ADULTS LEARNING MATHS NEWSLETTER No. 15 March 2002

From the Chair

Gender in Numeracy

A

A

Inge Henningsen, University of Copenhagen, Denmark

LM is an international research forum bringing together researchers and practitioners in adult mathematics/numeracy teaching and learning in order to promote the learning of mathematics by adults.

t ALM2 and ALM3 Paul Ernest and Mary Harris gave keynote addresses explicating the need for gender being considered an important, explicit and independent factor in adults learning mathematics. Because women and men approach mathematics from different perspectives, encounter different difficulties and bring different strengths to the classroom, and as well come from different cultures both at work and in their civil lives, there is a need for a gender specific notion of numeracy and a gender conscious teaching of mathematics.

The Charity’s objects are the establishment and development of an international research forum in the lifelong learning of mathematics and numeracy by adults by: • encouraging research into adults learning mathematics at all levels and disseminating the results of this research, for the public benefit; •

promoting and sharing knowledge, awareness and understanding of adults learning mathematics at all levels, to encourage the development of the teaching of mathematics to adults at all levels, for the public benefit.

May I invite you to think about the things ALM (= we all!) should do to reach these aims better? If you have any ideas please share them! I think we should try to make our communication and cooperation more intensive. We have: • this newsletter: do you want to write an article? •

our conferences: I hope we meet in Uxbridge, London, UK in July.

•

some publications: for example the cooperation with the Journal for Literacy and Numeracy Studies has started. A special ALM issue will be printed in Autumn.

•

some good ideas: for example a new ALM members communication listserv and local ALM groups.

The next step should be more (inter-)national cooperation: ALM has members from many countries. This provides very good chances for international collaborative research. Do you have any ideas? Please email them. Prof. Dr. Juergen Maasz, University of Linz, Austria.

Company No. 3901346/Charity No. 1079462

Numeracy is destined to be a key concept for adults learning mathematics. One strength of the concept is that it is lifted out of the eternal, disembodied realm of pure mathematics and grounded in application and practice, e.g. in time and place. This, however, points to a gender specific concept of numeracy related to the fact that women and men to a great extent inhabit different cultures both at work and in civil life. Gender is of course not the only aspect that needs to be taken into account when adults learn mathematics—class, age and social circumstances are others. On a theoretical level this diversity is in most cases readily conceded – even built into definitions of numeracy. Day to day practices tend, on the other hand, to forget or ignore the differences. Here gender seems to act as a useful reminder about the ever present inhomogeneities in society, a perspective that enables differences to be foregrounded and reinterpreted. In this paper I take a social constructivist approach to gender. On the learning of mathematics, I feel most at home with Evans’ ideas of “positioning”, softening the strong ideas of situated learning, and opening up the possibility of some measure of transfer. (Evans, 2000).

Two conflicting meanings of numeracy

▲

In this issue ! Gender in numeracy ! What Counts as Mathematics in Adult and Vocational Education? ! Personal mental methods ! Adult and Lifelong Education in Mathematics ! News and events ! About ALM

Numeracy is gendered

1 4 7 9 10 12

In the literature there is a number of (provisional) definitions of numeracy (Gal et al, 1999; Evans, 2000; Lindenskov & Wedege 2001). These definitions differ on the question of what one could term “the unity of numeracy”. The concept of numeracy has (at least) two sides. Numeracy as a theoretical concept and numeracy as used in government programs for mathematics education and comprehensive assessment activities. This accounts both for the complexity and the attraction of the concept of numeracy, the latter to a great extent stemming from the short distance between theory and continued on page 2

ALM NEWSLETTER Gender in Numeracy

No. 15, March 2002 from p. 1

practice in current educational activities. This duality is, however, not unproblematic since it has led to numeracy currently being used with two conflicting meanings. One is a research definition of numeracy that can be exemplified by Evans’ definition: Numeracy is the ability to process, interpret and communicate numerical, quantitative, spatial, statistical, even mathematical information, in ways that are appropriate for a variety of contexts, and that will enable a typical member of the culture or subculture (my italics) to participate effectively in activities that they value. (Evans, 2000) This concept of numeracy obviously allows for a plurality of numeracies and gender will almost inevitably enter into the delimitation of numeracy in various subcultures. The other can be exemplified by OECD (1996) stipulating that quantitative literacy on their level 3 is the minimum requirement to manage the complex demands from work and everyday life in the knowledge society. This represents a concept of a unified, common numeracy. Evans’ definition opens up the possibility of the simultaneous existence of a number of numeracies, whereas the OECD definition operates with a numeracy in principle common to all in a given society. Lindenskov and Wedege (2001) point out that numeracy is dependent on time and geographical localisation as shown in the following quotation: Numeracy changes in time and space along with social change and technological development. (Lindenskov and Wedege, 2001:5) It is my contention that the dissimilarities due to factors like age, social class or gender are of the same magnitude as temporal and geographical dissimilarities implying that these factors should enter into any definition of numeracy.

Research & reports

Gender, math and life history There is a growing body of scientific literature exploring gender and mathematics learning based on research on students in high schools and universities. (For an overview see Fennema, 1995; Hanna, 1996.) This research demonstrates that content, context and ways of instruction have gender implications: girls/women prefer problems with a people/nature content, women do better in internal/project oriented assessment than in traditional timed exams, women benefit from a teaching style stressing collaboration and open-ended problems. Life histories are important for the learning of mathematics1. Thus, adult women and men must be expected to approach mathematics from different perspectives, to encounter different difficulties and to bring different potentials to the classroom. With a highly gender segregated labour market mathematics in the work place will, on the average, be different for women and for men. Studies of possible interactions between gender, occupation and age could shed interesting light on adults and mathematics. In the context of Page 2

adults learning mathematics it might be fruitful to study how these factors interact with work performance and work demands and also study the correlation – if any – between math performance and work performance in a gender perspective.

Math anxiety and gender Many authors have explored math anxiety. Tobias (1978), one of the pioneers, found in her investigations that women had higher levels of math anxiety than men. Others have made the same findings. Math anxiety is significant for adults learning mathematics. Here one would expect men and women to have different problems if only because of the different expectations society imposes on men and women. Not being able to do mathematics is very identity threatening for a man, while women traditionally have been allowed to shrug their shoulders and do something else. Hence a failure in school mathematics is not as debilitating for women as for men. One would therefore expect male math anxiety to be more pervasive and more deep-rooted than with females. It is my contention that many studies could be interpreted or reinterpreted to point in that direction. Below are some examples. A study of medical students in Norway reported on striking differences between men and women. Some of the male students were very worried that their inferior results in mathematics in high school would impinge on their performance as doctors, while none of the women had similar concerns. Hence the men seemed to see qualifications in mathematics as a measure of their general ability, while the women had no such notion. Evans (2000:36-7) studied performance in school and practical mathematics in a population of polytechnic students. The women did consistently worse than men in test situations and many more women than men had low entrance qualifications in mathematics. One way of looking at this is that fewer women than men are deterred from higher education by bad grades or lack of qualifications in mathematics. (Another explanation might be, of course, that the women have chosen subjects with less demanding mathematics content.) These observations point to the understanding that mathematics represents a danger of failure for men and an opportunity for success for women. However, it must be noted that being good at mathematics might in some instances still be a problem for women.

Paradoxes of numeracy, democracy and empowerment Many authors have identified mathematics as a dominantly male science. (See Ernest 1995 for an overview.) At the same time knowledge of mathematics is claimed to be an important feature of democratic competence and also as a means of empowerment. (Niss, 1994; Skovsmose, 1998; Benn, 1997). Gal (1998) and Lindenskov & Wedege (2001) have in the

ALM NEWSLETTER

No. 15, March 2002 same vein linked numeracy with democracy and empowerment. These are very important observations with far-reaching consequences for education and research. They might, however, have unintended gender implications. If empowerment and democracy are linked with numeracy, this might in turn be taken to imply that innumeracy (or lack of mathematics) causes lack of power and lack of democratic competence. Hence with the widespread public belief that women are less numerate compared to men, women are by implication considered less democratic and an objective reason is given for their being less powerful in society. Here is the paradox: Although numeracy does contribute to democratic competence, there are no studies that have established a positive correlation between level of numeracy/ mathematics competence and democratic competence. We have no proof that other activities or competencies do not contribute the same or more to democratic competence. Speaking about numeracy and democracy together, we have to acknowledge different and possibly gendered ways we develop as democratic citizens, not privileging any one particular way. The same holds true for numeracy and empowerment. In Evans’ definition, acquiring numeracy, so to speak, automatically implies empowerment, since it “will enable a typical member of the culture or subculture to participate effectively in activities that they value”. On the other hand, a more or less voluntary education in (somebody else’s) numeracy might easily be disempowering. Therefore the theoretical definition of numeracy has very practical implications and it is important for the ALM community to distance itself from any unsubstantiated claims about mathematics education in itself leading to empowerment.

Building blocks for a gendered numeracy

Ernest (1995) describes the widespread public image of mathematics as “difficult, cold, abstract, theoretical, ultrarational …remote and inaccessible”. Later he notes the similarity to Gilligan’s (1982) “separated” stereotyped male values. Teaching mathematics consistent with Gilligan’s “connected”, female values should be based on, and valorise, relationships, connections, empathy, caring, feelings, intuition and tend to be holistic and human-centred in its concerns. Is such a mathematics possible without losing the “unreasonable effectiveness of mathematics”? Could numeracy be part of the solution? To me the construction of a gendered numeracy would here serve as a litmus test. So my

References: Benn, R (1997). Adults count too. Mathematics for empowerment. Leicester: NIACE. Ernest, P (1994). “Images of Mathematics, Values and Gender” in Coben, D (ed) Proceedings of ALM-3 the Third Conference of Adults Learning Mathematics. London: Goldsmiths University of London, 1-15. Fennema, E (1995). ”Mathematics, Gender and Research”. In Grevholm, B. and Hanna, G. (eds.), Gender and Mathematics Education: an ICMI Study, Sweden 1993, Lund: Lund University Press, 45-64. Gal, I (1998).”Numeracy Education and Empowerment: Research Challenges”. In Groenestijn, M. van & Coben, D. (eds.). Proceedings of the fifth international conference of Adults Learning Mathematics. (ALM-5). London: Goldsmiths University of London, 9-19. Gal, Iddo, Mieke van Groenestijn, Myrna Manly, Mary Jane Schmitt, Dave Tout, (1999). Numeracy Framework for the international Adult Literacy and Lifeskills Survey (ALL) http:// nces.ed.gov./surveys/all, Ottawa, Canada, Statistics Canada (internal publication) Gilligan, C (1982). In a different voice, Cambridge, Massachusetts: Harvard university Press. Hanna, G (ed.) (1996). Towards Gender Equality in Mathematics Education: An ICMI Study, Dordrecht: Kluwer. Harris, M (1994). ”Women, mathematics and work” in Coben, D. (ed.)Proceedings of ALM-2 the Second Conference of Adults Learning Math. London: Goldsmiths University of London. Evans, J (2000). Adults’ Mathematical Thinking and Emotions, London: Routledge Falmer. Lindenskov, L & Wedege, T (2001). Numeracy as an Analytical Tool in Mathematics Education and Research. Centre for Research in Learning Mathematics 2001/31. Roskilde:RUC. Niss, M (1994). ”Mathematics in Society”. In R. Biehler et.al. (eds) Didactics of Mathematics as a Scientific Discipline Dordrecht: Kluwer Academic Publishers 367-78. OECD (1995). Literacy, Economy and society. Results of the first International Adults Literacy Survey. Statistics Canada. Skovsmose, O (1998). ”Linking Mathematics Education and Democracy: Citizenship, Mathematical archaeology, Mathemacy and Deliberative Interaction.” Zentralblatt für Didaktik der Mathematik. 98/6 195-203 Tobias, S (1978). Overcoming math anxiety. New York: Norton. 1

In Evans (2000) investigations one women reacts negatively to a problem concerning tipping in a restaurants. It makes her feel dependent, since she is often being paid for by others in restaurants.

Inge Henningsen: [email protected]

▲ Page 3

Research & reports

It was my aim to give a number of concrete instances, where I saw gender as an important agent in adult women’s relationship with mathematics and numeracy. It turned out, however, that this was very difficult. There is considerable literature on what make women feel bad about mathematics. There is some research on what make women feel better about mathematics, but very little about what makes women feel good about mathematics.

recommendation is that since we have a strongly gendered learning of mathematics we should also have a strongly gendered teaching of mathematics.

ALM NEWSLETTER

What Counts as Mathematics in Adult and Vocational Education? Numeracy for empowerment and democracy? Gail E. FitzSimons, Monash University Keynote address for Adults Learning Mathematics conference [ALM 8] Roskilde University, Denmark, 28-30 June, 2001 NOTE: This is the second part of Gail’s address - the first part was in the previous issue of the ALM newsletter.

Democracy and Empowerment

Research & reports

Where do democracy and empowerment come into all of this? Following the work of Jacques Derrida1 (1994, 1997), democracy is not the present, lived reality of Western ‘liberal’ systems of government. Neither is it a regulative framework, a source of deduction or determinate judgement. Nor is it a Utopia — an idealised concept that we should aim for and might achieve some time in the future. Rather, it is an ethical demand or injunction, concerning concepts of friendship, community, and so forth. It must be understood in terms of democracy to come; not in the future, but in the sense of maintaining now “l’ici maintenant,” without presence. As I understand it, democracy is not a thing to be grasped, nor an absolute standard of moral judgement. In its pragmatic realisation through ‘representative democracy,’ power is distributed unequally. However, Barry Barnes (1988) comments that although the commonsense view of power is as an entity or attribute of things, processes, or agents, he sees power as a theoretical concept referring to capacity, potential, or capability. More strictly, it should be taken as referring to distributions of these. Any specific distribution of knowledge confers a generalized capacity for action upon those individuals who carry and constitute it, and that capacity for action is their social power, the power of the society they constitute by bearing and sharing the knowledge in question. (p. 57) In other words, “social power is the capacity for action [embedded] in a society, and . . . is possessed by those with discretion in the direction of social action” (p. 58). Taking knowledge to be accepted, generally held belief, routinely implicated in social action and, consonant with Pierre Bourdieu’s (1991) notion of cultural capital, Barnes asserts that the distribution of knowledge in society defines the distribution of power. In a similar vein, Mary Klein (2000a, 2000b) considers numeracy not as a thing to be possessed, but as a capacity for action. Thus, democratic power depends upon access to knowledge — information selectively derived from a range of possibilities and which is capable of being interpreted and understood — access to which is also unequally distributed. Page 4

No. 15, March 2002 As mentioned in the introduction, mathematical knowledge is said to be empowering. I return to the questions: What mathematics? How much mathematics? For whom? Who decides? Who should decide? These are in addition to the pedagogically- and administratively-oriented questions concerning how, where, when, and why.

The Concept of Technology Related to power in diverse and complex ways, the concept of technology is central to mathematics in adult and vocational education. Firstly, from an industrial perspective it is integrally linked with mathematics in production (in the sense of the totality of effort, physical, intellectual, symbolic, and so forth) in manufacturing, service, and symbolicanalytic sectors. Secondly, technology is utilised as a tool of management, both in industrial settings and throughout the adult and vocational education sector ¾ although here the education sector itself is being transformed in many countries from a public good to a competitive industry (FitzSimons, 2000). Andrew Feenberg (1995) argues that technology is one of the major sources of power in modern societies; the power wielded by masters of technical systems largely overshadows political democracy in the control they exert over, inter alia, experiences of employees and consumers. However, rather than accepting a thesis of technological determinism, he asserts that technology is but one important social variable. Feenberg continues that modern technology can only be understood against the background of the traditional technical world from which it developed. However, rather than a generic shift, he claims that there has been a significant shift in emphasis of features such as the use of precise measurements and plans, and the technical control over some people by others. Wolfgang Schlöglmann and Jürgen Maasz (Maaß & Schlöglmann, 1988; Maaß, 1998 have drawn our attention to the complex links between technology, mathematics, and society, particularly in adult and vocational education. Although each one of us can chose to operate in the sense of democracy, as I have outlined it above, within our own classrooms or sites of learning – that is, taking account of the multi-voicedness and the historicity of our students and ourselves, as well as the contradictions which inevitably accompany the work of adult and vocational education, within an ethical framework2 – it is always within a range of constraints.

Expansive Learning Engeström’s fifth principle proclaims the possibility of expansive cycles — “transformations in activity systems” (p. 5). In these relatively long cycles of qualitative transformations, questioning and deviation from established norms sometimes escalates into a deliberate collective change effort. According to Engeström (1999, p. 5) “a full cycle of

No. 15, March 2002

ALM NEWSLETTER

expansive transformation may be understood as a collaborative journey through zone of proximal development [ZPD] of the activity.” Or, as Ros Brennan (2000) expresses it, expansion from isolation to collaboration; learning from conversations and research.

Beach, K. (1999). Consequential transitions: A sociocultural expedition beyond transfer in education. In A. Iran-Nejad & P. D. Pearson (Eds.), Review of research in education, 24 (pp. 101-139). Washington, DC: The American Educational Research Association.

In the case of numeracy for empowerment and democracy, is it possible that Critical Mathematics (Skovsmose, 1994) may be a serious part of the intended and implemented curriculum for adult and vocational education? What might be a role for aesthetics? Is it possible to value both cognitive and affective development? How might alternative, research-based, curricular and pedagogical practices be incorporated into curriculum and pedagogy? How might alternative, more positive, public perceptions of mathematics/ mathematics education be generated and enhanced to encourage the uptake of lifelong education in ways that benefit all stakeholders?

Bessot, A., & Ridgway, J. (Eds.) (2000). Education for mathematics in the workplace. Dordrecht: Kluwer Academic Publishers.

More broadly, is it or might it be possible to involve some or all of the stakeholder groups in a collective change effort to serve their mutual needs? In order to benefit all stakeholders, I believe that an expansive cycle, as described by Engeström, is a necessary component. That is, there needs to be open, respectful dialogue between all Activity groups.

Conclusion

I believe that in order to support numeracy for empowerment and democracy, we actually need democracy for numeracy and empowerment.

References Bagnall, R. G. (2000). Lifelong learning and the limitations of economic determinism. International Journal of Lifelong Education, 19(1), 20-35. Barnes, B. (1988). The nature of power. Cambridge: Polity Press.

Bourdieu, P. (1991). Language and symbolic power (G. Raymond & M. Adamson, Trans.). Cambridge, UK: Polity. Brennan, R. (2000). Competing views on online delivery of education and training. In F. Beven, C. Kanes, & D. Roebuck (Eds.), Learning together: Working together: Building communities for the 21st century. Proceedings of the 8th Annual International Conference on Post-Compulsory Education and Training (Vol. 1) (pp. 200-208). Brisbane: Centre for Learning and Work Research, Griffith University. Buerk, D. (2000). What we say, what our students hear: A case for active listening. Humanistic Mathematics Journal, 22, 1-11. Butler, E. (1998). Persuasive discourses: Learning and the production of working subjects in a post-industrial era. In J. Holford, P. Jarvis, & C. Griffin (Eds.), International perspectives on lifelong learning (pp. 69-80). London: Kogan Page. Civil, M. (2001). Adult learners of mathematics: Working with parents. In G. E. FitzSimons, J. O’Donoghue, & D. Coben. (Eds.). Adult and life-long education in mathematics: Papers from Working Group for Action 6, 9th International Congress on Mathematical Education, ICME 9 (pp. 201210). Melbourne: Language Australia in association with Adults Learning Mathematics – A Research Forum (ALM). Clarke, D., & Helme, S. (1993, July). Context as construct. Paper presented at the 16th Annual Conference of the Mathematics Education Research Group of Australasia, Brisbane. Clements, M. A. (Ken), & Ellerton, N. F. (1996). Mathematics education research: Past, present and future. Bangkok: UNESCO Principal Regional Office for Asia and the Pacific. Coben, D. (2001). Fact, fiction and moral panic: The changing adult numeracy curriculum in England. In G. E. FitzSimons, J. O’Donoghue, & D. Coben. (Eds.). Adult and life-long education in mathematics: Papers from Working Group for Action 6, 9th International Congress on Mathematical Education, ICME 9 (pp. 125-153). Melbourne: Language Australia in association with Adults Learning Mathematics – A Research Forum (ALM). continued on page 6

Page 5

Research & reports

In this paper I have attempted to paint the bigger picture of what is, or what might be, in adult and vocational mathematics education — but not in an overly deterministic or pessimistic manner. Many practitioners are well aware of the power struggles associated with constant change. Engeström’s model of expansive learning allows for creativity and interaction between Activity Systems operating in relative isolation. Contradictions, which may be exploited for or against the interests of democracy, could then be negotiated. The ideal would be expansion from isolation to collaboration in design and implementation of curriculum (taken in its broadest sense) and engagement by all stakeholders with this curriculum — not forgetting the importance of members of each group being able to operate within their individual Zones of Proximal Development. Expansive Learning would emanate from conversations, analyses, and genuinely open research; and by stakeholders reflecting on alternative models of implementation.

Bourdieu, P. (1980). Le sens pratique. Paris: Les éditions de minuit.

ALM NEWSLETTER What counts as Mathematics

No. 15, March 2002 from page 5

Delors, J. (Chair). (1996). Learning: The treasure within. Report to UNESCO of the International Commission on Education for the Twenty-first Century. Paris: United Nations Scientific, Cultural and Scientific Organization (UNESCO). Derrida, J. (1994). Spectres of Marx : The state of the debt, the work of mourning, and the new international (transl. P. Kamuf). New York: Routledge. Derrida, J. (1997). Politics of friendship (trans. G. Collins). London: Verso. Engeström, Y. (1999). Expansive learning at work: Toward an activity-theoretical reconceptualization. Keynote address in Changing practice through research: Changing research through practice. Proceedings of the 7th Annual International Conference on Post-Compulsory Education and Training. Brisbane: Centre for Learning and Work Research, Griffith University. Evans, J. (1999). Building bridges: Reflections on the problem of transfer of learning in mathematics. Educational Studies in Mathematics, 39(1-3), 23-44. Evans, J. (2000). Adults’ mathematical thinking and emotions: A study of numerate practices. London: Routledge Falmer. FitzSimons, G. E. (1993, March). Constructivism and the adult learner: Jane’s story. Paper presented at the Constructivism: The Intersection of Disciplines Symposium, Deakin University, Geelong, Vic. FitzSimons, G. E. (2000). Mathematics in the Australian VET Sector: Technologies of power in practice. Unpublished doctoral dissertation, Monash University, Victoria. FitzSimons, G. E. (2001). Mathematics and Lifelong Learning: In whose interests? In G. E. FitzSimons, J. O’Donoghue, & D. Coben. (Eds.). Adult and life-long education in mathematics: Papers from Working Group for Action 6, 9th International Congress on Mathematical Education, ICME 9 (pp. 11-20). Melbourne: Language Australia in association with Adults Learning Mathematics – A Research Forum (ALM). Herremans, A. (1995). New training technologies. [Studies on Technical and Vocational Education No. 2.] Turin: International Labour Organisation in association with UNESCO [UNEVOC]. Jørgensen, C. H., & Warring, N. (2001, March). Learning in the workplace — The interplay between learning environments and biography. Paper presented at 29th Congress of the Nordic Educational Research Association (NFPF), Stockholm. Keitel, C., Kotzmann, E., & Skovsmose, O. (1993). Beyond the tunnel vision: Analysing the relationships between Page 6

mathematics, society and technology. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics education and technology (pp. 243-279). Berlin: Springer Verlag. Klein, M. (2000a). Is there more to numeracy than meets the eye? Stories of socialisation and subjectification in school mathematics. In J. Bana & A. Chapman (Eds.), Mathematics education beyond 2000. Proceedings of the 23rd Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1) (pp. 72-78). Perth: Mathematics Education Research Group of Australasia. Klein, M. (2000b). Numeracy for preservice teachers: Focusing on the mathematics and its discursive powers in teacher education. In J. Bana & A. Chapman (Eds.), Mathematics education beyond 2000. Proceedings of the 23rd Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1) (pp. 347-354). Perth: Mathematics Education Research Group of Australasia. Longoni, P., Riva, G., & Rottoli, E. (2001). Ethics of “delicatezza”. Mathematical literacy in the digital era. CIEAEM 53, Verbania – Italy (pp. 105-106). Verbania: Ghisetti e Corvi Editori. Maaß, J. (1998). Technology transfer – A useful metaphor for university level mathematics courses for engineers and scientists. Keynote address. In ALM-4. Proceedings of the Fourth Annual Conference of Adults Learning Mathematics – A Research Forum (pp. 58-62). London: Goldsmiths College, University of London. Maaß, J., & Schlöglmann, W. (1988). The mathematical world in the black box-significance of the black box as a medium of mathematizing. Cybernetics and Systems: An International Journal, 19, 295-309. Noss, R., Hoyles, C., & Pozzi, S. (1998). ESRC end of award report: Towards a mathematical orientation through computational modelling project. London: Mathematical Sciences Group, Institute of Education, University of London. Onstenk, J. (1998). New structures and new contents in Dutch vocational education. In W. J. Nijhof & J. N. Streumer (Eds.), Key qualifications in work and education (pp. 117132). Dordrecht: Kluwer Academic Publishers. Onstenk, J. (1999, August). Competence development and learning at work. Paper contributed to the European Association for Research in Learning and Instruction (EARLI 1999) symposium, Learning and Working, Göteborg, Sweden. Schlöglmann, W. (2001, June). Affect and cognition: Two poles of a learning process. Paper presented at the 3rd Nordic Conference on Mathematics Education, Kristianstad, Sweden. continued on page 8

No. 15, March 2002

Personal mental methods Janet Duffin, Hull University, UK.

I

have always been interested in people’s personal mental methods of calculating ever since I was involved concurrently with the CAN (Calculator Aware Number) Project in the UK (1986 - 1992) and with undergraduates in the University (1985 - 1999) studying subjects other than mathematics who, on approaching graduation, found that they had to face employers’ tests in numerical reasoning, something they thought they had left behind them on making their subject choices for university. During that time the striking difference that emerged between the two groups (CAN children and students) was that the children were supremely confident about their own mental calculating ability while the students were, like so many adult students, totally lacking in confidence in their own ability in mathematics of any kind. This was a curious finding and one of the things I always tried to do on first meeting the students was to investigate their mental methods for subtraction. What I found was that, while most had their own personal mental methods, they all believed that ‘it was not really the right way to do it’. Some even indicated that, at school, when asked to do subtraction by the laid down method, they did it in their head, using their own method, before attempting to fit their answer into the written method required of them.

And one important way in which we might be better able to achieve our aims seems to me to be that of bringing out into the open the mental calculating methods so many of them secretly have, hence giving them credence and value. If indeed, instead of using their methods secretly and believing them to be unacceptable, they could take pride in something they had done spontaneously and on their own, a really big step would have been taken towards the achievement of the goals of ALM. With this thought in mind, I ran a workshop session at ALM 8 on this very issue and I would like to share the outcomes of that workshop with readers of the Newsletter who did not attend it. I started by recounting the story of a colleague with whom I was discussing my work with undergraduates and I later told his story to my daughter, an artist, whose plea is that she

cannot do anything hard because she has never been any good at mathematics. During the conversation with my colleague, he told me that, given 17 x 17 to calculate he would say ‘twenty seventeens are 340 and then I would subtract fifty one but I always thought it was the wrong way to do it’. My daughter, in contrast said, ‘Oh, I wouldn’t do it like that; I’d say ten seventeens are a hundred and seventy, ten sevens are seventy and seven sevens are forty nine and add them all together’. I was initially quite surprised at these two quite different methods by people who appeared to share the feelings and attitudes I have been describing. Let us look at their two methods written down mathematically: 17 x 17 = (17 x 20) - (17 x 3) = 340 - 51 = 289 17 x 17 = (17 x 10) + (7 x 10) + (7 x 7) = 170 + 70 + 49 = 289 Notice that the first of these involves a method which makes use of the relationship of 17 to 20 in two distinct ways: to find an easy multiplier and then to use the fact that 17 being three less than 20 gives an easy way to get three times it (60 9). He uses rounding and distribution. The second, in contrast, uses the steps of the algorithmic method for ‘long multiplication’ but uses distribution to carry out the second calculation of 17 x 7. In other words both these people, claiming ineptitude in mathematics, used sound mathematical principles to find an answer to this calculation and, moreover, both did it mentally rather than by recording it. Several other methods emerged from the people present at the session and were followed by a lively discussion about the issue generally. An interesting element in that discussion came from the person chairing the session who told us about an earlier experience of his when he was working with some children in a school. When giving pupils mathematical problems he divided them into groups and encouraged them to devise their own calculating methods for solving the problems. He spoke about how difficult it could sometimes be to refrain from intervening by telling the children how to proceed. This temptation to intervene when people are in the process of trying to devise a calculating method for themselves can be a pitfall for us all as teachers because we are so keen that our students should get there and our instinct is to weigh in and help them. I well remember an occasion within the CAN project when I was working with a child who was having some difficulty in devising his own method for some procedure. Another child sitting nearby suddenly said ‘Perhaps he can’t understand what you are saying because he is thinking about it in a different way from you’. It was a salutory lesson for me the teacher but, if we are to achieve our goals of helping our adult students towards competence and confidence in their continued on page 8

Page 7

Research & Practice

As a result of the insight that came from such an investigation of students’ school experiences - an insight that I am sure is shared by many practitioners and researchers in ALM who spend their time trying to overcome the sense of inadequacy felt by so many adult learners - I became increasingly convinced that, if we could overcome this self-perception of inadequacy amongst our students, we would be better able to achieve what we are all setting out to do: improve the competence and confidence of adult learners of mathematics.

ALM NEWSLETTER

ALM NEWSLETTER Personal mental methods

No. 15, March 2002 from page 7

own abilities with numbers, we have to learn to hold back, give them time to try things out in order to find a satisfactory way for themselves. Such difficulties for the teacher are endemic in a learning situation in which learners are being encouraged to find and develop their own calculating methods rather than merely being the recipients of the teacher’s knowledge and methods. However it has its own strength in the confidence that acceptance of their own methods can bring to the learner, a confidence that is enhanced by the sharing of methods and hence their propagation amongst a whole group. Some strategies devised by learners can be appropriate in a variety of different circumstances and these are worth looking at in their own right. Thus the expedient of finding 8 + 7 by saying it is one less than 16 or one more than 14 (see the Proceedings of ALM 1) can be employed in a variety of number contexts. Similarly the idea of doubling and halving can be used to simplify a calculation like 36 x 17. 36 x 17 can be rewritten as 9 x 68 by dividing the 36 by 4 and multiplying the 17 by 4 to compensate, thus reducing it to a ‘short’ multiplication which can be done mentally.

then be in a better position to detect the many mathematical anomalies that occur daily in conversation and in the media. There is some evidence that children have always invented their own calculating methods, largely unknown to their teachers, as indicated earlier. Indeed, a re-examination of the APU (Assessment of Performance Unit) report of UK children’s mathematics in 19871 (in Research in Mathematics Education, eds. Candia Morgan and Keith Jones) acknowledged this saying ‘It is accepted that pupils can themselves invent mental calculating methods and teachers should be aware of these and build on them’. I endorse this. So my call is to value personal mental calculating methods so that adult learners can feel that what they have always done for themselves is valued. With this new value set upon their internal methods their self-esteem will be raised and, in building upon that valuation, they can become mathematically literate. 1

Note: the APU was a Government appointed unit set up in the UK to monitor mathematics performance.

Research & Practice

What counts as Mathematics

Note here that I am assuming that anyone doing this calculation knows what 4 x 17 is, something that is part of the repertoire I suggest learners equip themselves with to become good at calculating mentally. I advocate the learning of up to four times as many numbers greater than the usual 10 or 12 times for the normally learned ‘tables’. Thus I am suggesting knowing that the 13 times table goes 13, 26, 39, 52, by doubling the 13 to get 26, adding 13 to 26 to get 39 and doubling the 26 to get the 52. Doing this for all numbers as far as you can and noticing that, for numbers such as 19, 29, 39 etc the process can be made easier by using the expedient discussed above for 17 x 17 where note was taken of 17 being 3 less than 20. Being able to perform such mental calculating feats does wonders for your confidence and competence in mental calculating. The importance of the sharing of the mental calculating methods within a group is seen both in its value to the individual of seeing others accepting their method and to the group as a whole because it gives each the opportunity to see the methods of others. To the teacher the sharing gives the opportunity for helping students to refine and extend their own methods. And finally, for those who have difficulty in devising their own methods, the sharing offers a choice from the methods of others and perhaps, eventually, will aid their progress towards being able to devise their own. If learning can be done in this way competence and confidence will be improved and we may then have some hope that adults of the future will be able to approach mathematics less hopelessly, less deprecatingly, and might Page 8

▲

Janet Duffin: [email protected]

from page 6

References (continued) Schubring, G. (1998). Contribution to the ICMI discussion document: The role of the history of mathematics in the teaching and learning mathematics (1997-2000). Unpublished manuscript. Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Dordrecht: Kluwer Academic Publishers. Wake, G. D., Williams, J. S., & Haighton, J. (2000). Spreadsheet mathematics in college and in the workplace: A mediating instrument?. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th conference of the International Group for the Psychology of Mathematics Education (Vol. 4) (pp. 265-272). Hiroshima: Hiroshima University. Waterhouse, P. (1995, September). Exploring the tensions in assessment and the CGEA. Adult Literacy Research Network Node (ALRNN) Newsletter, 2. Wedege, T. (1999). To know – or not to know – mathematics, that is a question of context. Educational Studies in Mathematics, 39(1-3), 205-227. Wu, H. (1997). The mathematics education reform: Why you should be concerned and what you can do. American Mathematical Monthly, 104, 946-954. 1

I am indebted to Evan Kritikakos, from Monash University, for discussion on this issue. 2 See also, Longoni, Riva, & Rottoli (2001).

▲

ALM NEWSLETTER

No. 15, March 2002

Adult and Lifelong Education in Mathematics. Papers from the Working Group for Action (WGA) 6, 9th International Congress on Mathematics Education, ICME 9 FitzSimons, Gail E., O’Donoghue, John and Coben, Diana (eds), Adults Learning Mathematics- A research forum (ALM) and the Adult Education Resource & Information Service (ARIS), Language Australia, Melbourne, 2001 “Although values teaching and learning inevitably happen in all mathematics sessions, the teaching of values appears to be mostly implicit.” (p.37) This quotation is taken from the paper ‘Lifelong Learning and Values’, one of nineteen in this publication. Along with the other eighteen, the paper is a readable and enlightening contribution to the field of mathematics education. Published in three parts this collection, totalling 262 pages, is well presented with all graphics and tables reflecting attention to detail. Part 1 titled ‘What is it We’re Looking at? Conceptual Issues in Adult and Lifelong Education in Mathematics’ consists of five papers. Gail FitzSimons, one of the editors, questions whose interests are being served from the promotion of lifelong learning: “The rhetoric of lifelong learning has been adopted by governments of many countries around the world over the last quarter century.” (p.11) Artificiality in school texts and workplace training documents is one of a number of examples presented that cause FitzSimons concern as does the failure to: “theorise mathematics curriculum and pedagogy in any serious way….”. (p.19) Tine Wedege in the paper ‘ “Competence” as a construction in adult and mathematics education.’ analyses the term in detail from various perspectives and indicates that Danish adult education research has made several attempts at defining a concept of competence. A key feature of this paper is the author’s intent to distinguish between competence and performance, with an emphasis on the will to perform.

This paper also contains an extremely interesting concept of numeracy as elaborated by the numeracy component of the

•

To what extent can mathematics teachers gain control over their own values teaching?

•

Is it possible to increase the possibilities for more effective mathematics teaching through values education of teachers, and of teachers in training? and • How malleable are the values we learn in school? •

Which values learnt in school have a ‘long life’, and which are eroded by life experiences?

•

When and where are alternative values learnt out of school?

This article, as with others, should also appeal to ‘nonnumeracy’ readers. To complete Part 1, Ken Clements from Universiti Brunei Darussalam made a major contribution with his paper questioning the equity of mathematical education throughout the world. Part 2 consists of a further four papers gathered under the heading “What Do We Know About the Field of Mathematics Education and How Can We Find Out More About Adults’ Mathematical Abilities?” The detailed paper from Myrna Manly and Dave Tout, ‘Numeracy in the Adult Literacy and Lifeskills Project’, reports on the findings of the team that has been collaborating on this since 1998. An impressive feature of this work is the clear but comprehensive statement of numerate behaviour and its facets. These facets aid the accessibility of the statement and clarify its intent. The team also developed a scheme of five factors to account for the difficulty of different tasks, so as to allow observed performance to be explained based on cognitive factors. The factors are fully detailed. Based on this conceptual framework, items were developed and a strong correlation was found between the predicted and the actual performance. This article alone would be reason enough to obtain this book. continued on page 10

Page 9

Reviews

Dave Tout, who, along with a team from Language Australia assisted with producing this publication, contrasts Numeracy with Mathematics in his paper ‘What is numeracy? What is mathematics?’ Tout, possibly differing from the stance taken by FitzSimons, outlines discussions and positions from within adult education that have occurred over the last ten years or more. One such position is that: “numeracy is not less than mathematics but more”. (p.32)

International Adult Literacy and Lifeskills (ALL) Survey, more on this later. Part 1 also has the Lifelong Learning and Values paper, mentioned previously, which was produced by a group of researchers from Monash University and the Australian Catholic University. The thought provoking points raised throughout this document include a series of questions such as: • What are mathematics teachers’ understandings of their intended and implemented values?

ALM NEWSLETTER Adult &Lifelong Education in Mathematics.

No. 15, March 2002 from page 9

For some time in Australia, adult educators have believed that we have been one of the leaders in this field. Rightly or wrongly, one could be forgiven for thinking this after reading the paper by Katherine Safford-Ramus. She laments the fact that in 20 years (1980-2000) only 38 research reports into adult mathematics education were published in the USA. Silvia Alatorre, Universidad Pedagógica Nacional, Mexico City and fellow contributors have a sub-heading School mistakes in ‘Mexican Adults’ Knowledge about Basic School Mathematics’. Alarmingly, she asserts that the more schooling one had increases the probability of an incorrect answer to certain types of questions, even for those who have completed a college education. Two examples of this type of question are: “Which of these numbers is larger and which is smaller: 1.5, 1.30, 1.465?” and “Yesterday these trousers cost 200 pesos; today they cost 250 pesos. In what percentage were they increased?” “These were interpreted as school mistakes, that is, either originated in school or not corrected by it.” (p.102)

News & Book Reviews

For me, this made keen reading. In a similar vein, Mac McKenzie’s ‘Transfer of Mathematics Learning: You Can’t Get From A to B’ highlights other problems associated with traditional mathematics teaching. “A similar willingness to discard school learning has been observed elsewhere,…” (p.119) and furthermore, “Generally students with low mathematical backgrounds did not transfer skills learned on the programme to other situations.” (p.119) Part 3, ‘Some Issues in Teaching and Learning in Adult and Lifelong Mathematics Education’, has a diverse range of papers including : ‘Fact, Fiction and Moral Panic’; ‘An Educational Programme for Enhancing Adults’ Quantitative Problem Solving’and Decision Making’; ‘Working with Parents’; ‘An Analysis of “Mathematical Museums” from the viewpoint of lifelong education’ (particularly enjoyable with

The history of adult literacy, numeracy and ESOL education policy 1970-2002 This tremendous new project, funded by the UK Economic and Social Research Council is now underway at the University of Lancaster and City University, under the aegis of Professor Mary Hamilton and Dr Yvonne Hillier. The main questions for investigation are: • How have policy decisions in Adult Basic Education [ABE] and ESOL [English for Speakers of Other Languages] been made over this period and what are the rationales for them? •

How have practitioners engaged with the changing policy discourses of basic skills over the past three decades, in

Page 10

information on a beehive, a nautilus and a sunflower). Overall more than two dozen researchers from a dozen countries have had input into this book. The quality of the content will assist in expanding the theoretical knowledge of adult numeracy education in particular but it should not be limited to just that audience. The mathematical content as such is non-taxing and does not hinder the gleaning of the riches within. Policy makers, resource developers, teacher educators and teachers would benefit from exposure to the ideas and evidence presented throughout. This is not a student resource but student learning should be enhanced if those charged with the responsibility of providing suitable learning opportunities are able to implement the changes a reading of these materials demand. Chris Anderson, Holmesglen Institute of TAFE, Australia.

Availability: UK: Cost: £15.00 plus p&h. AVANTI BOOKS 8 Parsons Green, Boulton Road, Stevenage, SG1 4QG Tel: (01438) 350 155; Fax: (04138) 741 131; Email: [email protected] USA: Cost: $US20.00 plus p&h. Peppercorn Books & Press PO Box 693, Snow Camp, NC 27349 Phone: (336) 574-1634 Tollfree (USA only): (877) 574-1634 Fax: (336) 376-9099 WWW: www.peppercornbooks.com Australia: Cost: $Aus38.50 plus p&h. ARIS, Language Australia - NLLIA, GPO Box 372F, Melbourne VIC 3001. Phone: +61 3 9612 2600 Fax: +61 3 612 2601 Email: [email protected] Note: Review reprinted with permission from the ARIS Bulletin, Vol. 12, No. 4, Dec 2001.

▲

defining, analysing and carrying out their work? •

How far has ABE policy been relevant to the lives of adults in the ‘end user target groups’ of ABE

Three stakeholder groups will be investigated: policy makers; ABE practitioners ;and adults with basic skills needs and aspirations. The methods will include documentary analysis; stats analysis of existing datasets; surveys; and interviews with key players. This project will be particularly welcomed by those researchers and practitioners who feel that a ‘critical’ history of these fields of work is long overdue. It will provide valuable insights for those working in the new national Research and Development Centre. More news will be available shortly but in the meantime, do contact Mary, [email protected] or Yvonne, [email protected] for more details. ▲

No. 15, March 2002

ALM NEWS Policies and Practices for Adults Learning Mathematics: Opportunities & Risks ALM-9 Conference 9th International Conference on Adults Learning Mathematics 17 – 20 July 2002, Uxbridge, London, UK The Venue The conference will be run from Wednesday evening until Saturday afternoon. All sessions will take place at Uxbridge College (except Wednesday evening) in Uxbridge, Middlesex. Residential accommodation and evening meals will be on the nearby campus of Brunel University. Regular transport will be provided between the two venues. There is a direct public transport train link to central London from Uxbridge station on the Metropolitan and/or Piccadilly lines of the London Underground. Uxbridge is in West London and near to London Heathrow Airport.

Conference Organisation The members of the local organising committee are David Kaye (Chair), Valerie Seabright (venue coordinator), Alison Tomlin, Jeff Evans with financial advice from Sue Elliott (ALM Treasurer). The local committee is advised by a UK ALM9 group comprising Diana Coben, Dhamma Colwell, Pat Healy, Alice Lewis and Caz Randall.

Call for papers

Paper Presentations (45 minutes) These are traditional style research reports to inform participants about your research work and findings. It is expected that about 20 minutes at the end of the session will be made available for questions and discussion.

Workshops (45 minutes or 1 hour 30 minutes) These can be used to report on projects or work in progress in an interactive and participative manner. The expectation is that presenters will either make the whole session interactive or will enable participants to contribute through activities or discussion for at least half the allocated time.

Topic Groups (2 x 1 hour 15 minutes) Topic groups have met for 2 or more conferences so that discussions develop over time, they will have two meetings during the conference, giving participants the opportunity for more in-depth discussions. Each topic group will have a convenor, and participants may make short planned contributions (maximum ten minutes); longer contributions will be considered by the programme committee and the convenor. Topic groups are in the same time slots (that is, participants can contribute to only one topic group). Group A: Developing a theoretical framework on adult learning and teaching. Group B: Affective factors in adult mathematical learning. Group C: Mathematics education in the workplace.

Discussion Groups (1 hour 15 minutes) These will probably take place in the same time slots as the Topic Groups, and may have one or two meetings. If you would like to set up a discussion group, please outline the questions and how you would start off the discussion.

Poster Presentation (continuous display + 1 x 30 minute discussion time) Poster presentations will be on permanent display and in addition there will be programmed sessions for contributors to discuss their presentations informally. This is an ideal opportunity to present very visual material that should receive wider dissemination. It is also an opportunity to present new ideas or experimental material in a supportive environment. Deadline for all submissions 2 April 2002. A member of the organising committee will respond to your submission by 30th April 2002.

Further information Information about the conference program will be available at the ALM website: http://www.alm-online.org, where an initial call for papers will be posted in late November. The deadline for submitting proposals for the conference is 1 March, 2002 (abstracts or outlines of 100 – 200 words). For further details, please contact: [email protected] or Valerie Seabright, Uxbridge College, Park Road, Uxbridge, UB8 1NQ (tel. 00 44 1895 853613). ▲

Visit the ALM website The address is: http://www.alm-online.org ALM members are invited to send information about their publications for announcement on the ALM website.

Page 11

News & events

We are now inviting contributions to the ALM9 conference in July 2002. Contributions can be for one of five types of presentation. In all cases, collaborations involving more than one author/presenter or group leader are encouraged. The organising committee would particularly like to encourage joint presentation from researchers and practitioners and reports of the results of action research by practitioners.

ALM NEWSLETTER

ALM NEWSLETTER

About ALM

Company No. 3901346 Charity No. 1079462

Adults Learning Maths – A Research Forum (ALM) is an international research forum bringing together researchers and practitioners in adult mathematics/numeracy teaching and learning in order to promote the learning of mathematics by adults.

What is ALM? ALM was formally established at the Inaugural Conference, ALM-1, in July 1994 as an international research forum with the aim to promote the learning of mathematics by adults through an international forum which brings together those engaged and interested in research and developments in the field of adult mathematics/numeracy teaching and learning. ALM is a forum for experienced and first-time researchers to come together and share their ideas and their reflections on the process as well as the outcomes of research into hitherto neglected area of adults learning mathematics. ALM puts people in touch with each other, providing a framework for collaboration and helping to stimulate and develop research plans. We are especially keen to encourage practitioners to undertake research. Since 1994, ALM has gone from strength to strength and now has 140 members in 19 countries. In 2000, it was registered as a company and as a charity in England and Wales.

What does ALM offer? ALM membership brings with it opportunities to: • contribute to an international forum of researchers and practitioners in the field • share concerns, insights and research at ALM annual conferences, and to attend at a reduced rate • receive ALM newsletter (free) • receive ALM conference proceedings (free of charge to conference delegates). These proceedings constitute the most significant and authoritative collection of papers on adults learning mathematics available today • network, electronically and otherwise, with practitioners and researchers in the emerging field of adults learning mathematics.

ALM Officers Chair:

Prof. Dr. Juergen Maasz, University of Linz, Austria Secretary: David Kaye, London Treasurer: Sue Elliott, Sheffield Hallam University Membership Secretary: Prof. Sylvia Johnson, Sheffield Hallam University Page 12

No. 15, March 2002

Join ALM today! ALM is actively seeking to expand its membership worldwide. Membership is open to all individuals and institutions who subscribe to its aims. For details contact Sylvia Johnson, Membership Secretary at the Centre for Mathematics Education, Sheffield Hallam University, 25 Broomsgrove Road, Sheffield S10 2NA, UK email: [email protected] or your regional ALM membership agent: ARGENTINA Dr Juan Carlos Llorente, Fundacion PAIDEIA, Instituto de Investigacion Educativa, Mitre 862 (8332), Gral Roca, RN, Argentina. Email: [email protected] AUSTRALIA Dr Janet Taylor, OPACS, Uni. of Southern Queensland, Toowoomba, Australia. Email: [email protected] BRAZIL Eliana Maria Guedes, Dept. of Architecture, Mathematics and Computing, UNITAU, University of Taubaté, Sao Paulo, Brazil. Email: [email protected] DENMARK Dr. Tine Wedege, IMFUFA, Roskilde University, Box 260, 4000 Roskilde, Denmark. Email: [email protected] NEW ZEALAND Barbara Miller-Reilly, Student Learning Centre, The University of Auckland, Private Bag 92019, Auckland, N.Z. Email: [email protected] REPUBLIC OF IRELAND Prof. John O’Donoghue, Dept of Maths and Statistics, University of Limerick, Limerick, Ireland. Email: [email protected] THE NETHERLANDS Mieke van Groenestijn, Utrecht University of Professional Education, PO Box 14007, 3508 SB, Utrecht, The Netherlands. Email: [email protected] UNITED KINGDOM Sue Elliott, Centre for Mathematics Education, Sheffield Hallam University, 25 Broomsgrove Road, Sheffield S10 2NA, UK. Email: [email protected] USA Dr Katherine Safford, Saint Peter’s College, Kennedy Boulevard, Jersey City, NJ 07306, USA. Email: [email protected]

Membership fees Individual: Student/unwaged:

£15 £3

Institution:

£30

Editorial Committee Mieke van Groenestijn

Utrecht University of Professional Education, Netherlands Dave Tout Language Australia Tine Wedege Roskilde University, Denmark For more information email: [email protected] The views expressed in articles are those of the authors and do not necessarily represent the views of ALM or of the editorial committee. Printed on recycled paper.