- Email: [email protected]
Can research inform classroom practice?: The particular case of buggy algorithms and subtraction errors
Teaching & Teacher Education. Pnntd 8” Great Bntam
Vol.
7. No. 4. pp. 395403.
1991
07424t5lX:91 s3.oo+o.cm Q 1991 Pergmm Press pk
CAN RESEARCH INFORM CLASSROOM PRACTICE?: THE PARTICULAR CASE OF BUGGY ALGORITHMS AND SUBTRACTION ERRORS
DAVID McNAMARA and DEIRDRE PETTITT University of Durham, U.K.
Abstract-Discussions about the relevance of research to classroom practice are usually expressed in general terms. An alternative approach is taken and a specific body of psychological research which has investigated children’s errors when undertaking subtraction computations is critically reviewed in order to assess whether it yields information which may be of worth and relevance to the teacher who must teach subtraction and diagnose and remedy children’s errors within the classroom environment. It is concluded that the research has little to offer the teacher and that it is the teacher’s knowledge of methods of teaching subtraction together with her or his classroom expertise and personal qualities which are the factors which are likely to inform her or his teaching. Examples of what the teacher’s professional knowledge should consist of are illustrated.
One of the justifications for educational research which has its conceptual roots in social science is that it is of value for practising teachers in so far as it generates information which enables them to improve the quality and effectiveness of their teaching. This is a claim which is regularly contested as part of the enduring debate about the relevance of educational theory to classroom practice (see for example, Jefferys, 1975; Desforges & McNamara, 1977; Dearden, 1980). Arguments about the relevance of social science-based research to practice are usually couched in general terms and consider the issues at the level of principles and procedures. In what follows we take a different approach and focus upon a narrow area of the curriculum, namely the teaching of subtraction in primary classrooms beyond the early years introductory stage. This procedure is adopted because in order to investigate whether educational research informed by social science may have an effect upon those teachers’ classroom practices associated with pupils’ learning it is appropriate to examine specific aspects of classroom practice. Since we are concerned about the impact of research upon practice as it occurs in the classroom we do not enter into the debate about whether the essential purpose of primary mathematics is to develop pupils’ com-
putational skills or to foster their conceptual understanding (see for example, Cockroft, 1982; Lampert, 1986). We take the pragmatic view and note that progressive approaches to primary mathematics teaching have never been widely established in primary schools (Hughes, 1986) and that there is a substantial emphasis upon learning the four arithmetic rules with about 60% of the learning activities provided for children involving practice tasks (Bennett, Desforges, Cockbum, & Wilkinson, 1984). the mathematics National CurMoreover, riculum in England and Wales lays down targets for learning subtraction specific (D.E.S., 1989): For example, the proposals for testing children at key stage one include for “Attainment Target 3: Number” the specifications that children at Level 2 must know and use subtraction facts up to 10; at Level 3 know and use subtraction facts up to 20; and at Level 4 subtract mentally two digit numbers and subtract three digit numbers with the aid of a calculator. We argue that at the crux point where the teacher seeks to promote pupils’ understanding, deal with their errors and misconceptions, or diagnose the reasons for children’s failure to understand, the available research has very little to offer. We propose that the essential resource which teachers have to 395
DAVID
396
McNAMARA
draw upon is their own understanding of mathematics together with their experience which includes intangible qualities such as intuition, common sense, or discernment. It is not the formal knowledge drawn from teachers’ research evidence which may enable them to promote children’s learning of subtraction, it is rather their ability to deploy their personal qualities and tacit skills (Polanyi, 1969) within the time and resource constraints of busy classrooms combined with their personal knowledge of the relevant mathematics. Indeed, if the available research establishes anything, it is that the manner in which primary classrooms are organised and resourced makes it extremely difficult for the teacher to act in any other way (Desforges & Cockbum, 1987). The Research
and the Classroom
Context
We illustrate our argument and explore it in some detail with reference to research located within a psychological framework which has investigated the systematic errors which pupils make when doing subtraction calculations. The ability to compute is a basic arithmetical skill which children are expected to be able to perform at an early age. Teaching young children this skill presents a challenge since it is a procedure which will appear virtually meaningless to many children as they have no appreciation of its underlying semantics. It requires them to learn procedures which are unrelated to their everyday experience outside the classroom (see for example, Leinhardt, 1988). It is claimed that knowledge of buggy algorithms will be useful to teachers and inform their efforts to teach abstract procedures and, in addition, bugs may offer an explanation for children reaching an impasse in their learning (Lampert, 1986; Romburg & Carpenter, 1986). The notion of buggy algorithms refers to the systematic errors which children may make when doing computations (Romburg & Carpenter, 1986). Bugs are precise descriptions of errors and lead to predictions of what the errors will be, for example: BUG 207 -169 162
Child substracts the smaller from the larger number in each case.
and DEIRDRE
207 -169
PETIT7
Child puts a zero when there is a need to borrow.
100 207 - 169
Child substracts smaller number from larger instead of borrowing from zero.
42 It is claimed that children who make such mistakes have failed to follow or understand the routines (algorithms) which teachers have taught them. In order to make sense of and complete subtraction tasks they invent their own “buggy algorithms”. The aim of the research investigating this problem is to discover the children acquire buggy ways in which algorithms since this would provide valuable information for teachers which could help them to eliminate sources of misunderstanding and to take appropriate remedial action. (It should be noted that within the research the use of is limited to written algorithms, “algorithms” i.e., those which can be identified and expressed verbally and written down. It does not refer to other algorithmic procedures which learners may adopt or invent and which nevertheless may produce the correct answer.) An initial problem is that the examples illustrated above cannot be taken out of the social context of the classroom if one wishes to acquire a comprehensive understanding of the processes which generated the wrong answers. There are a variety of reasons why a child might not have produced the correct answer to the subtraction problem (Confrey, 1987) and the child’s response could be interpreted in other ways such as: I. The child may not have understood place value and had no conceptiion of borrowing, in which case the remedial action may be to return to the meaning of multi-digit numbers or simple addition with carrying. 2. The child may always have been used to subtraction processes set out differently, for example, written horizontally, and find the vertical format puzzling. 3. The child may have copied down the problem incorrectly, in which case the appropriate action would be to check for a careless error.
Can Research Inform Classroom Practice?
4. The child may have made a hurried, careless mistake, in which case the teacher would place the onus on the child to make the correction. 5. The child may have been idle, fooling around, not interested or emotionally upset. In such cases the teacher’s task may be to comfort or motivate the child in some appropriate way. Direct observation of children’s mathematical learning in the classroom (Desforges & Cockbum, 1987) reveals that young children use a variety of strategies besides attempting calculations in order to complete mathematics tasks. These include “recalling earlier work”, “looking back at an earlier solution”, “reading the teacher” and “memorising what the teacher said in her description of the task”. Errors in any of these strategies could also have led to a variety of wrong answers to the problem. Finally, and most importantly, we must consider the distinct possibility that the children have been taught badly and that embedded in the “buggy explanations” of children’s mistakes are prescriptions for further poor practice. Two of the bugs mentioned in the illustrated example assume that the appropriate way to proceed when doing subtraction is, when necessary, to borrow from the next column. Hence, we have a problem with the algorithm itself since it does not relate directly to the mathematics involved in doing subtraction. Subtraction algorithms which involve a reference to “borrowing” or “paying back” do not clearly indicate the relationship to the mathematics residing in algorithms based upon equivalence or equal addition procedures. Thus, a more powerful and arithmetically correct procedure may be to teach subtraction using language appropriate to the equivalence method, say: 42 -26
change 42 to 30 + 12
-
‘4212
change 26 to 20 + 6
-
26 16
Hence it is necessary for researchers to probe the very algorithms they are employing to explain the subtraction errors. Indeed, the current popular procedure for teaching subtraction in Britain is the decomposition or equivalence method (Hughes, 1986) which is illustrated
397
above and an important argument for doing so is that this algorithm reflects what is actually happening in the operation in a way which young children may be able to understand and model. Moreover, accepted good practice warns against the use of the misleading and incorrect term “borrow”. Terms such as “change” or “rearrange” are recommended. Yet the term “borrow” is consistently used in the psychological research literature. If children are being taught subtraction using the equivalence method with the appropriate terminologies the buggy explanations are inappropriate and misleading. The researcher’s explanation must map on to the teacher’s methods. The point to stress, therefore, is that when examples of children’s calculations are located in the social arenas in which they were generated, errors can be accounted for in a variety of ways and it can only be practising teachers with their knowledge of the children in their classrooms and their personal knowledge of the methods they use to teach subtraction who will be in a position to acquire the evidence which permits valid alternative accounts of children’s errors. Teachers’ available knowledge will include any formal records they may keep, it will be supplemented by their personal, somewhat intuitive knowledge of their pupils and depend in part upon the commitment, industry, and flair which they bring to bear in their teaching so as to understand their children and acquire information about them. Their personal knowledge may, for instance, enable them to judge whether a particular child’s error is, say, due to a slip in concentration which can be ignored or, say, a more substantial error which is worrying and requires remedial action on their part. What Does the Research Actually Discover? A close reading of the research itself casts doubts about whether what has been “discovered” about children’s use of buggy algorithms could be of value to teachers in any case, even if the information could be readily deployed in classroom contexts. In fairness to the research, it tends to be educationists who have argued for its relevance to teaching, whereas the central aim of the research is to understand how people learn cognitive skills in
398
DAVID McNAMARA
instructional settings (Van Lehn, 1990); within the original research, classroom application is suggested rather than prescribed. On the other hand, it is clear that the research is based upon a limited and somewhat uninformed appreciation of how subtraction is taught in schools and how children may approach learning subtraction (Van Lehn, 1990). A prodigious quantity of resource and research time has been devoted to investigating children’s buggy subtraction errors. Van Lehn (1982) reviews empirical studies which have involved 925 pupils and 4000 or 5000 hours of expert diagnosis of their subtraction errors. In terms of effort and expertise the scale of the research is more than an army of primary teachers could amass in a lifetime. To duplicate this endeavour for other procedural skills in other areas of, say, only the primary mathematics curriculum would be a daunting task. What has the available research discovered about children’s mistakes when doing subtraction? First, that at best “buggy algorithms” explain only about one third of the errors children make in subtraction. Hence, a comprehensive knowledge of the bugs children may employ when doing subtraction would enable teachers to cope with only a minority of the subtraction mistakes which children make in the classroom. Another notion, that of “repair theory*‘, is invoked to explain about another third of children’s subtraction errors. Repairs are defined as the local problem-solving strategies which pupils devise, such as skipping a bit of a calculation, which generates an answer that is not only incorrect but is also not the type of predictable incorrect answer generated by the systematic application of a buggy algorithm. Repairs provide explanations for about another third of children’s subtraction errors. Since, by definition, repairs are local problem solving strategies, it is only the class teacher who is in a position to find out what they are. The huge research effort can offer no diagnosis for the final third of all subtraction errors. There is a sense in which this research effort is of vital interest to teachers. In one narrow but important area of the curriculum in one subject a substantial research effort can offer information which may be of value in helping teachers diagnose children’s learning failure in one third of all cases, suggest that failure is caused by
and DEIRDRE PETTIm
pupils’ local strategies in one third of cases, and is unable to offer information concerning the remaining third. What, then, can one reasonably expect of the teacher in the classroom and by what criteria should we judge the teacher’s success in diagnosing children’s learning difficulties? Since in 27.5% of the instances where a buggy error was diagnosed the bug was a failure to borrow and subtract the lower digit from the upper one (an obvious error which any teacher is familiar with) (Van Lehn, 1990) any practical pay off is further reduced. It might seem, nevertheless, that in one third of cases in which children make systematic errors the research has something to offer teachers in that bugs provide an explanation which may be used to diagnose learning difficulties. This is unlikely to be the case. Various research teams working with the aid of considerable computing power have been able to identify over 100 subtraction bugs. (A full glossary of bugs is listed in Van Lehn, 1990). This is hardly information which may be of value to teachers with 30 or more children in their care; even if they could assimilate this information, how could they deploy it in their practical teaching? A final theme emerging from the research will at least strike a chord with teachers, even though it presents further difficulties conceming its practical application. It was found that some pupils tested a few days apart answered all problems correctly on the first occasion but made slips on the second and that children could manufacture bugs for the duration of one test and then discard them once they had helped them get through that particular exercise. This phenomenon has been designated as bug migration (Van Lehn, 1990, p. 16). Thus, we are reminded that children can be little “buggers” rather than bug users. Many of the usual problems facing teachers may be to do with children’s unsystematic errors and erratic performance from day to day and not to do with systematic errors encountered at specific stages in their learning. The Algorithm Learning
as an Explanation
for
A substantial research programme focussing upon one particular facet of learning fails to
Can Research
Inform
provide information for teachers which could inform and enhance their routine classroom practice. This could be because the theory underpinning the research is flawed with respect to its potential for practical application in the classroom due to the fact that the conceptual framework is based upon a model of leaming drawn from cognitive psychology and computer programming (Van Lehn, 1990). Algorithms are important and powerful heuristics which are part of the necessary intellectual equipment of practising mathe1990) and cognitive maticians (Penrose, psychologists. It is a mistake, however, to assume that conceptual tools which are of value to cognitive psychologists and mathematicians can also be employed as devices for explaining children’s learning failure in the primary mathematics class. This may be demonstrated by attempting to consider the problem of doing subtractions from the point of view of the child. It is necessary to enquire how children probably tackle subtraction tasks in the classroom. It must be accepted that such enquiries are necessarily speculative since we, like the psychologist, are denied direct access to the child’s mind. There can be no question that children do acquire procedures and strategies from teachers in order to cope with subtraction problems but if the upshot is that they can merely apply these procedures in parrot-like fashion we are unlikely to agree that the child or been “educated” (Ryle, has “learned” 1979). Learning involves more than the automatic carrying out of a procedure; it also entails such things as being able to apply the procedure in novel situations or scrutinise one’s own work and rectify mistakes made in following procedures. If the child could do no more than, metaphorically speaking, apply an algorithm in the manner of a calculating machine the teacher should be just as concerned with that child as with the child who makes erratic mistakes. Presumably it is only the teacher who is in a position to make the judgement about whether the child has learned a procedure in a manner which permits its application with understanding. There can be little doubt that children can invent their own algorithms and that these may be different from those taught in formal schooling (Carraher, Carraher, & Schliemann, 1985)
Classroom
Practice?
399
in so far as they can devise rule governed procedures which permit the correct computation of practical transactions. One may wonder, however, whether a significant proportion of children conscientiously invent mental procedures which are systematically incorrect and which, incidentally, can be expressed as written algorithms. Is it reasonable to assume that a child who makes use of a predictable procedure to get through a set of subtraction calculations which yield rule-governed wrong answers has independently devised and applied what we may term one of a large number of “buggy algorithms” and that this notion offers a way for diagnosing and remedying learning failure? Reconsider the original illustration, 207 -169
and the examples of buggy algorithms invoked to explain systematic errors such as “child takes smaller number from larger”. For many children there will be a stage in their mathematical learning when this sum presents a challenge. It will be more difficult than examples they have encountered previously. It is interesting to note that the illustrations of buggy algorithms all provide solutions which are easier than the correct one for coming up with the right answer. In such a circumstance it is difficult to accept the notion that the child has consciously invented a procedure, albeit an incorrect procedure and that knowledge of that buggy algorithm will aid the teacher in diagnosing learning difficulties and helping the child. Indeed, it is doubtful whether, with reference to the specific case of subtraction, children are actually able to apply a written algorithm model to match the rule (Hart, 1989). What the teacher needs to know is why, at the outset, the child did not or could not follow the harder correct procedure and decided upon a different easier strategy. Invoking knowledge about buggy algorithms introduces extra redundant information which is unlikely to be of value to the teacher. The reason why the child got the incorrect answer could be that the child lacked motivation or, more likely, did not understand place value (setting aside factors such as
400
DAVID McNAhfARA
carelessness or copying from a child also making mistakes). It is commonplace notions such as these which provide teachers with the information they need to help the child overcome its difficulties. The only person who can have direct access to plausible and usable explanations of the child’s learning difficulties is the teacher. In the classroom it is the teacher who has to decide how the child’s subtraction errors have been generated. This is part and parcel of teachers’ practical, professional expertise. There can be little point in recasting their expertise within a mathematical or psychological framework (Smedslund, 1982) and elaborating it beyond the feasibility of practical application. Teaching and learning involve a personal encounter between the teacher and learner and the only explanation for learning failure which can be used by the teacher is one which is expressed in ordinary language and invokes everyday notions. This, of course, places a burden upon the teacher but there is no one else upon whom the onus can be placed. The teacher who gets it wrong by deciding that the child has been “lazy” rather than appreciating that the child does not understand place value, and keeps the child in at playtime for extra practice makes a mistake which could have detrimental effects upon the child’s motivation.
Implications Discussions about children’s difficulties in learning mathematics are likely to address issues such as: conjectural knowledge about the development of children’s cognitive processes and its implications for structuring and sequencing the mathematics curriculum; problems entailed in crossing the divide between practical activities and formal representation; and the seeming disjunction between the mathematics of everyday life and experience and the decontextualised mathematics of the classroom. We do not engage in these debates. Our starting point is the recognition that the commercially produced mathematical schemes of work which are used regularly in most primary classrooms, the National Curriculum for mathematics and, indeed, parental expectations, all require primary teachers to teach what may crudely be termed “doing subtraction sums”. Moreover,
and DEIRDRE PETTIlT
there is compelling evidence that mathematical computation is given a high priority in the primary classroom (D.E.S., 1978; BarkerLunn, 1982; Sirotnik, 1983; Mortimore, Sammons, Stoll, Lewis, & Ecob, 1988); how may the teacher be best advised and prepared so as to undertake this activity? First it should be recognised that any explanation of a child’s failure to come up with the correct answer must, if it is to facilitate the teacher’s instruction, be expressed in terms of the familiar language which a teacher uses when talking with a young child. The teacher when teaching the child can do one thing not two. The teacher cannot both at the same time seek to explain to the child and also engage in some second order mental activity which draws upon different terminology and concepts so as to provide heuristics or data to inform practical activity (among the various discussions of this problem see for example: Ryle, 1949; Rorty, 1989). Thus, we must be circumspect if we seek to go beyond commonplace or common-sense accounts of why children make computational errors. Advice based upon informed professional experience coupled with the teacher’s understanding of the relevant mathematics is more likely to be of value to the practitioner than that which derives from research which is based upon conjectural notions about how the child’s mind works. It is a moot point whether the prescriptions for teaching subtraction using “carrying”, “equivalence”, or “equal addition” are derived from an appreciation of mathematics or “common sense”, but what is clear is that these methods for teaching subtraction are not derived from research. it should be appreciated that Second, teachers’ ability to teach subtraction to young children will depend upon their formal mathematical knowledge and upon those personal qualities and skills which enable them to become perceptive and sensitive observers of children’s learning. It is through watching children, talking with them about their computations, and assessing their work that the teacher is likely to diagnose their learning problems. This is an ability which, in part, can be developed through practical experience and beginning teachers can be tutored so as to help them develop these practical skills. The basis for such tutoring or training should include
Can Research
Inform
the accumulated professional wisdom of experienced practitioners. Finally, it is important to recognise the crucial importance of time as an essential resource in the teaching process. Both evidence and experience indicate that the best efforts of highly regarded teachers are routinely deflected by the demands of 30 or more children (see for example, Desforges & Cockbum, 1987) and that when a second party spends a substantial period of time in the class engaged in sustained observation of, and talk with, an individual child, it is a comparatively straightforward matter to assess the extent of a child’s learning and engage in the detailed diagnosis of difficulties (Bennett et al., 1984). What beginning teachers require in order to maximise the amount of their scarce time available for productive teaching is, first of all, advice about the organisation of the classroom and the management of the children’s learning. Teachers need information and suggestions not only about optimising the use of their own time but also that of their children (teachers sitting at their desks dealing with queues of inquiring children may be allocating all their time to teaching but they are also minimising the time individual children devote to learning). Hence sound and realistic suggestions from experienced practitioners who have tackled the problem of how best to organise the mathematical classroom should be made available to beginning teachers. Such advice should not be cluttered up with hortatory prescriptions for primary practice; advice must be considered on its merits in terms of the goal of maximising both the teacher’s and the children’s use of time. For example, teacher dominated whole class instruction is an efficient way of ensuring that it is at least possible for all children to benefit from the teacher’s teaching. The essential thrust of what we are saying is that in order to teach subtraction successfully in busy classrooms with many children teachers are very much thrown upon their own resources of perception, energy, and intuition. In order to become trained or better mathematics teachers they are more likely to require guidance, advice, and information drawn from professional practice and experience and from the subject itself rather than knowledge selected from research which has investigated learning in laboratory like settings which do not reflect the
Classroom
Practice?
401
classroom environments in which teachers must teach their pupils. Professional Advice We have sought to demonstrate, with reference to one particular arithmetic skill, that empirical research based upon large populations and involving a massive investment in investigators’ time and effort does not generate information which can be of assistance to teachers when helping children with subtraction in the classroom. The teachers’ task, however, remains; they must be responsive to the particular errors of particular children and take appropriate action within the resource and time constraints of the classroom (Leinhardt, 1988). This entails, among other things, assessing individual children and then taking appropriate action (Ausubel, 1968). Since research has little to offer, teachers must fall back on their knowledge and experience. There seems to be no reason why this should necessarily be their own personal experience; it may include accumulated professional wisdom which is passed on in the staffroom or included in, say, teacher training courses. We suggest that much more effort and attention should go into the collection and codification of professional expertise since it is this which may prove to be of distinctive benefit to the teacher (McNamara, in press). In this section we offer an indication of what this information could consist of. Experience indicates that it is quite common for children in classrooms to subtract the smaller from the larger digit (whatever the vertical order) when undertaking computational sums. The teacher’s first strategy should be to consider and, if appropriate, reject explanations such as copying or a moment’s carelessness. At this stage the teacher may decide that the preferred action is to ignore the error since, say, it may affect the child’s motivation or hold him or her back unnecessarily. On the other hand, the teacher may judge that the child has a shaky grasp of, say, place value and that some time must be spent engaging the child in conversation so as to assess the nature of the difficulty and then take appropriate remedial action. This demands time and at this point what is probably more important than the knowledge of mathematics or research is the teacher’s ability
402
DAVID
McNAMARA
to organise the class so as to create a learning environment in which the rest of the class remain engaged with their work while the teacher concentrates on an individual child. In the one-to-one situation several options are open to the teacher and the one which is likely to be employed is to revise the algorithmic procedure. We would hope that it would be one, such as the equivalence method, which has a clear relationship to the mathematics involved, especially with respect to place value. An alternative strategy might be to revive earlier regrouping and equivalence models of place value, perhaps using structural apparatus. A distinctive characteristic of teachers’ acquired expertise is that it does not offer the correct solution but alternatives which are open to professional discussion. Just as the members of the Gardener’s Question Time panel may offer different advice on how to grow bigger and better begonias, teachers may offer different suggestions for dealing with a learning problem. The first strategy we proposed may be objected to on the grounds that it is mechanistic and does not contribute to the child’s understanding. But teachers have to make professional judgements: Will they introduce a child to a practical exercise dealing with notions of exchange and equivalence so as to aid (hopefully) the child’s understanding or, say, persist with the algorithm in the expectation that the child will eventually realise why it works. Research cannot provide the answer; teachers must rely upon their personal knowledge of the child as to what is preferable in the particular case. Another aspect of the teacher’s knowledge is likely to be that many children find it difficult to understand place value (and that this understanding comes later than parents, say, may expect). Understanding place value is a gradual process which develops over time and the appropriate teaching strategy (given that the teacher has to teach subtraction) may be to re-teach the algorithm and pursue practice exercises until such time as the child appears to genuinely understand the higher order skills involved (Gelman & Gallistel, 1986). The teacher may, of course, attempt to supplement this approach by explaining why the algorithm works but this requires considerable expository skill and even if the teacher can provide an ade-
and DEIRDRE
PETTITT
quate account the explanation may fall on deaf ears. In our experience very few student teachers, all of whom have ‘0’ level standard mathematics, can explain why the equal addition or the equivalence algorithms work - just as they find it difficult to operate in number bases other than 10. It is not that they are deficient in mathematics; they have simply forgotten that understanding base 10 is easy because they have used it routinely all their lives. For young children, as primary teachers know, subtraction is difficult territory. Teachers also appreciate that requiring children to go back to an earlier stage and recover previous activity is not always a good idea. It is often resented by children and, for example, additional practical work does not necessarily facilitate understanding. Indeed, it has been suggested that structural apparatus is not useful until after a concept has been acquired (Hart, 1989). In a word, at the “chalk face” the teacher cannot legislate for learning subtraction, yet must take steps to foster learning. As such the teacher will be required to make professional decisions in circumstances which are not conducive to sound decision making, such as lack of time, lack of information, and competing The major factor influencing demands. decision-making is likely to be the teacher’s professional experience. This should, preferably, be experience which has been exposed to reflection and supplemented with professional expertise gained from other teachers. But in the final analysis the teacher acts as a human being making decisions about the behaviour of other, younger, human beings and it will be a subtle amalgam of professional skill, intuition, and expediency which inform these actions.
References Ausubel, D. P. (1968). Educorional psychology: A cognirive view. New York: Holt. Rinehart and Winston. Barker-Lunn, J. (1982). Junior school teachers: Their methods of practices. Educofional Research. 26. 178- 188. Bennett. N., Desforges, C., Cockbum, A., & Wilkinson, B. (1984). The quo&y of pupil learning experiences. London: Lawrence Earlbaum Associates. Carraher. T. N.. Carraher. D. W.. & Schliemann. A. D. (1985): Mathematics in the streets and in schools. Brirish Journal of Developmental Psychology, 3, 2 I -29.
Can Research
Inform
Cockroft, W. H. (1982). (Chairman). i%rhemcltics counts. London: H.M.S.O. Confrey, J. (1987). Bridging research and practice. Educarional 77wo~, 37, 383-394. Dearden, R. F. (1980). Theory and practice in education. Journal of Philosophy of Education, 14, 17-37. Department of Education and Science (D.E.S.) (1978). Primary educarion in England: A survey by H. M. Inspecfors of Schools. London: H.M.S.O. Department of Education and Science (D.E.S.) (1989). Mathemofics in the national curriculum. London: H.M.S.O. Desforges, C., & Cockbum. A. (1987). Understanding the mathematics reacher: A study ofpracrice in schools. London: Falmer Press. Desforges. C., & McNamara, D. (1977). One man’s heuristic is another man’s blindfold: Some comments on applying social science to educational practice. Brirish Journal of Teacher Education. 3, 27-39. Gelman. R. U.. & Gallistel, C. R. (1986). The child’s undersranding of number. Cambridge, MA: Harvard University Press. Han. K. (1989). There is little connection. In P. Ernest (Ed.), Marhematics teaching: The state of the arr, London: Falmer Press. Hughes. M. (1986). Children and number. Oxford: Basil Blackwell. Jefferys, D. (197.5). How psychology fails the teacher. British Journal of Teacher Education, 1, 63 -69. Lampert, M. (1986). Knowing. doing and teaching multiplication. Cognirion and Instruction, 3, 305 -342. Leinharht. G. (1988). Situated knowledge and expertise in teaching. In J. Calderhead (Ed.), Teachers’professional learning (pp. 146- 168). London: Falmer Press.
first
Classroom
403
Practice?
McNamara. D. (in press). Vernacular oedaeogv. British Journal of Educarjonal Studies. . - -. Mortimore, P.. Sammons, P.. Stall. L.. Lewis. D.. & Ecob, R. (1988). School matters: 7he junior years. Wells: Open Books. Penrose, R. (1990). 7he emperor’s new mind: Concerning compurers, minds, and rhe laws of physics. London: Vintage. Polanyi. M. (1969). Knowing and being. London: Routledge and Kegan Paul. Romburg. T. A., & Carpenter, T. P. (1986). Research on training and learning mathematics: Two disciplines of scientific enquiry. In M. C. Wittrock (Ed.). Hundbook of Research on Teaching (3rd. ed., pp. 850-873). New York: Macmillan. Rorty, R. (1989). Contingent?, irony and solidarity. Cambridge: Cambridge University Press. Ryle, G. (1949). The concept of mind. London: Hutchinson. Ryle, G. (1979). 01 rhinking. Oxford: Basil Blackwell. Sirotnik. K. A. (1983). What you see is what you get consistency, persistency, and mediocrity in classrooms. Hazard Educational Review. 53 (I ), 16 - 3 I. Smedslund, J. (1982). Revising explications of common sense through dialogue: Thirty six psychological theorems. Scandanavian Journal of Psychology, 23, 299 - 305. Van Lehn. K. (1982). Bues are not enouzh: Emoirical studies of bugs, impas& and repairs in proc’edural skills. Journal oJMarhemafica1 Behaviour. 3. 3-71. Van Lehn, K. (1490). Mind bugs: 7he origins of procedural misconceprions. London: Batsford Books. Received
2 May 1991 0
Vol.
7. No. 4. pp. 395403.
1991
07424t5lX:91 s3.oo+o.cm Q 1991 Pergmm Press pk
CAN RESEARCH INFORM CLASSROOM PRACTICE?: THE PARTICULAR CASE OF BUGGY ALGORITHMS AND SUBTRACTION ERRORS
DAVID McNAMARA and DEIRDRE PETTITT University of Durham, U.K.
Abstract-Discussions about the relevance of research to classroom practice are usually expressed in general terms. An alternative approach is taken and a specific body of psychological research which has investigated children’s errors when undertaking subtraction computations is critically reviewed in order to assess whether it yields information which may be of worth and relevance to the teacher who must teach subtraction and diagnose and remedy children’s errors within the classroom environment. It is concluded that the research has little to offer the teacher and that it is the teacher’s knowledge of methods of teaching subtraction together with her or his classroom expertise and personal qualities which are the factors which are likely to inform her or his teaching. Examples of what the teacher’s professional knowledge should consist of are illustrated.
One of the justifications for educational research which has its conceptual roots in social science is that it is of value for practising teachers in so far as it generates information which enables them to improve the quality and effectiveness of their teaching. This is a claim which is regularly contested as part of the enduring debate about the relevance of educational theory to classroom practice (see for example, Jefferys, 1975; Desforges & McNamara, 1977; Dearden, 1980). Arguments about the relevance of social science-based research to practice are usually couched in general terms and consider the issues at the level of principles and procedures. In what follows we take a different approach and focus upon a narrow area of the curriculum, namely the teaching of subtraction in primary classrooms beyond the early years introductory stage. This procedure is adopted because in order to investigate whether educational research informed by social science may have an effect upon those teachers’ classroom practices associated with pupils’ learning it is appropriate to examine specific aspects of classroom practice. Since we are concerned about the impact of research upon practice as it occurs in the classroom we do not enter into the debate about whether the essential purpose of primary mathematics is to develop pupils’ com-
putational skills or to foster their conceptual understanding (see for example, Cockroft, 1982; Lampert, 1986). We take the pragmatic view and note that progressive approaches to primary mathematics teaching have never been widely established in primary schools (Hughes, 1986) and that there is a substantial emphasis upon learning the four arithmetic rules with about 60% of the learning activities provided for children involving practice tasks (Bennett, Desforges, Cockbum, & Wilkinson, 1984). the mathematics National CurMoreover, riculum in England and Wales lays down targets for learning subtraction specific (D.E.S., 1989): For example, the proposals for testing children at key stage one include for “Attainment Target 3: Number” the specifications that children at Level 2 must know and use subtraction facts up to 10; at Level 3 know and use subtraction facts up to 20; and at Level 4 subtract mentally two digit numbers and subtract three digit numbers with the aid of a calculator. We argue that at the crux point where the teacher seeks to promote pupils’ understanding, deal with their errors and misconceptions, or diagnose the reasons for children’s failure to understand, the available research has very little to offer. We propose that the essential resource which teachers have to 395
DAVID
396
McNAMARA
draw upon is their own understanding of mathematics together with their experience which includes intangible qualities such as intuition, common sense, or discernment. It is not the formal knowledge drawn from teachers’ research evidence which may enable them to promote children’s learning of subtraction, it is rather their ability to deploy their personal qualities and tacit skills (Polanyi, 1969) within the time and resource constraints of busy classrooms combined with their personal knowledge of the relevant mathematics. Indeed, if the available research establishes anything, it is that the manner in which primary classrooms are organised and resourced makes it extremely difficult for the teacher to act in any other way (Desforges & Cockbum, 1987). The Research
and the Classroom
Context
We illustrate our argument and explore it in some detail with reference to research located within a psychological framework which has investigated the systematic errors which pupils make when doing subtraction calculations. The ability to compute is a basic arithmetical skill which children are expected to be able to perform at an early age. Teaching young children this skill presents a challenge since it is a procedure which will appear virtually meaningless to many children as they have no appreciation of its underlying semantics. It requires them to learn procedures which are unrelated to their everyday experience outside the classroom (see for example, Leinhardt, 1988). It is claimed that knowledge of buggy algorithms will be useful to teachers and inform their efforts to teach abstract procedures and, in addition, bugs may offer an explanation for children reaching an impasse in their learning (Lampert, 1986; Romburg & Carpenter, 1986). The notion of buggy algorithms refers to the systematic errors which children may make when doing computations (Romburg & Carpenter, 1986). Bugs are precise descriptions of errors and lead to predictions of what the errors will be, for example: BUG 207 -169 162
Child substracts the smaller from the larger number in each case.
and DEIRDRE
207 -169
PETIT7
Child puts a zero when there is a need to borrow.
100 207 - 169
Child substracts smaller number from larger instead of borrowing from zero.
42 It is claimed that children who make such mistakes have failed to follow or understand the routines (algorithms) which teachers have taught them. In order to make sense of and complete subtraction tasks they invent their own “buggy algorithms”. The aim of the research investigating this problem is to discover the children acquire buggy ways in which algorithms since this would provide valuable information for teachers which could help them to eliminate sources of misunderstanding and to take appropriate remedial action. (It should be noted that within the research the use of is limited to written algorithms, “algorithms” i.e., those which can be identified and expressed verbally and written down. It does not refer to other algorithmic procedures which learners may adopt or invent and which nevertheless may produce the correct answer.) An initial problem is that the examples illustrated above cannot be taken out of the social context of the classroom if one wishes to acquire a comprehensive understanding of the processes which generated the wrong answers. There are a variety of reasons why a child might not have produced the correct answer to the subtraction problem (Confrey, 1987) and the child’s response could be interpreted in other ways such as: I. The child may not have understood place value and had no conceptiion of borrowing, in which case the remedial action may be to return to the meaning of multi-digit numbers or simple addition with carrying. 2. The child may always have been used to subtraction processes set out differently, for example, written horizontally, and find the vertical format puzzling. 3. The child may have copied down the problem incorrectly, in which case the appropriate action would be to check for a careless error.
Can Research Inform Classroom Practice?
4. The child may have made a hurried, careless mistake, in which case the teacher would place the onus on the child to make the correction. 5. The child may have been idle, fooling around, not interested or emotionally upset. In such cases the teacher’s task may be to comfort or motivate the child in some appropriate way. Direct observation of children’s mathematical learning in the classroom (Desforges & Cockbum, 1987) reveals that young children use a variety of strategies besides attempting calculations in order to complete mathematics tasks. These include “recalling earlier work”, “looking back at an earlier solution”, “reading the teacher” and “memorising what the teacher said in her description of the task”. Errors in any of these strategies could also have led to a variety of wrong answers to the problem. Finally, and most importantly, we must consider the distinct possibility that the children have been taught badly and that embedded in the “buggy explanations” of children’s mistakes are prescriptions for further poor practice. Two of the bugs mentioned in the illustrated example assume that the appropriate way to proceed when doing subtraction is, when necessary, to borrow from the next column. Hence, we have a problem with the algorithm itself since it does not relate directly to the mathematics involved in doing subtraction. Subtraction algorithms which involve a reference to “borrowing” or “paying back” do not clearly indicate the relationship to the mathematics residing in algorithms based upon equivalence or equal addition procedures. Thus, a more powerful and arithmetically correct procedure may be to teach subtraction using language appropriate to the equivalence method, say: 42 -26
change 42 to 30 + 12
-
‘4212
change 26 to 20 + 6
-
26 16
Hence it is necessary for researchers to probe the very algorithms they are employing to explain the subtraction errors. Indeed, the current popular procedure for teaching subtraction in Britain is the decomposition or equivalence method (Hughes, 1986) which is illustrated
397
above and an important argument for doing so is that this algorithm reflects what is actually happening in the operation in a way which young children may be able to understand and model. Moreover, accepted good practice warns against the use of the misleading and incorrect term “borrow”. Terms such as “change” or “rearrange” are recommended. Yet the term “borrow” is consistently used in the psychological research literature. If children are being taught subtraction using the equivalence method with the appropriate terminologies the buggy explanations are inappropriate and misleading. The researcher’s explanation must map on to the teacher’s methods. The point to stress, therefore, is that when examples of children’s calculations are located in the social arenas in which they were generated, errors can be accounted for in a variety of ways and it can only be practising teachers with their knowledge of the children in their classrooms and their personal knowledge of the methods they use to teach subtraction who will be in a position to acquire the evidence which permits valid alternative accounts of children’s errors. Teachers’ available knowledge will include any formal records they may keep, it will be supplemented by their personal, somewhat intuitive knowledge of their pupils and depend in part upon the commitment, industry, and flair which they bring to bear in their teaching so as to understand their children and acquire information about them. Their personal knowledge may, for instance, enable them to judge whether a particular child’s error is, say, due to a slip in concentration which can be ignored or, say, a more substantial error which is worrying and requires remedial action on their part. What Does the Research Actually Discover? A close reading of the research itself casts doubts about whether what has been “discovered” about children’s use of buggy algorithms could be of value to teachers in any case, even if the information could be readily deployed in classroom contexts. In fairness to the research, it tends to be educationists who have argued for its relevance to teaching, whereas the central aim of the research is to understand how people learn cognitive skills in
398
DAVID McNAMARA
instructional settings (Van Lehn, 1990); within the original research, classroom application is suggested rather than prescribed. On the other hand, it is clear that the research is based upon a limited and somewhat uninformed appreciation of how subtraction is taught in schools and how children may approach learning subtraction (Van Lehn, 1990). A prodigious quantity of resource and research time has been devoted to investigating children’s buggy subtraction errors. Van Lehn (1982) reviews empirical studies which have involved 925 pupils and 4000 or 5000 hours of expert diagnosis of their subtraction errors. In terms of effort and expertise the scale of the research is more than an army of primary teachers could amass in a lifetime. To duplicate this endeavour for other procedural skills in other areas of, say, only the primary mathematics curriculum would be a daunting task. What has the available research discovered about children’s mistakes when doing subtraction? First, that at best “buggy algorithms” explain only about one third of the errors children make in subtraction. Hence, a comprehensive knowledge of the bugs children may employ when doing subtraction would enable teachers to cope with only a minority of the subtraction mistakes which children make in the classroom. Another notion, that of “repair theory*‘, is invoked to explain about another third of children’s subtraction errors. Repairs are defined as the local problem-solving strategies which pupils devise, such as skipping a bit of a calculation, which generates an answer that is not only incorrect but is also not the type of predictable incorrect answer generated by the systematic application of a buggy algorithm. Repairs provide explanations for about another third of children’s subtraction errors. Since, by definition, repairs are local problem solving strategies, it is only the class teacher who is in a position to find out what they are. The huge research effort can offer no diagnosis for the final third of all subtraction errors. There is a sense in which this research effort is of vital interest to teachers. In one narrow but important area of the curriculum in one subject a substantial research effort can offer information which may be of value in helping teachers diagnose children’s learning failure in one third of all cases, suggest that failure is caused by
and DEIRDRE PETTIm
pupils’ local strategies in one third of cases, and is unable to offer information concerning the remaining third. What, then, can one reasonably expect of the teacher in the classroom and by what criteria should we judge the teacher’s success in diagnosing children’s learning difficulties? Since in 27.5% of the instances where a buggy error was diagnosed the bug was a failure to borrow and subtract the lower digit from the upper one (an obvious error which any teacher is familiar with) (Van Lehn, 1990) any practical pay off is further reduced. It might seem, nevertheless, that in one third of cases in which children make systematic errors the research has something to offer teachers in that bugs provide an explanation which may be used to diagnose learning difficulties. This is unlikely to be the case. Various research teams working with the aid of considerable computing power have been able to identify over 100 subtraction bugs. (A full glossary of bugs is listed in Van Lehn, 1990). This is hardly information which may be of value to teachers with 30 or more children in their care; even if they could assimilate this information, how could they deploy it in their practical teaching? A final theme emerging from the research will at least strike a chord with teachers, even though it presents further difficulties conceming its practical application. It was found that some pupils tested a few days apart answered all problems correctly on the first occasion but made slips on the second and that children could manufacture bugs for the duration of one test and then discard them once they had helped them get through that particular exercise. This phenomenon has been designated as bug migration (Van Lehn, 1990, p. 16). Thus, we are reminded that children can be little “buggers” rather than bug users. Many of the usual problems facing teachers may be to do with children’s unsystematic errors and erratic performance from day to day and not to do with systematic errors encountered at specific stages in their learning. The Algorithm Learning
as an Explanation
for
A substantial research programme focussing upon one particular facet of learning fails to
Can Research
Inform
provide information for teachers which could inform and enhance their routine classroom practice. This could be because the theory underpinning the research is flawed with respect to its potential for practical application in the classroom due to the fact that the conceptual framework is based upon a model of leaming drawn from cognitive psychology and computer programming (Van Lehn, 1990). Algorithms are important and powerful heuristics which are part of the necessary intellectual equipment of practising mathe1990) and cognitive maticians (Penrose, psychologists. It is a mistake, however, to assume that conceptual tools which are of value to cognitive psychologists and mathematicians can also be employed as devices for explaining children’s learning failure in the primary mathematics class. This may be demonstrated by attempting to consider the problem of doing subtractions from the point of view of the child. It is necessary to enquire how children probably tackle subtraction tasks in the classroom. It must be accepted that such enquiries are necessarily speculative since we, like the psychologist, are denied direct access to the child’s mind. There can be no question that children do acquire procedures and strategies from teachers in order to cope with subtraction problems but if the upshot is that they can merely apply these procedures in parrot-like fashion we are unlikely to agree that the child or been “educated” (Ryle, has “learned” 1979). Learning involves more than the automatic carrying out of a procedure; it also entails such things as being able to apply the procedure in novel situations or scrutinise one’s own work and rectify mistakes made in following procedures. If the child could do no more than, metaphorically speaking, apply an algorithm in the manner of a calculating machine the teacher should be just as concerned with that child as with the child who makes erratic mistakes. Presumably it is only the teacher who is in a position to make the judgement about whether the child has learned a procedure in a manner which permits its application with understanding. There can be little doubt that children can invent their own algorithms and that these may be different from those taught in formal schooling (Carraher, Carraher, & Schliemann, 1985)
Classroom
Practice?
399
in so far as they can devise rule governed procedures which permit the correct computation of practical transactions. One may wonder, however, whether a significant proportion of children conscientiously invent mental procedures which are systematically incorrect and which, incidentally, can be expressed as written algorithms. Is it reasonable to assume that a child who makes use of a predictable procedure to get through a set of subtraction calculations which yield rule-governed wrong answers has independently devised and applied what we may term one of a large number of “buggy algorithms” and that this notion offers a way for diagnosing and remedying learning failure? Reconsider the original illustration, 207 -169
and the examples of buggy algorithms invoked to explain systematic errors such as “child takes smaller number from larger”. For many children there will be a stage in their mathematical learning when this sum presents a challenge. It will be more difficult than examples they have encountered previously. It is interesting to note that the illustrations of buggy algorithms all provide solutions which are easier than the correct one for coming up with the right answer. In such a circumstance it is difficult to accept the notion that the child has consciously invented a procedure, albeit an incorrect procedure and that knowledge of that buggy algorithm will aid the teacher in diagnosing learning difficulties and helping the child. Indeed, it is doubtful whether, with reference to the specific case of subtraction, children are actually able to apply a written algorithm model to match the rule (Hart, 1989). What the teacher needs to know is why, at the outset, the child did not or could not follow the harder correct procedure and decided upon a different easier strategy. Invoking knowledge about buggy algorithms introduces extra redundant information which is unlikely to be of value to the teacher. The reason why the child got the incorrect answer could be that the child lacked motivation or, more likely, did not understand place value (setting aside factors such as
400
DAVID McNAhfARA
carelessness or copying from a child also making mistakes). It is commonplace notions such as these which provide teachers with the information they need to help the child overcome its difficulties. The only person who can have direct access to plausible and usable explanations of the child’s learning difficulties is the teacher. In the classroom it is the teacher who has to decide how the child’s subtraction errors have been generated. This is part and parcel of teachers’ practical, professional expertise. There can be little point in recasting their expertise within a mathematical or psychological framework (Smedslund, 1982) and elaborating it beyond the feasibility of practical application. Teaching and learning involve a personal encounter between the teacher and learner and the only explanation for learning failure which can be used by the teacher is one which is expressed in ordinary language and invokes everyday notions. This, of course, places a burden upon the teacher but there is no one else upon whom the onus can be placed. The teacher who gets it wrong by deciding that the child has been “lazy” rather than appreciating that the child does not understand place value, and keeps the child in at playtime for extra practice makes a mistake which could have detrimental effects upon the child’s motivation.
Implications Discussions about children’s difficulties in learning mathematics are likely to address issues such as: conjectural knowledge about the development of children’s cognitive processes and its implications for structuring and sequencing the mathematics curriculum; problems entailed in crossing the divide between practical activities and formal representation; and the seeming disjunction between the mathematics of everyday life and experience and the decontextualised mathematics of the classroom. We do not engage in these debates. Our starting point is the recognition that the commercially produced mathematical schemes of work which are used regularly in most primary classrooms, the National Curriculum for mathematics and, indeed, parental expectations, all require primary teachers to teach what may crudely be termed “doing subtraction sums”. Moreover,
and DEIRDRE PETTIlT
there is compelling evidence that mathematical computation is given a high priority in the primary classroom (D.E.S., 1978; BarkerLunn, 1982; Sirotnik, 1983; Mortimore, Sammons, Stoll, Lewis, & Ecob, 1988); how may the teacher be best advised and prepared so as to undertake this activity? First it should be recognised that any explanation of a child’s failure to come up with the correct answer must, if it is to facilitate the teacher’s instruction, be expressed in terms of the familiar language which a teacher uses when talking with a young child. The teacher when teaching the child can do one thing not two. The teacher cannot both at the same time seek to explain to the child and also engage in some second order mental activity which draws upon different terminology and concepts so as to provide heuristics or data to inform practical activity (among the various discussions of this problem see for example: Ryle, 1949; Rorty, 1989). Thus, we must be circumspect if we seek to go beyond commonplace or common-sense accounts of why children make computational errors. Advice based upon informed professional experience coupled with the teacher’s understanding of the relevant mathematics is more likely to be of value to the practitioner than that which derives from research which is based upon conjectural notions about how the child’s mind works. It is a moot point whether the prescriptions for teaching subtraction using “carrying”, “equivalence”, or “equal addition” are derived from an appreciation of mathematics or “common sense”, but what is clear is that these methods for teaching subtraction are not derived from research. it should be appreciated that Second, teachers’ ability to teach subtraction to young children will depend upon their formal mathematical knowledge and upon those personal qualities and skills which enable them to become perceptive and sensitive observers of children’s learning. It is through watching children, talking with them about their computations, and assessing their work that the teacher is likely to diagnose their learning problems. This is an ability which, in part, can be developed through practical experience and beginning teachers can be tutored so as to help them develop these practical skills. The basis for such tutoring or training should include
Can Research
Inform
the accumulated professional wisdom of experienced practitioners. Finally, it is important to recognise the crucial importance of time as an essential resource in the teaching process. Both evidence and experience indicate that the best efforts of highly regarded teachers are routinely deflected by the demands of 30 or more children (see for example, Desforges & Cockbum, 1987) and that when a second party spends a substantial period of time in the class engaged in sustained observation of, and talk with, an individual child, it is a comparatively straightforward matter to assess the extent of a child’s learning and engage in the detailed diagnosis of difficulties (Bennett et al., 1984). What beginning teachers require in order to maximise the amount of their scarce time available for productive teaching is, first of all, advice about the organisation of the classroom and the management of the children’s learning. Teachers need information and suggestions not only about optimising the use of their own time but also that of their children (teachers sitting at their desks dealing with queues of inquiring children may be allocating all their time to teaching but they are also minimising the time individual children devote to learning). Hence sound and realistic suggestions from experienced practitioners who have tackled the problem of how best to organise the mathematical classroom should be made available to beginning teachers. Such advice should not be cluttered up with hortatory prescriptions for primary practice; advice must be considered on its merits in terms of the goal of maximising both the teacher’s and the children’s use of time. For example, teacher dominated whole class instruction is an efficient way of ensuring that it is at least possible for all children to benefit from the teacher’s teaching. The essential thrust of what we are saying is that in order to teach subtraction successfully in busy classrooms with many children teachers are very much thrown upon their own resources of perception, energy, and intuition. In order to become trained or better mathematics teachers they are more likely to require guidance, advice, and information drawn from professional practice and experience and from the subject itself rather than knowledge selected from research which has investigated learning in laboratory like settings which do not reflect the
Classroom
Practice?
401
classroom environments in which teachers must teach their pupils. Professional Advice We have sought to demonstrate, with reference to one particular arithmetic skill, that empirical research based upon large populations and involving a massive investment in investigators’ time and effort does not generate information which can be of assistance to teachers when helping children with subtraction in the classroom. The teachers’ task, however, remains; they must be responsive to the particular errors of particular children and take appropriate action within the resource and time constraints of the classroom (Leinhardt, 1988). This entails, among other things, assessing individual children and then taking appropriate action (Ausubel, 1968). Since research has little to offer, teachers must fall back on their knowledge and experience. There seems to be no reason why this should necessarily be their own personal experience; it may include accumulated professional wisdom which is passed on in the staffroom or included in, say, teacher training courses. We suggest that much more effort and attention should go into the collection and codification of professional expertise since it is this which may prove to be of distinctive benefit to the teacher (McNamara, in press). In this section we offer an indication of what this information could consist of. Experience indicates that it is quite common for children in classrooms to subtract the smaller from the larger digit (whatever the vertical order) when undertaking computational sums. The teacher’s first strategy should be to consider and, if appropriate, reject explanations such as copying or a moment’s carelessness. At this stage the teacher may decide that the preferred action is to ignore the error since, say, it may affect the child’s motivation or hold him or her back unnecessarily. On the other hand, the teacher may judge that the child has a shaky grasp of, say, place value and that some time must be spent engaging the child in conversation so as to assess the nature of the difficulty and then take appropriate remedial action. This demands time and at this point what is probably more important than the knowledge of mathematics or research is the teacher’s ability
402
DAVID
McNAMARA
to organise the class so as to create a learning environment in which the rest of the class remain engaged with their work while the teacher concentrates on an individual child. In the one-to-one situation several options are open to the teacher and the one which is likely to be employed is to revise the algorithmic procedure. We would hope that it would be one, such as the equivalence method, which has a clear relationship to the mathematics involved, especially with respect to place value. An alternative strategy might be to revive earlier regrouping and equivalence models of place value, perhaps using structural apparatus. A distinctive characteristic of teachers’ acquired expertise is that it does not offer the correct solution but alternatives which are open to professional discussion. Just as the members of the Gardener’s Question Time panel may offer different advice on how to grow bigger and better begonias, teachers may offer different suggestions for dealing with a learning problem. The first strategy we proposed may be objected to on the grounds that it is mechanistic and does not contribute to the child’s understanding. But teachers have to make professional judgements: Will they introduce a child to a practical exercise dealing with notions of exchange and equivalence so as to aid (hopefully) the child’s understanding or, say, persist with the algorithm in the expectation that the child will eventually realise why it works. Research cannot provide the answer; teachers must rely upon their personal knowledge of the child as to what is preferable in the particular case. Another aspect of the teacher’s knowledge is likely to be that many children find it difficult to understand place value (and that this understanding comes later than parents, say, may expect). Understanding place value is a gradual process which develops over time and the appropriate teaching strategy (given that the teacher has to teach subtraction) may be to re-teach the algorithm and pursue practice exercises until such time as the child appears to genuinely understand the higher order skills involved (Gelman & Gallistel, 1986). The teacher may, of course, attempt to supplement this approach by explaining why the algorithm works but this requires considerable expository skill and even if the teacher can provide an ade-
and DEIRDRE
PETTITT
quate account the explanation may fall on deaf ears. In our experience very few student teachers, all of whom have ‘0’ level standard mathematics, can explain why the equal addition or the equivalence algorithms work - just as they find it difficult to operate in number bases other than 10. It is not that they are deficient in mathematics; they have simply forgotten that understanding base 10 is easy because they have used it routinely all their lives. For young children, as primary teachers know, subtraction is difficult territory. Teachers also appreciate that requiring children to go back to an earlier stage and recover previous activity is not always a good idea. It is often resented by children and, for example, additional practical work does not necessarily facilitate understanding. Indeed, it has been suggested that structural apparatus is not useful until after a concept has been acquired (Hart, 1989). In a word, at the “chalk face” the teacher cannot legislate for learning subtraction, yet must take steps to foster learning. As such the teacher will be required to make professional decisions in circumstances which are not conducive to sound decision making, such as lack of time, lack of information, and competing The major factor influencing demands. decision-making is likely to be the teacher’s professional experience. This should, preferably, be experience which has been exposed to reflection and supplemented with professional expertise gained from other teachers. But in the final analysis the teacher acts as a human being making decisions about the behaviour of other, younger, human beings and it will be a subtle amalgam of professional skill, intuition, and expediency which inform these actions.
References Ausubel, D. P. (1968). Educorional psychology: A cognirive view. New York: Holt. Rinehart and Winston. Barker-Lunn, J. (1982). Junior school teachers: Their methods of practices. Educofional Research. 26. 178- 188. Bennett. N., Desforges, C., Cockbum, A., & Wilkinson, B. (1984). The quo&y of pupil learning experiences. London: Lawrence Earlbaum Associates. Carraher. T. N.. Carraher. D. W.. & Schliemann. A. D. (1985): Mathematics in the streets and in schools. Brirish Journal of Developmental Psychology, 3, 2 I -29.
Can Research
Inform
Cockroft, W. H. (1982). (Chairman). i%rhemcltics counts. London: H.M.S.O. Confrey, J. (1987). Bridging research and practice. Educarional 77wo~, 37, 383-394. Dearden, R. F. (1980). Theory and practice in education. Journal of Philosophy of Education, 14, 17-37. Department of Education and Science (D.E.S.) (1978). Primary educarion in England: A survey by H. M. Inspecfors of Schools. London: H.M.S.O. Department of Education and Science (D.E.S.) (1989). Mathemofics in the national curriculum. London: H.M.S.O. Desforges, C., & Cockbum. A. (1987). Understanding the mathematics reacher: A study ofpracrice in schools. London: Falmer Press. Desforges. C., & McNamara, D. (1977). One man’s heuristic is another man’s blindfold: Some comments on applying social science to educational practice. Brirish Journal of Teacher Education. 3, 27-39. Gelman. R. U.. & Gallistel, C. R. (1986). The child’s undersranding of number. Cambridge, MA: Harvard University Press. Han. K. (1989). There is little connection. In P. Ernest (Ed.), Marhematics teaching: The state of the arr, London: Falmer Press. Hughes. M. (1986). Children and number. Oxford: Basil Blackwell. Jefferys, D. (197.5). How psychology fails the teacher. British Journal of Teacher Education, 1, 63 -69. Lampert, M. (1986). Knowing. doing and teaching multiplication. Cognirion and Instruction, 3, 305 -342. Leinharht. G. (1988). Situated knowledge and expertise in teaching. In J. Calderhead (Ed.), Teachers’professional learning (pp. 146- 168). London: Falmer Press.
first
Classroom
403
Practice?
McNamara. D. (in press). Vernacular oedaeogv. British Journal of Educarjonal Studies. . - -. Mortimore, P.. Sammons, P.. Stall. L.. Lewis. D.. & Ecob, R. (1988). School matters: 7he junior years. Wells: Open Books. Penrose, R. (1990). 7he emperor’s new mind: Concerning compurers, minds, and rhe laws of physics. London: Vintage. Polanyi. M. (1969). Knowing and being. London: Routledge and Kegan Paul. Romburg. T. A., & Carpenter, T. P. (1986). Research on training and learning mathematics: Two disciplines of scientific enquiry. In M. C. Wittrock (Ed.). Hundbook of Research on Teaching (3rd. ed., pp. 850-873). New York: Macmillan. Rorty, R. (1989). Contingent?, irony and solidarity. Cambridge: Cambridge University Press. Ryle, G. (1949). The concept of mind. London: Hutchinson. Ryle, G. (1979). 01 rhinking. Oxford: Basil Blackwell. Sirotnik. K. A. (1983). What you see is what you get consistency, persistency, and mediocrity in classrooms. Hazard Educational Review. 53 (I ), 16 - 3 I. Smedslund, J. (1982). Revising explications of common sense through dialogue: Thirty six psychological theorems. Scandanavian Journal of Psychology, 23, 299 - 305. Van Lehn. K. (1982). Bues are not enouzh: Emoirical studies of bugs, impas& and repairs in proc’edural skills. Journal oJMarhemafica1 Behaviour. 3. 3-71. Van Lehn, K. (1490). Mind bugs: 7he origins of procedural misconceprions. London: Batsford Books. Received
2 May 1991 0