Memes and Mathematics Education

MEMES AND MATHEMATICS EDUCATION

Robert Ward-Penny

University of Warwick

R.M.Ward-Penny(at)warwick.ac.uk

The theme of reproduction is frequentlycentral 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 contemplatereproduction 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 individual’s 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 jinglescompete to be held in an individual’s 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 subsequentsupporting 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 certainfashionable 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 mathematicsexist at present.Research such as that of Lim and Ernest (2000) has demonstrated that many of these beliefs relate toeither 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, suchideas areneither uniform nor discrete; at an individual level beliefs might consist of vague understandings or well-formed arguments, anda person’s 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, ofcertainideas 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 child’s 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 children’s 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 children’s competencewere shown to be strong predictors for their children’s 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 children’s 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 teacher’s 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 aformalist 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 promulgatea 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 pupil’s 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 influenceof 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.

Advantagesof 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 behavioursof 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 teacher’s choice of pedagogic strategy (Lerman, 1983). Whilst such a strategy mightinvolve resources such as card-matching activities, on-screen quizzes or jigsaw puzzles, possession of the meme will ensure that thecorrect use and manipulation of symbols and syntax will remain central to the presentation of mathematical activity, and through exposure and reinforcement thepupils will likely develop a similar perception of mathematics; in this casetheformalist 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 pupil’s 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 thusalso 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.