GIRLS' ATTITUDES TOWARDS THE TEACHING AND LEARNING OF MATHEMATICS AND SCIENCE Aletta Zietsman, University of the Witwaterstand, Johannesburg and Stephen Sproule, Florida State University, Tallahassee Central to the teaching and learning of mathematics and science are the perceptions, aspirations, beliefs and motives the leather brings to the classroom. These affective elements are influences by social interactions and expectations as well previous classroom experiences. In turn they influence the learner's relationship with the teacher, their peers and their science and maths experiences. The following central issues of the affective domain have been discussed by other authors. Firstly, the impact of society on the role of girls in mathematics and science is well documented (Line and Hyde, 1989; Isaacson, 1989; Fennemma 1990; Jngwith, 1991). It would however appear that the influence of society on the scientific and mathematical success and access of African women has been afforded too central a role. Clearly limited access and secondary schooling is problematic in the African context, while wholesale adoption of European and American curriculum does nothing to improve the success of African learners. The study of mathematics and science classrooms in the West has elucidated the essential role that classroom practice, implemented curriculum, assessment and teachers can have on the development of attitudes and beliefs (Leder, 1987; Jungwith, 1991;Feidhusen and Willard-Holt, 1993). The possibility that mathematics and science classrooms can have a similar effect on African girls attitudes towards the subjects should not be dicounted (Sayers,1994). The number of African girls failing mathematics and science suggests that it may be school mathematics and science that are actually ÒfailingÓ the girls. There is a considerable need for the revaluation of these curricula it is for these reasons that then current study has chosen to consider the influence of the classroom on the attitudes and beliefs of girls. Before the issue of success can be effectively addressed the nature of African girls' access to mathematics and science education must be considered. Thus, the study of girls choosing to continue mathematics and science is the central consideration of the paper. It was the purpose of the study to evaluate girls' attitude about what they perceive (1) as effective classroom practice, (2) their role in classroom discourse and (3) why they choose to continue with mathematics and science past the ninth year of schooling . In the paper we first present a review of pertinent literature on the nature of gender differences in achievement, participation and classroom interaction will be considered. These are the central themes contained in the research we will report on. Secondly, a discussion of preliminary results highlighting possible factors influencing girlsÍ achievement, participation and classroom interaction are presented. This analysis is not complete statistical and qualitative analyses. ACHIEVEMENT The study of gender difference in mathematics and science achievement has received much attention (Kaiser-Messner, 1993; Fennemma and Carpenter, 1981; AAUW, 1992, isaacson, 1989). At the elementary level there appear to be few difference in performance (Leder, 1990), in the past gender difference in performance developed during the secondary phase of education and tender to favor boys (Fennemma and Carpenter, 1981). However, in AAUW (1992) report illustrates that these differences, in the case of the United States, have shown a marked decrease and are at present negligible. Similar results are found in Europe where girls perform as well as the boys (Isaacson, 1989). The intergender difference in performance are at present significantly smaller than the intragender differences (AAUW, 1992; Leder, 1990) What is of concern is the difference in attribution of these comparable results. Jungwith (1991) shows that boys achievement is generally ascribed to natural ability whereas the girlsÍ achievement is ascribed to their hard working nature. Kloosterman (1991) discusses at length the studentsÍ attribution of success and failure. Often girls ascribe their success to external influences such as peer assistance of the simplicity of the exam, at the same time attributing failure to lack of ability. This is an enthuses of the attribution style of the boys. A number of factors influencing discrepancies in ability have been proposed. Fennemma and Tatre (1986) illustrated that girls with low spatial visualization ability also scored in mathematical tests. The difference in mathematical and classroom experience at the elementary level are offered as possible causes for the few difference in achievement that persist (Isaacson, 1989). This is likely to include the role of single-sex classrooms as opposed to co-educational classrooms. Kaiser-Messmer (1993) argues that girls in single -sex classrooms display greater confidence in themselves as well as their ability to solve problems. Similar results are found by Mallam (1993) in Nigeria. This may have to do with the quantity and quality of time the teacher is able to spend with the girls in the single-sex classrooms. Hanna (1989) notes that differences in mathematics and achievement between countries is far than gender differences within a country. However results from African nations illustrate that a notable gender difference in mathematics and science achievement persists (Mallam,1993; Christie, 1988). The above options intimate that African women are mathematically and scientifically the least proficient. The African results illustrates a further problem since the number of girls discarding the subjects remains substantially higher than the number of boys. Thus in the African context a more important question is what can be done to keep girls interested in mathematics and science? It is prudent to note the concern expressed by Fennama (1990) when she claims that much of the research into gender difference in mathematics education portrays the standard as the benchmark. The benchmark is often the attainment of good test results. There may be an alternate paradigm where test scores and achievement are not the standards by which we measure success. However one could extend this concern and suggest that is the Western male standard which is often implicitly established, in the literature, as the benchmark standard. It is noted that female learners portray a greater propensity for collaborative work. This may offer an alternative paradigm from which to gauge success. This should not be restricted to women but these properties are certainly more common among African communities than those communities described in the majority of the research. PARTICIPATION Reports from Western nations again illustrate that at least at school level there is relative equality in, mathematics and science participation across gender lines. However at higher levels of schooling the number of men outweighs the number of women participating. In South Africa the number of students participating in the subjects is of concern (Christie, 1987). A similar problem is described by Sayer (1994) in Zambia. The problem goes beyond gender issues but that makes the participation of women in mathematics and science even more fragile. Factors influencing participation are diverse and often very personal, however there are a number of factors more commonly perceived as influential. The students' belief about the usefulness of mathematics or science in later life is perceived as a major contributing factor to their continued participation in mathematics and science (Kaiser-Messmer, 1993; Leach, 1994). Thus social expectations of suitable careers contribute to the girlsÕ choice to pursue mathematics and science. The real world examples used by teachers are often taken from areas of study described by boys as areas of interest. This may contribute to the girls not seeing mathematics and science as useful elements in their choice of career. A second factor that contributes to continued participation is the students' perception that the subjects are enjoyable. It appears that this attitude is influenced by the studentsÕ view of their ability. Students who enjoy maths and science but perceive themselves as less competent are also more likely to discontinue with more advanced courses. A third factor is encouragement by teachers, role models and parents to pursue mathematics and science. This usually has a positive influence on participation (Leach, 1994). The role of confidence in continue participation in the subjects requires further study. Linn and Hyde (1989) illustrate that confidence does influence participation but this is primarily restricted to advanced courses. It is notable that during the elementary school years gender differences in confidence are non-existent. To date participation more than achievement or the nature of classroom interactions has been ascribed to broader social influences. However it can be suggested that positive classroom experiences can be more influential than societal expectations in encouraging girls to take mathematics. CLASSROOM INTERACTION Two determinant elements of classroom interaction, the quantitative and qualitative nature of discourse are elaborated in the discussion. The role of teacher-student and student-student interaction is a further complication to this already complex and highly debated element of research. The complication is established because as Jungwirth (1991) illustrates the entire classroom interaction cannot be conceived in terms of the sum of its parts. The following discussion is primarily related to co-educational classrooms. Quantitatively the boys are more actively engaged in classroom discourse. This involves more interaction with teachers as well as dominating group discussion in such settings as laboratory activity (Barba and Cardinare, 1991; Jugwirth, 1991; Isaacson, 1989; Kaiser-Messmer,1993). The boys seek more attention ask more questions of the teacher and call out answers more than the girls. However what is of more concern is that the teachers direct more questions to the boys (Leader, 1987). This applies irrespective of the teacher's gender (Jungwirth,1991). Such evidence infers that teachers may not be aware of the attention seeking and unknowingly (or in some cases knowingly) contribute to the further exclusion of girls. Further evidence for this is that teachers are also more involved in reprimanding boys while the girls who conform to the classroom discipline code tend to receive the least attention. The above analysis of classroom interaction is primarily related to whole group discourse. Leader and Fennama (1990) show very little difference in the amount of time teachers spend with individuals of different gender. It is here that a qualitative difference interaction offers us greater insight. Qualitatively the differences in discourse are more marked (Kaiser-Messmer, 1993: Feidhusen and Willard-Holt, 1993). Teachers ask the boys more difficult questions (Barba and Cardinal, 1991). Boys are also usually more involved in answering questions that are ambiguous or involve reference to knowledge outside of classroom (Jungwirth,1991). The teacherÕs response to students' answers also impact on the students perceived success and results in changes in belief about their ability to undertake problems. Achievement and ability although discussed above are not only a result of test performances but are communicated to the students during classroom interactions. Leach (1994) claims that boys are more often exposed to praise for academic accomplishments white girls receive praise for behavior. The influence of the student-teacher interactions may be consequential since the teacher is communicating more than the subject knowledge by how and when they communicate their approval of disapproval. Teachers tend to 'unpack' self-contained and complete answers. Jungwirth (1991) suggests that girls answers are usually more complete and therefore are subject to teacher 'unpacking'. This tends to imply that their response is not wholly correct. Secondly, boys diminish the severity of failure by affirming that they understand the teachers explanation. This is inclined to reduce failure to a 'little mistake' in contrast girls' failures tend to have a more negative effect on their future performance (Jungwirth, 1991; Leader and Fennama, 1990). Thirdly boys who are unable to answer a difficult question will most likely be coached with leading questions. In contrast girls will most likely receive the answer from the teacher (Fedhusen and Willard - Holt, 1993; Leach, 1994). Leader and Fennama (1990) argue that withholding help from the girls may be more beneficial. This tends to leave the boys more accustomed to problem solving, an area where gender differences in achievement remain consequential (Buchanan, 1987). The teacher behaviors listed above do not only influence the students' learning but more importantly it affects how the students' perceive their ability. Studies of boys in social settings demonstrate a propensity for jesting, mockery, posturing and challenges during interaction, Fedhusen and Willard-Holt (1993) suggest that this type of interaction is more common in the classroom and therefore boys are more comfortable in classroom communication. It is however uncisar whether the boys domination of classroom interaction influences the nature of the interaction or whether the nature of the subjects' discourse requires conjecture and debate and therefore the boys are more comfortable speaking out. Girls often find the role of companion studying useful. This is corroborated by studies that suggest girls learn better in co-operative environments. A consequence of collaborative studying is that the high achieving girls are more likely to concern themselves with peer pressure and perceptions than high achieving boys (Feldhusan and Willard- Holt, 1993). In contrast boys are driven by the need for high achievement (Kasier- Messner,1993). This is the dominant paradigm in secondary schools. It may be that a shift from a competitive learning environment is required to encourage more girls to participate in mathematics. This may be one of the reasons girls achieve more in single sex schools. In conclusion, the complexity of gender issues in mathematics and science education is apparent. A concern is that in South Africa gender discrepancies in achievement and participation remain. At present the challenge is perceived as getting girls to participate and success in secondary school mathematics. One way this may be achieved is through the improved interaction of girls in classroom. This will require a change in teachers' classroom practices as well as a review of the dominant paradigm defining achievement in maths and science. RESEARCH DESIGN 1. Classroom observations and interviews South African students (male, N-19; and female, N-23) identified during classroom observations were interviewed about situations in the classrooms that are constructive or destructive as far as their participation is concerned. The identification was made by the researcher involved (Zietsman) based on three criteria: 1students who participated fully in the classroom discussions, 2. students who never participated in the discussion, and 3. students regard by the teacher as being ÒinterestingÓ in terms of their participation, but not chosen by the researcher. The interviews were conducted with students from different socio-economic backgrounds, and at different levels of schooling (St. 7, age 14 - 15 years; St. 9, age 19 - 17 years), and first year university age 18 -19 years ). These interviews used the questions designed by the researchers Gillgan and Belenky, Chinchy, Goldbenger and Tarule in their studies that examined women's alienation in academic settings and women's experiences of 'formal' education as peripheral or irrelevant to their central interests and development (Gillgan, 1982, Belenky, Clinchy, Goldbenger and Tarule, 1986.) Examples of the questions asked are : * What does being a woman/girl/boy/man mean to you? * Do you think there are any important differences between the way boys/men and girls/women think about/physics/biology? * What do you think, will stay with you about year experiences here in this physics course (maths class etc.)? [Probe for specific academic and non-academic experiences, good and bad teachers; good and bad assignments.] Our analyses show that all the females interviewed, without an exception. were intimidated or 'silenced' by the mere presence of males. Remarks ranged from Tina, an articulate, very successful student in her first year at university feeling threatened, and thus remaining silent Transcripts conventions used: interviewer -| prolonged silence -// speech omitted -... speech overiap -{} researcherÕs comment -[] Tina: I myself have experienced feeling threatened by guys. They [guys] seem to look down on you, on females. I dunno why. I: what gives you that feeling? Tina: I dunno! It's just a feeling, I guess. The way they talk to you, the way they look at you. It's // just generally a feeling that they don't like of you as an equal, they think of you as someone lower than themselves. I: Uhhm? Tina: I dunno! It threatens me to sit in a whole group of people, with a lot of males and the try and explain my point of view and get a whole lot of feedback from these guys, you know. Totally rejecting the idea you know, and making you feel about this big,[ Laughs, show inch high with thumb and forefinger.] Elisia and Mabusane (14 - years old, standard 7) find the boys intimidating and childish. I. Okay, tell me something- does it matter to you that there are so many boys in the class? M: No. {The boys ask the questions. E: {The boys are terrible in this class. They are so intimidating. I: Terrible? M: they put up their hands and sort of scream. If I have no opinion they make a whole lot of remarks and // they are terrible. E: Ja, it's most irritating, they are childish [thumps desk in irritation]. I: That matters to you, doesn't it? E: Most of the time bothers me, in these activities. M: Ja, sometimes they have no interest in their group activities. 2. Design of survey questions These transcript analyses suggested to us the questions are included in the pilot survey. Questions that emerged from the protocol analyses focus on (amongst others): reasons for choosing the subjects; what constitutes good teaching; what constititues good 'learning'; asking questions in classroom/lectures. Analyses of the pilot survey data resulted in the refining of the final survey questions. The final survey questionnaire seeks out factors that may influence female participation in maths and science classrooms and lecture halls, See the results section for the questions in the survey. 3. Pilot survey The pilot questionnaire was administered to students at the 8 level (15 - 16 years of age) in two South African schools. The schools were in the Gautenq (Johannesburg area). one mixed gender and ones girls' school. The sample included 341 girls and 257 boys. 4. Administering the survey Schools in South Africa and Malawi were chosen in the rural and urban areas (gauteng and Kwa- Zulu in south Africa, and the city of Zomba and the rural areas surrounding Zomba in Malawi). Co-educational schools, boys' school and girls' schools, two in each country, in rural and urban areas, were selected on the results, but we view this research as a first effort in looking at the effect of classroom culture on access and success for girls in maths and science. The survey was administered directly following a lesson in their science or mathematics, since questions referred specifically to participation in the classroom activities. 4. Participants The survey was administered in South African and Malawian schools in October 1994. We have decide to focus on students in schools just after they chosen to continue with maths or science. This is the St. 8 (15 - 16 year old) scholars in south Africa And their peers in Malawi (Form III). Both mixed race and mixed gender schools were included. There were 2 140 girls and 2 342 boys in the population surveyed. ANALYSES OF THE SURVEY We have not yet performed complete analysis of all the data. The study's analyses is in progress, and we have just obtained the results from some sites. We aim to report at the conference in more detail (e.g. factor analyses) with regards to the girls' expectations and attitudes toward classroom activities, learning and teaching. For this pre-proposal for the conference we present only the % responses on the different items, as well as a summary of written responses to open ended questions on the science questionnaire. Open ended options with multiple chock utopians are summarized directly below the table. The first two questions are not analyzed these were information collection questions about variables such as gender and language. RESULTS The data for each of the multiple choice questions are presented in Tables 1 to 10. A short discussion of possible implication beneath each table is presented when deemed necessary. Table 1: Questions 3 I chose to do science because a. science is important for my career, but I don't like science b. science is important for my career, and I love science c. simply because I love science d. my parents told me to do science e. my teacher told me to do science f. I have to g. other reasons........ option %girls %boys a. 16 16 b. 42 51 c. 0 8 d. 0 0 e. 0 0 f. 27 20 g. 15 5 Table 2: questions 4 How many science questions did you ask in class today? a. none b. one c. two d. I asked a lot if questions - can't remember how many. option %girls %boys a. 90 47 b. 6 45 c. 0 5 d. 4 3 TABLE 3: Question 5 How many science questions did you ask friends in class today? a. none b. one c. two d. I asked a lot of questions - can't remember how many. option % girls % boys. a. 13 34 b. 23 17 c. 26 19 d. 39 30 Table 4: Question 6 How many questions did the science teacher ask you today? a. none b. one c. two d. asked me a lot of questions, can't remember how many. options % girls % boys. a. 70 13 b. 5 23 c. 2 26 d. 23 39 A confounding factor here may be that the question did not establish whether the class was a laboratory lesson or a 'lecture' type lesson. The AAUW (1992) reports that girls asked more questions and were asked more questions in interactive lessons, such as laboratory lessons. We will have to consider including questions that established the type of lesson the children are referring to in their responses in future surveys. Table 5: Question 7 I often talk about science or ask questions in the lectures because a. Some students do, so I am not different b. I know most of the answers to the lecturerÍs questions c. I really want to understand the work option % girls % boys. a. 3 9 b. 6 16 c. 74 75 There may be a conflict here with responses to question 4 and 5 above, the responses imply that most students asked a lot of questions in class. On the other hand. the question's phrasing (Oreally want to understand.O) is too leading - most students would probably choose this option. Table 6: Question 8 I do not ask questions in the science class because a. I am too shy b. the guys in the class make rude remarks or tease me c. the girls the class make rude remarks or tease me d. other reason........... option % girls % boys. a. 39 35 b. 29 2 c. 6 5 d. 25 57 Although there are no tests performed to establish the significance of the differences between boy/girl answers. Inspection seems to suggest that the responses to (b) present a contradiction to our interview data. Most of the girls (18 of 23) interviewed mentioned boys (rudeness, interruptions. 'looking down on' girls) as a reason for their silences. All the boys' 57% responses in 'other' indicated that they 'asked a lot of questions', again conflict with question 4 above. Table 7: Question 13 I enjoy a science class where I haveto try out problems before the teacher explains how to do it a. I agree strongly b. I agree c. I do not care whether I have try first or whether the teacher explains first d.I do not like to try problems without the teacher doing an example first. option % girls % boys. a. 26 38 b. 52 41 c. 16 12 d. 6 9 Table 8: Question 14 A good science teacher will explain very carefully how to do a problem and then gives us lots of the same problems to try a. I agree strongly b I agree c. I want to try problems myself first, but the teacher must then show us the right way d.I want to try problems myself first, and find out for myself whether I am right or wrong. options % girls % boys a. 35 39 b. 16 18 c. 45 34 d. 3 8 Table 9: Question 15 I enjoy a science Claus where I have to work with my class mates to solve problems a. I agree strongly b. I agree c. I like working with my classmates, but the teacher must show us whether we were right or wrong d. I like it best to work on my own. option % girls % boys a. 39 30 b. 26 27 c. 26 25 d. 9 18 Table 10: Question 16 I do not mind making mistakes when trying science problems, because I think that I can lean from mistakes a. I agree strongly b. I agree c. I get confused when I make mistakes, the teacher must show me the right answer d. I cannot carry on with problems when I make mistakes. options % girls % boys a. 42 57 b. 32 31 c. 6 3 d. 19 8 SUMMARY OF OPEN ENDED QUESTION RESPONSES (GIRLS ONLY) As far as possible on statement that summarizes most responses is noted first. The order in which the responses are given also indicates the frequency with which the response appeared. Thus, response I appeared most equally response 2 second highest frequency etc. Question 9 How do you study for a science test ? Say in a few sentences what you do when you study for a science test. -I read through my notes, and then through my book and then try the questions/problems in the book (94%) -I read through my book, and try to make a summary of all the work, and solve problems (5%) -I try to understand the work and relate it to everyday life (2%) Question 10 Is the way you study for a science test different from the way you study for other subjects? Please give an example. - No (96%) -Yes, it is different for biology and history. I do not solve problems. I learn biology like a parrot. (4%) Question 11 What does it mean to you to 'learn'? Please give an example of your learning. 1. Understanding and application (81%) Response types are: To read and understand the work. To be able to use the ideas, to do the problems. To apply the ideas in real life. To Understand and apply the work. 2. Learning and remembering (12%) Response types are: Learn to me means to read, understand and try not to forget what you went through. To learn is to be able to understand something and being able to do the work from your head offer you understand the work. 3. Knowledge acquisition (4%) Response types are: To acquire knowledge. Learn means to increase the knowledge you have. To develop my knowledge and generally. To read what I know and knowing what I don't [know?]. 4. Processes in Learning (3%) Response types are: It is to ask questions and read your text book, and solve lots and lots of problems and have debates about whatever you are learning. To discuss and ask questions. Question 12 Do you think science will be useful to you one day? Please give an example. 1. Useful in everyday life (80%) Examples of responses are: - Yes, for general knowledge. - For everyday examples. - For everyday life like cooking - It is applicable in everyday life so it is useful to understand everyday life. - Yes, maybe I'll be able to explain why certain changes occurs and the reasons for it. - On the social side of my life maybe. - Yes, I can apply it for the benefit of mankind, and it will reward me. - Yes, we live in a physics world, so one must be familiar with physics and in times of danger apply it, it might help. - Yes, it's useful even now because from science I learn how things are like in this world. - Yes, there is always a science answer to something. - Yes, I know how to connect a light bulb! 2. Ueful for my career (12%) Examples of responses are: - Yes because engineers use a lot of science ideas to solve problems in industry. - Yes, I want to become a physicist. - Yes as a physicist that I want to be. - Probably, if I decide to study medicine. 3. Useful for my thinking (1%) Examples of responses are: - Yes, it already helps me to think objectively and I know that in the future it will help me too. -Yes, science also encourage logical thinking, which I will need one day. 4. Not useful (1%) Examples of responses are: - Not really. - Not that much. - I am not sure. REFERENCES The American Association of University Women(AAUW). (1992). How schools shortchange girls. Washington, DC: The American Association of University Women Educational Foundation. Barba, R. and Cardinare,L. (1991). Are females invisible students? An investigation of teacher student questioning interactions school science and mathematics 91(7). 306-310. Belenky, M.F. Clinchy, B.M., Goldbenger, N.R. and J. M. Tarule (1986). Women's ways of knowing: The developing of self, voice and mind. New York; Basic Books. Buchanan, N. K. (1987). Factors contributing to mathematical problem-solving performance: an exploratory study. Educational studies in mathematics 18. 399-415. Christie, P. (1988). The right to learn. Johannesburg: Raven Press Publications. Feldhusen, J. F. and Willard-holt, C. (1993). Gender differences in classroom interactions and career aspirations of gifted students. Contemporary Educational Psychology 18. 355-362. Fennama, E. and Carpenter, T. (1981). The second national assessment the sex related differences in mathematics, Mathematics Teacher, 74. 554-559. Fennama, E. (1990). Justice, equity and mathematics education, In E. Fennama & G. C. Leder (Eds.), Mathematics and Gender (pp.1-9). New York: Teacher College Press. Gillgan C. (1982). In a different voice: Psychological theory and women's development. Cambridge, MA: Harvard University Press. Hanna, G. (1989). Mathematics achievement of girls and boys in grade eight: results from twenty countries. Educational Studies in Mathematics 20. 225-232. Isaacson, Z. (1989). The marginalisation of girl in mathematics : some causes and some remedies, in D. Pimm (Ed.) Mathematics teachers and children (pp. 95-108). London: Oxford University Press. Jungwirth, H. (1991). Interaction and gender - findings of a microethnographicals approach to classroom discourse. Educational Studies in Mathematics 22. 263-284. Kaiser-Messmer, G. (1993). Results of an imperil study into gender differences in attitudes towards mathematics Educational Studies in Mathematics 25. 209-233. Kloosterman, P. (1991). Beliefs and achievement in seventh- grade mathematics. Focus on Learning Problems in Mathematics 13(3). 3-15. Leach, L. S. (1994). Sexism in the classroom : a self-quiz for teachers. Science scope March, 54-59. Leder, G. C. (1990) Gender differences in mathematics an overview, in E. Fennama & G.C. Leder (Eds.), Mathematics and Gender (pp. 10-26). New York: Teacher College Press. Leder, G.C. and Fennama, E. (1990) Gender differences in mathematics: a synthesis, in E. Fennama & G.C. Leder (Eds.), Mathematics and Gender (pp. 188-199). New York: Teacher College Press. Linn, M. C. and Hyde, J. S. (1989). Gender, mathematics and science Educational Researcher 18(8). 17-19, 22-27. Mallam, W.A. (1993). Impact of school-type and sex of the teacher on female students' attitudes toward mathematics in Nigerian Secondary schools. Educational Studies in Mathematics 24(2). 223-239. Sayers, R. (1994). Gender differences in mathematics education in Zambia Educational Studies in Mathematics 26. 389-403.