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1、FORMS OF KNOWLEDGE IN MATHEMATICS AND MATHEMATICS EDUCATION: PHILOSOPHICAL AND RHETORICAL PERSPECTIVESPaul ErnestExeter University, UK, Oslo University, Norway, Brunel University, UKp.ernest(at)ex.ac.ukABSTRACTNew forms of mathematical knowledge are growing in importance for mathematics and educatio
2、n, including tacit knowledge; knowledge of particulars, language and rhetoric in mathematics. These developments also include a recognition of the philosophical import of the social context of mathematics, and are part of the diminished domination of the field by absolutist philosophies. From an epi
3、stemological perspective, all knowledge must have a warrant and it is argued in the paper that tacit knowledge is validated by public performance and demonstration. This enables a parallel to be drawn between the justification of knowledge, and the assessment of learning. An important factor in the
4、warranting of knowledge is the means of communicating it convincingly in written form, i.e., the rhetoric of mathematics. Skemps concept of logical understanding anticipates the significance of tacit rhetorical knowledge in school mathematics. School mathematics has a range of rhetorical styles, and
5、 when one is used appropriately it indicates to the teacher the level of a students understanding. The paper highlights the import of attending to rhetoric and the range of rhetorical styles in school mathematics, and the need for explicit instruction in the area.BACKGROUND In the past decade or two
6、, there have been a number of developments in the history, and philosophy and social studies of mathematics and science which have evoked or paralleled developments in mathematics (and science) education. I shall briefly mention three of these that have significance for the main theme of this paper,
7、 the import of rhetoric and justification in mathematics and mathematics education. Even though all of the developments I mention below are continuing sites of controversy, I merely list them rather than offer extended arguments in support of the associated claims, since this would draw me away from
8、 the main theme. Anyway such arguments can be found elsewhere (e.g., Ernest 1997).An important background development has been the emergence of fallibilist perspectives in the philosophy of mathematics. These views assert that the status of mathematical truth is determined, to some extent, relative
9、to its contexts and is dependent, at least in part, on historical contingency. Thus a growing number of scholars to question the universality, absoluteness and perfectibility of mathematics and mathematical knowledge (Davis and Hersh 1980, Lakatos 1976, Tymoczko 1986, Kitcher 1984, Ernest 1997). Thi
10、s is still controversial in mathematical and philosophical circles, although less so in education and in the social and human sciences. One consequence of this perspective is a re-examination of the role and purpose of proof in mathematics. Clearly proofs serve to warrant mathematical claims and the
11、orems, but from a fallibilist perspective this warranting can no longer be taken as the provision of objective and ironclad demonstrations of absolute truth or logical validity. Mathematical proofs may be said to fulfil a variety of functions, including showing the links between different parts of k
12、nowledge (pedagogical), helping working mathematicians develop and extend knowledge (methodological), demonstrating the existence of mathematical objects (ontological), and persuading mathematicians of the validity of knowledge claims (epistemological), see, e.g., Hersh (1993) and Lakatos (1976). Be
13、low I elaborate further on the persuasive, epistemological role of proofs in mathematics.The impact of these developments on education are indirect, as they do not lead to immediate logical implications for the teaching and learning of mathematics or the mathematics curriculum without the addition o
14、f further deep assumptions (Ernest 1995). Nevertheless fallibilist philosophies of mathematics are central to a variety of theories of learning mathematics including radical constructivism (Glasersfeld 1995), social constructivism (Ernest 1991), and socio-cultural views (Lerman 1994) which can have
15、classroom consequences. The second development is the emerging view that the social context and professional communities of mathematicians play a central role in the creation and justification of mathematical knowledge (Davis and Hersh 1980, Kitcher 1984, Latour 1987). These communities are not mere
16、ly accidental or contingent collections or organisations of persons incidental to mathematics. Rather they play an essential role in epistemology in two ways: their social organisation and structure is central to the mechanisms of mathematical knowledge generation and justification, and they are the
17、 repositories and sites of application and transmission of tacit and implicit knowledge (Ernest 1997, Lave and Wenger 1991, Restivo 1992). In education, the vital roles played by the social and cultural contexts (Bauersfeld 1992, Cobb 1986, 1989), and the centrality of tacit and implicit knowledge i
18、n the mathematics classroom do not need to be argued, as they are already widely recognised (Bishop 1988, Hiebert 1986, Saxe 1991, Tirosh 1994).Third, there is a move in the sociology and philosophy of science mathematics to focus on communicative acts and performances of scientists and mathematicia
19、ns, and in particular on their rhetorical practices (Woolgar 1988, Simons 1989, Fuller 1993, Kitcher 1991). In mathematics the parallel concern has been with writing genres and proof practices (Ernest 1997, Livingston 1986, Rotman 1993). While there has been attention to the role of language in math
20、ematics education for some time (Aiken 1972, Austin and Howson 1979, Durkin and Shire 1991, Pimm 1987, Skemp 1982) it is only recently that an awareness of the significance of genres and rhetoric for the field are emerging (Ernest 1993, Morgan 1998, Mousley and Marks 1991).These background developme
21、nts raise a number of issues concerning the form or forms of mathematical knowledge and the role and function of mathematical texts and proofs within the discipline itself and in the teaching and learning of mathematics. Whereas traditionally mathematical knowledge was understood as a collection of
22、validated propositions, i.e., a set of theorems with proofs, a number of philosophers such as Ryle (1949) Polanyi (1959) Kuhn (1970) and Kitcher (1984) have argued that not all knowledge can be made explicit. The claim that know how and tacit knowledge are important in all areas of human thought inc
23、luding mathematics. The argument for including tacit know how as well as propositional knowledge as part of mathematical knowledge is that it takes human understanding, activity and experience to make or justify mathematics. Much that is accepted as a sign that persons are in possession of mathemati
24、cal knowledge consists in their being able to carry out symbolic procedures or conceptual operations. To know the addition algorithm, proof by induction or definite integrals is to be able to carry out the operations involved, not merely to be able to state certain propositions. Thus what an individ
25、ual knows in mathematics, in addition to publicly stateable propositional knowledge, includes mathematical know how. Kuhn (1970) argues that part of such knowledge in the empirical sciences consists of “the concrete problem-solutions that students encounter from the start of their scientific educati
26、on, whether in laboratories, on examinations, or at the ends of chapters in science texts and technical problem-solutions found in the periodical literature.” (Kuhn, 1970: 187). Thus Kuhn claims that the experience of problem solving and of reading through various problem solutions leads to tacit kn
27、owledge of problem types, solution strategies, and acceptable modes of presentation of written work, i.e., tacit rhetorical norms (learned via instances). Kitcher extends the argument to mathematics and argues that both explicit propositional and tacit knowledge are important in mathematical practic
28、e, listing “a language, a set of accepted statements, a set of accepted reasonings, a set of questions selected as important, and a set of meta-mathematical views (including standards for proof and definition and claims about the scope and structure of mathematics).” (Kitcher 1984: 163). This list i
29、ncludes two knowledge components which are mainly tacit, namely language and symbolism, and meta-mathematical views, both of which have a strong bearing on the written, rhetorical aspects of mathematics. The underlying language of mathematics is a mathematical sub-language of natural language (such
30、as English or German) supplemented with specialised mathematical symbolism and meanings. It comes equipped with an extensive range of specialist linguistic objects, including mathematical symbols, notations, diagrams, terms, definitions, axioms, statements, analogies, problems, explanations, method
31、applications, proofs, theories, texts, genres and rhetorical norms for presenting written mathematics. Mathematics could not be expressed without knowledge of its language, and most would argue more strongly that mathematics could not exist at all if mathematicians did not have knowledge of its lang
32、uage (Rotman 1993, Thom 1986). Although this knowledge includes explicit elements, as with any language, knowing how to use it is to a large extent tacit.The set of meta-mathematical views includes a set of standards, the norms and criteria that the mathematical community expect proofs and definitio
33、ns to satisfy if they are to be acceptable. Kitcher claims that it is not possible for the standards for proof and definition in mathematics to be made fully explicit. Exemplary problems, solutions, definitions and proofs serve as a central means of embodying and communicating the accepted norms and
34、 criteria. Like Kuhn, he argues that proof standards may be exemplified in texts taken as a paradigm for proof (as Euclids Elements once did), rather than in explicit statements. Thus mathematical knowledge not only encompasses a tacit dimension but also a concrete dimension, including knowledge of
35、instances and exemplars of problems, situations, calculations, arguments, proofs, models, applications, and so on. This is not widely acknowledged, although the importance of knowledge of particulars has been recognised in a number of significant areas of research in mathematics education. For examp
36、le, Schoenfeld (1985, 1992), in his research on mathematical problem solving, argues that experiences of past problems leads to an expanding knowledge-base which underpins successful problem solving. Current research on the situatedness of mathematical knowledge and learning also emphasises the role
37、 of particular and situational knowledge (Lave and Wenger 1991, Saxe 1991). More generally, in mathematics education the importance of implicit knowledge has been recognised for some time, and the categories of instrumental understanding (Skemp 1976, Mellin-Olsen 1981), procedural knowledge (Hiebert
38、 1986) and implicit knowledge (Tirosh 1994) have been developed and elaborated to address it. These categories go beyond know-how, for as Fischbein (1994) argues, other forms of implicit knowledge such as tacit models are also important.MATHEMATICAL KNOWLEDGE AND ITS JUSTIFICATIONDrawing on Kitcher
39、(1984) I have been proposing an extended concept of mathematical knowledge that includes implicit and particular components, but without reference to justification. Berg (1994) argues that this is illegitimate and that implicit knowledge is a misnomer, for what passes under this name is either tacit
40、 belief (including misconceptions) or implicit method, since it lacks the robust justification that epistemologists require of knowledge. He is right that from an epistemological perspective knowledge only deserves its title if it has some adequate form of justification or warrant. However I reject
41、his main critique because I believe that adequate warrants can be provided for tacit knowledge. Explicit knowledge in the form of a theorem, statement, principle or procedure typically has a mathematical proof or some other form of valid justificatory argument for a warrant. Of course the situation
42、is different in mathematics education, as in other scientific, social or human science research, where an empirical warrant for knowledge is needed. Nevertheless, tacit knowledge can only be termed knowledge legitimately, in the strict epistemological sense, if it is justified or if there are other
43、equivalent grounds for asserting it. However, since the knowledge is tacit, then so too its justification must be at least partly tacit, on pain of contradiction. So the validity of some tacit knowledge will be demonstrated implicitly, by the individuals successful participation in some social activ
44、ity or form of life. Not in all cases, however, need the justification be tacit. For example, an individuals tacit knowledge of the English language is likely to be justified and validated by exemplary performance in conformity with the publicly accepted norms of correct grammar, meaning and languag
45、e use, as related to the context of use. Thus a speakers production of a sufficiently broad range of utterances appropriately in context can serve as a warrant for that speakers knowledge of English. This position fits with the view of knowledge in Wittgensteins (1953) later work, according to which
46、 to know the meaning of a word or text is to be able to use it acceptably, i.e., to engage in the appropriate language games embedded in forms of life. Practical know how is also validated by public performance and demonstration. Thus to know language is to be able to use it to communicate (Hamlyn 1
47、978). As Ayer says “To have knowledge is to have the power to give a successful performance” (Ayer 1956: 9). Such a validation is to all intents and purposes equivalent to the testing of scientific theories in terms of their predictions. It is an empirical, predictive warrant. It is a weaker warrant
48、 than a mathematical proof, for no finite number of performances can exhaust all possible outcomes of tacit knowledge as a disposition (Ryle 1949), just as no finite number of observations can ever exhaust the observational content of a scientific law or theory (Popper 1959). Thus tacit knowledge of
49、 mathematics can be defended as warranted knowledge provided that it is supported by some form of justification which is evident to a judge of competence. On this basis, what an individual knows in mathematics, in addition to publicly stateable propositional knowledge (provided it is warranted), includes her tacit knowledge. However what is warranted in this case is not the tacit knowledge, but the individual as possessor of that knowledge. We can then assert that Gerhard knows German, or Alicia knows proof by mathematica
