used
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Pythagoras' Theorem or, indeed, any piece of mathematics they were first taught in secondary school. Probing more deeply will often reveal that they use quite a lot of elementary mathematics in a quite sophisticated way - a key feature of ML -but they do not see it as mathematics. The powerful mathematical tools they learnt in secondary school where they spent at least a thousand hours on them, remain unused. To see what really functional mathematics can equip you to do, consider the following tasks. (Assessment tasks are a compact and powerful way of exploring the meaning of general statements) These examples are chosen as tasks that any welleducated person should, and could, be able to tackle sensibly by the time they leave school at age 18. The comments on each task illustrate current deficiencies. Primary teachers In a country with 60 million people, about how many primary school teachers will be needed? Estimate a sensible answer, using your own everyday knowledge about the world. Write an explanation of your answer, stating any assumptions you make. This kind of back-of -the-envelope calculation is an important life skill. Here it requires choosing appropriate facts (6 years in primary out of a life of 60-80 years, one teacher for 20-30 kids), and recognizing and using proportional relationships giving 60*(6/70)/25 = 0.2 million primary teachers (to an accuracy appropriate to that of the data). This kind of linkage with the real world, common in the English curriculum, is rare in school Mathematics (and absent in most tests). The following task reflects key features of some high-profile miscarriages of justice, where an 'expert' witness's gross errors went unchallenged by lawyers or judge. Sudden Infant Deaths = Murder? In the general population, about 1 baby in 8,000 dies in an unexplained "cot death". The cause or causes are at present unknown. Three babies in one family have died. The mother is on trial. An expert witness says: "One cot death is a family tragedy; two is deeply suspicious; three is murder. The odds of even two deaths in one family are 64 million to 1" Discuss the reasoning behind the expert witness' statement, noting any errors, and write an improved version to present to the jury. What we expect here from a non-specialist educated citizen is not a full statistical analysis, which would also need more information, but a recognition that the reasoning presented is deeply flawed. There are two elementary mistakes in the statement, as well as others that are more subtle. It would be correct to say, for example: The chance of these deaths being entirely unconnected chance events is very small indeed - if there has been one death, the chance of two more unconnected deaths is about 64 million to one. What can the connection be? It may be that the mother killed the children; on the other hand, particularly since the cause(s) of cot death are still unclear, there may be other explanations. For many conditions (cancer and heart disease, for example), genetic andor environmental factors are known to affect the probabilities substantially.
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Any lawyer or judge with functional mathematics should see problems with the witness statement. Their failing was not lack of basic skills (they could, no doubt, calculate the chance of a double six on throwing two dice) but a lack of understanding of the basic and necessary assumption of independence. Ice cream van
You are considering driving an ice cream van during the Summer break. Your friend, who “knows everything”, says that “It’s easy money”. You make a few enquiries and find that the van costs f600 per week to hire. Typical selling data is that one can sell an average of 30 ice creams per hour, each costing 50p to make and each selling for f 1S O . How hard will you have to work in order to make this “easy money”? Explain your reasoning. This task, with two more like it, was used in a research study (Treilibs et al, 1980) of the performance of 120 very able 17 year old students. Many solved the tasks, using arithmetic and, sometimes graphs. None used algebra, the natural language for formulating such problems. Their algebra was non-functional, despite 5+ years of great success in the standard imitative algebra curriculum The mathematical concepts and skills needed to tackle these and similar problems are all taught and ‘learnt’by age 14 - yet they are non-functional for many well-educated adults. This should not be surprising, since they have not been taught how to use their mathematics in tackling everyday life problems as they arise. The skills of modelling non-routine problems are not trivial, but they can be taught; currently they are not. As the following outline (and other papers in this book) will show, there are established methods, accessible to typical teachers and their students, for teaching people to use their mathematics on real problems. Any curriculum that delivers functional mathematics will have to include them. 2. CURRICULA FOR FUNCTIONAL. MATHEMATICS: PRINCIPLES What changes in current curricula seem likely to be needed to deliver mathematical literacy - mathematics that is really functional in people’s lives? First, let us review the various key questions. What does ML/FM involve? Functional mathematics involves all the key aspects of ‘doing mathematics’ (see for example, Schoenfeld 1992): knowledge of concepts and skills strategies and tactics for modelling with this knowledge metacognitive control of one’s reasoning disposition to think mathematically, based on beliefs about maths. These are not, of course, independent elements but must be integrated into coherent mathematical practices for tackling whatever problem is at hand. (Lynn Steen (2002) has pointed out that FM involves “The sophisticated use of (often) elementary mathematics whereas current curricula have it the other way round.) ”
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Of the standard phases of modelling process: formulation of a mathematical model of the practical situation solution of the model by mathematical manipulation interpretation of the model solution in terms of the practical situation evaluation of the model. reporting of the results and reasoning to those concerned. Only solution gets serious attention in most c w en t school mathematics curricula. The following three similar-looking problems show the need to address formulation : a) Joe buys a six-pack of coke for $3 to share among his friends. How much should he charge for each bottle? b) It takes 30 minutes to bake 6 potatoes in the oven. How long will it take to bake one potato? c) It takes 30 minutes to play Beethoven's 6th Symphony. How long will Beethoven's 1st Symphony take to play? a) is typical of problems on proportional reasoning; students have been shown the model and know that every problem in those lessons will be of that kind. While b) and c) are superficially similar in structure, for b) either constant (30 mins) or proportional (-5 mins) is appropriate, depending on the type of oven (conventional or microwave). For c) there is, of course, no predictable connection. The essential point here is that, in current curricula, students get no teaching or experience in formulation - in choosing a model that fits the problem. Interpretation, evaluation, and reporting are similarly neglected. Moving beyond imitation Modelling everyday life situations is at the heart of functional mathematical literacy. Young people and the adults they become will face a range of problems where the tools of mathematics can help - common challenges in everyday life and more specialised problems in their work. These cannot all be anticipated, let alone covered, in school; we have to develop students' ability to use their mathematics in tackling non-routine problems - problems they have not analysed before or been shown how to do, like those in Q 1. In contrast, mathematics teaching in most classrooms at all levels is based on imitative learning - the 'EEE' model of explanation - example - exercises. This works in the short term for many students learning specific skills. There is much research showing that, for many students, EEE does not lead to long-term learning, because the essential conceptual foundation of the skills is not developed. EEE does not develop students' ability to use them in other situations. For this a richer pattern of learning activities is needed. This 'teaching by transmission' is not enough. The main extra elements needed are: active modelling with mathematics of non-routine practical situations diverse types of task, in class and for assessment students taking responsibility for their own reasoning classroom discussion in depth of alternative approaches and results teachers with the skills needed to handle these activities. Table 1 (Burkhardt et al, 1988) shows the changes in roles and mutual expectations:
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Table 1. Teacher and Student Roles. for imitative learning for modelling, add Directive roles
Facilitative roles
Manager Explainer Task setter with students as Imitator Responder
Counsellor Fellow student Resource with students as Investigator Manager Explainer
3. FUNCTIONAL MATHEMATICS IN CLASSROOMS: TASK EXEMPLARS What do we know about how to achieve these changes, and what more do we need to know? What resources will be needed? How far have they already been developed? How can mathematics teachers be enabled to handle the challenges, that are new to most of them? The situation is encouraging, though there are obstacles. The last 40 years have seen the explicit teaching of modelling with mathematics move forward (see Burkhardt with Pollak 2006) from small scale explorations in the 196Os, through pilot developments in the 1970s, to developments for typical classrooms in the 1980s, to established courses, albeit in a small minority of mathematics classrooms worldwide (many of these are described in this series of ICTMA books). I shall not attempt a detailed review but seek to show through working examples what is involved. $1 included tasks that reflected the goals of functional mathematics for adults, here I shall give some insight into what is involved in the classroom through examples of tasks that work well there. They illustrate the learning goals explicitly. The pattern of classroom activities will be exemplified in the next section. Functional mathematics involves a much broader range of task types than current curricula, asking students to investigate, plan, design, evaluate and recommend, critique and improve, or simply interpret. (These assessment tasks are much richer in a lesson activity form - but need more space to show than I have!). Planning tasks, like A day out in Derbyshire, do not have to involve the whole of the planning process. This process sequence critique - improve - create reflects a scale of challenge, and of time needed for meaningful work, that is useful for task and curriculum designers. In an IT-rich world the data would be provided via the web rather than on paper. We found that students enjoy finding the many faults in such a plan - especially when it comes from the examination board!
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Alison and two friends have planned a cycling trip around Derbyshire on Saturday. Here is their plan for the day >>>
Meet at Loughborough station at 7.23 am. Buy tickets and then catch the train to Derby. This arrives at 7.51 am. At Derby, catch the 8.20 am train to Cromford. This arrives at 8.41 am.
Read through the plan and the information sheets (map, timetables, brochures supplied). -_ If you find a mistake, or realise something has been forgotten, write it down and say how they change the plan.
Here are the instructions for getting to the Cycle Hire centre: “Turn left as you come out of Cromford station, walk along by the river and down Mill road. Cross over the A6. Walk up Cromford hill for about 1/2 mile and you will see.
........... (theplan continues) Design is an area that is motivating for students and important in practice. Design a Tent
A
\ \
These ends should zip together at night
c
Your task is to design this tent, sketching patterns for the plastic waterproof base, for the fabric of the tent, and for the two vertical tent poles. Give all dimensions. It must hold two adults with backpacks, with space to move around on their knees. As usual with FM/ML, this design task needs elementary mathematics, used in a flexible and reliable way. The visualisation challenge of relating three dimensions to two is not trivial.
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Making a case The spreadsheet contains 2 sets of reaction times, 100 each for Joe and Maria. Using this data, construct two arguments: A: that Joe is quicker than Maria and B: that Maria is quicker than Joe This kind of evaluate and recommend task (spreadsheet not shown here) builds understanding and intelligent scepticism, showing how political and marketing data is used - a crucial part of mathematical literacy. Likewise, a lot of information is now presented in graphical form interpretation tasks (see for example, Swan et al, 1986) develop and test understanding.
4. LEARNING FUNCTIONAL MATHEMATICS What do these changes in tasks mean for the pattern of classroom activities needed to enable students to meet such performance targets. I shall illustrate this by describing a curriculum and assessment development that has enabled some fairly typical teachers in English secondary schools to develop the mathematical literacy of their students across the age range 1 1-16. Their achievement was assessed by a public examination, administered by a major examination board. This development, Numeracy through Problem Solving (NTPS, Shell Centre 1987-89) sets out to support teachers without closing down the essentially open challenge. Developed through a process of creative design and systematic refinement in a sequence of trials, it worked well with typical teachers and students across the full ability range. It illustrates the power of well-engineered tools to help teachers (as with others) to tackle challenging tasks more successfully than they could without such help. NTPS offers a sequence of five modules that develop students‘ ability to use mathematics, together with other skills, in tackling problems of concern or situations of interest in everyday life. Each module provides a theme within which the students take responsibility for planning, organizing or designing. The students work on a group-project basis, primarily guided by a student booklet, with the teacher playing a consultant role. Each module is designed to take between 10 and 20 classroom hours to complete, and has four Stages. 1. Explore the context - students explore the domain by working on and evaluating exemplars provided. 2. Brainstorm approaches - generating and sifting ideas, which are developed and implemented in detail in Stage 3. 3. Detailed design and development of the analysis and the product 4. Implementation and evaluation - each group evaluates the things that the other groups have produced. For each module these stages take a form which fits the context. We outline this for one module, Be a Shrewd Chooser. Here students research and provide expert consumer advice for ‘clients’ in their class.
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1 Students listen to a radio show on audiotape which contains a number of
interviews with people who have just bought different products, and an interview with two students who have been involved in producing a consumer report on choosing orange drinks. As students reflect on and analyse the tape and the report, they begin to consider important factors that are taken into account when making a choice and different methods of making consumer decisions. 2 Students in a group now begin to work on their consumer report. They have to choose a product and decide on their research aims and methods. 3 Students develop their plan. They will be involved in conducting surveys, writing questionnaires and carrying out experiments in the classroom. They will also be considering how best to present their findings. This could involve posters and oral presentations in addition to written reports. 4 All the written reports are circulated around the other groups, and any group making an oral presentation does so. The reports are evaluated by the rest of the class, and then each group improves its own report taking into account these comments. The four other modules - Be a Paper Engineer, Design a Board Game, Produce a Quiz Show, Plan a Trip - take a similar four-stage approach. Extracts from each module can be found at www.mathshell.com . Two kinds of assessment are provided. Formative assessment, built into the teaching materials, is designed to check that each student in a group understands all aspects of the work, not simply those for which they have been responsible. A final examination at the end of the module assesses how well students can transfer what they have learned to more or less closely related problem situations. (The project outcomes, though evaluated by other students in the class, were not included in the formal assessment, reflecting concerns over how to assign individual credit for group work)
5. POLICY AND PROGRESS: HOW DO WE GET THERE? If a school system wants to make functional mathematics an important goal of its implemented and achieved mathematics curricula what must it do? It is necessary to make it a well-defined part of the intended curriculum, as set out in policy - the description and exemplification above provide a basis for that. But this is far from sufficient; most curriculum specifications, including the National Curriculum in England, contain important elements that are completely absent in all-but-a-few classrooms. For successful implementation, everyone involved must be both motivated and enabled to make the changes. Teachers, as ever, are the key. The system challenge that this presents is generally underestimated. It requires: pressure and support: Teachers and other professionals involved are busy people working under pressure; that pattern of pressure must be changed to give appropriate priority to functional mathematics. In systems with high-stakes assessment for accountability, this will only happen if the tested curriculum (that is, the high-stakes tests) includes FM. Equally, teachers and others have to be enabled to respond effectively; for most teachers this requires wellengineered teaching materials and professional development support. (Pressure
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is less expensive than support, which tends to be neglected by politicians encouraged by the “we can handle whatever is required’’ posture of most professions.) digestible pace of change: The professional development required of teachers and others to move beyond an imitative curriculum in the way described above is profound. It will only happen through a well-supported sequence of successful experiences in their own classroom. This requires well-engineered tools and live professional development over a period of years, encouraged by recognizable achievements at each stage. One model that works well in practice involves (see Swan et al, 1986): one new task type each year in the high-stakes tests, embodied in a well-engineered 3-4 week curriculum unit of teaching and in-class assessment materials, supported by well-engineered professional development tools that work well on an in-school do-it-yourself basis on which to build the essential but expensive (and alwaystoo-limited) live professional development. The development of such modules, so that they work well with typical teachers and students in realistic circumstances off support, is a challenging design and development task. Thus a substantial research and development effort over a period of years will be needed to get good assessment, encouraging good learning that produces functional mathematical literacy in all students. The pay-offs in student performance in, attitude to, and take-up of mathematics as a whole will be substantial. The challenge of developing a model of change that works is not yet recognised at policy level. Politicians and senior civil servants generally assume that once “difficult decisions” on what should be done have been made and legislated, faithful implementation will follow. In practice, outcomes that resemble the intentions are rare. The profound changes needed to make school mathematics functional will only happen if they are well-engineered - that is, skilfully designed to take into account the factors sketched here and carefully developed on a limited scale first. We now know how to teach the full range of modelling and other skills involved; it is the system implementation problems that are the difference between success and failure. To conclude, Bill Gates has said that “the three R’s for the 2 1st Century are Rigour, Relevance and Relationships.” Learning functional mathematics, built on modelling with mathematics, develops them all. Any mathematics curriculum for all students should give it priority. ACKNOWLEDGEMENTS
The vitality of mathematics in most classrooms rests on the work of designers; the examples quoted here are largely the work of the Shell Centre team, particularly Malcolm Swan, who has led the work on functional mathematics.
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REFERENCES Burkhardt, H. (198 1) The Real World and Mathematics, Blackie-Birkhauser; reprinted (2000) Nottingham: Shell Centre Publications http://www.mathshell.com/scp/index.htm . Burkhardt, H., Fraser, R., Coupland, J., Phillips, R., Pimm, D. and Ridgway, J. (1988). Learning activities and classroom roles with and without the microcomputer. Journal of Mathematical Behavior, 6,305-338. Burkhardt, H . with contributions by Pollak, H.O. (2006) Modelling in Mathematics Classrooms: reflections on past developments and the future. Zeitschrijit fur Didaktik der Mathematik, 38 (2). OECD (2003) The PISA 2003 Assessment Framework: Mathematics, Reading, Science and Problem Solving Knowledge and Skills. Paris: https://www .pisa.oecd.org/dataoecd/3815 1/3 3707 1 92 .pdf . Schoenfeld, A. H. (1992) Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (ed.), Handbook for Research on Mathematics Teaching and Learning. New York: Macmillan, 334-370. Steen, L. A. (ed) (2002) Mathematics and Democracy: the case for quantitative literacy. The National Council on Education and the Disciplines (NCED), USA. http://www.maa.org/qI/mathanddemocracy.html . Shell Centre: Swan, M., Binns, B., Gillespie, J. and Burkhardt, H. (1987-89) Numeracy through Problem Solving. Harlow: Longman, revised (2000) Nottingham: Shell Centre Publications. Swan, M., Pitts, J., Fraser, R., Burkhardt, H. et al. (1986). The Language of Functions and Graphs Manchester: Joint Matriculation Board, reprinted (2000) Nottingham: Shell Centre Publications. http://www.mathshell.com/scp/index.htm. Treilibs, V., Burkhardt, H. and Low, B. (1980) Formulation processes in mathematical modelling, Nottingham: Shell Centre Publications.
1. FUNCTIONAL MATHEMATICS - A DESCRIPTION WITH EXAMPLES Mathematical literacy (ML) under various more-or-less equivalent names, including functional mathematics, quantitative literacy, and numeracy, is now becoming a major concern of mathematics curriculum improvement programmes in many countries. This movement is epitomised by the emergence of PISA (OECD 2003), which aims to assess ML, as a prime international comparison of standards of performance in mathematics. PISA is both a symptom of, and a support for, this new emphasis on mathematical literacy for all citizens as a curriculum responsibility. Functional mathematics (FM) is mathematics that nonspecialist adults, if they are taught how, will benefit from using in their everyday lives better to understand the world they live in, and to make better decisions. This reflects the PISA definition but may be less susceptible to misinterpretation as "business as usual". FM is distinct from specialist mathematics that prepares people for certain professions like physics, engineering, economics or accounting. It is also much more than applicable mathematics - that is, mathematics that is potentially useful in applications and modelling (as most mathematics is). Current curricula do address both these important needs; however, they lack key elements so that, for most people, secondary school mathematics is almost entirely nonfunctional. If you doubt this, perhaps surprising, assertion, ask any adult with a job that doesn't involve mathematics when they last solved a quadratic equation, used
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Pythagoras' Theorem or, indeed, any piece of mathematics they were first taught in secondary school. Probing more deeply will often reveal that they use quite a lot of elementary mathematics in a quite sophisticated way - a key feature of ML -but they do not see it as mathematics. The powerful mathematical tools they learnt in secondary school where they spent at least a thousand hours on them, remain unused. To see what really functional mathematics can equip you to do, consider the following tasks. (Assessment tasks are a compact and powerful way of exploring the meaning of general statements) These examples are chosen as tasks that any welleducated person should, and could, be able to tackle sensibly by the time they leave school at age 18. The comments on each task illustrate current deficiencies. Primary teachers In a country with 60 million people, about how many primary school teachers will be needed? Estimate a sensible answer, using your own everyday knowledge about the world. Write an explanation of your answer, stating any assumptions you make. This kind of back-of -the-envelope calculation is an important life skill. Here it requires choosing appropriate facts (6 years in primary out of a life of 60-80 years, one teacher for 20-30 kids), and recognizing and using proportional relationships giving 60*(6/70)/25 = 0.2 million primary teachers (to an accuracy appropriate to that of the data). This kind of linkage with the real world, common in the English curriculum, is rare in school Mathematics (and absent in most tests). The following task reflects key features of some high-profile miscarriages of justice, where an 'expert' witness's gross errors went unchallenged by lawyers or judge. Sudden Infant Deaths = Murder? In the general population, about 1 baby in 8,000 dies in an unexplained "cot death". The cause or causes are at present unknown. Three babies in one family have died. The mother is on trial. An expert witness says: "One cot death is a family tragedy; two is deeply suspicious; three is murder. The odds of even two deaths in one family are 64 million to 1" Discuss the reasoning behind the expert witness' statement, noting any errors, and write an improved version to present to the jury. What we expect here from a non-specialist educated citizen is not a full statistical analysis, which would also need more information, but a recognition that the reasoning presented is deeply flawed. There are two elementary mistakes in the statement, as well as others that are more subtle. It would be correct to say, for example: The chance of these deaths being entirely unconnected chance events is very small indeed - if there has been one death, the chance of two more unconnected deaths is about 64 million to one. What can the connection be? It may be that the mother killed the children; on the other hand, particularly since the cause(s) of cot death are still unclear, there may be other explanations. For many conditions (cancer and heart disease, for example), genetic andor environmental factors are known to affect the probabilities substantially.
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Any lawyer or judge with functional mathematics should see problems with the witness statement. Their failing was not lack of basic skills (they could, no doubt, calculate the chance of a double six on throwing two dice) but a lack of understanding of the basic and necessary assumption of independence. Ice cream van
You are considering driving an ice cream van during the Summer break. Your friend, who “knows everything”, says that “It’s easy money”. You make a few enquiries and find that the van costs f600 per week to hire. Typical selling data is that one can sell an average of 30 ice creams per hour, each costing 50p to make and each selling for f 1S O . How hard will you have to work in order to make this “easy money”? Explain your reasoning. This task, with two more like it, was used in a research study (Treilibs et al, 1980) of the performance of 120 very able 17 year old students. Many solved the tasks, using arithmetic and, sometimes graphs. None used algebra, the natural language for formulating such problems. Their algebra was non-functional, despite 5+ years of great success in the standard imitative algebra curriculum The mathematical concepts and skills needed to tackle these and similar problems are all taught and ‘learnt’by age 14 - yet they are non-functional for many well-educated adults. This should not be surprising, since they have not been taught how to use their mathematics in tackling everyday life problems as they arise. The skills of modelling non-routine problems are not trivial, but they can be taught; currently they are not. As the following outline (and other papers in this book) will show, there are established methods, accessible to typical teachers and their students, for teaching people to use their mathematics on real problems. Any curriculum that delivers functional mathematics will have to include them. 2. CURRICULA FOR FUNCTIONAL. MATHEMATICS: PRINCIPLES What changes in current curricula seem likely to be needed to deliver mathematical literacy - mathematics that is really functional in people’s lives? First, let us review the various key questions. What does ML/FM involve? Functional mathematics involves all the key aspects of ‘doing mathematics’ (see for example, Schoenfeld 1992): knowledge of concepts and skills strategies and tactics for modelling with this knowledge metacognitive control of one’s reasoning disposition to think mathematically, based on beliefs about maths. These are not, of course, independent elements but must be integrated into coherent mathematical practices for tackling whatever problem is at hand. (Lynn Steen (2002) has pointed out that FM involves “The sophisticated use of (often) elementary mathematics whereas current curricula have it the other way round.) ”
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Of the standard phases of modelling process: formulation of a mathematical model of the practical situation solution of the model by mathematical manipulation interpretation of the model solution in terms of the practical situation evaluation of the model. reporting of the results and reasoning to those concerned. Only solution gets serious attention in most c w en t school mathematics curricula. The following three similar-looking problems show the need to address formulation : a) Joe buys a six-pack of coke for $3 to share among his friends. How much should he charge for each bottle? b) It takes 30 minutes to bake 6 potatoes in the oven. How long will it take to bake one potato? c) It takes 30 minutes to play Beethoven's 6th Symphony. How long will Beethoven's 1st Symphony take to play? a) is typical of problems on proportional reasoning; students have been shown the model and know that every problem in those lessons will be of that kind. While b) and c) are superficially similar in structure, for b) either constant (30 mins) or proportional (-5 mins) is appropriate, depending on the type of oven (conventional or microwave). For c) there is, of course, no predictable connection. The essential point here is that, in current curricula, students get no teaching or experience in formulation - in choosing a model that fits the problem. Interpretation, evaluation, and reporting are similarly neglected. Moving beyond imitation Modelling everyday life situations is at the heart of functional mathematical literacy. Young people and the adults they become will face a range of problems where the tools of mathematics can help - common challenges in everyday life and more specialised problems in their work. These cannot all be anticipated, let alone covered, in school; we have to develop students' ability to use their mathematics in tackling non-routine problems - problems they have not analysed before or been shown how to do, like those in Q 1. In contrast, mathematics teaching in most classrooms at all levels is based on imitative learning - the 'EEE' model of explanation - example - exercises. This works in the short term for many students learning specific skills. There is much research showing that, for many students, EEE does not lead to long-term learning, because the essential conceptual foundation of the skills is not developed. EEE does not develop students' ability to use them in other situations. For this a richer pattern of learning activities is needed. This 'teaching by transmission' is not enough. The main extra elements needed are: active modelling with mathematics of non-routine practical situations diverse types of task, in class and for assessment students taking responsibility for their own reasoning classroom discussion in depth of alternative approaches and results teachers with the skills needed to handle these activities. Table 1 (Burkhardt et al, 1988) shows the changes in roles and mutual expectations:
Burkhardt
181
Table 1. Teacher and Student Roles. for imitative learning for modelling, add Directive roles
Facilitative roles
Manager Explainer Task setter with students as Imitator Responder
Counsellor Fellow student Resource with students as Investigator Manager Explainer
3. FUNCTIONAL MATHEMATICS IN CLASSROOMS: TASK EXEMPLARS What do we know about how to achieve these changes, and what more do we need to know? What resources will be needed? How far have they already been developed? How can mathematics teachers be enabled to handle the challenges, that are new to most of them? The situation is encouraging, though there are obstacles. The last 40 years have seen the explicit teaching of modelling with mathematics move forward (see Burkhardt with Pollak 2006) from small scale explorations in the 196Os, through pilot developments in the 1970s, to developments for typical classrooms in the 1980s, to established courses, albeit in a small minority of mathematics classrooms worldwide (many of these are described in this series of ICTMA books). I shall not attempt a detailed review but seek to show through working examples what is involved. $1 included tasks that reflected the goals of functional mathematics for adults, here I shall give some insight into what is involved in the classroom through examples of tasks that work well there. They illustrate the learning goals explicitly. The pattern of classroom activities will be exemplified in the next section. Functional mathematics involves a much broader range of task types than current curricula, asking students to investigate, plan, design, evaluate and recommend, critique and improve, or simply interpret. (These assessment tasks are much richer in a lesson activity form - but need more space to show than I have!). Planning tasks, like A day out in Derbyshire, do not have to involve the whole of the planning process. This process sequence critique - improve - create reflects a scale of challenge, and of time needed for meaningful work, that is useful for task and curriculum designers. In an IT-rich world the data would be provided via the web rather than on paper. We found that students enjoy finding the many faults in such a plan - especially when it comes from the examination board!
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Functional Mathematics and Teaching Modelling
Alison and two friends have planned a cycling trip around Derbyshire on Saturday. Here is their plan for the day >>>
Meet at Loughborough station at 7.23 am. Buy tickets and then catch the train to Derby. This arrives at 7.51 am. At Derby, catch the 8.20 am train to Cromford. This arrives at 8.41 am.
Read through the plan and the information sheets (map, timetables, brochures supplied). -_ If you find a mistake, or realise something has been forgotten, write it down and say how they change the plan.
Here are the instructions for getting to the Cycle Hire centre: “Turn left as you come out of Cromford station, walk along by the river and down Mill road. Cross over the A6. Walk up Cromford hill for about 1/2 mile and you will see.
........... (theplan continues) Design is an area that is motivating for students and important in practice. Design a Tent
A
\ \
These ends should zip together at night
c
Your task is to design this tent, sketching patterns for the plastic waterproof base, for the fabric of the tent, and for the two vertical tent poles. Give all dimensions. It must hold two adults with backpacks, with space to move around on their knees. As usual with FM/ML, this design task needs elementary mathematics, used in a flexible and reliable way. The visualisation challenge of relating three dimensions to two is not trivial.
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Making a case The spreadsheet contains 2 sets of reaction times, 100 each for Joe and Maria. Using this data, construct two arguments: A: that Joe is quicker than Maria and B: that Maria is quicker than Joe This kind of evaluate and recommend task (spreadsheet not shown here) builds understanding and intelligent scepticism, showing how political and marketing data is used - a crucial part of mathematical literacy. Likewise, a lot of information is now presented in graphical form interpretation tasks (see for example, Swan et al, 1986) develop and test understanding.
4. LEARNING FUNCTIONAL MATHEMATICS What do these changes in tasks mean for the pattern of classroom activities needed to enable students to meet such performance targets. I shall illustrate this by describing a curriculum and assessment development that has enabled some fairly typical teachers in English secondary schools to develop the mathematical literacy of their students across the age range 1 1-16. Their achievement was assessed by a public examination, administered by a major examination board. This development, Numeracy through Problem Solving (NTPS, Shell Centre 1987-89) sets out to support teachers without closing down the essentially open challenge. Developed through a process of creative design and systematic refinement in a sequence of trials, it worked well with typical teachers and students across the full ability range. It illustrates the power of well-engineered tools to help teachers (as with others) to tackle challenging tasks more successfully than they could without such help. NTPS offers a sequence of five modules that develop students‘ ability to use mathematics, together with other skills, in tackling problems of concern or situations of interest in everyday life. Each module provides a theme within which the students take responsibility for planning, organizing or designing. The students work on a group-project basis, primarily guided by a student booklet, with the teacher playing a consultant role. Each module is designed to take between 10 and 20 classroom hours to complete, and has four Stages. 1. Explore the context - students explore the domain by working on and evaluating exemplars provided. 2. Brainstorm approaches - generating and sifting ideas, which are developed and implemented in detail in Stage 3. 3. Detailed design and development of the analysis and the product 4. Implementation and evaluation - each group evaluates the things that the other groups have produced. For each module these stages take a form which fits the context. We outline this for one module, Be a Shrewd Chooser. Here students research and provide expert consumer advice for ‘clients’ in their class.
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1 Students listen to a radio show on audiotape which contains a number of
interviews with people who have just bought different products, and an interview with two students who have been involved in producing a consumer report on choosing orange drinks. As students reflect on and analyse the tape and the report, they begin to consider important factors that are taken into account when making a choice and different methods of making consumer decisions. 2 Students in a group now begin to work on their consumer report. They have to choose a product and decide on their research aims and methods. 3 Students develop their plan. They will be involved in conducting surveys, writing questionnaires and carrying out experiments in the classroom. They will also be considering how best to present their findings. This could involve posters and oral presentations in addition to written reports. 4 All the written reports are circulated around the other groups, and any group making an oral presentation does so. The reports are evaluated by the rest of the class, and then each group improves its own report taking into account these comments. The four other modules - Be a Paper Engineer, Design a Board Game, Produce a Quiz Show, Plan a Trip - take a similar four-stage approach. Extracts from each module can be found at www.mathshell.com . Two kinds of assessment are provided. Formative assessment, built into the teaching materials, is designed to check that each student in a group understands all aspects of the work, not simply those for which they have been responsible. A final examination at the end of the module assesses how well students can transfer what they have learned to more or less closely related problem situations. (The project outcomes, though evaluated by other students in the class, were not included in the formal assessment, reflecting concerns over how to assign individual credit for group work)
5. POLICY AND PROGRESS: HOW DO WE GET THERE? If a school system wants to make functional mathematics an important goal of its implemented and achieved mathematics curricula what must it do? It is necessary to make it a well-defined part of the intended curriculum, as set out in policy - the description and exemplification above provide a basis for that. But this is far from sufficient; most curriculum specifications, including the National Curriculum in England, contain important elements that are completely absent in all-but-a-few classrooms. For successful implementation, everyone involved must be both motivated and enabled to make the changes. Teachers, as ever, are the key. The system challenge that this presents is generally underestimated. It requires: pressure and support: Teachers and other professionals involved are busy people working under pressure; that pattern of pressure must be changed to give appropriate priority to functional mathematics. In systems with high-stakes assessment for accountability, this will only happen if the tested curriculum (that is, the high-stakes tests) includes FM. Equally, teachers and others have to be enabled to respond effectively; for most teachers this requires wellengineered teaching materials and professional development support. (Pressure
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is less expensive than support, which tends to be neglected by politicians encouraged by the “we can handle whatever is required’’ posture of most professions.) digestible pace of change: The professional development required of teachers and others to move beyond an imitative curriculum in the way described above is profound. It will only happen through a well-supported sequence of successful experiences in their own classroom. This requires well-engineered tools and live professional development over a period of years, encouraged by recognizable achievements at each stage. One model that works well in practice involves (see Swan et al, 1986): one new task type each year in the high-stakes tests, embodied in a well-engineered 3-4 week curriculum unit of teaching and in-class assessment materials, supported by well-engineered professional development tools that work well on an in-school do-it-yourself basis on which to build the essential but expensive (and alwaystoo-limited) live professional development. The development of such modules, so that they work well with typical teachers and students in realistic circumstances off support, is a challenging design and development task. Thus a substantial research and development effort over a period of years will be needed to get good assessment, encouraging good learning that produces functional mathematical literacy in all students. The pay-offs in student performance in, attitude to, and take-up of mathematics as a whole will be substantial. The challenge of developing a model of change that works is not yet recognised at policy level. Politicians and senior civil servants generally assume that once “difficult decisions” on what should be done have been made and legislated, faithful implementation will follow. In practice, outcomes that resemble the intentions are rare. The profound changes needed to make school mathematics functional will only happen if they are well-engineered - that is, skilfully designed to take into account the factors sketched here and carefully developed on a limited scale first. We now know how to teach the full range of modelling and other skills involved; it is the system implementation problems that are the difference between success and failure. To conclude, Bill Gates has said that “the three R’s for the 2 1st Century are Rigour, Relevance and Relationships.” Learning functional mathematics, built on modelling with mathematics, develops them all. Any mathematics curriculum for all students should give it priority. ACKNOWLEDGEMENTS
The vitality of mathematics in most classrooms rests on the work of designers; the examples quoted here are largely the work of the Shell Centre team, particularly Malcolm Swan, who has led the work on functional mathematics.
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REFERENCES Burkhardt, H. (198 1) The Real World and Mathematics, Blackie-Birkhauser; reprinted (2000) Nottingham: Shell Centre Publications http://www.mathshell.com/scp/index.htm . Burkhardt, H., Fraser, R., Coupland, J., Phillips, R., Pimm, D. and Ridgway, J. (1988). Learning activities and classroom roles with and without the microcomputer. Journal of Mathematical Behavior, 6,305-338. Burkhardt, H . with contributions by Pollak, H.O. (2006) Modelling in Mathematics Classrooms: reflections on past developments and the future. Zeitschrijit fur Didaktik der Mathematik, 38 (2). OECD (2003) The PISA 2003 Assessment Framework: Mathematics, Reading, Science and Problem Solving Knowledge and Skills. Paris: https://www .pisa.oecd.org/dataoecd/3815 1/3 3707 1 92 .pdf . Schoenfeld, A. H. (1992) Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (ed.), Handbook for Research on Mathematics Teaching and Learning. New York: Macmillan, 334-370. Steen, L. A. (ed) (2002) Mathematics and Democracy: the case for quantitative literacy. The National Council on Education and the Disciplines (NCED), USA. http://www.maa.org/qI/mathanddemocracy.html . Shell Centre: Swan, M., Binns, B., Gillespie, J. and Burkhardt, H. (1987-89) Numeracy through Problem Solving. Harlow: Longman, revised (2000) Nottingham: Shell Centre Publications. Swan, M., Pitts, J., Fraser, R., Burkhardt, H. et al. (1986). The Language of Functions and Graphs Manchester: Joint Matriculation Board, reprinted (2000) Nottingham: Shell Centre Publications. http://www.mathshell.com/scp/index.htm. Treilibs, V., Burkhardt, H. and Low, B. (1980) Formulation processes in mathematical modelling, Nottingham: Shell Centre Publications.