On What Isn't Learned in School

Introduction

At first glance, anyone can say what is meant to be learned at school: the three R's, Spanish, woodwork, etc., etc. These are the sorts of thing on the timetable. More recently students of education have been sensitized to a vast number of other things that are supposedly learned, and which are often thought to be essential to the school's continued place in the social formation: the web of values, norms, and expectations that Dreeben (1968) focussed on in a book whose title I have adapted; sex-roles (Delamont, 1980); addictive consumerism (Illich, 1971); etc., etc. So many 'hidden' curricula have been revealed that one is hardly surprised the schools do so badly at the three R's and the rest of the explicit and acknowledged curricula! Between them, these two bodies of content seem to exhaust the attention of curriculum theorists and other persons concerned with education.

One aim of this paper is to draw attention to a variety of skills, of possible curricular contents, that are frequently invoked by educators as important aims, often to justify the teaching of subjects or particular topics, but which are in fact neither explicitly taught, nor, more importantly, properly learned in the schools.

I shall focus on reasoning skills, in particular the skills involved in appreciating deductive relations and recognizing logical fallacies.¹ A well-known example can illustrate the kind of concern: from the two premises, All men are mortal, and Socrates is a man, one can deductively infer the conclusion, Socrates is mortal. The validity of this inference is independent of the truth or falsity of the premises or the conclusion (change mortal to alive today in both sentences and the argument remains equally valid). Similarly the invalidity of other arguments is independent of the truth or falsity of their component statements (change all to most and the argument is invalid, however certain we might be of Socrates' mortality). Deductive relations are of course only a part of what must be grasped to reason sensibly or to have a fair chance of successfully solving problems, but they are a sine qua non for such abilities, and they are also somewhat easier to study than the broader skills.

The Educational Paradox of Reasoning

I have said educators invoke the promotion of reasoning but do virtually nothing about it. It is worth noting the extent of this paradoxical neglect. Any number of subjects have had their curricular space justified on the grounds that they would help people to think better. In the Caribbean, the C.X.C. claims that "one of the aims of the Secondary School curriculum is to produce students who can think clearly, reason and follow a logical argument" (C.X.C., 1979) and the syllabi for Integrated Science, Mathematics, and English, and the draft syllabi for Physics and Chemistry, make explicit mention of some skills in deductive reasoning among their other aims.

Part of the problem lies in the jump from grand, unspecific aims to detailed objectives. Logical inference, argument evaluation, hypothesis testing, etc. get an acknowledgement in the ritual listing of aims, but rarely survive nearer the chalk-face world of objectives. It is rare to find educationalists specifying areas and topics such as logic, semantics, data versus hypothesis, the need for experimental controls, etc., as do Hillman and Sattory (1980) in their recommendations for the teaching of biology. It is instructive that a prominent philosopher of education who thinks that education involves an awareness of logical distinctions and that it is in this area "that we have most obviously failed, to date" (Barrow, 1981, p. 44) is content to leave it unclear "what is implied for the curriculum" (p. 112).

So, despite the grand aims, English teachers teach English, Chemistry teachers teach Chemistry, and it must be very rare even for a philosopher of education to explicitly teach trainee teachers anything about reasoning as such. In the schools and teacher training colleges, neither teachers nor taught ever study reasoning, logic, or informal logic.² Points about reasoning are of course sometimes made, but reasoning skills are most definitely to be 'caught' rather than taught: an Australian researcher sums up various studies of classroom interaction by saying that "the crucial fact about teacher talk ... is that teachers do most of the higher level cognitive work and they do it implicitly" (Young, 1983, p. 11).

Implicitness is a strongly conservative device. When things are taught explicitly there is a fair chance that students can make progress, and can often surpass their teachers. But when something remains tacit, not only is acquisition harder but further progress beyond the standards of the home and school milieu is particularly difficult (giving rise, according to some, to the ideology of giftedness (Bourdieu and Passeron, 1977)). Imagine where we would be if no one were taught even the rudiments of arithmetic explicitly, or at least not until they got into tertiary education.

The rationality of conservatism depends crucially upon the value of what is conserved. In the case of our tacit 'common-sense' ideas about good and bad reasoning, that value is unfortunately very low. Data concerning reasoning skills will be presented later, but it is worth noting also that what is tacitly transmitted regarding the evaluation of argument leaves much to be desired. Scriven comments on a problem found in an English project in San Francisco - "the incapacity of experienced judges from the composition field to separate logic from style considerations, in particular their tendency to overgrade stylish nonsense and undergrade crude but basically correct criticism" (1980, p. 153). I think the C.X.C. English syllabus and examination would yield evidence of similar confusions. So we cannot excuse the inattention to reasoning skills in the day-to-day work of schooling on the grounds that tacit transmission is doing an adequate job.

Deductive Competence in the Caribbean

I have said we do not do very well when it comes to teaching people to reason, either here or in the metropolitan countries that have been more extensively studied. I should like to present a few figures to support this claim, though this is not the place to go into the details of the evidence (for some of which, see Nolan (1984) or Nolan and Brandon (1984); some of the data has not been presented elsewhere). The Cornell Conditional Reasoning Test, Form X, (Ennis, Gardiner, Guzzetta, Morrow, Paulus, and Ringel, 1964), a test of twelve principles of conditional logic, was administered to various groups of secondary school children in Jamaica. Another shorter test, the Mona Reasoning Test, Form A, as I have named it, which is based on the Cornell test, was administered to Jamaican school children and also to trainee teachers in St. Lucia. Both tests are scored in such a way that wild guessing is penalized and the overall total score feels like a percentage, although in fact the range on the Cornell test is from -9 to 99 and on the Mona test from -8 to 100.

Consider first the overall mean scores found on the Cornell test both here and in the U.S. (with groups of somewhat above average students, as reported in Ennis and Paulus, 1965), as presented in Figure 1.

Two points are particularly important. The above average Jamaican comprehensive school groups end up fairly close to the U.S. mean at a similar age and grade; but notice where they start - way below the U.S. group. In fact the grade 7 U.S. mean score is over 1½ standard deviations above the grade 7 Jamaican mean score. It would seem that students in Jamaica and the U.S. can 'catch' about the same amount of these reasoning skills from their teachers and environment, at least in the academic streams, but the Jamaicans enter secondary school with very, very little. (And note that there is no significant difference between the mean score for the grade 7s who passed the Common Entrance and the grade 7s who failed it.)

But for the other side of the coin, consider the groups labelled 3 and 4. The groups labelled 3 belong to a vocational stream in the same Jamaican comprehensive. They end up, at the start of grade 11, below the mean U.S. grade 5, and below the Jamaican academic stream's grade 9. The groups labelled 4 simply get older. In these three grades of the 'secondary' segment of a rural All-Age school, the children seem to have caught absolutely nothing.

One insistent question is: how much was there to catch? This question cannot be given a direct answer, but some light can be thrown on it by the results obtained on the Mona test. At one Kingston high school, three grade 10 groups averaged 48.1 (SD = 15.2) (n = 86), at another a selection of two good grade 1O groups (the same as the groups labelled 6 in Figure 1) gave an average of 55.3 (SD = 16.7) (n = 35). In St. Lucia, 100 trainee teachers gave an average of 55.4 (SD = 13.8). When these trainees are separated into those preparing for secondary schooling and those preparing to teach younger children, the results are 59.6 (SD = 14.0) (n = 23) for the secondary group and 53.4 1 (SD = 13.5) (n = 68) for the others (9 respondents could not be classified). This is a quite insignificant difference, t (89) = 1.87, p = 0.065.

The same picture of virtually identical performance from grade 10 high school students and trainee teachers is found if one turns to the different logical principles in the test. The mean facility, for instance, of one set of items testing modus ponens was 74 for the grade 10 students and 81 for the trainees, for contraposition, 48 and 52, and for denying the antecedent, 25 and 24 respectively.

This last example brings home one very important finding of this and much other research. People, both pupils and their teachers, are hopeless when it comes to recognizing invalid arguments. 30% of the trainee teachers had not mastered modus ponens,³ which is hardly encouraging; but only 6% had mastered the skill of recognizing that one cannot deny the antecedent of a conditional and infer the negation of its consequent. The general weakness in simple forms of inference is displayed in Table 1 below.

Table 1: Percentage Mastery of Four Logical Principles, Jamaican Secondary Students and St. Lucia trainees

St. Lucia is not Jamaica; trainee teachers in 1984 are not much like experienced teachers in the primary and All-Age sector. But for all that, there is nothing in the data to suggest that teachers will far outshine their pupils in logical reasoning. In St. Lucia the younger teacher trainees tend to do better than the older ones (r (93) = -0.22, p = 0.018; a few did not reveal their ages); it is not unreasonable to expect a similar situation in Jamaican primary schools. Add to that the possibility, documented in the U.S. (Hansen and Pearson, 1983, and references therein), that teachers spend even less time on reasoning and higher-order skills with poor achievers than with academically more successful children, and the failure of the All-Age children and of those in the vocational stream to catch much, if anything, of fairly simple reasoning skills becomes somewhat less surprising, if no less deplorable.

Obstacles to the Teaching of Reasoning

The educational treatment of reasoning is deeply anomalous. There is much to be done to improve the learning of logical reasoning skills both here and elsewhere. But while I would like to see action to improve the skills and eliminate the anomaly, I do not imagine such action will be easy to initiate or sustain. I wish now to examine some of the difficulties in the way of informal logic in the schools, difficulties that are for the most part at one and the same time inherent as facts about logical reasoning and also the foundation of possible or actual social pressures against putting such reasoning into the syllabus.

The first difficulty has already been adumbrated: teachers in general, and as they are currently trained, have little or no more expertise in this area than their pupils. People sometimes seem to doubt whether there is expertise here to be had, or whether it is needed. Flew reminds us of Locke's dictum that "God has not been so sparing to men, to make them barely two-legged creatures and left it to Aristotle to make them rational" (Flew, 1975, pp. 20-21, quoting from Locke, 1961, IV xvii 4); but Flew's own book belies much of this point. One would hardly write, or publish, books specifically focussing on reasoning if there were not some grounds for thinking that such study makes a difference. And one has only to teach formal logic to discover that people can come to learn that certain arguments that they may have thought perfectly acceptable are in fact fatally flawed, and that others, whose complexity makes intuitive evaluation uncertain, are logically impeccable. There is, then, expertise to be had; there are techniques that can be taught and which will at least go a long first step towards an informed judgment about a particular argument. As things are, one can hardly expect teachers, whose self-respect depends in large measure on having something their pupils don't, to plunge in to the study of argument with much enthusiasm; but this situation is at least alterable - if collectively we have the will to alter it.

But here one should note a danger that appears in one form or another in probably all subjects, but which is particularly noxious in the case of reasoning. To have something their pupils don't, teachers do well to have some esoteric learning; they also want something that lends itself to the making of obstacles (i.e. examinations). For these reasons, among others, many subjects that enter curricula are perverted away from their intended focus - we want to keep fit, but we learn the physiology and anatomy of our bodies; we want to read Chaucer with enjoyment and understanding, but we learn the sound changes of Middle English; we want to improve our curricula, but we learn models of curricular decision making; we want to learn to reason better, but we learn Godel's incompleteness theorems. Of course, academics are adept at making a case for such divagations from the simple, practical goal; and often they are right enough; but the danger is obvious and commonly we suffer from it. It is the main reason I do not overmuch lament the C.X.C. decision to scrap the "Reasoning and Logic" option: in the context of mathematics, logic so easily becomes simply another bit of arcane symbolism, unrelated to human argument, and producing only "puzzles".⁴ The notorious question that even made the Barbados newspapers⁵ is not really likely to help one confront the confused argumentation of such newspapers' opinion columns, or even the evidence of normal informants. But if reasoning as taught becomes insulated from the real world of argument and discussion, there seems to me little point in teaching it in school. So in advocating the study of reasoning I am not ipso facto advocating courses in formal logic, though pace Flew I do not think it likely for there to be good coaching in logical reasoning without some acquaintance on the teacher's part with formal and theoretical matters.

But if we do try to keep the study of reasoning in touch with actual argument we must face some further problems. To use logical techniques on an actual argument, you must first catch your argument; you must know what the argument is that you want to evaluate. But while this might be easily dismissed as merely a matter of comprehension, that would be too facile a response. Comprehension is in remarkably short supply, at least if my own experiences both as examiner of other people and as misunderstood writer, are at all typical. Academics are notoriously bad at grasping each other's point of view, though this is, I suspect, more a matter of misrepresentation than of sheer inability to understand or simple carelessness (though Guthrie, who had every reason to know, commented "it is remarkable how often one scholar's reference to another needs to be checked by first-hand inspection" (1975, p. 460 ftn 1)). But I doubt that ordinary people do much better.

One major problem in identifying the argument you are judging is that in real life people leave a great deal unsaid. Implicit assumptions must therefore be suggested to get to grips with most arguments in real life; but there can often be genuine disagreements about what should be restored, about what the original arguer had in mind, or about what would yield the strongest argument from the fragments actually given (see Ennis, 1982, for a clear discussion). No doubt teachers of literature have learned to live with similar uncertainties, but they are worth noting when we are looking at the dangers of introducing the teaching of informal logic to teachers who may be used to more cut-and-dried subjects.

A second point here is that formal logic offers idealizations, schemata, or simplifications. These can serve very useful purposes (cf. Kielkopf's (1984) discussion of the use of "target forms", for instance) but it should not be surprising that they often fail to measure up to the nuanced complexity of actual argument.

In fact, it is not only complex argument that creates a problem. Consider a very simple inference: All birds lay eggs, so there are some birds that lay eggs. Is it valid? Modern logic says that it isn't, until you add an extra premise such as There are birds. Many people find this counter-intuitive; they think that All A are B tells them about how many As are B and not just about how many As are not B. There are even greater difficulties with translating the fundamental connective if ... then ... into a formal logical language. Depending on one's theory of English, these may be cases of logic's failure to measure up to the real world at the very start of the enterprise.

Logicians try to extend the range of their weaponry, but diverse theories are suggested. We do not have an agreed account of how a word like large works, or of how best to characterize the inference from John is walking sadly to John is walking (cf. Platts, 1979, chs VII and VIII) and while there is usually no disagreement about what phenomena to account for, even this cannot always be said. And again, even in the fundamental core areas of logic, there are alternative approaches whose differences run pretty deep (e.g. Quine and Prior on the understanding of quantification); and this is to ignore those who would scrap classical logic itself. This is not the place to enter into or even to explain these differences, but they do indicate yet another reason why even expertise in logic will not guarantee one access to an exclusive set of right answers. Perhaps it is sufficient for most purposes that such expertise ought to enable one to pick out wrong ones.

A final point against informal logic is that it is likely to be disruptive. Our beliefs tend to be erected upon much shakier foundations than we like to think. It is no accident that the citizens of Athens thought Socrates made young men into cynical destroyers of what was taken for granted. There are, of course, powerful counter-pressures that help to keep critical reflection within its usual bounds (cf. my 1982 for a few of them), but the practice of evaluating argument is by nature an explosive discipline; teachers can never be sure what might be tackled next.

These considerations, and no doubt many others, suggest that the explicit study of actual reasoning, guided by the tools and techniques of logical theory, is not likely to stand much chance in the cut and thrust of curricular power politics, unless it is given self-conscious and very determined support. Scholastic elaborations, or even mathematical logic, cut adrift from actual argument, might stand a chance; but there is already an excess of superfluous, supposedly intrinsically worthwhile pursuits in the curriculum, and these out-growths of applied logic tend to be pretty difficult, mostly theory with few of the interminable data to clog up children's minds.

Does it Matter?

This exhumation of one of education's notable skeletons is not intended simply for show. I think we should do something about our failure to take reasoning skills seriously. We pay lip service to conceptions of education that stress reasoning power and critical insight; let us practise at least this bit of our preaching.

Sometimes when one points to the prevalence of logical error, people reply that anyway the people making the mistakes still manage to survive quite cheerfully. True, in many cases they do. But this type of consideration would undermine virtually all efforts to get people to think or do things differently. Grasping the implications of one's situation or stock of information may require a lot more than logical acumen, but it does require such skill; and I would hope that one does not need to argue much for the importance of such a grasp. As things are, some people do acquire such a grasp; but as I have argued, the absence of direct attention to reasoning leaves most people almost wholly at the mercy of their home and school communities: not exactly a recipe for development.

A more sophisticated response to the same point is to challenge the claim that there is in fact widespread logical error. I have already said that real arguments are not ready-made for the application of formal techniques; this allows for the possibility that what superficially look like fallacies may really be interpretable as good, but more complex, arguments. And in fact this is sometimes an acceptable approach. It is anyway always possible: any invalid argument can be made valid somehow by various additions, though not always by making otherwise plausible additions. Some ways of 'excusing' bad arguments are more plausible than others; one particularly fruitful method appeals to ideas offered by H.P. Grice about the rules of polite, and not so polite, conversation. To illustrate the idea with a kind of invalid argument mentioned already, denying the antecedent:⁶ suppose someone says to you "If you move, I'll shoot"; surely you can infer that if you don't move, he won't shoot? Well you can if the man with the gun is playing by the rules of polite hold-ups; but of course he might be unscrupulous. In general, the argument is invalid. I assume no one would think that if it were true that if we ask for 300% we won't get 40% then it would be true that if we don't ask for 300% we will get 40% - I could wish negotiating were that simple.

People are in fact very charitable with statements, their own and even other people's (see Fillenbaum, 1978, for some evidence here). Many students trying to justify their answer that If the West Indies win the next test, they'll win the series, so if the West Indies don't win the next test, they won't win the series is a valid argument told me of enormously involved assumptions that would indeed allow the conclusion to follow from the premise, though no reason was ever offered why the bare argument should be thought to involve such a set of assumptions. (Of course the ingenuity used here, in contrast to cases where the argument actually is valid, no doubt testifies to some sneaking and implicit awareness that all is in fact not well with the argument.) When such people lose their bets, it is perhaps a matter of indifference whether we say that they cannot tell an invalid argument or whether we say they made certain false assumptions; they lose their bets either way. Using Kielkopf's terminology, we might say that since they lack the target that says "patently invalid" for the bare argument, if they are required to consider the validity of the argument they are all but forced into supposing some assumptions or other which would allow the argument to go through on the mistaken presumption that arguments usually are valid. If they had the correct target, then they could be rather more circumspect in looking explicitly for unstated assumptions that might save the inference, and they would be prepared to allow for the possibility that there aren't any.

Education has usually been conceived as centrally concerned with promoting reasoning, or more grandly, rationality (this is a thread running through the contributions in Hollins, 1964, and it is easy to find many other remarks to the same effect). Even vocational education in the U.S. was "introduced to improve the learning of abstract material through practical experiences" (Brickell, 1984, p. 64) rather than to become the narrow inculcation of limited skills that we see around us. If, as I suppose obvious, grasping the implications of what you believe, testing possible answers, and guarding against overconfident inference, are as important for an auto-mechanic as for an educational researcher, then the present disdain for the direct study of argument and reasoning is not going to help our nonacademic students one little bit. As we saw earlier, without the stimuli of the academic stream, such students intellectually vegetate at a little above the chance level of performance on the reasoning tests.⁷ Sharing Brickell's doubt that vocational education can become, in the foreseeable future, what its original proponents wished, it seems to me imperative that we somehow overcome the obstacles I have outlined to the study of argument in the classroom, and begin, for all students, to do something about all the protestations for improved reasoning skills with which we started.

FOOTNOTES

1. These topics are treated by many authors: for accessible introductions see Flew, 1975, Scriven, 1976, or even Brandon, 1983 and for somewhat more technical approaches, Geach, 1979, or Hodges, 1977. It might also be worth making two caveats: I speak deliberately of logical fallacies since several moves in argument are traditionally regarded as fallacious although there need be nothing wrong as far as deductive logic goes; and I give an example of what is known as a syllogism but this should not encourage the false belief that all valid arguments are, or can be transformed into, such syllogisms.

2. This is of course a slight exaggeration. The University of London has had an O-level in Logic for a good time; C.X.C. Mathematics has tried, but abandoned, an option in "Reasoning and Logic"; and of course there have been innovations in various places, even in the Mona School of Education. But for the general picture all these qualifications are insignificant.

3. Each principle is tested by 6 items; getting 5 or 6 right is regarded as mastery; less than 4 is lack of mastery; exactly 4 is on the borderline. The form of the logical principles and their mnemonics are: modus ponens (MODPON) - if p then q, p, therefore q; contraposition (CONTRA) - if p then q, therefore if not q then not p; modus tollens (MODTOL) - if p then q, not q, therefore not p; and denying the antecedent (DENYAN) - if p then q, not p, therefore not q. Of these principles only denying the antecedent is invalid.

4. In the context of school mathematics there is also a strong likelihood that logic will be done sloppily in at least one crucial respect. In the nineteenth century Frege attacked prominent mathematicians for their syntactic and semantic confusions. I cannot say whether specialist mathematicians have taken his strictures to heart, but at the level of school maths things continue to be grim. Thus in the 1982 C.X.C. Maths paper students are asked about two statements, "p unless q" and "if q does not occur then p will", where in the first example p and q are used correctly as sentence-variables but in the second they must be understood as variables for the names of events (the second statement-form should have been "if not q then p"). The Mona test results support what the teaching of logic suggests, viz. that people find it hard to grasp sentence variables, while name variables are fairly straightforward. But when their examiners are utterly at sea, how can we hope for students to grasp these distinctions?

5. On the 1980 C.X.C. Maths paper: The country of Enon is inhabited by two types of people: liars and truthers. Liars always tell lies and truthers always tell the truth. A visitor to Enon held a conversation with three citizens of Enon: Alan, Beryl and Carole. During the conversation Alan remarked, "Beryl and Carole are liars". Beryl heatedly replied, '"I am not a liar," and Carole quietly added, "Beryl is indeed a liar". (i) Which one of the three citizens of Enon must be a liar? (ii) How many of these three citizens are liars? In each case justify your answer.

I must admit that I have a personal dislike, amounting to loathing, for such logical puzzles; but I don't think this makes me misjudge their utility.

6. This example is suggested by Wexler's (1978) discussion of Fillenbaum's examples. It should be noted that denying the antecedent is the argument pattern: if p then q, not p, therefore not q. What I discuss in the text is the pattern: if p then q, therefore if not p then not q. This distinction seems too nice for most people to notice.

7. The chance total score on the Cornell test is 27, and 28 on the Mona test, but it should not be presumed that the low scoring groups are simply answering at random: their performance on the different logical principles, and their facility scores for the different questions, reflect the same overall patterning as higher achieving groups; they certainly tried on the whole to answer the questions.

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