Showing papers in "for the learning of mathematics in 1988"

Unlearning to Teach Mathematics.

Abstract: A constructivist perspective holds that children's learning of subject matter is the product of an interaction between what they are taught and what they bring to any learning situation. This view is based on increasing evidence from cognitive science research that pupils' prior knowledge and beliefs powerfully affect the way they make sense of new ideas [see, for example, Anderson, 1984; Davis, 1983; diSessa, !982; Posner, Strike, Hewson, & Gertzog, 1983; Schoenfeld, 1983] Although children are usually the focus of these studies, this article is based on the premise that teacher education could be improved by taking this perspective on teacher learning

Math Anxiety and Elementary Teachers: What Does Research Tell Us?.

Abstract: It is now ten years since Sheila I obias published her book, Overwming Math Anxiety. At the time of the book's release, the phrase ''math anxiety\" had been taken up as a battle cry by the media as the reason and rationale fOr many of the problems that mathematics educatms faced So pervasive was the media attention, that math anxiety became one of the few mathematics education issues to appear as the focus of certain popular cartoon strips [see, for example, Larson, 1986] In spite of the fact that the popular press no longer appears to consider this issue a newsworthy one, the educational community is still very much interested in the topic It is precisely because the issue of mathematics anxiety is no longer clouded by the media attention previously given to it that an analytical look at what ten years of research on the subject has been able to find out is now possible The main goal of this paper is to try to synthesize what the research literature tells us about the importance of mathematics anxiety to elementary teachers at both the preservicc and the inservice levels It is not possible, however, to restrict the discussion to this focus alone because the whole field of mathematics anxiety has other dimensions that affect how the research must be interpreted Consequently, the paper will consider five geneml areas of impmtance:

Towards New Customs in the Classroom.

Abstract: Diagnosis of the problem The mathematical productions of many students at the beginning of their first year in the University often seem to mimic the writing of the teacher: the control of meaning does not appear to be a primary purpose of the texts. Syntactic characteristics often seem to prevail over semantic characteristics. Meanings are not used as a means of controlling the results of algorithms. Another observation is that the students have no interest

A Linguistic Basis for Student Difficulties with Algebra.

Abstract: Student difficulties with algebra are common, not only when students are first introduced to the subject in secondary school, but also when they are presumed to have mastered the subject and are beginning to study calculus. One need only utter the phrase "word problems" to elicit a sympathetic response, though this is certainly not the only trouble calculus students have with algebra. It is the thesis of this paper that a major component of student difficulty with algebra is the inability to make sense of the algebraic symbol system as a language, and accordingly that remedies should be sought by considering algebra in a linguistic

A Widespread Decorative Motif and the Pythagorean Theorem.

Abstract: In their already classical study of the mathematics learning difficulties of the Kpelle (Liberia), Gay and Cole [1967, p 6] concluded that there do not exist any inherent difficulties What happened in the classroom was that the contents did not make any sense hom the point of view of Kpel!eculture Subsequent research and analyses reinforced this conclusion and recognised that in view of the "educational failure" of many children from Third World countries and from ethnic minmity communities in industrialised countries like Great Britain, France and U.S A, it is necessary to (multi)wlturalise the school cuniculum in order to improve the quality of mathematics education [cf. e.g Bishop, 1988; D'Ambrosio, 1985 a, b; Eshiwani, 1979; Gerdes 1985 a, b, 1981 a, 1988 a, b; Ginsburg & Russell, 1981; Mellin-Olsen, 1986; Nebres, 1983; Njock 1985] In other words, the mathematics curriculum has to be "inbedded" into the cultural environment of the pupils Not only its ethnomathematics, but also other culture element5, may serve as a starting point for doing and elabmating mathematics in the classroom [cf D'Ambrosio, 1985 a, b; Gerdes, 1986 b, 1988 a, b] In this article we explore the mathematical-educational potential of such a cultural element: a widespi"ead decorative motif

Mathematics and society: ethnomathematics and a public educator curriculum

Abstract: Our primary purpose in this paper is to generate a discussion about the potential role of a "Mathematics and Society" curriculum in formal education. The paper is probably most relevant to formal education systems in Western industrialised countries though not exclusively so. The development of such a curriculum immediately raises questions about the sort of educational philosophies underpinning the curriculum. What might be the purpose of such a curriculum? Implicitly, a "Mathematics and Society" curriculum suggests that the study of the relationship between mathematics and society will be involved. Furthermore, questions concerning both the nature of mathematics and the nature of society need to be addressed. Thus a "Mathematics and Society" curriculum needs to be informed by models/theories of how mathematics is socially produced and of the role of mathematics in society as well as an underlying educational philosophy. Much can be learned from debates about the purpose of mathematics education, both past and present. In fact, school science educators were the first to grapple with the problem of how the nature of the subject (in this case science) is related to educational philosophy and viceversa. The nineteenth-centry advocates of the "science-ofcommon-things" curriculum have much in common with the proponents of modern day ethnomathematics, for instance.

A Week Has Seven Days. Or Does It? On Bridging Linguistic Openness and Mathematical Precision.

Abstract: I o many, educators as well as laymen, it would probably appear to be a considerable advantage if everyday concepts had the same degree of precision as do scientific ones Concepts that are used in what Berger and l uckmann [ 1966] refer to as "finite provinces of meaning", such as science, are refined and specified to refer in an as explicit way as possible to a particular class of phenomena. This process of achieving referential exactness is an essential ingredient of scientific progress and a prerequisite for theorising and increased conceptual mastery of reality. However, as Rommetveit [1974] and others have pointed out, the more open semantic structure of everyday linguistic expressions is not to be conceived as a defect in our everyday language In fact, it is an essential asset making possible communication across social, cultural, and professional boundaries and between individuals and groups that do not share experiential backgrounds Everyday, spoken, language thus serves many purpos~s and is structured to make possible communication in situations where the communicative priorities are different from those that apply to scientific work However, in specific situations, the openness of everyday linguistic expressions causes difficulties. Many words, concepts, and even sentences, have to be given a context before they become intelligible. The famous example within cognitive psychology of the sentence, "The notes were sour because the seams were split", appears to have been generated at random by a computer from a corpus of the English language until one is informed that the remark refers to a bagpipe. In the context of learning to quantify and to solve work pwblems, much research has pointed to the significance of the fact that solving word problems ''draws upon the processes of language understanding and strategy seeking as well as upon basic computational skills" [Resnick & Ford, 1984, p. 95] Consequently, manipulation of wording, sentence order, problem context, and many other aspects that, from a strictly formal mode of thinking, do not affect the mathematical structure of the problem at hand, have been shown to influence problem difficulty [cf. e.g. Resnick & Ford, 1984, p. 22 ff] Similarly, Nesher [1982; cf Nesher & Kairiel, 1978] divides '"the semantic component" into two different aspects; "the contextual constituent" (refeuing to how aver bal problem as a whole is understood) and "the lexical constituent" [p.

Cartesian and Euclidean Rhetoric.

Abstract: Is there a best way of communicating mathematics? Those who have spent their formative years attending lectures had a good opportunity to reflect on the variety of communication styles. In one time-hallowed teaching mode, the learner sat day after day, watching the back of someone writing up on the board, for fifty minutes at a stretch, a string of things like Lemma, Theorem 4.2. 1, Proof, Remark and so on. The idea was that what the lecturer copied from her notes onto the board should be recopied into the student's notebook (and there stored, perhaps, until it was committed to memory shortly before the examination). By contrast, another style of lecturing is to enthuse and inspire, stimulating the student to believe that this is a really exciting subject and fun to learn. This style leaves the learner with a warm, happy feeling, but carries the risk of imparting not enough of substance for consolidation later; or giving a feeling that things are easy, which in the privacy of one's room turns out to present unexpected difficulties, if possible at all. Each of these styles of communication can also be found in written mathematical texts. Indeed, we can see as the paradigms of these extremes two highly influential texts from the past: Euclid's Elements of about 300 B.C., and Descartes' Geometry, published in 1637. I describe mathematical communication as having a Euclidean or a Cartesian rhetoric, according as a text looks and reads more like the Elements or the Geometry.[l] To speak of rhetoric in connection with mathematics may at first sound a somewhat strange notion. But what is meant is just a concern for how language is used in communicating mathematics; I have in mind some triangle of writer, text and reader, whereby the writer is taken to be trying to communicate something to the reader via a text. It is the rhetorical form of this text, the result of the choices the writer can be presumed to have made, which is our focus of interest.

Three Hungry Men and Strategies for Problem Solving.

Abstract: Sometimes a problem starts out innocently, like the prover­bial mustard seed; but then it starts to grow and, Zap, you are engulfed by a mustard tree! So it was with the Three Hungry Men problem: Three tired and hungry men went to sleep with a bag of apples. One man woke up, ate 1/3 of the apples, then went back to sleep Later a second man woke up and ate 1/3 of the remaining apples, then went back to sleep Finally, the third man woke up and ate 1/3 of the remaining apples. When he was finished there were 8 apples left How many apples were in the bag originally?

Mathematics Education and the Future: A Long Wave View of Change.

Abstract: Mathematics education is a concern of the whole community in that the generation of wealth requires a workforce equipped with an appropriate level of scientific abilities. On the other hand sub-standard levels of numeracy contribute to problems of unemployment and hence to a drain on welfare systems. Enquiries such as [1,2] reflect the importance with which mathematics continues to be regarded at national levels. One problem with mathematics education is its lack of a macro theory. In consequence, while much research and development occurs this tends to be piecemeal and uncoordinated. As evidence of this, from time to time state of the art documents appear [3,4,5,6] which attempt to summarize what has been achieved across a wide variety of contexts and to suggest key areas for renewed effort. But the search for a set of principles to explain the past and to direct future efforts remains unfulfilled. The future is usually viewed within some context of extrapolation from the past and present. This paper is an attempt to contribute to the development of guiding principles for mathematics education by taking a frame of reference outside mathematics education itself. It can be viewed as the development of a scenario that has its basis in the literature of the economic long wave [7,8,9,10,11,12,13]. This paper examines the implications of long wave theory for mathematics education. It addresses some of the traditional issues associated with educational research, interprets some historical events as outcomes rather than causes and points some directions towards which efforts should be directed. In this sense, and from a provocative perspective it attempts to identify priorities for future developments in mathematics education.

Didactics, Mathematics, and Communication*

Abstract: This is an extended version of some lectures given at universities in the USA. It contains basic deas of the author's view on mathematics and (mathematical) education with a special emphasis on the relationship of mathematics to communication. Most of this paper is a free translation of parts of the book "Mensch und Mathematik: Eine Einfflhrungin didaktisches Denken und Handeln" by R. Fischer, G. Malle and H Bdrger, and of the article "Zum VerhSltnis von Mathematik und Kommunikation" by R. Fischer.

Two Theories of "Theory" in Mathematics Education: Using Kuhn and Lakatos to Examine Four Foundational Issues.

Abstract: Meaningful inquiry is always guided by a theory. The theory may be a refined, highly predictive calculus, as it is in physics, or it may be a rough, tentative collection of hunches, as it often is in education. When a mathematics educator studies the effects of lax and restrictive learning environments on children of different anxiety levels, she presumably has a theory that relates achievement to both anxiety and the structure of the learning environment. Or when a cognitive psychologist examines classification and sedation tasks in the learning of early number concepts, the psychologist most likely has a hunch as to how these tasks are related. Or, when a doctoral candidate designs an experiment in which children are taught several different problem solving heuristics, she presumably has a theory that predicts which of these treatments will be the most effective.

Two Kinds of "Elements" and the Dialectic between Synthetic-deductive and Analytic-genetic Approaches in Mathematics.

Abstract: axiomatics by means of which concepts like group, field, topological space, ordered group, etc., are defined as types of structures [see Steiner, 1964]. Structures are given by sets of (unspecified) elements together with one or several relations into which these elements enter; they belong to a special type if the characteristic type-properties listed as "axioms" are fulfilled. So a typical feature of building up mathematics in Bourbaki's "Elements" could be described by Bourbaki himself by saying: From this point of view, mathematical structures become, properly speaking, the only objects of mathematics. [Bourbaki 1950, p. 226 f.; see also Dieudonne, 1939; Cartan, 1943] Concerning the rooting of his "Elements" in the logical foundations of mathematics, Bourbaki developed a differentiated attitude. On the one hand he took the naive position of a working mathematician, using the language of structures and morphisms as it grew out of previous developments in topology and modern algebra in a more unified and generalized way, especially by the consequent application of set-theoretical concepts. On the other hand, he was seriously concerned with the logical and foundational status of his theory of sets and structures and therefore carefully investigated at an early stage of his work the place of his position with respect to formal logic and metamathematics, the results of which appeared in the "Description of formal mathematics" in the first chapter of the first book of his "Elements" [see Dieudonne, 1939; Cartan, 1943; Bourbaki, 1949, 1950]. With respect to the logical completeness and the consistency of his foundations he made the famous pragmatic claim: On these foundations I state that I can build up the whole of mathematics of the present day; and if there is anything original in my procedure, it lies solely in the fact that, instead of being content with such a statement, I proceed in the same way as Diogenes proved the existence of motion; and my proof will become more and more complete as my treatise grows. [Bourbaki, 1949] If a science is not only understood as a body of knowledge represented in books, but rather as a social institution with human beings involved who in a cooperative way produce, organize, and change knowledge, it is quite natural to see the didactical problems related to a science as constitutive components rather than second order questions which only come up if one deals with institutionalized teaching. This internal didactics of a science is concerned with an economic and transparent ordering of existing knowledge, with the development, choice and standardization of concepts and language, with problems of communication and distribution of knowledge, solutions, tools, etc. In this sense, E. Peschl in the discussion following Cartan's talk commented: "Thus Bourbaki's work as a whole is a necessary task from general didactical considerations as well and in this way represents a great achievement by itself. Cartan himself argued: While the members of Bourbaki considered it their duty to elaborate all mathematics according to a new approach, they did this with the hope and expectation of putting into the hands of future mathematicians an instrument which would ease their work and enable them to make further advancements. Concerning this point, I believe that their goal has already been reached as I have frequently observed that concepts elaborated by us with so much effort are now being practiced easily and artfully by the young who have learned them from Bourbaki's books. [Cartan, 1959: transi, by the author] In a rejoinder to strong criticisms of "modern mathematics", including Bourbaki's work, put forward by R. Thorn [1971], Jean Dieudonne, another founding member of Bourbaki, especially emphasized the role of communication within the scientific community. Thorn had questioned deeply the particular emphasis on rigor, axiomatization,

Context and Creation in the Learning of Computer Programming

Abstract: "'Chris Hancock is a project associate at Harvard's Educational Technology Center (ETC), a federally funded research center which investigates the uses of computers in education. Some of the ideas discussed here were developed in collaboration with other members of ETC's programming research group. Teaching and learning computer programming at the introductory level have both turned out to be difficult. Many students in high schools and colleges have had unrewarding experiences in programming courses and come away thinking "I'm not the kind of person who can do this." It's hard for those of us with a lot of programming background to appreciate the difficulties that novice programmers confront. Perhaps it's partly because we've worked with the basic concepts of programming for long enough that they now seen trivial. But another reason may be that there are some important ideas in programming that no-one made explicit to us when we learned to program. We picked these ideas up without realizing what we were learning, and even now we don't have words for some of our most fundamental knowledge.