The National Curriculum or Mathematics Education? - Ncm

7 downloads 1920 Views 26KB Size Report
mathematics as a series of 296 separate ... Richard Noss, Department of Mathematics, Statistics and Computing ..... ce task that he and his class had done . They.
The National Curriculum or Mathematics Education? Richard Noss, Department of Mathematics, Statistics and Computing Institute of Education, University of London, gav denna bild av läroplansarbetet i England och Wales vid ett lärarutbildarseminarium i Borlänge i september 1990.

for example, a written constitution, but that does not prevent there being clearly expressed views about what is, and what is not constitutional. Much the same has been true in the teaching profession. For example we have a national system of certification examinations at age 16. The syllabus for these examinations is tightly specified by the Examination Boards, and naturally, teachers take this very seriously; from the age of 12 or 13, most pupils in most schools throughout the United Kingdom are studying broadly the same mathematics with broadly the same approach. So the first point I want to make is that we are not in a situation of anarchy where pupils are failing to learn mathematics because there is no clarity about what to learn. The second point is that I think it is clear just how much the advent of a ’National’, centrally-controlled curriculum has taken us by surprise and why we have been forced to consider a range of educational issues which we had – perhaps for too long – taken for granted. Before going any further, I should point to one crucial aspect of the National Curriculum: it is not national at all. It only applies to those children who do not attend private schools. In the UK, something like 91 or 92% of children attend state-funded schools and around 8 or 9% of children attend private schools. Every member of the Government has educated his or her child in a private school. This is important background information, and it may be

The UK Parliament has decreed that teachers in England and Wales will teach mathematics as a series of 296 separate ’statements of attainment’. This is now the law of the land. To us in the UK, this is a rather astonishing phenomenon: apart from anything else, the statements of attainment are very finely specified. So for example, one of them says ”understand and apply Pythagoras’ Theorem” [Attainment Target 10, level 7 (D.E.S. 1989)] and another states ”solve linear equations” [Attainment Target 6, level 6 ibid]. Note that this second one is therefore defined as easier than the first. As far as I know, it is unique to the UK that such curricular specificity is enshrined as law. Even in those countries where we have traditionally thought the education system to be most centralized, for example in France, such fine detail is not specified, and certainly not as the law of the land. It is interesting to speculate on what is going to happen to teachers who break the law by, for example, teaching Pythagoras’ theorem before they teach linear equations. As you probably know, British Primary and Secondary teachers have traditionally had a high degree of autonomy in their classrooms: they have been relatively free to teach in a style and with a method which is more their own. There have, of course, been implicit curriculum goals in mathematics education as well as in other subjects. This seems to be a characteristic of the United Kingdom: we don’t even have,

16

about mathematical education. (Many of these questions are discussed further in DOWLING P. & NOSS R. (1990)). I would like to begin with some issues related to mathematics itself. Here is a statement about mathematics from one of the ’Working Party’ reports whose responsibility it was to design the National Curriculum:

helpful in interpreting the interesting fact that private schools do not have to follow the National Curriculum. If it is the case, as the UK Government argues, that the problem of mathematics education is that it is too anarchic and that we need some centralization in order to ’improve standards’, does it not seem odd that private schools are absolved from the responsibility of obeying the law? It is interesting to speculate on the number of laws of this or any other country which are entered into the statute book of its Parliament but which do not apply to 8% of the population. So much for the background. What I would like to do is to use the National Curriculum documents as an excuse to think much more broadly about mathematics education. So for example, I am going to ask questions about the nature of mathematics and the nature of mathematics teaching which seem to me to be traditionally the kinds of issues that people like ourselves might discuss over coffee, but which have not adequately been discussed in the research literature or in teachers’ journals. I want to ask some fundamental questions about the nature of assessment in mathematics education. I will argue that the story of the National Curriculum is the story of evaluation. All of us who have been teachers in the classroom have images of what it means to evaluate mathematical performance; but few of us have spent time wondering about what precisely it is that we are evaluating and why we should evaluate it. I also want to ask some questions about the ways in which mathematicians and mathematics educators, should, or should not be involved in what are essentially questions of politics. The National Curriculum proposals are political proposals and have been introduced by a politically astute government: there is a debate to be had about the extent to which people like us should be involved in these kinds of debates. So as you can see, what I am going to do is to use the National Curriculum as a kind of hook on which to hang some rather fundamental questions

Mathematics .... should also be a source of delight and wonder, offering pupils intellectual excitement, for example, in the discovery of relationships, the pursuit of rigour and the achievement of elegant solutions. Pupils should .also appreciate the essential creativity of mathematics: it is a live subject which is continuously evolving as technology and the needs of society evolve. (DES, 1988, § 2.2) There is not much to disagree with in this statement (although I think I disagree with the last sentence: it seems to me not true at all that as technology evolves we need more mathematics; actually as technology evolves ”we” need less mathematics, but I cannot afford to digress). By and large, anybody with the smallest grain of mathematical sensitivity, the smallest amount of mathematical background would feel happy that such a sentence should be written in a governmental report on Mathematics. A few paragraphs later the same report states: ’We have taken it as axiomatic that the mathematics which pupils learn at school should support the mathematics which they actually need to use in later life, particularly at work’. (DES, 1988, §3.15). There has been a substantial research effort in various parts of the world which has shown that the amount of mathematics which anybody actually needs to use in later life particularly at work is almost zero. Of course I am not talking about mathematicians or engineers or computer scientists, as they are such an insignificant fraction of the world’s population that we can safely ignore them for the moment. In

17

so far as ordinary people need to use mathematical knowledge at all, the kind of mathematical knowledge they need is about that of the average 10 year old in this or any other ’developed’ country. So the question is whether these two statements are compatible. Is it really the case that if we focus our attention on the mathematics which pupils actually need to use in later life, ”particularly at work”, we can at the same time stress the essential creativity, beauty and aesthetics of mathematics? Are these two things not somewhat contradictory? Even if these two positions are compatible, do we really want to devise an educational system where we decide what to teach because it is what is useful in later life particularly at work? If that is so, is this only the case in mathematics? Do we apply this criterion to the study of art or music? Perhaps mathematics is special? Perhaps mathematics is only worthy of study because it is useful? There is a very interesting view in a slightly later report, which tells us something about the vision that those responsible for the introduction of the National Curriculum have of mathematics; it seems to me that it is not necessarily a vision which we ought to accept uncritically. When talking of mathematics, they say, pupils should ”apply ” it [mathematics] sensibly and efficiently”, they should ”try alternative strategies” if needed, ”check on progress at appropriate states” and ”analyse the final results to ensure that the initial requirements have been met” (DES. 1988a, § 4. 7). It is certainly true that this is how some people have to use mathematics. For example, if you are working in a factory and receive a blueprint from the manager for cutting a particular tool or for making a particular electronic circuit to a given specification, then this is precisely how you will have to use mathematics. But this is a long way from creativity, beauty, aesthetics and so on. It is a long way from mathematics being ”a live subject, essentially evolving as technology and society evolve”. This is a very different kind of metaphor. This is

mathematics as a tool to solve somebody else’s problems, this is mathematics almost as a weapon. The metaphor here is as a set of tools, like a spanner, a hammer and a screwdriver: when you see a problem the test of whether you have understood the necessary mathematics is to see whether you know how to choose the right tool, and then when you have chosen it, how to use it properly to solve the problem that you have been given. If that is how we would like mathematics education to be focussed then it is right that we should say so explicitly; but it is not the same as regarding mathematics as a beautiful and creative subject. Let us consider the sort of problems the National Curriculum developers were thinking of, which invoke the tool metaphor. There are very few examples on which to draw – indeed there is very little serious consideration of any of these issues in any of the National Curriculum documents. Nevertheless, I have found one. This is referred to as a real problem by its authors: 7.22 This problem is appropriate across the age range 7 to 16. ”If British Airways runs flights between each of 8 major airports in Europe, how many routes is that? What happens for a different number of airports? 7.33 The mathematics of the above example is exactly the same type (sic) as that needed in a variety of other contexts. For example: a) The offices of 15 cabinet ministers are linked by direct telephone lines. How many lines are needed? b) If 30 people at a party shake hands with each other, how many handshakes will there be? ”This use of the same mathematical content in widely different situations illustrates the power and beauty of mathematics”. (DES, 1988, § 7.23; my emphasis).

18

as we know – but teachers are rather sensitive to the needs of charting their pupils’ progress and they are equally sensitive about the need to understand what the pupil understands before they can plan the next educational steps. Now we turn to the second extract:

”Problems” like this are part of the ritual of mathematics teaching; problems which are not really about anything, yet masquerading as if they were – in this case about airports, shaking hands or going to a party. It is not clear why we do this but it is hardly uncommon. Do we make it sound as if the problem is about airports because it might make it easier? Maybe yes. Do we do it because we think it might motivate students to identify with the problem, whereas if we cast the question in terms of permutations they would say it is just ”boring mathematics”; maybe that is what we are doing. Or is there something deeper? Maybe we believe that all questions should be about something concrete and that asking questions about something concrete is somehow better than asking questions about something abstract? We ought to discuss these questions, and as I said earlier, for us in the UK the publication of the National Curriculum and its associated documents has made some of us think a little bit more deeply about these kinds of issues. Whatever else can be said about the above example, I think we can agree that it is not a real problem. Now I want to turn to the crucial issue of assessment. I am going to begin with two statements from a recent document, The TGAT report (see references), which acted as a precursor to the National Curriculum: In fact the publication of this document started a chain of events which has ended in the National Curriculum itself. Here is the first extract:

’At the heart of the system will be the new School Examinations and Assessment Council and the National Curriculum Council, to coordinate the development of the national curriculum and the programme of assessment’. (DES 1987, para 201). Everything I want to say today can be traced back to the dichotomy between these two statements. Because if it is really true that evaluation and assessment are about improving our sensitivity to pupils’ understandings, charting their progress, opening dialogue between teachers and so on, then it is not at all clear why a Government which is committed to cutting thousands of millions of pounds off the education budget, has at the same time, created two completely new quasi-governmental organizations each with a budget of millions of pounds a year to centralize the assessment procedure. Why is it necessary to have a national centralized system of evaluation if the purpose of this evaluation is for teachers to understand their students and to facilitate dialogues between them? I want to distinguish between private and public assessment. It is rare to encounter a teacher who is opposed to private assessment. In fact it could be argued that the teaching of mathematics (or any other subject) necessarily involves private assessment. You cannot really teach somebody something unless you have some idea of what they already understand. Of course private assessment does not have to be strictly private between pupil and teacher: it can involve parents or other teachers. But this is very different from assessment whose results determine the future pattern of an individual’s life opportunities, or assessment which is used to make schools compete against each other or to use the results of

’Promoting children’s learning is a principal aim of schools. Assessment lies at the heart of this process. It can provide a framework in which educational objectives may be set, and pupils’ progress charted and expressed’ (DES 1987, para 3). This paragraph seems to me to sum up rather nicely the kinds of things all teachers know instinctively about evaluating their pupils. It sounds comfortable, it sounds very teacher, it is what teachers regularly do with their pupils – not always perfectly

19

the tests to determine the performance (and pay?) of teachers; this is public assessment and it needs to be strongly differentiated from the private kind.(See Noss, Goldstein and Hoyles (1989) for further discussion of this distinction.) I understand that there are plans to introduce more comprehensive assessment procedures in Sweden at least in mathematics. So let me try to encapsulate my views more generally by challenging four hidden assumptions about assessment which often appear to implicitly underpin the arguments of those who advocate it as a mechanism for ’improving’ mathematics education.

Assumption 1 It is impossible to present abstract mathematics to all types of children and expect them to get something out of it. This is a quote from Hart K. 1981, p. 210. I am not so sure. I am not suggesting that we should all immediately teach Ring Theory to 5 year-olds, but it is surprising what children can understand and even more surprising if we enlarge the meaning of the word ’understand’. The whole idea of ”getting something out of it” seems to me quite a good one. Getting something out of it is not the same as understanding. One of the beautiful aspects of mathematics is that we never completely understand everything about anything, because the whole power of the subject is based on building layers of abstractions on top of one other. We should be wary of stating categorically what it is that children will or will not be able to ’get something out of’. Who, for example, would have believed five years ago that a book about the history of the development of nonlinear dynamical systems would be a best seller? James Gleick’s wonderful history of chaos is a best seller list in the UK – I don’t know about Sweden. Suddenly we can generate incredibly

beautiful images on computers. I do not think anybody will disagree with me if I say that we can show these things to children and expect them to get something out of it. They won’t understand – neither do I. But we can all get something out of it. So it seems to me that Kath Hart’s statement is rather misleading. I turn a blind eye to the remark about ”all types of children” as I have no idea what she means.

Assumption 2 Mathematical knowledge is organized hierarchically. I do not believe this at all. Of course it is true that there are local hierarchies. It helps to know what a group is before you define a ring; it makes sense to know what an integer is before working with rationals. But these hierarchies are local, and it just does not seem possible to arrange mathematics into a global hierarchy. On the contrary the image I have is much less like a genuine hierarchy and much more like a set of blobs which are all connected up in some complex way. I am not saying mathematical knowledge is unstructured, far from it; but I am saying that it is not organized hierarchically and if it is not organized hierarchically then it does not make sense, for example, as the National Curriculum does, to number the statements of attainment from 1 to 10 down each side of the page and pretend that number 7 is slightly more difficult than number 6 but slightly easier than number 8.

Assumption 3 Mathematical learning is organized hierarchically. Let’s pretend that mathematical knowledge is organized hierarchically. Does that imply that mathematical learning should be structured in the same way as mathematical knowledge? The answer is no. We can all think of examples where it is actually a

20

is replaced by ’music’. What would assumption 4 mean in this case? It would mean that we would start by listening to a single note, then to two notes, then three; after a considerable time perhaps one would be allowed to listen to a sonata. In fact it is clear that a powerful way to feel what it means to make music is to listen to a piece of music. Of course this does not mean that the listener can immediately conduct (or write) a symphony, but it means that he or she might develop some feeling of what they would like to do and the power of musical expression they would like to acquire. I want to raise two more general issues concerning assessment. In any attempt to organize teaching according to a hierarchy of mathematical understanding, we encounter two insoluble problems of which the National Curriculum document appears to be completely oblivious. Consider the two statements:

very good pedagogical trick to introduce the general case before the particular. Mathematics is the classic example of where it simply does not work to follow the epistemological structure of what is being taught. Let me take a very obvious example. Consider the following set of equations: a) x + 3 = 5 b) 3 + x = 5 c) 2x + 1 = 5 d) 3x – 2 = 4 A common error which beginning teachers make is to give pupils a carefully sequenced set of examples of this kind. They are inevitably surprised when, at some point in the sequence, pupils are unable to apply their system to a slightly more complex example, say: e) 3x – 2 = 5. Why? Is it because example (e) is structurally more complex than (d)? The problem is that we have organized a mathematical hierarchy of equations based on their epistemological structure. And we have made an assumption along the way that the pupils are organizing in their minds a similar sequence of understanding. In general this is false. The student has not necessarily been using the system that the teacher thought he/she had been using; with the result that when an example appears which actually requires some generalizable strategy (as, for example, the production of a nonsolution in e)), the student is at a loss.

i) 4.5 + 0.5 ii) John saved £3.70 and his Mother gave him £1.50. How much would he have in all? The mathematical structure is similar. In the National Curriculum there is a statement which says: Solve addition or subtraction problems using numbers with no more than two decimal places (AT3, Level 4). Yet, when substantial numbers of pupils are tested on these items, 63% get the first right while 82% answer the second correctly.(APU 1986, p. 836). As far as the children are concerned, they are not the same problem despite the similarity of the mathematical structure. The meaning that the child gives to the problem is not the same; it does not make sense to say that the child can add two decimal numbers and produce a third number. We have to ask: what is the setting of the question? The second general point is the influen-

Assumption 4 Mathematical teaching should be organized hierarchically. This is really a corollary of assumption 3. The idea is that it is possible to predetermine the sequence of ideas which will be appropriate for every individual. I don’t believe this. As justification I offer the following heuristic: given any principle of mathematics education, check whether it still applies when the word ’mathematics’ 21

ce of question structure. Even minor differences such as replacing multiplication signs by ’of’ make massive differences to performance. Let us compare these two questions: i) How many halves are there in two and a half? ii) What is two and a half divided by a half?(APU, 1986, pp. 840–1) According to the Assessment of Performance Unit, 30% more pupils get the first question right than the second. If we were to try to write a list of attainment statements which were sensitive to these kinds of issues then we would end up, not with 296 statements but with 296 thousand: it is an impossibility. Let us leave aside the technicalities, and turn to questions of pedagogy. Here I think we can discern three further arguments which are commonly used by those who believe as an act of faith, that assessment will enhance education. The first argument is that ”it increases motivation”. In fact, this is an open question. From a commonpoint of view assessment improves motivation for the children who succeed and has a reverse effect for those who fail. In fact the situation is more complex than this, and there is an interesting research literature on this issue. I cannot discuss its detailed findings here: all I want to say is that the simple equation ”assessment = motivation” is entirely unproven. The second argument is: ”If activities are assessed then everybody knows where they stand”. This might be true on one level; but we need to consider it more carefully. Let us do a thought experiment. Imagine a beautiful assessment question for some mathematical topic or concept. It is open. It allows the pupils to be creative. It allows the teacher and the pupil to interact in a productive way. It is such a good question that a very bright pupil can work for six weeks developing this project; yet a weak pupil will ”get something out of it” as well. Now my question is this. If the Swe-

dish Parliament passed a law that every pupil in every school must do this question at a specific stage of their schooling, would this change the nature of the activity? I think it would. That is why I think so many of my colleagues are mistaken in believing they can dilute the adverse effects of the National Curriculum by devising interesting and investigative tasks. The point is that the most exciting task becomes drudgery once it is part of the paraphemalia of testing and competitive judgements. As an illustration, let me give an example of this phenomenon, which simultaneously highlights the question of the relationship between pupil and teacher. My 6 year-old son came home six months ago and told me about a most interesting science task that he and his class had done. They took several different kinds of food, crisps and apples etc., they buried them in the ground and they dug them up every couple of days to look at the rate of decomposition of the food. This seems to me an excellent activity. I only discovered that this was a National Assessment pilot test because I accidentally discovered several other children in different local schools doing exactly the same task. The problem is this: When my son did the task the teacher had no particular time pressure on her, my son wasn’t assessed, and it was all very relaxed. In fact, one of the points my son made was that ”Miss said that she was a scientist and she really loves doing this kind of thing.” But imagine if she had to do several dozen of these tests within a tightly specified time? It seems to me that the teacher/ pupil relationship is something rather fragile and that we cannot afford to pressurize it in this way. When this activity becomes a compulsory task for every 7 year-old then the relationship between the child and the teacher must change. As it is clear that I do not believe that the purpose behind the National Curriculum is to improve performance, I want to end by asking what its true purpose might be? For the last 15 years in England it has become slowly accepted that children should not

22

tional Curriculum. Of course, there is a long history of the view that mathematics is a mechanism for social control, a means of instilling what, over 100 years ago, were called habits of ’promptitude’ and ’exactness and order’ (quoted in Howson, Keitel and Kilpatrick p. 24). This is, of course, only a special case of the view made famous in the remark in the British Parliament, made in 1867 after the town workers of England obtained the vote, that ’We must educate our masters’ (quoted in Sarup, 1982). Now I do not mind if we really want to use mathematics as a mechanism for social discipline, or as a means to force children into a particular way of thinking; I think it would be a pity; but if it is so, I think it would be as well to recognize that this is the case, and to bury once and for all the idea that the mathematics we teach children has intrinsic use-value beyond that of elementary arithmetic.

waste any time at all learning how to do things like multiplying 857 by 942 with pencil and paper. In 1974, when the first electronic hand-held calculator came onto the market, I remember debating whether schools would ever be able to afford them in bulk. Now there are, at least in the UK, very few teachers who would seriously argue against the calculator. Of course, there is an argument about when pupils should use a calculator and so on. I am quite aware that this question is not closed, but very few people would say that children must never see calculators. So it came as quite a shock to us when in 1989 it became law in England and Wales that children of a certain age had to learn how to multiply a 3 digit number by a 2 digit number, (that’s bad enough) and to divide a 3 digit number by a 2 digit number (which is worse). What we have here is an interesting phenomenon: at precisely that moment in history when nobody will ever again need to perform such calculations (remember the criterion for teaching mathematics is to teach things that people actually need in their work and in later life), every child will have to learn how to do it.(Unless they attend a private school.) Now it seems to me that this raises an important and interesting question: I do not have time to outline a complete answer; but it is clear that the question needs answering because we certainly know that long division is not creative or aesthetic and we know that it is not actually needed in later life and at work. Why then has it become the law of England and Wales? To point towards an answer, I would like to end by offering a quotation from another report from the same series, which makes clear that the Mathematics National Curriculum is concerned to: ... seek to draw out each child’s full potential through the development of sound work habits, self-decipline and industry together with good personal qualities. (DES, 1988, § 8.4). Recall that this is the Mathematics Na-

References ASSESSMENT OF PERFORMANCE UNIT(APU) (1986) A review of monitoring in Mathematics, 1978 – 1982. London: Dept. of Education and Science. DES & WELSH OFFICE, (1987) Task Croup on Assessment and Testing, London: DES. DES & WELSH OFFICE, (1988) Mathematics for ages 5–16, London: HMSO. DES & WELSH OFFICE, (1989) Mathematics in the National Curriculum, London: HMSO. Dowling, P. & Noss, R. (1990) (Eds.) Mathematics versus the National Curriculum. Basingstoke: Falmer Press. Hart, K. (1981) Children’s Understanding of Mathematics 11–16NFER-Nelson. Howson, G., Keitel, C. & Kilpatrick, J., (1981) Curriculum Development in Mathematics, Cambridge: CUP. Noss, R., Goldstein, H. & Hoyles, C. (1989) Graded Assessment and Learning Hierarchies in Mathematics. British Education Research Journal, 15, 2, 109–120 Sarup, M. (1982) Education, State and Crisis. London: RKP

23