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This chapter discusses the system of whole numbers and its development and properties, proving either in the exposition or exercises that the properties are valid. Certainly, a child's natural approach to the system of whole numbers is the best for him. When children learn to multiply whole numbers, they know how to add and multiplication is often approached from the standpoint of repeated addition. This approach to multiplication is extremely teachable but has one stumbling block: What does multiplication by zero mean? Is 0 × 3 the sum of zero threes? Numbers allow to make various size comparisons. These comparisons are based on the ordering properties of the whole numbers. This ordering is an important aspect of elementary school arithmetic.

Given some iterative process, the natural approach is to start with a more or less accurate initial value u l 0 and to perform several steps of the iteration.

These notes originated from a seminar on dynamical systems held at the University of Louvain-la-Neuve (Belgium) in the spring of 1985. Our guide for that seminar was the book Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields by Guckenheimer and Holmes [9]. This otherwise excellent book has one disadvantage for mathematicians: it contains very few proofs, and hence one is forced to go searching in the literature if one wants to fill in the details. When I tried to do this for chapter 3 of the book (on centre manifolds, normal form theory, and codimension one bifurcations), I rapidly got frustrated by the rather sketchy way in which most texts deal with the more technical parts of this theory. Around the same time I found in the PhD thesis of S. Van Gils [36] the idea of using spaces of exponentially growing functions in order to formulate and prove the centre manifold theorem. This seemed (at least for me) to be a more natural approach, and later we found out that also others had already used this idea to some extent. Stimulated by a few colleagues I then started on the project of writing some notes which would contain a reasonably complete account of the theory, which would use the new approach as its main guiding principle, and which would be suitable for use in seminars and graduate courses. Of course I had seriously underestimated the efforts needed to finish such a project, and it was only with a considerable delay that I was able to produce a first version during the summer of 1986. Since then this version has circulated among some colleagues and was tried out at a few seminars. The many remarks and suggestions which I received were taken into consideration when I wrote the version presented here; the main difference with the earlier version is the use of the fibre contraction theorem to prove the differentiability of the centre manifold. I leave it to the reader to judge to what extent the text still reflects its original goals.

This chapter presents a general theory of approximation methods. In the case of boundary value problems, there are variational methods, and methods similar to these, and also difference methods and interpolation methods. To judge the effectiveness of these methods, and the extent to which they can be justified, the methods need to be investigated theoretically. Such an investigation gives rise to three problems. In order of increasing specificity and difficulty, these are: (1) to establish that the algorithm is feasible, and that it converges; (2) to investigate its speed of convergence; and (3) to give an effective estimate of the error. Different techniques for tackling these problems have been used for each class of equations and each approximation method, and they have frequently encountered considerable difficulties, which in a number of cases have remained insuperable up to the present day. The problem of combining these investigations and constructing a unified theory of approximation methods is an important one. A natural approach to its solution is afforded by the ideas of functional analysis.

The distinction between ‘natural’ and medical models of childbirth has been a common theme in the sociological literature on reproduction (Comaroff, 1977; Nash and Nash, 1979; Graham and Oakley, 1981; Macintyre, 1981; Nelson, 1983; Oakley, 1984b). Whereas the medical model is said to view the event as inherently problematic, requiring medical intervention and control in order to be accomplished successfully, the natural model regards it as a normal biological process over which the women themselves should exert active control and in which medical intervention should be undertaken only in exceptional circumstances. These two models, in turn, imply very different measures of success. With the medical model, success is measured in accordance with narrowly defined medical criteria, whereas with the natural model, success is assessed in holistic terms. From the perspective of the natural approach, a ‘successful’ birth is one that is accomplished without the assistance of drugs or other forms of intervention and in which the woman herself has retained control over the process.

The first modest trials of replacing several mechanical computing elements in conventional desk top calculators by electrical components have already been attempted in Japan round about 1939 by H. Yamashita. The principle of program control, however, was not applied until the American automatic computer developments became known after the end of World War II. The application of the theory of switching to relay networks has been brought to a high standard at an early stage in Japan. Therefore, it was but a natural approach to utilize in practice the results obtained by the theoretical investigations for the design of computers. We started with the construction of relay computers. Here, the large-scale relay computer ETL Mark II installed at the Electrotechnical Laboratory (ETL) of the Ministry of International Trade and Industry, belongs to the most significant developments. This computer the design principles of which are described in the first part of this contribution is, since it has been completed in November 1955, actively used for the solution of various kinds of mathematical and scientific problems such as logical mathematics, atomic power reactor calculations, numerical weather forecasting, etc.

With numerous shared aspects, philosophy and law are two distinct but closely interrelated disciplines. With that said, the present paper aims to address the impact of Lockean philosophical inclinations on the process of the US Constitution development by means of textual and historical analysis. In so doing, attention is directed towards the influence of Lockean philosophy upon the Founding Fathers, particularly through scrutinizing the writings of the figures playing leading roles in the procedure of drafting the Constitution. Locke’s natural approach to law, and to citizen rights and liberties in particular, is among the most outstanding sphere of influence put upon the examined figures. From adoption of the contract theories for justifying their break from England, to taking a natural approach in their dealing with the rights to life, liberty, and property, the Founding Fathers have been under influence of Locke’s epistemological principles and conceptions. This study concludes based upon the investigated materials and historical context that the American Founding Fathers have extensively employed Lockean notions at legal philosophy level in drafting the Constitution. Fecha de envio / Submission Date: 24/04/2015 Fecha de aceptacion / Acceptance Date: 17/05/2015

Consensus is a central problem in fault tolerant distributed computing. A vast number of (positive and negative) results for consensus in various system models have been established. In this paper we isolate three features that all these system models share, and we show that inappropriate modelling choices have led to overcomplicate the approaches to studying the consensus problem, thus yielding too restrictive solutions for real systems. It is hard to question these modelling choices, as they have gained the status of dogmas. Nevertheless, we propose a simpler and more natural approach that allows us to get rid of these dogmas, and to handle all types of benign fault, be it static or dynamic, permanent or transient, in a unified framework.

Large-scale, multi-agent systems are becoming extremely complex due to the rapid advances in computation and communication. A natural approach to deal with the increased complexity of such systems is the use of abstractions: given a complicated model and some properties of interest, extract simpler models of the original system that propagate the desired properties to the abstracted model, while hiding details that are of no interest. In this paper, we review our methodology for extracting hybrid systems out of continuous control systems while preserving timed languages. This allows us to extract high level models that can be used for real time scheduling while ensuring that high level plans have feasible implementations at the lower level model. Our methodology, is then fully illustrated by a search and rescue case study.

In this paper we propose a new method for sharpening and refinements of some trigonometric inequalities. We apply these ideas to some inequalities of Wilker-Cusa-Huygens's type.

Agent technology is often claimed to be the most natural approach for automating e-commerce business processes. Despite these claims, up till now, the most successful e-commerce systems are still based on humans to make the most important decisions in various stages of an e-commerce transaction. Consequently, it is difficult to find successful actually implemented and working large-scale agent-based e-commerce applications to confirm agents superiority. Here, we discuss an abstract e-commerce environment that allows agents of different types to interact with each other and operate with an overarching goal of supporting an e-commerce transaction. A prototype system that implements this vision using JADE agent platform is also described. Finally, we report on experiments with the implemented system skeleton.

Massively parallel computers are increasingly being used to solve large, I/O intensive applications in many different fields. For such applications, the I/O requirements quite often present a significant obstacle in the way of achieving good performance, and an important area of current research is the development of techniques by which these costs can be reduced. One such approach is collective I/O, where the processors cooperatively develop an I/O strategy that reduces the number, and increases the size, of I/O requests, making a much better use of the I/O subsystem. Collective I/O has been shown to significantly reduce the cost of performing I/O in many large, parallel applications, and for this reason serves as an important base upon which we can explore other mechanisms which can further reduce these costs. One promising approach is to use threads to perform the collective I/O in the background while the main thread continues with other computation in the foreground. In this paper we explore the issues associated with implementing collective I/O in the background using threads. The most natural approach is to simple, spawn off an I/O thread to perform the collective I/O in the background while the main thread continues with other computation. However our research demonstrates that this approach is frequently the worst implementation option, often performing much more poorly than just executing collective I/O completely in the foreground. To improve the performance of thread-based collective I/O, we developed an alternate approach where part of the collective I/O operation is performed in the background, and part is performed in the foreground. We demonstrate that this new technique can significantly improve the performance of thread-based collective I/O, providing up to an 80% improvement over sequential collective I/O (where there is no attempt to overlap computation with I/O). Also, we discuss one very important application of this research which is the implementation of the split-collective parallel I/O operations defined in MPI 2.0.

This paper explores the use of decomposition-based methods for aircraft family design. The traditional approach in multidisciplinary design optimization is to decompose a problem along disciplinary lines. For aircraft family design problems, a more natural approach is decomposition by individual aircraft. This decomposition facilitates the concurrent development of several aircraft variants, providing substantial autonomy to individual aircraft development programs. Two decomposition-based methods are applied to the aircraft family problem: collaborative optimization and analytical target cascading. This paper marks the beginning of a collaborative eort to clarify the distinctions between these two methods, and to identify how these dierences impact the relative performance and applicability of these methods. Initial product family results illustrate how decomposition-based methods can be applied to the aircraft family problem.

Summarization: Visual queries assist non-expert users to extract information from spatial databases in an intuitive and natural approach, making Geographic information systems comprehensive and efficient for a wide range of applications. A common visual means of querying takes the form of drawings or graphs, under which many spatial ambiguity and translation errors rise. In this study, common query attributes extracted from user graphs such as spatial topology, size, cardinality, and proximity are regarded under a conceptual moderation scheme. Thus, the system/user may concentrate on various conceptual combinations of information. Furthermore, time is incorporated to support spatiotemporal queries for changing scenes and moving objects. Arbitrary, relative, and absolute scaling is possible according to the data-set and application at hand. The theoretic approach is implemented under a prototype user interface system, called ShapeController. Under this prototype, a user may extract scene-based relations in an automatically inferred fashion, or include single object-oriented relations when all possible relations seem redundant. Finally, a natural language description of the query is extracted upon which the user may select the desired query relations. Experimentation on a spatial database demonstrates the concepts of predefined draw objects, scaling relaxation, conceptual abstraction, and scene, object- and textual-oriented transitions that promote query expressiveness and restrain ambiguities. Presented on: Geo-Spatial Information Science

Traditional analytic methods have not been powerful enough to solve for strongfield time-dependent solutions of the nonlinear Einstein field equations. For this reason, numerical solution of the equations seemed to be a natural approach. Building on the earlier work inspired by DeWitt & Misner, my colleagues and I have developed a practical algorithm for constructing space-times. The point of view is that space-time is decomposed into spacelike time slices and a congruence of curves threading these slices. The time coordinate is that scalar function which is constant on each time slice, while spatial coordinates are constant along the members of the congruence. The systematic 3+1 Hamiltonian approach was exhaustively studied by Arnowitt et al . (1962); the application to solution of the field equations was recently stressed by Smarr & York (1978 a, b ).

CA: Resources for Teachers. Krashen, S.D., & Terrell, T.D. (1983). The natural approach. Oxford: Pergamon/ Alemany. McCarthy, B. (1981). The 4mat system: Teaching to learning styles with right/left mode techniques. Barrington, IL: Excel. Mohan, B.A. (1986). Language and content. Reading, MA: Addison-Wesley. Piaget, J. (1966). The origins of intelligence in children (M. Cook, Trans.). New York: International Universities Press. (Original work published 1936) Ramirez, M., & Castafieda, A. (1974). Cultural democracy, bicognitive development and education. London: Academic Press. Smith, F. (1973). Psycholinguistics and reading. New York: Holt, Rinehart and Winston.

Over two decades ago the spoken message was characterized as the most prized of the four language skills. The emphasis on communicative competence in recent years where second-language (L2) instruction has focused on the primacy of speech from the very onset of instruction would seem to indicate that priorities have not changed. Entire chapters, even volumes, have been written on this topic.2 With the Report of the President's Commission on Foreign Language and International Studies advocating that second-language instruction should concentrate on speaking and understanding before other skills are developed, and with the advent of methodologies such as the natural approach to L2 acquisition, it is likely that a strong emphasis on speaking will continue in our classrooms for years to come.3 Furthermore, the notion of proficiency-based foreign language teaching, which stresses the importance of the productive skills, is rapidly gaining momentum.4 Although the goal of producing a comprehensible spoken message has remained the same, discussions of this topic rarely devote much attention to the teaching of pronunciation. (The rating scales developed for the ACTFL-ETS oral proficiency interviews are a notable exception, since they attempt to provide a graduated sequence of learning goals for teachers and learners, including the quality of sounds produced at the various levels of proficiency.5) This is ironic, since the comprehensibility of a message depends, to a large degree, on the speaker's mastery of the prosodic features of a language. Encounters with persons who have mastered the structure and lexicon of a second language, yet produce all but unintelligible speech due to inadequacies in pronunciation, are not rare. With appropriate training, such speech habits could certainly be improved. The same holds true for our L2 classrooms. Clearly, if we expect reasonable accuracy in structure and vocabulary, then our expectations should be no lower in pronunciation. Apparently, little or no time is devoted to the creative teaching of pronunciation in many classrooms today.6 One way to remedy this problem might be to spend a minimal amount of time (even as little as five minutes daily) on the improvement of pronunciation in four-skills courses at the elementary and intermediate levels. On the post-secondary level, a solution to the problem might be a course devoted to the teaching of pronunciation, such as the one described here.7

By now, it is a commonplace among language teaching theorists that the focus of instruction, even at the beginning levels, should be on communicative skills-though the debates continue on which methods attain this goal most effectively, and to what degree fluency should be valued over accuracy. Nonetheless, from the Natural Approach of Krashen and Terrell to the skill-getting/skill-using concepts of Wilga Rivers, from Brumfit's notional/functional syllabus to the dynamics of group interaction delineated by Kramsch (1983), the message remains clear: Students must learn to negotiate meaning in contexts "motivated by real communicative needs" (Littlewood 70). While many of the resulting theoretical and

An intergenerational model of the income distribution, including educational effects, is presented. It is used to analyze issues in the debate over equity effects of subsidies for higher education. In particular, the analysis supports Pechman's view that conventional net benefit calculations are poor indicators of such equity effects. In this Journal [4, 7, 8, 9, 11, 13, 14] and elsewhere [16, references there], Hansen and Weisbrod, Pechman, and others have debated the equity effects of subsidized tuition for higher education. At the heart of the debate are net benefit calculations presented by Hansen and Weisbrod [7]. Pechman's critical view of these calculations may be summarized in his own words [16, p. S256]: The traditional way of looking at the burdens and benefits of higher education is to distribute the net benefits received by students by the income classes of parents (taxpayers). This sweeps the problem created by the intergenerational nature of the benefit transfer under the rug. It seems to me that a more useful way of looking at the problem is to acknowledge that the benefits of public higher education are received by one generation while the costs are paid by another and that there is no way of merging the benefits and costs in one distribution to evaluate the equity of the system. If such net benefit calculations do not measure the equity effects of subsidized tuition, what calculations do? Intuitively, the issue involves comparison of two societies identical in all respects except that one subsidizes higher education more generously. As time passes, will a difference in distributional The author is Professor of Economics, University of California, San Diego. * Richard Attiyeh and two anonymous referees provided most helpful comments. This research was supported by NSF grant SOC74-15238. [Manuscript received May 1975; accepted June 1976.] The Journal of Human Resources * XII * 2 This content downloaded from 157.55.39.207 on Thu, 20 Oct 2016 04:05:27 UTC All use subject to http://about.jstor.org/terms 148 1 THE JOURNAL OF HUMAN RESOURCES equality develop between the societies, and in which direction? Analytically, a natural approach to this question is to construct a complete intergenerational model of the size distribution of income, including higher education variables and tuition subsidy parameters; then to derive income distribution indicators and analyze the sensitivities of the indicators to changes in the subsidy parameters, and, in particular, to see whether distributional inequality increases or decreases with an increase in the rate of subsidization. Sections I and II present an appropriate intergenerational model. In principle, a sufficiently elaborated version of the model, fit to appropriate data, might yield a respectable estimate of the equity effects of higher educational subsidies. However, the appropriate data are not available, and the needed elaborations of the model would be major projects in themselves. Therefore, this paper sets two more modest goals. The first is to propose a promising approach for analyzing the subsidization issue. The second is to use the model in sorting out the logic of the net benefit calculations referred to in the Pechman quotation. Sections III and IV concern the second goal. The model will yield a measure-call it the equity measure-of the sensitivity of income inequality to the level of higher educational subsidy. The equity measure will represent what we would like to know, but cannot estimate directly due to lack of data. The model will also yield a net benefit measure of the sort referred to in the Pechman quotation. The net benefit measure will represent what we can directly estimate from available data (as Hansen and Weisbrod [7] have done). The Hansen-Weisbrod-Pechman debate will be interpreted as a debate over the following proposal: that the net benefit measure be used as a proxy for the equity measure. Hansen and Weisbrod might be identified with the affirmative, Pechman with the negative. It turns out in the model that the equity measure is not well proxied by the net benefit measure, thus favoring Pechman's side. Of course, it should be emphasized that the true debate (as opposed to my characterization of it) is about equity and net benefits in real life, not in a highly simplified model. However, if the logical connection between equity and net benefits does not hold up in a simple economy, it seems improbable that it will hold up in a complicated one.

Aller Anfang ist schwer... Beginning language instruction based on communicative interaction calls for its participants to negotiate a substantial transition during the first few minutes of each class session. First, and most obvious, students must move from the secure language environment of their native tongue to a session conducted entirely in the target language. They encounter a new set of teaching and learning strategies: most of their other classes will depend on a highly cognitive learning posture, while language acquisition, following Krashen and Terrell, calls for the development of comprehension skills and gradual assimilation of structures. And finally, language students are asked to participate in a learning environment that stresses group interaction, where the processes of communicative exchange are as much a part of the course as the "material" to be covered (Brumfit: 76-82; Kramsch 1983). Kramsch is not overstating the case when she observes: "By entering a foreignlanguage classroom, students leave behind the social reality created by their native tongue and start constructing a new reality, which is potentially very different from the one they just left" (1987: 17). It follows that the strategies adopted by the teacher to ease this transition require special consideration. Certainly no teacher wishing to steer students into meaningful discourse will begin by announcing in English: "Today, class, we want to look at. ."; nor would a post-Skinnerian language instructor be likely to pick up the book and launch into the pattern drill for the day. But how should one begin? While there is, of course, no single theoretical formula for building an effective bridge into a communicative environment, a pragmatic view of the situation suggests that each day's opening activities should focus on four fundamental goals: (1) to communicate nonverbally the teacher's expectation of energetic and alert participation; (2) to establish German as the accepted mode of verbal communication [I do not say "exclusive," in defense to numerous theorists, notably Krashen and Terell of the Natural Approach, who opt for allowing students-at least at first-to answer in their native language; though my experience suggests that the expectation at the very outset of simple German responses poses no problem to students and in fact enhances a positive sense of challenge]; (3) to aid students in focusing on comprehension of entirely German input; and (4) to establish contacts for later small-group work. In short, the opening activities should serve to set a distinctly interactive tone, to foster dialogue "motivated by real communicative needs" (Littlewood: 70), in order that acquisition and eventually more sophisticated speech production can follow.

One important issue confronting language educators today is not "Which single method is best?", but, rather, "Which of several methods are best employed at what stages of the teaching/learning process?" Thus it is not a question of whether James Asher's Total Physical Response (TPR) is superior to Tracy Terrell's Natural Approach (or whether either of these is superior, or inferior, to Canadianstyle immersion education, audiolingualism, grammar-translation, cognitive code, Community Language Learning, the Silent Way, and whatever methodologies are to be derived from such currently popular emphases as communicative competence, functional/notionalism, and the oral proficiency movement). The question is when to utilize what. Real-time empirical investigation that covers at least 500 hours of a large group of students' instructional experiences with a variable pick of methodologies variously sequenced has yet to be undertaken. But if it ever is, I predict that the winning combination will feature the following sequence: TPR first, then a natural approach, and, subsequently, a judicious mixture of subject matter taught via the target language (immersion education), drill (audiolingual), grammar (with translation), communicative exercises, and especially functional/notionalism, whose usefulness at the more advanced levels of instruction has yet to be appreciated. TPR's chief virtues are that it gives students an immediate sense of accomplishment and participation while keeping the effective filter miraculously unclogged. Natural approaches ease students into producing simple speech sequences almost unconsciously. Immersion techniques provide learners with vast quantities of subject-matter input, which, if Stephen Krashen is right, will constitute the stored material needed to foster the oral production which in due course is prompted by conversationally-generative techniques whether communicatively competent, orally proficient, or functional. Grammar and drilling will doubtless have their place as well, since there is some support for the notion that comprehensible input can be derived from sources of formal presentation as well as informal. In sum, our profession should be thankful that after so many years of invention and experimentation it is free to pick from and combine such a rich harvest of methods. Yet we must recall that many of these methods especially the less traditionalwere brought to life as hopeful panaceas purportedly enabling their users to shave scores or even hundreds of hours off the language-learning process. Though time-reduction has been achieved and efficiency enhanced, it is nevertheless crucial to pay close attention to some recent comments by Renate Schulz ("Foreign Language Proficiency for the Real World: Wedding the Ideal to Reality, " in John Joseph, ed., Applied Language Study ... [Lanham MD: University Press of America, 1984], pp. 3-17). Dr. Schulz, a professor of German and foreign language education at the University of Arizona, is a leading expert in her fields and, for years, a well-known figure at many conventions. For Schulz, a major task faced by us as professionals is to change "the totally unrealistic expectations of the lay public regarding what it takes to gain proficiency in a second language." Our problem is a monumental lack of classroom time (the consecrated "two years of high school" remaining an all-too-typical average). Lack of time can only be circumvented by immersion experiences abroad and/or intensive hundreds-of-hours-on-track programs at home. "Two eight to ten-week intensive summer courses could provide from 640 to 720 hours of language instruction, approaching the minimum number" that experts say is needed to obtain the respectable "3" rating on the government proficiency scale. Anything less is insufficient and (perhaps worse) self-deceptive. No method is immune from the dictates of the timepiece. As practitioners of natural approaches are discovering, their students fail to transcend an uninhibitedly fluent but flawed and fossilized "learners' dialect" unless their subsequent work in the language is the equivalent of an undergraduate major in it, preferably complemented by a year abroad. In his call for comments, Editor Sackett notes there is still another issue he wishes

Virginia C. Gathercole's recent article in the TESOL Quarterly (Vol. 22, No. 3, September 1988) reviews a wealth of research and raises a number of interesting points. In these respects, the article is a valuable addition to our understanding of first and second language acquisition in the area of theory. However, the implications for teaching that Gathercole draws from her review of the research are disturbing. The first assumption Gathercole questions is the idea that comprehension precedes production. She claims that this "is an explicit element in the Input Hypothesis of Krashen (1982), who argues that we acquire language by understanding language containing structures just beyond our productive capacity-at a level he calls 'i + 1' " (p. 409). In fact, by "i + 1" Krashen refers to an acquirer's ability to comprehend, not produce, language containing structures slightly beyond his or her present stage. Stronger support for the claim that production may precede and facilitate comprehension comes from research in emergent literacy. Clay (1975) and others have pointed to the fact that some students produce written language before they develop reading comprehension. These researchers argue that it is not necessary to delay writing until reading is mastered. In the same way, in both Asher's Total Physical Response (1977) and Terrell's Natural Approach (Krashen & Terrell, 1983), students are not required to produce language at the initial stages. Nevertheless, some students do attempt to produce the target

The keystone to effective language acquisition is assured comprehension. Students who can identify people in white shirts when asked "Wer hat ein weil3es Hemd?" are clearly on their way to comprehension. The language models provided in the classroom should enable the teacher to create a framework for communicative exchanges, and must ensure that the students understand. Within such a framework, for example, students will realize, affectively as well as intellectually, that "Tisch" means "the four-legged item of furniture upon which we place meals, papers, etc.," rather than making the conventional assumption that "Tisch" means "table," an English word, and is trying to usurp its place. The legitimacy of the target language should be self-evident in the classroom, not merely for the sake of mimicry and memorization, but for the more primary purpose of communication. As Leonard Newmark comments: "(B)ecause attaching meaning to something is an experiential cognitive act, the listener must be attending to the string (of sounds in the second language) while experiencing its meaning."' Since the meaning of the subjunctive centers on the nonreality of something, e.g., "If he were wearing sandals (which he is not)," the absence of that reality poses a challenge to the teacher to create speaking situations in the classroom that will facilitate student comprehension. The Natural Approach (NA) to language acquisition, which has gained increasing approval among language teachers in recent years, is particularly well-suited for presenting the subjunctive. This approach turns difficulties into advantages by presenting students with activities in which the individual statements are unambiguous, but which collectively demonstrate the "contrary-to-fact" situations, wishes, fears, and indirect discourse that typify the German subjunctive. Precisely because the early phases of instruction in the NA utilize tangible items such as classroom props and graphic illustrations through pictures and gestures, the sudden absence of these items can be useful in demonstrating the nonreality, the conjectural nature of the subjunctive. For example, the NA normally progresses from presenting objects that can somehow be shown, like "Tisch" or "Papier," to more abstract qualities2 that can be shown by extension from those objects: "Der Tisch ist schwer; das Papier ist leicht." Even with a very elementary familiarity with the modal verbs, miniature dramas can be created by the lack of some object for which students already know the name

Nevertheless, the fact that we do admit some virtue to these questions suggests that choice must have at least a subsidiary role to play in what it is that we do with works of art. I shall try to establish the role of choice in the realm of aesthetics because I believe that showing what can be expected of it and what cannot must inevitably suggest avenues along which we may make a freer and more natural approach to the teaching of art and appreciation.