ZDM Mathematics Education (2007) 39:395–403 DOI 10.1007/s11858-007-0041-4

ORIGINAL ARTICLE

Problem solving in the United Kingdom Hugh Burkhardt Æ Alan Bell

Accepted: 1 June 2007 / Published online: 14 August 2007 FIZ Karlsruhe 2007

Abstract We trace the development of problem solving in UK school mathematics over the last century or so, illuminating our descriptions with task exemplars. This is an informative and cautionary tale of mutual incomprehension between leaders in mathematical education and the public they seek to serve. Intelligent and energetic pioneers have developed and improved ways to teach problem solving, though often with little attention to the challenges these methods present to typical teachers. Political decision makers, despite rhetorical support for ‘‘real-world’’ problem solving, have failed to understand the need for the changes proposed. Currently, there are some hopeful signs but it is doubtful if they will be realized in classroom practice. The challenge of modifying the system dynamics so as to yield large-scale improvements remains an unsolved problem in the UK, as elsewhere; at least, it is now recognized and being worked on.

1 Overview Mathematics has always had two surface aspects, often called skills and problem solving. Skills are seen as a set of well-defined procedures for transforming numbers, symbols or shapes; the role of the ideal performer is that of an automaton, and the role of the student that of learning to be

H. Burkhardt (&) A. Bell Shell Centre for Mathematical Education, School of Education, University of Nottingham, Nottingham NG8 1BB, UK e-mail: [email protected]

a reliable one. Problem solving, in contrast, involves tackling tasks that are significantly different from those one has learned ‘‘by heart’’; a major part of the challenge lies in deciding how to tackle the problem, and which bits of one’s toolkit of mathematical skills will help. A problem solver needs a rich, connected understanding of mathematics and the ability to see patterns of similarity and association, as well as the skills to carry out the planned attack, and to check that the results make sense in the context of the problem. No wonder these are called ‘‘higher level skills’’. Historically, different people needed very different kinds of mathematical capability. Skills were enough to guarantee ‘‘skilled employment’’, as bookkeepers or surveyors for example, with the higher pay and status that this brings. Apprenticeships covered all the kinds of task that the profession involved, and the procedures for dealing with them. Expertise was a matter of applying ‘‘best practice,’’ as it still largely is—one prefers one’s brain surgeon or car mechanic to employ well-learned skills rather than to be ‘‘problem solving.’’ Equally there have always been problem solvers. Every theorem was a problem once, and every new algorithmic procedure an improved solution. But problem solving was seen as a specialized skill, only for mathematicians or others who saw the power of mathematics in opening up new fields. Thus geometry developed from the needs of surveying. Newton invented calculus to enable him to prove that an inverse square law of gravitation, along with his laws of mechanics, lead to closed elliptical orbits for the planets. (He then translated the proof into Euclidean Geometry so his contemporaries could understand it.) Mathematicians worked to extend the understanding and power of each innovation. When new procedures emerged, as in navigation, they became part of

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professional training. In short, problem solving was the domain of a few.1 In such a situation, it is not surprising that mathematics education at school level concentrated on developing skills, leaving problem solving to more specialized contexts, mostly in graduate schools. However, practical circumstances have changed, with human automata losing their jobs all over the world to reliable, and much less expensive, calculators and computers. Life and work have become more mathematical. Employers, facing ever-changing demands, require flexibility that precludes teaching all the necessary skills in advance. These changes have produced pressures for mathematical literacy, and for problem solving in school mathematics education. We will outline what has happened in the UK, particularly England and Wales.

2 Early history During the nineteenth Century it gradually became recognized that the rote learning of procedures was inadequate, and that mathematical reasoning was important. A Royal Commission, reporting on the state of mathematics teaching in nine leading Public (i.e., private) Schools in 1837, noted that the typical two weekly hours of mathematics consisted of Arithmetic, a little manipulative Algebra, and ‘‘Euclid’’, learned by rote. By 1861, about half the boys leaving the sixth form at Harrow School had done six books of Euclid, some trigonometry and quadratic equations, and a little Conic Sections and Mechanics. Of the subjects examined by the London Matriculation Board, ‘‘the Arithmetic was heavily commercial, the Algebra formal, and for Geometry there was Euclid’’. Euclid was supposed to provide a training in logic, but it was ‘‘not well taught,’’ and little logical thinking was acquired. In 1870, Rawdon Levett, the only mathematics master teaching the 400 boys of King Edward’s School, Birmingham, proposed the founding of an anti-Euclid association. This was named The Association for Improvement of Geometry Teaching (AIGT), and was the forerunner of the current Mathematical Association. This movement gradually led to a new kind of task in Geometry. Students were first asked to prove a theorem they had been taught. This was followed by a ‘‘rider’’—a non-routine problem related to the proof. The following example comes from a 1947 examination in ‘‘Elementary Mathematics’’:

H. Burkhardt, A. Bell

Two chords AB and CD of a circle, when produced, cut each other at a point O outside the circle. Prove that OA.OB = OC.OD.

Two unequal circles touch each other externally at a point R. The diameter through R of the larger circle is RP, and that of the smaller circle is RQ. An exterior common tangent touches the larger circle at S and the smaller circle at T, and it cuts PQ produced at X. Prove that

XP XS 2 = XQ XT 2

The rider was recognized as testing mathematical reasoning, rather than simply recall of learned proofs of theorems. Note that, in modern terms, the transfer distance of the rider from the theorem was under the control of the task designer. In this case, the rider is a direct adaptation of the proof of the theorem—some thought rather than just memory is required but the transfer distance is relatively short. In other questions, it was sometimes greater. Arithmetic and Algebra textbooks contained ‘‘exercises’’, the hardest of which could demand relatively complex reasoning:

Find the maximum rectangle which can be inscribed in a given semicircle.

Though expressed geometrically, this involves formulating and transforming an algebraic quadratic. In this way, non-routine problem solving entered the English school curriculum—at least for the more able students. As we look at the development of problem solving up to the present, it is salutary to ask how current non-routine tasks compare with those from the past. From about 1940 onwards, the design of mathematics curricula moved from an accepted historical fact to become a subject of discussion, and of controversy. Innovators in mathematics education began to point out that problem solving is an integral part of ‘‘doing mathematics,’’ and should play a similar role in school education. In 1938 the Spens Report on Secondary Education, in a critical comment on current mathematics teaching, asserted that ‘‘it should be taught as Art and Music and Physical Science, as one of the main lines followed by the creative spirit of man’’. Elsewhere, it suggests that ‘‘the topics should be chosen to develop a grasp of mathematical ideas.... and practical questions which have been of urgent interest and utility to man in his affairs.’’ These complementary views of mathematics as: •

1

It is said that the Japanese Government once decided that only 2% of students need to learn problem solving.

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•

an important aspect of culture to be studied for its own sake a powerful toolkit for solving practical problems

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persist to this day. Some mathematicians and teachers regard the former as sufficient2, and utility as an inferior justification for learning mathematics, reflecting G.H. Hardy’s famous toast: ‘‘To pure mathematics; may it never be applied.’’ However, it is power in application that is persuasive to policy makers and to many users of mathematics who, while enjoying its charms per se, live by applying its power to other problems.3

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4. 5.

The following example, Consecutive Sums, illustrates these distinctions. •

3 The coming of ‘‘investigation’’ An important strand of development after World War II focused on the classroom experiences of learners, in particular on engaging students in activities more nearly like those of practicing mathematicians. Like the Mathematical Association in the previous century, the Association of Teachers of Mathematics (ATM) was formed4 in 1951 to push back the frontiers of school mathematics. Influences from psychology, stemming from Piaget and Gattegno, were important. Piaget showed how the concepts actually held by children differed markedly and unexpectedly from the corresponding adult ideas—for example, that a row of beads spread out had ‘‘more’’ than when closer together, and that in a bunch of many poppies and a few lilies there were ‘‘more poppies’’ than flowers— showing the difficulties of simultaneously recognizing the whole and the part, and of technical versus informal language. Gattegno’s emphasis was on creating and manipulating situations in which children could see or make mathematics. They were led to ‘‘read’’ the mathematics shown by patterns of colored (Cuisenaire) rods, for example a red and a yellow rod matching a black rod, showing that 2 + 5 = 7—but also that 5 + 2 = 7, and that 7 could be made up in other ways from smaller numbers. Further research into learning mathematics showed that rote-learned procedures remain reliable only if they are practiced regularly, and that long term learning depends on conceptual understanding of the basis of the procedure. The learning of mathematics was coming to be seen as working at various levels, according to various similar schemes; we choose to describe them as: 1. 2. 3.

2

Rote learning, memorization and skill practice Meaningful learning of concepts and skills Learning to solve problems

Though can one, on cultural grounds alone, justify giving mathematics more curriculum time than, say, music? 3 Alas for both these perspectives, interview data suggest that the overwhelming majority of students see school mathematics as just a route to a necessary qualification. 4 As the Association for Teaching Aids in Mathematics.

Posing problems in making investigations of open problem situations Describing mathematical structures embodied in a practical situation (applied maths) and relationships within and between these structures (pure maths).

Investigate the properties of numbers that can be represented as sums of consecutive whole numbers as, for example: 9¼4þ5 ¼2þ3þ4 proving your conjectures where you can.

This is level 4, while a level 3 problem version is: •

Prove that the sum of 5 consecutive whole numbers is divisible by 5 and, similarly, for any odd number. Related tasks at level 2 and level 1 are: • Explain why the sum of an arithmetic progression is the average of the first and last terms multiplied by the number of terms • Write the formula for the sum of an arithmetic progression. Note how the level of mathematical thinking decreases through this sequence of tasks. In contrast, a level 5 explanation would, at least, connect the number sequences, the algebraic formulae and the geometric ‘‘staircase’’ representations of this rich mathematical microworld5—and, perhaps, the number theory that underlies the most interesting question: •

Why are the powers of 2 the only numbers that cannot be so represented?

The investigative approach set the style for a flowering of methods and materials by members of the young ATM, who ran conferences during the 1950s and 1960s where there were often demonstration lessons, and always ‘‘workshops’’ in which members could make and explore their own mathematics starting from the multiplicity of materials made available. Some Lessons in Mathematics (Fletcher 1964) brought this approach to the wider mathematics education community. The ATM conference activities continue to this day, sometimes in conjunction with the Mathematical 5

I am grateful to Susie Groves and Kaye Stacey who first introduced me to Consecutive Sums as an investigation in ‘The Burwood Box’, their pioneering work on problem solving in Australian teacher education. HB.

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Association. Similar activities are reported and discussed in the quarterly journal Mathematics Teaching, together with plenty of reportage from the classroom, including conversations with and among children. This group represents a seed bank for ideas and methods of development; however, the seeds have not always found fertile ground. Such groups of exceptional people do not always address the difficulties others may have in adopting the same methods. Handling problem solving in the classroom requires a broader range of teaching skills and a different ‘‘classroom contract’’ (Brousseau 1997) of mutual expectations, unwritten but understood by both teacher and students. Much of the work on problem solving since the 1960s has gone into designing and developing support that enables typical teachers to meet these challenges.

4 Modeling the real world Applications have played a larger role in mathematics and mathematical education in England than elsewhere since the time of Newton. Indeed his influence can still be seen in the emphasis on algebra and calculus and, particularly, in the inclusion of Newtonian Mechanics in the standard Advanced Level Mathematics syllabus for students age 16–18. However, non-routine problems were not on the agenda. A fairly arbitrary collection of standard applications were taught and learned either as illustrative applications of pure topics (compound interest or radioactive decay for exponentials,...) or in Mechanics (projectiles, ladders on rough floors leaning against smooth walls, light inextensible strings over smooth weightless pulleys,...). In 1957 a seminal conference in Oxford brought together the heads of mathematics from many prestigious schools and users of mathematics from industry and research. Their concern was that the mathematics taught in schools should reflect more closely the many uses of the subject in the wider world. Much of the mathematics presented at this meeting by the users was more advanced than could be assimilated into the school curriculum, but one successful outcome was the establishment of a new examination syllabus entitled Mathematics in Education and Industry (MEI); this was mainly taken by pupils in the schools involved in the eponymous project. The increased interest in the relevance of mathematics led to some fine publications showing mathematics in genuine real world applications; Sawyer’s Mathematics in Theory and Practice, Menninger’s Mathematics in Your World (1961) and the BBC’s Mathematics Miscellany (1966), with its associated Schools Broadcasts, are prominent examples. This corresponded with a sense that pupils should not be learning mathematics simply as ‘a trade’, but with a more

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liberal awareness of its place in the world outside school. The focus in this work was on explaining applications rather than developing problem solving by students. In the early 1960s, paralleling the work started by Polya (1945) in pure mathematics, researchers in applied mathematics began to reflect on the processes involved, and to explore the teaching of mathematical modeling at school level. In a recent special issue of ZDM (Burkhardt with Pollak 2006, see also Blum et al 2007) this development was reviewed—from explorations in the 1960s through experimental curricula in the 1970s to established methods of teaching modeling at school and undergraduate level in the 1980s and later. Here we shall only note that the modest success in achieving large-scale implementation at school level has been comparable with mathematical problem solving in general, in no way matching the current need for mathematically literate citizens or flexible professional applied mathematicians.

5 The Cockcroft era In 1976 the Prime Minister, James Callaghan, responded to concerns expressed by some leaders of business and industry with a speech setting out a broad range of goals for improving education. As usual, Mathematics was prominent and, soon after, he appointed a Committee of Inquiry under W. H. Cockcroft, a pure mathematics professor turned university vice-chancellor (president). Trevor Fletcher, Her Majesty’s Staff Inspector for Mathematics, was the committee’s adviser. Fletcher, also a core member of the ATM, was widely respected for his team’s work with schools. The Cockcroft Committee received evidence6 from a wide range of sources and commissioned reviews of the existing research in mathematics education (Bell et al. 1983). The Cockcroft report (1982) was unusually careful and wide ranging, covering principles for assessment as well as curriculum. [‘‘Examinations should enable students to show what they know, understand and can do (as distinct from what they cannot do)’’] Its most quoted paragraph, number 243, set out the principles of curriculum in disarmingly simple terms: ‘‘Mathematics teaching at all levels should include opportunities for • • • •

6

Exposition by the teacher; Discussion between teacher and pupils and among pupils themselves; Appropriate practical work; Consolidation and practice of fundamental skills and routines;

‘‘Evidence’’ to British government committees is opinion, with or without evidential support beyond anecdotes.

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• •

Problem solving, including the application of mathematics to everyday situations; Investigational work.’’

The Cockcroft Report was warmly welcomed by both policy makers and the mathematics education community, internationally as well as in the UK. It led to a decade of solid progress in mathematics education. Advisory teachers7 were appointed to help groups of schools implement the report, complementing the work of mathematics advisers in local school systems. Examination boards sought to improve their examinations to reflect Cockcroft recommendations. The Shell Centre for Mathematical Education at the University of Nottingham worked with the largest board to introduce new types of task year-by-year to the examination for age 16, developing teaching materials and professional development support aligned with each new task type. Problems with Patterns and Numbers (Shell Centre 1984) introduced problem solving in pure mathematics. The Language of Functions and Graphs (Swan et al. 1986) featured the interpretation of line graphs, using non-routine tasks like Hurdles Race.

Progress was, as always, patchy and misunderstanding remained common. For example, the following year every other examination board had a question just like Hurdles Race, missing the point of its design: asking students to make a novel connection between a familiar piece of mathematical technique and a familiar practical situation. 7

Informally called ‘‘Cockcroft Missionaries’’, they formed a cadre of expertise whose influence can still be found.

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But progress there was. Problem Solving was chosen as a Theme for the 1984 Fifth International Congress in Mathematical Education in Adelaide, Australia; the lively state of both pure problem solving and modeling is set out in the proceedings (Burkhardt et al. 1986) A year later, Alan Schoenfeld (1985) in Mathematical Problem Solving moved the field on from Polya’s reflective work to an empirical research footing, building on the research in cognitive science that had developed since the 1960s. Thinking mathematically (Mason et al. 1982) set out a clear, heuristic programme for developing one’s problem solving skills. Progress continued though the 1980s. The ATM and the Mathematical Association were active and influential. New forms of high-stakes assessment, more closely linked with curriculum, were explored. The mathematics education group at Kings College London developed Graded Assessment in Mathematics, which provided a library of test items classified according to the concept tested. The Shell Centre, again with the Joint Matriculation Board, developed Numeracy through Problem Solving (Shell Centre 1987–89), a modular scheme of real world problem solving projects assessed through transfer examinations. On a broader front Cockcroft, now Sir William and heading the Secondary Examinations Council, moved to integrate the CSE and the more academic GCE examinations at age 16 into the General Certificate of Secondary Education (GCSE), embodying the principles of the Cockcroft Report. As with all administrative changes in high-stakes assessment, the move to GCSE absorbed a great deal of energy. Further problem solving was introduced, notably through ‘‘coursework’’ in which students were required to produce reports on their extended investigations of rich problems. One important feature was a framework whereby, subject to approval, innovative syllabuses and examinations could be introduced in a limited number of schools. The ATM developed such a syllabus and Numeracy through Problem Solving was converted into a GCSE Syllabus, but with less problem solving. The imposition of the common framework of the GCSE was not without loss. It effectively killed off some of the more interesting and hopeful explorations of alternative approaches to assessment, more closely linked to curriculum. Coursework proved burdensome to teachers and students and became more routine, reflecting poor design of the constraints and the limited support offered to teachers. However, these setbacks were nothing compared to what followed.

6 The national curriculum: an accidental disaster For the 1987 election, the introduction of ‘‘a national curriculum’’ was not only a headline policy of the Thatcher

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Government; it was also supported by both main opposition parties. Reflecting the views of some vocal industrialists, policy makers decided that standards could be raised by giving ‘‘best practice’’ the standing of ‘‘the law of the land.’’ The model chosen was to legislate a set of Attainment Targets at eight levels for students across age range 5–16. The typical child would be at Level 2 at age 7, at Level 4 at age 11, and so on. (The UK, in contrast with the US where equity concerns are stronger, has always seen attainment as a ladder, up which students move at different speeds, rather than a standard that all should be able to reach.) Programmes of Study would specify how students were to reach these targets, while Assessment and Testing would determine how far they had progressed. The question, of course, was how these elements were to be specified. This turned out very differently in different subjects. In the outcome, the skill ladders in Mathematics were described in largely procedural terms (other subject descriptions were generally more open and flexible). The simplicity of this naı¨ve but deeply flawed model for mathematics was attractive to policy makers, probably reflecting their experience of mathematics at school and limited use thereafter. The senior civil servants steering the group saw a simple relationship between the three elements, with Programmes of Study and Assessment and Testing effectively determined by the Attainment Targets. ‘‘The Attainment Targets specify what they need to know, the Programmes of Study describe how teach it, and the Assessment tests how much of it they know’’ is close to a direct quote. The model was imposed on the National Curriculum Mathematics Working Group, of which one of us (HB) was a member. Our job was to fill in the details—to make a list of Statements of Attainment (‘‘things that a pupil can or can not do’’) for each Level under each Attainment Target: Number, Algebra, Shape and Space, Data and Probability, plus ‘‘Using and Applying.’’ That ‘‘using and applying’’ should have been an aspect of study in each content area, and that tasks involving content from more than one area should have been included, was recognized; it was not taken seriously because it did not fit the model. The following are typical statements of attainment8: Work out fractional and percentage changes and related calculations (Level 6)

8

Research on criterion referencing shows that thousands of much more precise statements are needed to define levels unambiguously. The policy makers saw that this was absurd but, not recognizing that the problem is fundamental, restricted the number to ten or so per level by making the statements broad and imprecise.

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Understand and use formulae, functions, equations and inequalities expressed in words (Level 5); and expressed symbolically (Level 6). The checklist approach was reinforced by the requirements of legislative draftsmen—the law of the land must be clear and unambiguous—and by a ‘‘back to basics’’ approach taken by some influential members of the working group who came from outside education. This naı¨ve view of criterion-referencing ignores the fact that the difficulty of a task is not just that of its separate parts, but that it depends on a combination of factors including: complexity, familiarity, conceptual and technical demand. The Attainment Targets describe elements of performance that could be tested in many ways: separately as short items, or as part of large integrated projects, or by many types of task in between. These constitute very different kinds of performance in the subject—as different as a spelling test and an extended essay in English.9 Ironically, these three aspects, if specified separately and fully exemplified, can provide a reasonable model for specifying a curriculum (Burkhardt 1987). The Working Group attempted to do this by including in its report (DES 1988) a 40-page appendix of assessment tasks exemplifying the recommended range of performance targets. They are mostly non-routine problems, for example: A milk crate holds 24 bottles and is shaped like this:

The crate has four rows and six columns. Can you put 18 bottles of milk in the crate so that each row and each column of the crate has an even number of bottles in it? Is there only one way to do it?

Contrast this with example, below, of the fragmented test items that emerged. This is not a coincidence. Levels belong to tasks, not to the constituent elements of tasks. The only way that elements of performance can be given a consistent Level is if they are always tested separately. That this does not 9

In contrast, English was allowed process-based attainment targets: Reading, Writing, Speaking and Listening. This probably reflected the deeper understanding of both politicians and civil servants, for whom English is their working toolkit while, in this land of C.P.Snow’s Two Cultures, their mathematical literacy is limited (see Sect. 7).

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represent performance in Mathematics, insofar as it was understood, was merely seen by civil servants and politicians as unfortunate10; it was ignored to sustain the criterion-referenced model, which those with decision-making power found plausible. The examination boards were instructed to assess the statements of attainment, with a specified number of marks (score points) on a test for each level. Each score point must directly relate to one statement of attainment. In consequence problem solving, indeed any extended chain of reasoning, was eliminated from the tests and, since these had high stakes attached, from the implemented curriculum in most classrooms. Ironically, the Statements of Attainment were soon recognized as a naive view of criterion-referenced assessment and abolished a few years later; however, by then the pattern of fragmentary item design was established. It remains to this day—the average reasoning length11 is about 90 seconds. Thinking with mathematics, like writing essays, is not like that. There has been some continuing activity in developing problem solving. For example, the Government funded the development, aimed at gifted students of ‘‘World Class Tests’’ of Problem Solving in Mathematics, Science and Technology with associated teaching material (MARS 2005). Research continues, but without impact on most classrooms.

10

The appendix of tasks was removed from subsequent revisions of the report on the spurious grounds that there was a different, crosssubject working group on assessment. 11 Reasoning length is the time a competent student spends on a prompted piece of a task. It differs from task length for tasks with several parts.

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We have related this story in some detail because it illustrates the way policy decisions, taken on plausible grounds by people with good intentions but limited understanding, frequently have unintended consequences that undermine the very goals they seek to advance.

7 A new opportunity: ‘‘Functional Mathematics’’ In 2001, in response to another wave of concern from industry and the universities at the levels of performance of UK students and the falling numbers of students who choose to study mathematics throughout their schooling, the Government appointed two committees. Adrian Smith (Smith Report 2004), a distinguished statistician turned university president, chaired a committee on secondary mathematics, while the Tomlinson Committee (Tomlinson Report 2004) re-examined the whole pattern of education from age 14 to 19. Out of their reports and subsequent discussion a consensus emerged on one thing—the importance of ‘‘Functional Mathematics’’. Government decided it should be one of the educational goals for every student. But what is it? There is broad agreement that Functional Mathematics is the ability to use mathematics to think about problems in the real world—essentially what the rest of the world calls mathematical literacy (see e.g. PISA 2003): Mathematical literacy is an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen.

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However, there are divergent views on what this should mean in curriculum and assessment terms. On the one hand, there are those who want to focus directly on mathematical literacy, teaching students how to model real world problems with mathematics, and assessing them with such tasks. The good news is that this seems to be the dominant view at policy level, and among most of the leaders from industry who speak on this matter. Even more surprising, given our fissiparous tendencies, this seems to be a consensus within the UK mathematics education community. On the other hand, there are still many who see the cause of mathematical illiteracy as lack of reliable skills in arithmetic and wish to focus on that, leaving problem solving to other people or other times. Since 1989 a new teaching area has emerged called ‘‘basic skills,’’ ‘‘key skills,’’ or ‘‘life skills.’’ Those who teach the numeracy12 aspects of this are rarely mathematically trained. They are fiercely protective of their students and believe that the challenges of real problem solving would make it even harder for them to get an essential qualification. The bad news is that, being closer to the status quo and presenting fewer challenges to teachers (and politicians), this view is more likely to prevail. The mechanisms for implementing Functional Mathematics are not in place. In particular, Government has given the job of designing pilot tests to the examination boards, whose designers of mathematics tests have experience only with short imitative items. Further, the thinking on assessment design is dominated by the multiplicity of

constraints, noted above, that preclude the kind of substantial tasks that real world problem solving inevitably involves. Given the dominant influence of tests on curriculum, the outlook is bleak if these tests are adopted. However, in the past the Government has sometimes recognized that new challenges need a more powerful approach to design and development. Their key advisers are advocating it. There is British expertise and experience in designing curriculum and assessment in functional mathematics. It is still an open question as to whether this research-based approach to implementing functional mathematics will be chosen. Otherwise, the initiative of 2004 will go the same way as those of 1976 and 1988—for another school generation there will be no problem solving in UK Mathematics curricula and most adults will still not use any of the mathematics they were taught after age 10 in their everyday lives. We remain cautiously hopeful—but we are not holding our breath!

8 Some lessons from history? We hope this story largely speaks for itself. The occurrence of various forms of problem solving in English schools is summarized in the diagram below. In conclusion, we will point out a few points that may not be obvious.

Non-routine problem solving in the UK In many innovative classrooms In most classrooms at some ages 1870

1950s

Mathematical Association 1870 Non-routine 'riders' and 'exercises' ATM 1951 Mathematical investigation Cockroft report 1982 Non-routine problems Modeliing 'real world' problems National Curriculum 1989 Functional mathematics 2005 Mathematical literacy

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‘‘Numeracy’’ has become a deeply ambiguous term. Its original definition (Crowther 1959) as ‘‘the mathematical equivalent of literacy’’ has been corrupted until, for most people, it means reliable skills in arithmetic (as though literacy were only spelling and grammar).

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1960s

1970s

1980s

1990s

2000s

2010s

?

•

The story set out above (and the references) show that enough is known, both of the importance of learning to solve non-routine problems and how to teach the skills involved; the causes of its absence lie elsewhere.

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•

•

•

•

•

Failures in introducing novel elements into the implemented curriculum largely arise from the challenge being underestimated. Governments tend to assume that policy decisions will be implemented, and on time, independent of the level of support they provide. The education professions, though they often complain about government initiatives, never say ‘‘we can’t do this’’—however far outside their current expertise ‘‘this’’ may be.13 The mathematical education community needs to be more effective in communicating to policy makers and the public changes that are needed, so as to overcome the general view that ‘‘mathematics is what I had at school’’. The importance for successful innovation of human capacity, skilled in research-based design and development, is not yet recognized, let alone developed and supported (Burkhardt and Schoenfeld 2003). ‘‘The road to hell is paved with good intentions.’’

References BBC (1966). Mathematics miscellany. London: BBC Publications. Bell, A. W., Costello, J., & Ku¨chemann, D. E. (1983). A review of research on learning and teaching. Windsor: NFER-Nelson. Blum W., Galbraith P., Henn W., & Niss M. (Eds.) (2007). Modelling and applications in mathematics education. Heidelberg: Springer Academics (former Kluwer Academics). Brousseau, G. (1997). Theory of didactical situations in mathematics (Didactique des mathe´matiques), 1970–1990. In N. Balacheff (Ed. and trans., Dordrecht) Netherlands: Kluwer. Burkhardt, H. (1987). On specifying a curriculum. In I. Wirszup, & R. Streit (Eds.), Developments in school mathematics around the world (pp. 3–30). Reston, VA: National Council of Teachers of Mathematics. Burkhardt, H., with contributions by Pollak, H.O. (2006). Modelling in mathematics classrooms: reflections on past developments and the future. Zeitschrift fu¨r Didaktik der Mathematik, 38(2), 178– 195. Burkhardt, H., Groves, S., Schoenfeld, A. H., & Stacey K. (1986). Problem solving: a world view. Nottingham: Shell Centre Publications.

403 Burkhardt, H., & Schoenfeld, A. H. (2003). Improving Educational Research: towards a more useful, more influential and better funded enterprise. Educational Researcher, 32, 3–14. Cockcroft, W. H. (1982). Mathematics Counts (Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, under the chairmanship of Dr. W. H. Cockcroft.). London: HMSO. Crowther Report 15–18 (1959). A report of the central advisory council for education. London: HMSO. DES (1988). Mathematics for ages 5 to 16: proposals of the secretary of state for education and science and the secretary of state for wales. London: HMSO. Fletcher, T.J. (Ed.) (1964). Some lessons in mathematics. Cambridge: Cambridge University Press. MARS (2005). The MARS Shell Centre Team: Swan, M., Crust, R., Pead, D., Burkhardt, H. et al. World class tests and developing problem solving. London: Nelson. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. Upper Saddle River, Prentice Hall, NJ Menninger, K. (1961). Mathematics in your world. London: G Bell. PISA (2003). The PISA 2003 assessment framework: mathematics, reading, science and problem solving knowledge and skills. Paris: OECD. Po´lya, G. (1945; 2nd edition, 1957). How to solve it. Princeton: Princeton University Press. Sawyer, W.W. (Ed.) (1948). Mathematics in Theory and Practice. London: Odhams. Schoenfeld, A.H. (1985). Mathematical problem solving. Orlando, FL: Academic. Shell Centre (1984). Swan, M., Pitt, J., Fraser, R. E., & Burkhardt, H., with the Shell Centre team, Problems with Patterns and Numbers. Manchester, UK: Joint Matriculation Board; revised 2000. Nottingham, U.K.: Shell Centre Publications. Shell Centre (1987–89). Swan, M., Binns, B., Gillespie, J., & Burkhardt, H., Numeracy through problem solving, five modules for curriculum and assessment in functional mathematical literacy. Harlow: Longman, revised 2000, Nottingham, U.K.: Shell Centre Publications. Smith Report (2004). Post-14 Mathematics Inquiry Steering Group. Making mathematics count. U.K. Department For Education and Skills, London: HMSO. Swan, M. with the Shell Centre team (1986). The language of functions and graphs. Manchester, U.K.: Joint Matriculation Board, reprinted 2000, Nottingham, U.K.: Shell Centre Publications. Tomlinson Report (2004). Working Group on 14–19 Reform. 14–19 curriculum and qualifications reform. Department For Education and Skills. London: HMSO.

13

Imagine: ‘‘To reduces greenhouse gases, we will go over to nuclear fusion for electric power next year.’’ ‘‘We will now provide effective cures for cancer to all who need it.’’ Why has education no sense of feasibility?

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ORIGINAL ARTICLE

Problem solving in the United Kingdom Hugh Burkhardt Æ Alan Bell

Accepted: 1 June 2007 / Published online: 14 August 2007 FIZ Karlsruhe 2007

Abstract We trace the development of problem solving in UK school mathematics over the last century or so, illuminating our descriptions with task exemplars. This is an informative and cautionary tale of mutual incomprehension between leaders in mathematical education and the public they seek to serve. Intelligent and energetic pioneers have developed and improved ways to teach problem solving, though often with little attention to the challenges these methods present to typical teachers. Political decision makers, despite rhetorical support for ‘‘real-world’’ problem solving, have failed to understand the need for the changes proposed. Currently, there are some hopeful signs but it is doubtful if they will be realized in classroom practice. The challenge of modifying the system dynamics so as to yield large-scale improvements remains an unsolved problem in the UK, as elsewhere; at least, it is now recognized and being worked on.

1 Overview Mathematics has always had two surface aspects, often called skills and problem solving. Skills are seen as a set of well-defined procedures for transforming numbers, symbols or shapes; the role of the ideal performer is that of an automaton, and the role of the student that of learning to be

H. Burkhardt (&) A. Bell Shell Centre for Mathematical Education, School of Education, University of Nottingham, Nottingham NG8 1BB, UK e-mail: [email protected]

a reliable one. Problem solving, in contrast, involves tackling tasks that are significantly different from those one has learned ‘‘by heart’’; a major part of the challenge lies in deciding how to tackle the problem, and which bits of one’s toolkit of mathematical skills will help. A problem solver needs a rich, connected understanding of mathematics and the ability to see patterns of similarity and association, as well as the skills to carry out the planned attack, and to check that the results make sense in the context of the problem. No wonder these are called ‘‘higher level skills’’. Historically, different people needed very different kinds of mathematical capability. Skills were enough to guarantee ‘‘skilled employment’’, as bookkeepers or surveyors for example, with the higher pay and status that this brings. Apprenticeships covered all the kinds of task that the profession involved, and the procedures for dealing with them. Expertise was a matter of applying ‘‘best practice,’’ as it still largely is—one prefers one’s brain surgeon or car mechanic to employ well-learned skills rather than to be ‘‘problem solving.’’ Equally there have always been problem solvers. Every theorem was a problem once, and every new algorithmic procedure an improved solution. But problem solving was seen as a specialized skill, only for mathematicians or others who saw the power of mathematics in opening up new fields. Thus geometry developed from the needs of surveying. Newton invented calculus to enable him to prove that an inverse square law of gravitation, along with his laws of mechanics, lead to closed elliptical orbits for the planets. (He then translated the proof into Euclidean Geometry so his contemporaries could understand it.) Mathematicians worked to extend the understanding and power of each innovation. When new procedures emerged, as in navigation, they became part of

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professional training. In short, problem solving was the domain of a few.1 In such a situation, it is not surprising that mathematics education at school level concentrated on developing skills, leaving problem solving to more specialized contexts, mostly in graduate schools. However, practical circumstances have changed, with human automata losing their jobs all over the world to reliable, and much less expensive, calculators and computers. Life and work have become more mathematical. Employers, facing ever-changing demands, require flexibility that precludes teaching all the necessary skills in advance. These changes have produced pressures for mathematical literacy, and for problem solving in school mathematics education. We will outline what has happened in the UK, particularly England and Wales.

2 Early history During the nineteenth Century it gradually became recognized that the rote learning of procedures was inadequate, and that mathematical reasoning was important. A Royal Commission, reporting on the state of mathematics teaching in nine leading Public (i.e., private) Schools in 1837, noted that the typical two weekly hours of mathematics consisted of Arithmetic, a little manipulative Algebra, and ‘‘Euclid’’, learned by rote. By 1861, about half the boys leaving the sixth form at Harrow School had done six books of Euclid, some trigonometry and quadratic equations, and a little Conic Sections and Mechanics. Of the subjects examined by the London Matriculation Board, ‘‘the Arithmetic was heavily commercial, the Algebra formal, and for Geometry there was Euclid’’. Euclid was supposed to provide a training in logic, but it was ‘‘not well taught,’’ and little logical thinking was acquired. In 1870, Rawdon Levett, the only mathematics master teaching the 400 boys of King Edward’s School, Birmingham, proposed the founding of an anti-Euclid association. This was named The Association for Improvement of Geometry Teaching (AIGT), and was the forerunner of the current Mathematical Association. This movement gradually led to a new kind of task in Geometry. Students were first asked to prove a theorem they had been taught. This was followed by a ‘‘rider’’—a non-routine problem related to the proof. The following example comes from a 1947 examination in ‘‘Elementary Mathematics’’:

H. Burkhardt, A. Bell

Two chords AB and CD of a circle, when produced, cut each other at a point O outside the circle. Prove that OA.OB = OC.OD.

Two unequal circles touch each other externally at a point R. The diameter through R of the larger circle is RP, and that of the smaller circle is RQ. An exterior common tangent touches the larger circle at S and the smaller circle at T, and it cuts PQ produced at X. Prove that

XP XS 2 = XQ XT 2

The rider was recognized as testing mathematical reasoning, rather than simply recall of learned proofs of theorems. Note that, in modern terms, the transfer distance of the rider from the theorem was under the control of the task designer. In this case, the rider is a direct adaptation of the proof of the theorem—some thought rather than just memory is required but the transfer distance is relatively short. In other questions, it was sometimes greater. Arithmetic and Algebra textbooks contained ‘‘exercises’’, the hardest of which could demand relatively complex reasoning:

Find the maximum rectangle which can be inscribed in a given semicircle.

Though expressed geometrically, this involves formulating and transforming an algebraic quadratic. In this way, non-routine problem solving entered the English school curriculum—at least for the more able students. As we look at the development of problem solving up to the present, it is salutary to ask how current non-routine tasks compare with those from the past. From about 1940 onwards, the design of mathematics curricula moved from an accepted historical fact to become a subject of discussion, and of controversy. Innovators in mathematics education began to point out that problem solving is an integral part of ‘‘doing mathematics,’’ and should play a similar role in school education. In 1938 the Spens Report on Secondary Education, in a critical comment on current mathematics teaching, asserted that ‘‘it should be taught as Art and Music and Physical Science, as one of the main lines followed by the creative spirit of man’’. Elsewhere, it suggests that ‘‘the topics should be chosen to develop a grasp of mathematical ideas.... and practical questions which have been of urgent interest and utility to man in his affairs.’’ These complementary views of mathematics as: •

1

It is said that the Japanese Government once decided that only 2% of students need to learn problem solving.

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•

an important aspect of culture to be studied for its own sake a powerful toolkit for solving practical problems

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persist to this day. Some mathematicians and teachers regard the former as sufficient2, and utility as an inferior justification for learning mathematics, reflecting G.H. Hardy’s famous toast: ‘‘To pure mathematics; may it never be applied.’’ However, it is power in application that is persuasive to policy makers and to many users of mathematics who, while enjoying its charms per se, live by applying its power to other problems.3

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4. 5.

The following example, Consecutive Sums, illustrates these distinctions. •

3 The coming of ‘‘investigation’’ An important strand of development after World War II focused on the classroom experiences of learners, in particular on engaging students in activities more nearly like those of practicing mathematicians. Like the Mathematical Association in the previous century, the Association of Teachers of Mathematics (ATM) was formed4 in 1951 to push back the frontiers of school mathematics. Influences from psychology, stemming from Piaget and Gattegno, were important. Piaget showed how the concepts actually held by children differed markedly and unexpectedly from the corresponding adult ideas—for example, that a row of beads spread out had ‘‘more’’ than when closer together, and that in a bunch of many poppies and a few lilies there were ‘‘more poppies’’ than flowers— showing the difficulties of simultaneously recognizing the whole and the part, and of technical versus informal language. Gattegno’s emphasis was on creating and manipulating situations in which children could see or make mathematics. They were led to ‘‘read’’ the mathematics shown by patterns of colored (Cuisenaire) rods, for example a red and a yellow rod matching a black rod, showing that 2 + 5 = 7—but also that 5 + 2 = 7, and that 7 could be made up in other ways from smaller numbers. Further research into learning mathematics showed that rote-learned procedures remain reliable only if they are practiced regularly, and that long term learning depends on conceptual understanding of the basis of the procedure. The learning of mathematics was coming to be seen as working at various levels, according to various similar schemes; we choose to describe them as: 1. 2. 3.

2

Rote learning, memorization and skill practice Meaningful learning of concepts and skills Learning to solve problems

Though can one, on cultural grounds alone, justify giving mathematics more curriculum time than, say, music? 3 Alas for both these perspectives, interview data suggest that the overwhelming majority of students see school mathematics as just a route to a necessary qualification. 4 As the Association for Teaching Aids in Mathematics.

Posing problems in making investigations of open problem situations Describing mathematical structures embodied in a practical situation (applied maths) and relationships within and between these structures (pure maths).

Investigate the properties of numbers that can be represented as sums of consecutive whole numbers as, for example: 9¼4þ5 ¼2þ3þ4 proving your conjectures where you can.

This is level 4, while a level 3 problem version is: •

Prove that the sum of 5 consecutive whole numbers is divisible by 5 and, similarly, for any odd number. Related tasks at level 2 and level 1 are: • Explain why the sum of an arithmetic progression is the average of the first and last terms multiplied by the number of terms • Write the formula for the sum of an arithmetic progression. Note how the level of mathematical thinking decreases through this sequence of tasks. In contrast, a level 5 explanation would, at least, connect the number sequences, the algebraic formulae and the geometric ‘‘staircase’’ representations of this rich mathematical microworld5—and, perhaps, the number theory that underlies the most interesting question: •

Why are the powers of 2 the only numbers that cannot be so represented?

The investigative approach set the style for a flowering of methods and materials by members of the young ATM, who ran conferences during the 1950s and 1960s where there were often demonstration lessons, and always ‘‘workshops’’ in which members could make and explore their own mathematics starting from the multiplicity of materials made available. Some Lessons in Mathematics (Fletcher 1964) brought this approach to the wider mathematics education community. The ATM conference activities continue to this day, sometimes in conjunction with the Mathematical 5

I am grateful to Susie Groves and Kaye Stacey who first introduced me to Consecutive Sums as an investigation in ‘The Burwood Box’, their pioneering work on problem solving in Australian teacher education. HB.

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Association. Similar activities are reported and discussed in the quarterly journal Mathematics Teaching, together with plenty of reportage from the classroom, including conversations with and among children. This group represents a seed bank for ideas and methods of development; however, the seeds have not always found fertile ground. Such groups of exceptional people do not always address the difficulties others may have in adopting the same methods. Handling problem solving in the classroom requires a broader range of teaching skills and a different ‘‘classroom contract’’ (Brousseau 1997) of mutual expectations, unwritten but understood by both teacher and students. Much of the work on problem solving since the 1960s has gone into designing and developing support that enables typical teachers to meet these challenges.

4 Modeling the real world Applications have played a larger role in mathematics and mathematical education in England than elsewhere since the time of Newton. Indeed his influence can still be seen in the emphasis on algebra and calculus and, particularly, in the inclusion of Newtonian Mechanics in the standard Advanced Level Mathematics syllabus for students age 16–18. However, non-routine problems were not on the agenda. A fairly arbitrary collection of standard applications were taught and learned either as illustrative applications of pure topics (compound interest or radioactive decay for exponentials,...) or in Mechanics (projectiles, ladders on rough floors leaning against smooth walls, light inextensible strings over smooth weightless pulleys,...). In 1957 a seminal conference in Oxford brought together the heads of mathematics from many prestigious schools and users of mathematics from industry and research. Their concern was that the mathematics taught in schools should reflect more closely the many uses of the subject in the wider world. Much of the mathematics presented at this meeting by the users was more advanced than could be assimilated into the school curriculum, but one successful outcome was the establishment of a new examination syllabus entitled Mathematics in Education and Industry (MEI); this was mainly taken by pupils in the schools involved in the eponymous project. The increased interest in the relevance of mathematics led to some fine publications showing mathematics in genuine real world applications; Sawyer’s Mathematics in Theory and Practice, Menninger’s Mathematics in Your World (1961) and the BBC’s Mathematics Miscellany (1966), with its associated Schools Broadcasts, are prominent examples. This corresponded with a sense that pupils should not be learning mathematics simply as ‘a trade’, but with a more

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liberal awareness of its place in the world outside school. The focus in this work was on explaining applications rather than developing problem solving by students. In the early 1960s, paralleling the work started by Polya (1945) in pure mathematics, researchers in applied mathematics began to reflect on the processes involved, and to explore the teaching of mathematical modeling at school level. In a recent special issue of ZDM (Burkhardt with Pollak 2006, see also Blum et al 2007) this development was reviewed—from explorations in the 1960s through experimental curricula in the 1970s to established methods of teaching modeling at school and undergraduate level in the 1980s and later. Here we shall only note that the modest success in achieving large-scale implementation at school level has been comparable with mathematical problem solving in general, in no way matching the current need for mathematically literate citizens or flexible professional applied mathematicians.

5 The Cockcroft era In 1976 the Prime Minister, James Callaghan, responded to concerns expressed by some leaders of business and industry with a speech setting out a broad range of goals for improving education. As usual, Mathematics was prominent and, soon after, he appointed a Committee of Inquiry under W. H. Cockcroft, a pure mathematics professor turned university vice-chancellor (president). Trevor Fletcher, Her Majesty’s Staff Inspector for Mathematics, was the committee’s adviser. Fletcher, also a core member of the ATM, was widely respected for his team’s work with schools. The Cockcroft Committee received evidence6 from a wide range of sources and commissioned reviews of the existing research in mathematics education (Bell et al. 1983). The Cockcroft report (1982) was unusually careful and wide ranging, covering principles for assessment as well as curriculum. [‘‘Examinations should enable students to show what they know, understand and can do (as distinct from what they cannot do)’’] Its most quoted paragraph, number 243, set out the principles of curriculum in disarmingly simple terms: ‘‘Mathematics teaching at all levels should include opportunities for • • • •

6

Exposition by the teacher; Discussion between teacher and pupils and among pupils themselves; Appropriate practical work; Consolidation and practice of fundamental skills and routines;

‘‘Evidence’’ to British government committees is opinion, with or without evidential support beyond anecdotes.

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• •

Problem solving, including the application of mathematics to everyday situations; Investigational work.’’

The Cockcroft Report was warmly welcomed by both policy makers and the mathematics education community, internationally as well as in the UK. It led to a decade of solid progress in mathematics education. Advisory teachers7 were appointed to help groups of schools implement the report, complementing the work of mathematics advisers in local school systems. Examination boards sought to improve their examinations to reflect Cockcroft recommendations. The Shell Centre for Mathematical Education at the University of Nottingham worked with the largest board to introduce new types of task year-by-year to the examination for age 16, developing teaching materials and professional development support aligned with each new task type. Problems with Patterns and Numbers (Shell Centre 1984) introduced problem solving in pure mathematics. The Language of Functions and Graphs (Swan et al. 1986) featured the interpretation of line graphs, using non-routine tasks like Hurdles Race.

Progress was, as always, patchy and misunderstanding remained common. For example, the following year every other examination board had a question just like Hurdles Race, missing the point of its design: asking students to make a novel connection between a familiar piece of mathematical technique and a familiar practical situation. 7

Informally called ‘‘Cockcroft Missionaries’’, they formed a cadre of expertise whose influence can still be found.

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But progress there was. Problem Solving was chosen as a Theme for the 1984 Fifth International Congress in Mathematical Education in Adelaide, Australia; the lively state of both pure problem solving and modeling is set out in the proceedings (Burkhardt et al. 1986) A year later, Alan Schoenfeld (1985) in Mathematical Problem Solving moved the field on from Polya’s reflective work to an empirical research footing, building on the research in cognitive science that had developed since the 1960s. Thinking mathematically (Mason et al. 1982) set out a clear, heuristic programme for developing one’s problem solving skills. Progress continued though the 1980s. The ATM and the Mathematical Association were active and influential. New forms of high-stakes assessment, more closely linked with curriculum, were explored. The mathematics education group at Kings College London developed Graded Assessment in Mathematics, which provided a library of test items classified according to the concept tested. The Shell Centre, again with the Joint Matriculation Board, developed Numeracy through Problem Solving (Shell Centre 1987–89), a modular scheme of real world problem solving projects assessed through transfer examinations. On a broader front Cockcroft, now Sir William and heading the Secondary Examinations Council, moved to integrate the CSE and the more academic GCE examinations at age 16 into the General Certificate of Secondary Education (GCSE), embodying the principles of the Cockcroft Report. As with all administrative changes in high-stakes assessment, the move to GCSE absorbed a great deal of energy. Further problem solving was introduced, notably through ‘‘coursework’’ in which students were required to produce reports on their extended investigations of rich problems. One important feature was a framework whereby, subject to approval, innovative syllabuses and examinations could be introduced in a limited number of schools. The ATM developed such a syllabus and Numeracy through Problem Solving was converted into a GCSE Syllabus, but with less problem solving. The imposition of the common framework of the GCSE was not without loss. It effectively killed off some of the more interesting and hopeful explorations of alternative approaches to assessment, more closely linked to curriculum. Coursework proved burdensome to teachers and students and became more routine, reflecting poor design of the constraints and the limited support offered to teachers. However, these setbacks were nothing compared to what followed.

6 The national curriculum: an accidental disaster For the 1987 election, the introduction of ‘‘a national curriculum’’ was not only a headline policy of the Thatcher

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Government; it was also supported by both main opposition parties. Reflecting the views of some vocal industrialists, policy makers decided that standards could be raised by giving ‘‘best practice’’ the standing of ‘‘the law of the land.’’ The model chosen was to legislate a set of Attainment Targets at eight levels for students across age range 5–16. The typical child would be at Level 2 at age 7, at Level 4 at age 11, and so on. (The UK, in contrast with the US where equity concerns are stronger, has always seen attainment as a ladder, up which students move at different speeds, rather than a standard that all should be able to reach.) Programmes of Study would specify how students were to reach these targets, while Assessment and Testing would determine how far they had progressed. The question, of course, was how these elements were to be specified. This turned out very differently in different subjects. In the outcome, the skill ladders in Mathematics were described in largely procedural terms (other subject descriptions were generally more open and flexible). The simplicity of this naı¨ve but deeply flawed model for mathematics was attractive to policy makers, probably reflecting their experience of mathematics at school and limited use thereafter. The senior civil servants steering the group saw a simple relationship between the three elements, with Programmes of Study and Assessment and Testing effectively determined by the Attainment Targets. ‘‘The Attainment Targets specify what they need to know, the Programmes of Study describe how teach it, and the Assessment tests how much of it they know’’ is close to a direct quote. The model was imposed on the National Curriculum Mathematics Working Group, of which one of us (HB) was a member. Our job was to fill in the details—to make a list of Statements of Attainment (‘‘things that a pupil can or can not do’’) for each Level under each Attainment Target: Number, Algebra, Shape and Space, Data and Probability, plus ‘‘Using and Applying.’’ That ‘‘using and applying’’ should have been an aspect of study in each content area, and that tasks involving content from more than one area should have been included, was recognized; it was not taken seriously because it did not fit the model. The following are typical statements of attainment8: Work out fractional and percentage changes and related calculations (Level 6)

8

Research on criterion referencing shows that thousands of much more precise statements are needed to define levels unambiguously. The policy makers saw that this was absurd but, not recognizing that the problem is fundamental, restricted the number to ten or so per level by making the statements broad and imprecise.

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Understand and use formulae, functions, equations and inequalities expressed in words (Level 5); and expressed symbolically (Level 6). The checklist approach was reinforced by the requirements of legislative draftsmen—the law of the land must be clear and unambiguous—and by a ‘‘back to basics’’ approach taken by some influential members of the working group who came from outside education. This naı¨ve view of criterion-referencing ignores the fact that the difficulty of a task is not just that of its separate parts, but that it depends on a combination of factors including: complexity, familiarity, conceptual and technical demand. The Attainment Targets describe elements of performance that could be tested in many ways: separately as short items, or as part of large integrated projects, or by many types of task in between. These constitute very different kinds of performance in the subject—as different as a spelling test and an extended essay in English.9 Ironically, these three aspects, if specified separately and fully exemplified, can provide a reasonable model for specifying a curriculum (Burkhardt 1987). The Working Group attempted to do this by including in its report (DES 1988) a 40-page appendix of assessment tasks exemplifying the recommended range of performance targets. They are mostly non-routine problems, for example: A milk crate holds 24 bottles and is shaped like this:

The crate has four rows and six columns. Can you put 18 bottles of milk in the crate so that each row and each column of the crate has an even number of bottles in it? Is there only one way to do it?

Contrast this with example, below, of the fragmented test items that emerged. This is not a coincidence. Levels belong to tasks, not to the constituent elements of tasks. The only way that elements of performance can be given a consistent Level is if they are always tested separately. That this does not 9

In contrast, English was allowed process-based attainment targets: Reading, Writing, Speaking and Listening. This probably reflected the deeper understanding of both politicians and civil servants, for whom English is their working toolkit while, in this land of C.P.Snow’s Two Cultures, their mathematical literacy is limited (see Sect. 7).

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represent performance in Mathematics, insofar as it was understood, was merely seen by civil servants and politicians as unfortunate10; it was ignored to sustain the criterion-referenced model, which those with decision-making power found plausible. The examination boards were instructed to assess the statements of attainment, with a specified number of marks (score points) on a test for each level. Each score point must directly relate to one statement of attainment. In consequence problem solving, indeed any extended chain of reasoning, was eliminated from the tests and, since these had high stakes attached, from the implemented curriculum in most classrooms. Ironically, the Statements of Attainment were soon recognized as a naive view of criterion-referenced assessment and abolished a few years later; however, by then the pattern of fragmentary item design was established. It remains to this day—the average reasoning length11 is about 90 seconds. Thinking with mathematics, like writing essays, is not like that. There has been some continuing activity in developing problem solving. For example, the Government funded the development, aimed at gifted students of ‘‘World Class Tests’’ of Problem Solving in Mathematics, Science and Technology with associated teaching material (MARS 2005). Research continues, but without impact on most classrooms.

10

The appendix of tasks was removed from subsequent revisions of the report on the spurious grounds that there was a different, crosssubject working group on assessment. 11 Reasoning length is the time a competent student spends on a prompted piece of a task. It differs from task length for tasks with several parts.

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We have related this story in some detail because it illustrates the way policy decisions, taken on plausible grounds by people with good intentions but limited understanding, frequently have unintended consequences that undermine the very goals they seek to advance.

7 A new opportunity: ‘‘Functional Mathematics’’ In 2001, in response to another wave of concern from industry and the universities at the levels of performance of UK students and the falling numbers of students who choose to study mathematics throughout their schooling, the Government appointed two committees. Adrian Smith (Smith Report 2004), a distinguished statistician turned university president, chaired a committee on secondary mathematics, while the Tomlinson Committee (Tomlinson Report 2004) re-examined the whole pattern of education from age 14 to 19. Out of their reports and subsequent discussion a consensus emerged on one thing—the importance of ‘‘Functional Mathematics’’. Government decided it should be one of the educational goals for every student. But what is it? There is broad agreement that Functional Mathematics is the ability to use mathematics to think about problems in the real world—essentially what the rest of the world calls mathematical literacy (see e.g. PISA 2003): Mathematical literacy is an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen.

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However, there are divergent views on what this should mean in curriculum and assessment terms. On the one hand, there are those who want to focus directly on mathematical literacy, teaching students how to model real world problems with mathematics, and assessing them with such tasks. The good news is that this seems to be the dominant view at policy level, and among most of the leaders from industry who speak on this matter. Even more surprising, given our fissiparous tendencies, this seems to be a consensus within the UK mathematics education community. On the other hand, there are still many who see the cause of mathematical illiteracy as lack of reliable skills in arithmetic and wish to focus on that, leaving problem solving to other people or other times. Since 1989 a new teaching area has emerged called ‘‘basic skills,’’ ‘‘key skills,’’ or ‘‘life skills.’’ Those who teach the numeracy12 aspects of this are rarely mathematically trained. They are fiercely protective of their students and believe that the challenges of real problem solving would make it even harder for them to get an essential qualification. The bad news is that, being closer to the status quo and presenting fewer challenges to teachers (and politicians), this view is more likely to prevail. The mechanisms for implementing Functional Mathematics are not in place. In particular, Government has given the job of designing pilot tests to the examination boards, whose designers of mathematics tests have experience only with short imitative items. Further, the thinking on assessment design is dominated by the multiplicity of

constraints, noted above, that preclude the kind of substantial tasks that real world problem solving inevitably involves. Given the dominant influence of tests on curriculum, the outlook is bleak if these tests are adopted. However, in the past the Government has sometimes recognized that new challenges need a more powerful approach to design and development. Their key advisers are advocating it. There is British expertise and experience in designing curriculum and assessment in functional mathematics. It is still an open question as to whether this research-based approach to implementing functional mathematics will be chosen. Otherwise, the initiative of 2004 will go the same way as those of 1976 and 1988—for another school generation there will be no problem solving in UK Mathematics curricula and most adults will still not use any of the mathematics they were taught after age 10 in their everyday lives. We remain cautiously hopeful—but we are not holding our breath!

8 Some lessons from history? We hope this story largely speaks for itself. The occurrence of various forms of problem solving in English schools is summarized in the diagram below. In conclusion, we will point out a few points that may not be obvious.

Non-routine problem solving in the UK In many innovative classrooms In most classrooms at some ages 1870

1950s

Mathematical Association 1870 Non-routine 'riders' and 'exercises' ATM 1951 Mathematical investigation Cockroft report 1982 Non-routine problems Modeliing 'real world' problems National Curriculum 1989 Functional mathematics 2005 Mathematical literacy

12

‘‘Numeracy’’ has become a deeply ambiguous term. Its original definition (Crowther 1959) as ‘‘the mathematical equivalent of literacy’’ has been corrupted until, for most people, it means reliable skills in arithmetic (as though literacy were only spelling and grammar).

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1960s

1970s

1980s

1990s

2000s

2010s

?

•

The story set out above (and the references) show that enough is known, both of the importance of learning to solve non-routine problems and how to teach the skills involved; the causes of its absence lie elsewhere.

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•

•

•

•

•

Failures in introducing novel elements into the implemented curriculum largely arise from the challenge being underestimated. Governments tend to assume that policy decisions will be implemented, and on time, independent of the level of support they provide. The education professions, though they often complain about government initiatives, never say ‘‘we can’t do this’’—however far outside their current expertise ‘‘this’’ may be.13 The mathematical education community needs to be more effective in communicating to policy makers and the public changes that are needed, so as to overcome the general view that ‘‘mathematics is what I had at school’’. The importance for successful innovation of human capacity, skilled in research-based design and development, is not yet recognized, let alone developed and supported (Burkhardt and Schoenfeld 2003). ‘‘The road to hell is paved with good intentions.’’

References BBC (1966). Mathematics miscellany. London: BBC Publications. Bell, A. W., Costello, J., & Ku¨chemann, D. E. (1983). A review of research on learning and teaching. Windsor: NFER-Nelson. Blum W., Galbraith P., Henn W., & Niss M. (Eds.) (2007). Modelling and applications in mathematics education. Heidelberg: Springer Academics (former Kluwer Academics). Brousseau, G. (1997). Theory of didactical situations in mathematics (Didactique des mathe´matiques), 1970–1990. In N. Balacheff (Ed. and trans., Dordrecht) Netherlands: Kluwer. Burkhardt, H. (1987). On specifying a curriculum. In I. Wirszup, & R. Streit (Eds.), Developments in school mathematics around the world (pp. 3–30). Reston, VA: National Council of Teachers of Mathematics. Burkhardt, H., with contributions by Pollak, H.O. (2006). Modelling in mathematics classrooms: reflections on past developments and the future. Zeitschrift fu¨r Didaktik der Mathematik, 38(2), 178– 195. Burkhardt, H., Groves, S., Schoenfeld, A. H., & Stacey K. (1986). Problem solving: a world view. Nottingham: Shell Centre Publications.

403 Burkhardt, H., & Schoenfeld, A. H. (2003). Improving Educational Research: towards a more useful, more influential and better funded enterprise. Educational Researcher, 32, 3–14. Cockcroft, W. H. (1982). Mathematics Counts (Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, under the chairmanship of Dr. W. H. Cockcroft.). London: HMSO. Crowther Report 15–18 (1959). A report of the central advisory council for education. London: HMSO. DES (1988). Mathematics for ages 5 to 16: proposals of the secretary of state for education and science and the secretary of state for wales. London: HMSO. Fletcher, T.J. (Ed.) (1964). Some lessons in mathematics. Cambridge: Cambridge University Press. MARS (2005). The MARS Shell Centre Team: Swan, M., Crust, R., Pead, D., Burkhardt, H. et al. World class tests and developing problem solving. London: Nelson. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. Upper Saddle River, Prentice Hall, NJ Menninger, K. (1961). Mathematics in your world. London: G Bell. PISA (2003). The PISA 2003 assessment framework: mathematics, reading, science and problem solving knowledge and skills. Paris: OECD. Po´lya, G. (1945; 2nd edition, 1957). How to solve it. Princeton: Princeton University Press. Sawyer, W.W. (Ed.) (1948). Mathematics in Theory and Practice. London: Odhams. Schoenfeld, A.H. (1985). Mathematical problem solving. Orlando, FL: Academic. Shell Centre (1984). Swan, M., Pitt, J., Fraser, R. E., & Burkhardt, H., with the Shell Centre team, Problems with Patterns and Numbers. Manchester, UK: Joint Matriculation Board; revised 2000. Nottingham, U.K.: Shell Centre Publications. Shell Centre (1987–89). Swan, M., Binns, B., Gillespie, J., & Burkhardt, H., Numeracy through problem solving, five modules for curriculum and assessment in functional mathematical literacy. Harlow: Longman, revised 2000, Nottingham, U.K.: Shell Centre Publications. Smith Report (2004). Post-14 Mathematics Inquiry Steering Group. Making mathematics count. U.K. Department For Education and Skills, London: HMSO. Swan, M. with the Shell Centre team (1986). The language of functions and graphs. Manchester, U.K.: Joint Matriculation Board, reprinted 2000, Nottingham, U.K.: Shell Centre Publications. Tomlinson Report (2004). Working Group on 14–19 Reform. 14–19 curriculum and qualifications reform. Department For Education and Skills. London: HMSO.

13

Imagine: ‘‘To reduces greenhouse gases, we will go over to nuclear fusion for electric power next year.’’ ‘‘We will now provide effective cures for cancer to all who need it.’’ Why has education no sense of feasibility?

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