ࡱ>  ,bjbjVV .<<|8Q4"(AWWW2>p p"r"r"r"r"r"r"$$)'"22"WW"vWWp"p"W (6\""0"'<$''D""`"' : MEMES AND MATHEMATICS EDUCATION Robert Ward-Penny University of Warwick R.M.Ward-Penny(at)warwick.ac.uk The theme of reproduction is frequently central to sociological narratives of education. Analyses such as Bourdieu and Passeron (1990) have argued that education plays a critical role in the reproduction of social stratification, and work such as Noyes (2004) contains evidence of reproduction at the level of individuals, in particular demonstrating that new mathematics teachers have a tendency to replicate many of the values and strategies that they themselves observed as learners. The field of memetics draws upon such evidence and analyses but goes further, suggesting that it is useful instead to contemplate reproduction at the level of ideas. In this way, memetics offers a novel and challenging framework with which to consider the interplay of thoughts and behaviours in the mathematics classroom. What is a Meme? The term meme was first coined by the evolutionary biologist Richard Dawkins in 1976 to describe a unit of cultural transmission, or a unit of imitation. The term was chosen so as to evoke a deliberate parallel with biological genes: just as genes propagate themselves in the gene pool by leaping from body to body via sperms or eggs, so memes propagate themselves in the meme pool by leaping from brain to brain via a process which, in the broad sense, can be called imitation (Dawkins 1989, p.192). A simple example of meme is therefore a catchy tune; for when a tune frequently repeats in an individuals mind, they frequently end up singing it, whistling it, or even overtly discussing it. Each of these behaviours leads to other people becoming aware of the tune, and thus facilitates propagation. In a manner reminiscent of biological natural selection, the catchiest tunes tend to spread faster, wider and last for longer in the public consciousness. More developed examples can be drawn from a wide range of fields of interest to the sociologist. Lynch (1996) offers the Amish religious taboo against modern farm machinery as an example of a self-perpetuating thought contagion (pp.1-2). Possession of the taboo gives rise to a greater need for manual labour which is met by taboo holders having large numbers of children. In this way the taboo is passed onto a large number of descendants and continues to spread and replicate. Dawkins, Lynch and other writers in the field have gone on to argue that many beliefs about religious observance, sexual practices and other social norms can be understood in this way. Equally, stories from folklore and urban legends can be considered mimetically, as can advertisements. These are often presented in a style that aids memorisation and encourages the listener to perpetuate their spread. Many slogans and jingles compete to be held in an individuals memory, but only the most memorable will survive. Memes can also manifest explicitly through physical behaviours, whenever an individual sees a physical display or achievement which they are keen to emulate; Dawkins here offers as examples clothes fashions, (and) ways of making pots or building arches (1989, p.192). Indeed, it might even be suggested that mathematics itself could be considered as a collection of memes, or a memeplex, since mathematical behaviours are observed and imitated by successive chains of learners. This is an intriguing notion, although it invokes the question as to whether Dawkins qualifier that imitation takes place in a broad sense includes instances of reading, writing and direct instruction. Blackmore (2000, p.28) argues that this is indeed the case: we may not wish to count these as forms of imitation, but I would argue that they build on the ability to imitate and could not occur without it. Dawkins criteria that imitation happens in a broad sense is perhaps both potently interdisciplinary and problematically vague; Blackmore (1994, p.42) notes, it is all too easy to get carried away with enthusiasm and to think of everything as a meme. In order to narrow the focus of discussion, and thus facilitate an initial exploration of the concept of a meme as it might apply to mathematics education, this paper will adopt the following criteria: that in order for an idea to qualify as a meme, or at least a successful one, it must be possible to argue that possession of this idea encourages behaviour which in turn leads to an increased propensity for others to adopt or reinforce a form of the same idea. In this way the processes of imitation and replication remain intrinsic to the concept of meme. Memes about Mathematics The concept of a meme as outlined above has a wide range of application within the mathematics classroom. For instance, it could be argued that the popular algebra mnemonic swap sides swap sign qualifies as a very successful meme. The limited but immediate achievement that often results from following this rule encourages the initial transmission of the phrase from teacher to pupil and subsequent supporting transmission between peers, whilst the alliterative composition aids both the retention and accurate replication of the meme. This meme is similar in many ways to certain fashionable weight-loss memes, where short-term advantage acts as a potent motivator for spreading a meme, and succinct but unusual instructions encourage the individual to accurately retain their own copy. Beyond simple examples such as mnemonics, the meme concept also calls for a consideration of the wider ideas that operate within the teaching and learning of mathematics. In particular, might some of the observed behaviours within the mathematics classroom be understood as being consequent of the replication processes of competing memes? In order to begin to answer this question it is necessary to start by considering the beliefs which surround mathematics. Whilst the public perception of mathematics is both fluid and diverse, it is apparent that a number of dominant and identifiable beliefs about mathematics exist at present. Research such as that of Lim and Ernest (2000) has demonstrated that many of these beliefs relate to either the nature or the characteristics of mathematics. The first of these categories includes perceptions of mathematics as a toolkit, a problem-solving tool or an absolute body of truth, whilst the second includes beliefs such as mathematics is difficult, or that mathematics is only for clever people. Of course, such ideas are neither uniform nor discrete; at an individual level beliefs might consist of vague understandings or well-formed arguments, and a persons opinions are also likely to be both interdependent and multifaceted. However, there is enough homogeneity in recorded responses to identify trends and categories that suggest the existence, in a phenomenological sense at least, of certain ideas that concern mathematics and the learning of mathematics. To argue that any of these ideas could indeed be considered as memes, it is necessary next to identify channels through which their reproduction might occur. Cavalli-Sforza and Feldman (1981) differentiate between three types of channels of memetic transmission: vertical, horizontal and oblique. Vertical Transmission There is a considerable amount of evidence for the vertical transmission between parents and children of ideas relating to mathematics education. For example, Chouinard, Karsenti and Roy (2007) explored the influence of social agents on pupils developing beliefs and found that whilst teachers actions influenced pupils beliefs about their own competence, it was the pupils perception of parental support that best explained measured variables relating to the value of mathematics. This relationship can be explained by memetic mechanisms. If a parent sees the study of mathematics as valuable, they are likely to take a greater interest in their childs attainment and effort in mathematics, paying more attention to achievements and reports when compared to other subjects. A pupil will interpret this effort and conclude similarly that the study of mathematics must be valuable. Conversely, if a parent sees mathematics as abstractly hard, they might be more forgiving if their child has low levels of attainment in the subject, and encourage their child to adopt a similar attitude through well-trod discourse such as I was never any good at maths when I was at school. There is also some evidence for subconscious replication; for instance Else-Quest, Hyde and Hejmadi (2008) found that the emotions of mothers and 11 year-old children were closely correlated when they were solving mathematics problems together. There is further a growing body of evidence that suggests that parents hold and pass on specific gender-related memes about their childrens mathematical performance, centred on the tenet that mathematics is predominantly a male pursuit. For instance, Herbert and Stipek (2005) report on an experiment that suggested not only that parents typically underestimate girls mathematical performance, but also that this bias is passed down to their children at an early age; the parents judgements of their childrens competence were shown to be strong predictors for their childrens self-evaluations. This result is consonant with older research; for instance Yee and Eccles (1988) found that parents attributed male child success in mathematics to talent and female child success in mathematics to effort. They argued that this could influence childrens emergent identities as mathematicians: talent is a stable attribute whilst effort is an unstable one while both are seen as important reasons for math success, that parents rate their relative importance differently for boys and girls may contribute indirectly to the inferences that their children develop regarding their own math talent (p.330). Gender stereotyping has also been shown to be perpetuated in discussions between parents and children about course selection (Tenenbaum, 2008). It is highly likely, then, that certain memes do indeed propagate through parent-child interactions. Vertical transmission of memes can also be argued to occur between teachers and pupils, with a teachers view about mathematics being inculcated in their pupils through the pupils interpretation of intermediary behaviours. If all mathematical pedagogy rests, however loosely, on a philosophy of mathematics (Thom, 1973) then the manner in which a teacher presents the subject inevitably betrays this philosophy, and in turn suggests to the pupils how they might position themselves with respect to the subject (Ernest, 2008). For instance, if a teacher believes mathematics to be a valuable real-world problem-solving tool, then they are likely to favour explanations and tasks that are contextualised and have clear relevance. Conversely, a teacher with a formalist philosophy of mathematics might present a mathematical technique as a game that has intrinsic value (Hersh, 1979; Lerman, 1983). Consistent, repeated exposure to either of these approaches would inevitably colour pupils perceptions, encouraging them to arrive at the same philosophical position, and thus adopt the same memes as their teachers. Whenever this happens, memetic reproduction has occurred. Lim and Ernest (2000) observe that peoples images of mathematics are closely related to their images of learning mathematics; this offers further support for the influence of pedagogy on learners wider perception of mathematics. Horizontal and Oblique Transmission There is perhaps less empirical evidence that supports the transmission of ideas about mathematics horizontally, or between peers. However there is vast extant literature concerning the wider subject of peer relationships and interactions, and in particular it has been demonstrated that pupils consciously and tactically vary their efforts in response to perceived peer norms (Juvonen and Murdock, 1995). This observation supports the notion that peer perception, together with concerns about self-esteem, could promulgate a thorough horizontal transmission of certain memes which concern affective aspects of learning mathematics. For instance, consider the meme maths is hard. If a pupil is struggling with classroom mathematics and holds an entity view of intelligence (Dweck, 2000), it is in their own self-interest first to adopt this meme themselves, then to behave in a way that convinces others of its veracity; for if mathematics is abstractly difficult, then lower levels of achievement can be tolerated without necessitating a challenge to the pupils sense of self-worth. This could lead to a cycle of reduced effort and lowered attainment. Conversely, if a pupil is performing highly in mathematics, then they too might benefit from spreading the maths is hard meme to their peers through their behaviour, as this stance compounds their existing achievement, albeit with a potential for related social disadvantage through claiming a marker of superiority. The third channel of memetic spread, oblique transmission, could be argued here to include the influence of the media on its audience. Picker and Berry (2000) contend that the media is a particularly salient influence on pupils images of mathematics: as far as the pupil is concerned, mathematicians are invisible. Stereotypes have filled this void (p.87). Unfortunately, although Furinghetti (1993) notes that images from outside the community of mathematicians can portray mathematics as a synonym for truth, integrity and justice (p. 36) it is also true that mathematics is often presented in some media as highly abstract practice that is dominated by males, and which can even act as a path to madness (Schoffer, 2002). It is outside the scope of this paper to examine comprehensively the media profile of mathematics, but it is enough to note that media presentations of mathematics, however peripheral, might serve both to propagate new memes and to reinforce existing memes that have been previously established through vertical and horizontal modes of transmission. Advantages of a Memetic Analysis The examples and evidence offered above make the outline of an argument which holds that definite ideas exist regarding the practice of mathematics; ideas which, when adopted by an individual, often lead to behaviours which in turn encourage others to adopt a similar idea. However, reproductive mechanisms have long been a feature of sociological readings of the mathematics classroom, and it is thus necessary to ask what advantages and new insights a memetic approach might offer a researcher. The developing and contested nature of memetics as a field (see for example Aunger, 2000) precludes a full answer to this question, but three promising arguments can be advanced at this stage. First, a memetic model of classroom interactions motivates a holistic approach which includes many different social actors and multiple potential channels of reproduction. In particular, it allows recognition of the fact that separate social actors may be motivated to act in significantly different ways by the same fundamental meme. Various behaviours of teachers, pupils and even parents may stem from the possession of similar ideas about the teaching and learning of mathematics, and thus may together enable further proliferation of the same memes. For example, it has been noted above how a formalist reading of mathematics as an abstract game which permits only certain logical moves can influence a teachers choice of pedagogic strategy (Lerman, 1983). Whilst such a strategy might involve resources such as card-matching activities, on-screen quizzes or jigsaw puzzles, possession of the meme will ensure that the correct use and manipulation of symbols and syntax will remain central to the presentation of mathematical activity, and through exposure and reinforcement the pupils will likely develop a similar perception of mathematics; in this case the formalist meme has spread vertically. A pupil who possesses this same meme may, however, consequently develop a depersonalised view of mathematics as a discipline, and thus there is a risk of an emergent quiet disaffection (Nardi and Steward, 2003). The pupils consequent withdrawal from active learning strategies is likely to be recognised and perhaps questioned by their peers; in some cases there is mutual support and emulation, and thus also arguably horizontal transmission of the formalist meme. In this way the efforts of the teacher and the lack of effort on the part of these pupils are superficially contradictory but profoundly connected. This is not to say that puzzle or matching activities are harmful or that they are in and of themselves a cause of disaffection; instead it suggests that memes may serve as a common causative factor which could offer valuable insight to readings of classroom dynamics. Further, a holistic perspective arguably addresses some of the bias implicit in other accounts of classroom interactions. This advantage stems in part from a redistribution of agency in the analysis. It is proper to note, however, that this is perhaps philosophically problematic. On the one hand, Dawkins holds that whilst both genes and memes behave as if they were purposeful (1989, p.196), this is only an illusion of agency which we adopt in order to facilitate discussion and circumvent clumsy patterns of speech. Conversely, other writers propose that this illusion is in fact closer to ontological truth, and that memes in fact offer a valuable window onto the nature of human consciousness itself (for example Blackmore, 1999). Again, these issues lie outside of the scope of this paper, but without endorsing either side of this argument, it is sufficient to note here that the location of agency with memes directs our attention to both individuals and institutions, as both contribute to the reproductive process. In this way, a memetic approach has some resonance with aspects of poststructuralist sociology. A second advantage of a memetic approach is that it recognises explicitly that social actors are not always fully conscious of the forces that shape their behaviour and decisions. Writers such as Bibby (2009) have noted that analyses of pedagogy and mathematical relationships tend to valorise conscious processes whilst rendering unconscious workings as insignificant. Conceptualising a scenario in terms of memes, which often remain unvoiced or disguised, might serve partially to redress this balance without resorting to a full psychoanalytic approach. This is particularly relevant when considering psychological issues such as stereotype threat. Research that shows that attainment is lowered when learners are exposed to pejorative ideas about the mathematical capabilities of their own gender or race (Beilock, Rydell and McConnell, 2007; Aronson, Lustina, Good and Keough, 1999) could be easily interpreted as evidence of how memes might influence cognising individuals at a subconscious level. Finally, the framework of memetics encourages researchers to explore familiar questions in mathematics education from a new perspective. Instead of asking what beliefs people hold about mathematics, memetics begins with specific, identified beliefs and asks why these have been so successful at replicating; rather than asking how people acquire ideas, it asks how ideas acquire people (Lynch, 1996, pp.17-8). Memetic reproduction is thus centred on selection and competition. A consideration of the motivational advantage associated with each meme that is, the benefit the learner obtains by accepting this meme over others forces a re-evaluation of the conditions of the learning environments which underlie these gains. If memes with negative messages about mathematics are as widespread as some reports suggest, then which aspects of the current educational environment have allowed them to proliferate so readily? A Memetic Model of Conflict in the Classroom As a brief example, consider a hypothetical classroom and the meme mathematics is a difficult subject, accessible only to the clever elite. If a teacher holds this meme they are prone to expect, at least on a subconscious level, that not all of their pupils will be able to fully engage with the subject matter. Conversely, the teacher will label those pupils that do achieve consistently highly as mathematically able. If a low-achieving pupil in this classroom holds the same meme as their teacher, they will see their low achievement as an indication that continued effort is pointless, and will most likely gradually withdraw active learning behaviours. Although this can lead to superficial conflict and disruptions in terms of their conduct, the teacher is unlikely to address this issue at its fundamental root; their own, consonant beliefs mean that the relegation of some pupils within most groups is not only tolerated but tacitly expected. By way of contrast, if a high-achieving pupil holds the same meme as this teacher, their achievements are likely to feed into a positive academic self-concept. Perhaps ironically, both sets of pupils benefit from subscribing to the same belief as their teacher; the high-achiever is labelled as special, whilst the low-achiever is given a framework with which to understand, and perhaps excuse, their difficulties. This is of course a fragile caricature of a real mathematics classroom, albeit one that will resonate with many peoples experience. Despite its simplicity, however, it demonstrates how a memetic approach can both unify different social actors motivations and recognise subconscious aspects. Note that, in this situation, possession of this meme by the pupil and the teacher encourages behaviour that might not only lead to memetic reproduction in other individuals, but also mutually supports and strengthens the meme in the minds of the two main actors. Further consolidation of such views could also be drawn from the discourses present in textbooks, the organisation of the school (for example, setting practices), and the similar actions and reactions of the pupils peers. The model presented above could be extended by questioning what might happen if our theoretical teacher and pupil held different memes. Such a situation could result in an intractable conflict of aims, but it is also possible that the stronger meme (or at least the meme which is most strongly enabled through the established power relations) would subdue the weaker one, and the respective party would change their mind. There is a potential parallel here with the conflicts envisioned by Skemp (1976) in his discussion of different types of understanding; there is also potentially a shift in unspoken behavioural and didactic contracts. However, a full conceptualisation of social interactions as a product of meme transmission and competition is perhaps more subtle. The examples offered above have all been crude, and centred on a single meme. Genuine interactions would involve complex memeplexes, with sets of inter-related ideas acting in combination. A more authentic situation might involve the mathematics is difficult idea being tempered and enhanced by other memes, perhaps algebra is hard or I am not any good with numbers. These memes would in turn be supported by memes encoding broader beliefs about intelligence, such as an incremental or fixed model of intelligence (Dweck, 2000). It is possible to speculate that one of the reasons that issues such as the debate about the nature of intelligence are so important in mathematics education (Lee, 2009) is that their corresponding memes are in some sense foundational, supporting a plethora of other related memes. The Value of the Meme as an Idea in Mathematics Education The idea of the meme is potentially a fruitful one, and it has the promise to give rise to many research questions within mathematics education. However, there are a number of outstanding issues which deserve consideration. First, there is not a single agreed definition of a meme, or uniform agreement on what would constitute a properly academic memetic analysis (Aunger, 2000). This disagreement, together with practical limitations which frustrate a reductionist engagement with the concept, leads to a second issue: it would be challenging to construct an ethical empirical test which directly measured the memetic relationships proposed above; further, if no test exists which could disprove the existence of memes, then memetics would merely qualify as a pseudo-science under Popperian criteria. Additional theoretical issues might also arise when connecting memetics to wider theories of mathematical learning. On the one hand, the mentalist conceptualisation of the meme as a notional mental entity is perhaps most consonant with constructivism as a learning theory. However, the central focus on the transmission of ideas is perhaps more comfortably resonant with behaviourism; a constructivist approach would at the very least demand a re-evaluation of what is meant by accurate replication in memetic propagation. Despite these concerns, even if we only accept the idea of a meme as a philosophical abstraction, it gives rise to a complementary reading of the teaching and learning of mathematics which has many potential benefits: a more holistic perspective on classroom dynamics, recognition of the potential role of the subconscious and a fresh perspective on well-trod questions. It also suggests new ways in which attitudes and beliefs towards mathematics and the learning of mathematics might be memetically engineered, or improved. One approach would be to actively spread memorable memes that promote the public understanding and awareness of mathematics, such as mathematics is the science of patterns (Devlin, 1997). Another would be to actively support the promotion of memes in the mathematics classroom which encourage resilient learning behaviours. Potentially injurious ideas such as the notion that there is always one right way to solve a mathematics problem, or that speed is paramount in mental calculation might be subverted by memes which extolled the merit of sustained effort, and recognised the place of making mistakes in mathematical learning (Johnston-Wilder and Lee, 2010). A final strategy would be to recognise that modern modes of communication mean that horizontal and oblique transmission may overtake vertical transmission as dominant channels of memetic spread, then to take advantage of this shift, using media channels such as the internet to promote the practice and appreciation of mathematics. None of these approaches are novel; however, an appreciation of memetics may inform them in a fresh and productive way. In the meantime we might all benefit from recognising in a new way that the mathematics classroom is full of ideas, and thus move towards a mapping of our own mathematical memeplexes. Bibliography Aronson, J., Lustina, M.J., Good, C. and Keough, K. (1999) When White Men Cant Do Math: Necessary and Sufficient Factors in Stereotype Threat, Journal of Experimental Social Psychology, 35: 29-46. Augner, R. (ed.) (2000) Darwinizing Culture: The Status of Memetics as a Science, Oxford: Oxford University Press. Beilock, S., Rydell, R. and McConnell, A. (2007) Stereotype Threat and Working Memory: Mechanisms, Alleviation and Spillover, Journal of Experimental Psychology: General, 136 (2): 256-276. Bibby, T. (2009) How Do Pedagogic Practices Impact on Learner Identities in Mathematics? A Psychoanalytically Framed Response. In L. Black, H. Mendick and Y. Solomon (eds.) 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Howson, e.d., (1973) Developments in Mathematics Education, Cambridge: Cambridge University Press, pp.194-209. Yee, D.K. and Eccles, J.S. (1988) Parent Perceptions and Attributions for Childrens Math Achievement, Sex Roles, 19(5-6): 317-333. Robert Ward-Penny holds an Economic and Social Research Council (ESRC) funded studentship at the University of Warwick Institute of Education.      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