Abstract: This paper looks at recent research dealing with uses of the equal sign and underlying notions of equivalence or non-equivalence among preschoolers (their intuitive nitions of equality), elementary and secondary school children, and college students. The idea that the equal sign is a “do something signal” This expression was first coined by Behr, Erlwanger and Nichols in their 1976 PMDC Technical Report (S. Erlwanger, personal communication, June 1980). (an operator symbol) persists throughout elementary school and even into junior high school. High schoolers' use of the equal sign in algebraic equations as a symbol for equivalence may be concealing a fairly tenuous grasp of the underlying relationship between the equal sign and the notion of equivalence, as indicated by some of the “shortcut” errors they make when solving equations.

Abstract: 116 Foundation Year Engineering Students, at the University of Technology, Lae, Papua New Guinea, were given a battery of mathematical and spatial tests; in addition, their preferred modes of processing mathematical information were determined by means of an instrument recently developed in Australia by Suwarsono. Correlational analysis revealed that students who preferred to process mathematical information by verbal-logical means tended to outperform more visual students on mathematical tests. Multiple regression and factor analyses pointed to the existence of a distinct cognitive trait associated with the processing of mathematical information. Also, spatial ability and knowledge of spatial conventions had less influence on mathematical performance than could have been expected from recent relevant literature.

Abstract: In statistics, and in everyday life as well, the arithmetic mean is a frequently used average. The present study reports data from interviews in which students attempted to solve problems involving the appropriate weighting and combining of means into an overall mean. While mathematically unsophisticated college students can easily compute the mean of a group of numbers, our results indicate that a surprisingly large proportion of them do not understand the concept of the weighted mean. When asked to calculate the overall mean, most subjects answered with the simple, or unweighted, mean of the two means given in the problem, even though these two means were from different-sized groups of scores. For many subjects, computing the simple mean was not merely the easiest or most obvious way to initially attack the problem; it was the only method they had available. Most did not seem to consider why the simple mean might or might not be the correct response, nor did they have any feeling for what their results represented. For many students, dealing with the mean is a computational rather than a conceptual act. Knowledge of the mean seems to begin and end with an impoverished computational formula. The pedagogical message is clear: Learning a computational formula is a poor substitute for gaining an understanding of the basic underlying concept.

Abstract: A case is presented for the importance of focusing on (1) average ability students, (2) substantive mathematical content, (3) real problems, and (4) realistic settings and solution procedures for research in problem solving. It is suggested that effective instructional techniques for teaching applied mathematical problem solving resembles “mathematical laboratory” activities, done in small group problem solving settings. The best of these laboratory activities make it possible to concretize and externalize the processes that are linked to important conceptual models, by promoting interaction with concrete materials (or lower-order ideas) and interaction with other people. Suggestions are given about ways to modify existing applied problem solving materials so they will better suit the needs of researchers and teachers.

Abstract: This article examines the conceptual problems that children experience when attempting to solve verbal problems which contain decimal numbers. After discussing the existing research, we then describe an experiment that was designed in three phases:

Abstract: A sample of less successful students (aged 12–15) was tested on the topic of fractions. The test paper presented diagrams, word problems, and computational questions. The analysis was designed to find out the specific difficulties and deficiencies of these children. It is shown that most of them are only able to apply remembered rules to the solution of the problems without knowing whether the rule works, i.e., their understanding is at most ‘instrumental’, but not ‘relational’.

Abstract: A clinical methodology was used to investigate the perceptions which pupils of secondary school age have concerning modes of mathematical argument which have an agreed status within the world of mathematics. The analysis of data obtained from three extended contexts led to the identification of clusters of characteristic response types. Differences were found to exist between the agreed meaning of some mathematical terms and procedures and the meaning ascribed to them by students. By considering levels of performance it was possible to identify particular components, the presence or absence of which consistently determined the capacity to structure or follow proofs and explanations.

Abstract: Experiences of teachers give evidence that the usal distinction between instrumental and relational understanding, as defined by the psychologist, is insufficient to interpret learning in an educational context. The learner often possesses relational understanding of some knowledge, for which he sees no use, outside its importance as “schoolknowledge”. The author analyzes a more general concept of instrumentalism. He defines it as a rationale for learning, connected to the role school has as an instrument for future schooling and employment. Examples of use of the project method are given, which can help to establish another rationale for learning.

Abstract: The main purpose of the present research was to check the possibility of measuring the feeling of “intuitive acceptance”, experienced by a subject when he offers an intuitive solution to a problem. It was postulated that two dimensions have to be considered and combined: The level of confidence and the degree of obviousness. Almost all the questions asked referred to the notion of infinity. The subjects were pupils belonging to grades 8 and 9. Three main categories of problematic situations have been identified: (a) Problems which got high percentages of correct solutions and high levels of intuitive acceptance. (b) Problems which got two types of contradictory solutions, each of them being accepted with moderate intuitiveness. (c) Problems which got low frequencies of correct solutions and high frequencies of typical incorrect solutions, the second category presenting higher levels of intuitive acceptance than the first (counter-intuitive problematic situations).

Abstract: In this paper we examine typical difficulties involved in solving a combinatorial problem, and demonstrate them throughout the solution of the “Problem of the misaddressed letters”. We focus on difficulties of two sorts: Those which stem from the fact that a combinatorial problem is usually an infinite set of problems, and others which are involved in finding a systematic approach to their solution.

Abstract: An earlier research project, the Concepts in Secondary Mathematics and Science (Mathematics) project, identified both a hierarchy of levels of understanding in different areas of secondary mathematics, and a number of particular errors which were made by significant proportions of the children tested. Preliminary consideration of these errors and the strategies which appear to have given rise to them suggests that the use of informal ‘naive’ methods which are limited in their applicability is widespread even at fourth-year level. The suggestion is made that there may be two ‘systems’ of mathematics coexisting in the secondary school classroom: the formal taught system, and a system of child-methods which are based upon a ‘counting’, ‘adding-on’ or ‘building-up’ approach, and by which children attempt to solve mathematical problems within a ‘human-sense’ framework. The difficulties which some children appear to experience in mathematics is suggested to be due in part to these children's non-initiation into the formal taught system. The implication of such a view for teaching and research are indicated.

Abstract: Recent advances in the way that adults perform computation in our society require reconsideration of the assumptions underlying current elementary mathematics instruction. The widespread use of calculators and computers for situations requiring precise calculation removes much of the motivation for teaching the current addition, subtraction, multiplication, and division algorithms. Yet precisely this use of computing technology now puts a premium on the exercise of estimation techniques for verifying the reasonableness of computations. These techniques, especially those that can be used “mentally” (without the use of any external tools), have been used informally for years, but never formalized for instruction. This paper discusses a range of estimation techniques, and presents in detail a series of mental estimation procedures based on the concepts of measurement and real numbers rather than on counting and integers. A set of techniques for teaching these procedures is described. These estimation techniques are evaluated against the multiple functions that elementary mathematics instruction needs to serve.

Abstract: This study attempts to explain certain difficulties which ninth grade students face in tackling geometrical “problems to prove”, by relating them to general and to specific rigidity and cognitive style variables. The specific Geometrical Rigidity (GR) construct was conceived as comprising a perceptual component named Geometrical Functional Fixedness (GFF) and a conceptual component named Geometrical Method Embeddedness (GME). The general rigidity constructs were SDI and BRT that were derived within the Field Theory of K. Lewin and the Gestalt Theory respectively. The cognitive style construct was articulated-global style (measured by EFT). The results show that (a) GFF and GME are mutually independent (b) GR and its components have small negative correlations with SDI (c) GR and its components have insignificant correlations with BRT (d) GR and its components have strong negative correlations with articulated-global style (e) school geometry achievement has strong negative correlations with GR and its components, positive correlations with SDI and EFT, and insignificant correlations with BRT (f) GR is a potent and efficient predictor of future failure in school geometry learning. These results confirm the conceptual analysis of GR and indicate that GR has an independent existence as a cognitive style construct rather than a personality trait.

Abstract: Items from ten mathematics tests developed at Chelsea College by members of the SSRC sponsored research programme “Concepts in Secondary school Mathematics and Science‘ are examined to see whether general dimensions can be found to explain differences and similarities in facility within and across tests.

Abstract: The article suggests that correlational studies between English language proficiency and success in mathematics have not been specific. English is composed of many proficiencies which are not always closely related. The article examines correlations between mathematics success and various English language skills, including vocabulary, reading comprehension, reading speed, listening, structure, and expression. Tests of these skills were given to incoming science students at the National University of Lesotho in both 1978 and 1979, and the results were compared with the students' success in first-year mathematics. In all cases, correlations were low. But correlations in 1978 were higher than in 1979. Attempts are made to explain these differences. The possibility is explored that speakers of different languages may learn mathematics (in English) in various ways. Finally, implications for the teaching of Use of English courses are discussed.

Abstract: In this paper a brief description and analysis is given of mathematics education in different phases of the history of Mozambique, in the feudal and colonial times, during the National Liberation Struggle and after the Independence in 1975. The successes and Problems that still have to be resolved of the post-independence period constitute the second part of the article, where particular attention has been given to teacher training and the first National Seminar on the Teaching of Mathematics.

Abstract: In the series ‘Change in Mathematics Eduation Since the Late 1950's-Ideas and Realisation’ which appeared in Education Studies in Mathematics, Vol 9, Ohuche (1978) surveyed the achievements, trials and tribulations of school mathematics in Nigeria for the period 1957 to 1977 Here we complement his paper by considering the current problems of school and university mathematics as they relate to teachers

Abstract: The link between personality variables and performance in mathematics has not been systematically and extensively covered even though it would be reasonable to assume that personality factors play a crucial role, particularly with issues such as motivation, attitudes and cognitive style. In surveying the literature the author concludes that a fuller understanding can only come through the use of formal models of personality development in childhood and adolescence.

Abstract: A multi-disciplinary team research project is described. The project is being undertaken in Papua New Guinea under the auspices of the Ministry of Education and UNESCO. The major goal is to document the relationship between environmental and cultural features, which vary widely in the country, and cognitive development. A second goal is the documentation of mathematics learning and instruction. The paper describes the various components of the project and some of the preliminary findings.

Abstract: In the article three parts of classroom conversation are studied with respect to the linguistic interference between teacher and pupils. Special attention is given to the reference of words and symbols of teacher and pupils. Some conclusions are: - In classroom conversation we may consider the references of speech act as one of the basic features of learning. - For succesful transfer of references between teacher and pupil, it is necessary that they are handling the same referential frameworks. Those frameworks are very fundamental in every day speech (‘form’ and ‘color’). - Differences in references between teacher and pupils may lead to blockages in the learning. Sometimes they result in learning that is an imitation of teacher behaviour. - To avoid these unwanted effects of learning, the teacher has to ensure that learning takes place within the everyday language of the pupils. So he will have to stimulate the pupils to explicate by themselves what they perceive and what they think.

Abstract: This article is a report on work done in developing open-ended investigations as a component in an undergraduate mathematics degree course. The context and organisation of the work is described, and sources of fruitful problems are discussed, along with issues of supervision and assessment. An important part of the article is an account of some particular student's investigations, with a report on student attitudes.

Abstract: In the past twenty years many countries have implemented new approaches to the teaching of geometry at school level. In England, most schools seem to have changed towards an ‘experimental science’ approach to geometry, whilst a significant number appear to have continued the traditional geometry of Euclid. The new approach uses a great variety of techniques, and thereby loses coherence: it is based largely on practical work and seems to have forsaken the deductive approach entirely.

Abstract: This paper arose from research on numerical errors by 6 and 7 year-olds particularly in subtraction. The present investigation covers four categories: 11 year-olds: school-leavers: student nursery nurses: bus conductor trainees. The central questions were whether earlier errors recurred at these later stages and how far feelings towards the subject (sometimes formed when young) affected performance. The sample was small (N=93) but the results at least suggest questions and directions for further work. In each category, for example, over 25 per cent got a very basic subtraction ‘sum’ wrong. This indicates that some young children are asked to do sums which are beyond their comprehension and the resulting confusion is difficult to remedy later.

Abstract: We propose the introduction of certain topics in the theory of machines and languages into school and college mathematics courses in place of the more usual discussions of groups and formal logic. We hold the belief that the theory of machines can be more easily motivated and is more obviously related to the real world than either of the above mentioned subjects, yet, it still provides an excellent framework in which to develop algebraic concepts and logical skills. We outline some examples of machines and languages and their interconnections suitable for such courses.

Abstract: Situations are often regarded as the panacea which guarantees ‘concretisation’ of abstract notions. Recent research results have led us to qualify this opinion. After analysing briefly the notion of situation, we raise various issues on the basis of the responses of children aged 11–15 to the ‘racing car problem’. We observed various limits brought about by the situation in the abstraction process, and also that the task involved more abstract notions (even pictorial) than expected. The conclusion singles out some principles ensuring a better use of situations and points out key problems such as the role of mental images and the importance of the ‘spoken’ language.

Abstract: In mathematics education literature the term ‘hierarchy’ is used in a number of ways It is important that the mathematics educator consider the usefulness of the hierarchies presented by various researchers and theorists, in the light of their application to teaching Current works on mathematical learning hierarchies are illustrated and in particular the work of the mathematics team of the research project ‘Concepts in Secondary Mathematics and Science’ is examined